0,\n \\end{split}\n \\end{equation*}\nwhere \n\\begin{equation}\\label{eq7.4}\n \\begin{split}\n & \\hat{J}_1(n; \\vartheta, \\tau) = \\int_0^1 G'_{\\vartheta n} \\left(\\mu_1\\| \\xi^n\\|^2+\\mu_2\\|\\nabla\\xi^n\\|^2 \\right) [\\vartheta^2\\|\\bar{\\delta}_t\\xi^n\\|^2+\\vartheta(\\bar{\\delta}_tw_h(t_n), \\bar{\\delta}_t\\xi^n)] d\\vartheta, \\\\\n & \\hat{J}_2(n; \\vartheta, \\tau) = \\int_0^1 G'_{\\vartheta n}\\Big( \\mu_1(w_h(t_n), \\xi^n) + \\mu_2(\\nabla w_h(t_n), \\xi^n)\\Big)[\\vartheta \\|\\bar{\\delta}_t\\xi^n\\|^2 +\\bar{\\delta}_tw_h(t_n), \\bar{\\delta}_t\\xi^n)] d\\vartheta,\n \\end{split}\n \\end{equation}\nwith $G'_{\\vartheta n}=G'(\\varphi_{\\vartheta}(t_n))$. Below, we shall analyze the terms $\\hat{J}_1$ and $\\hat{J}_2$ in \\eqref{eq7.4}. First, we apply Theorem \\ref{theorem2.4} to obtain $\\|\\bar{\\delta}_tU^n\\|\\leq C$, and \\eqref{eq5.7} gives\n\\begin{equation}\\label{eq7.5}\n \\begin{split}\n \\|\\bar{\\delta}_t w_h(t_n)\\| \\leq \\Big\\| \\frac{1}{2\\tau} \\int_{t_{n-1}}^{t_{n+1}} w_h'(t)dt \\Big\\| \\leq \\frac{C}{\\tau} \\int_{t_{n-1}}^{t_{n+1}}\\|w_h'(t)\\|dt \\leq C.\n \\end{split}\n \\end{equation}\nThen we employ the triangle inequality to get\n\\begin{equation}\\label{eq7.6}\n \\begin{split}\n \\|\\bar{\\delta}_t\\xi^n\\|\\leq \\|\\bar{\\delta}_tU^n\\|+\\|\\bar{\\delta}_t w_h(t_n)\\| \\leq C, \\quad n\\geq 2. \n \\end{split}\n \\end{equation}\nBased on \\eqref{eq7.5}--\\eqref{eq7.6} and the assumption ($\\mathbf{A2}$), we have for $n\\geq 2$\n\\[\\|\\hat{J}_1(n; \\vartheta, \\tau)\\| \\leq C\\left(\\| \\xi^n\\|^2+\\|\\nabla \\xi^n\\|^2 \\right) \\leq C\\|\\nabla \\xi^n\\|^2.\\]\nThen we use \\eqref{eq7.5}--\\eqref{eq7.6}, Poincar\\'{e} inequality and Young inequality to get\n\\[\\|\\hat{J}_2(n; \\vartheta, \\tau)\\| \\leq C\\left(\\| \\nabla \\xi^n\\|+\\|\\xi^n\\| \\right) \\|\\bar{\\delta}_t\\xi^n\\| \\leq C\\| \\nabla \\xi^n\\| \\|\\bar{\\delta}_t\\xi^n\\| \\leq \\frac{g_0}{8}\\|\\bar{\\delta}_t\\xi^n\\|^2 + C\\| \\nabla \\xi^n\\|^2.\\]\nWe further estimate the second term of the right-hand side of \\eqref{eq7.1}. We apply \\eqref{eq6.3} to get\n\\begin{equation}\\label{eq7.11}\n \\begin{split}\n &\\left(G_0\\left(u^n,\\nabla u^n\\right) \\bar{\\delta}_tu^n - G_0\\left( w_h(t_n),\\nabla w_h(t_n)\\right) \\bar{\\delta}_tw_h(t_n), \\bar{\\delta}_t\\xi^n \\right)\\\\\n &\\leq \\left| \\left( G_0\\left(u^n,\\nabla u^n\\right) [\\bar{\\delta}_tu^n - \\bar{\\delta}_tw_h(t_n)], \\bar{\\delta}_t\\xi^n\\right) \\right| \\\\\n & + \\left| \\left( [G_0\\left(u^n,\\nabla u^n\\right) - G_0\\left( w_h(t_n),\\nabla w_h(t_n)\\right)] \\bar{\\delta}_tw_h(t_n), \\bar{\\delta}_t\\xi^n\\right) \\right| \\\\\n & \\leq \\mathcal{L}\\left[ \\mu_1(\\|u^n\\|^2-\\|w_h(t_n)\\|^2)+\\mu_2(\\|\\nabla u^n\\|^2-\\|\\nabla w_h(t_n)\\|^2)\\right]\\\\\n & \\qquad \\times \\|\\bar{\\delta}_tw_h(t_n)\\|\\|\\bar{\\delta}_t\\xi^n\\| + g_1 \\left\\|\\bar{\\delta}_t\\eta^n\\right\\|\\|\\bar{\\delta}_t\\xi^n\\| \n \\leq C\\left(\\|\\nabla \\eta^n\\| + \\left\\|\\bar{\\delta}_t\\eta^n\\right\\| \\right) \\frac{\\|\\xi^n\\|_A+\\|\\xi^{n-1}\\|_A}{2}.\n \\end{split}\n \\end{equation} \nWe invoke \\eqref{eq7.2}--\\eqref{eq7.11} in \\eqref{eq7.1}, sum the resulting equation multiplied by $2\\tau$ for $n$ from 1 to $m$ and use \\eqref{eq6.2}--\\eqref{eq6.7} to get\n\\begin{multline*}\n \\|\\xi^m\\|^2_A \\leq \\|\\xi^0\\|^2_A\n + C \\tau \\sum\\limits_{n=1}^{m} \\Big(\\|\\xi^n\\|_A^2 + \\|\\delta_t^2 \\eta^n\\| \\frac{\\|\\xi^n\\|_A+\\|\\xi^{n-1}\\|_A}{2} \n \\\\\n + \\left\\|u'(t_n)-\\bar{\\delta}_tu^n\\right\\| \\frac{\\|\\xi^n\\|_A+\\|\\xi^{n-1}\\|_A}{2} \n + \\sum\\limits_{j=1}^{4}\\|R_j^n\\| \\frac{\\|\\xi^n\\|_A+\\|\\xi^{n-1}\\|_A}{2} \\\\\n + \\left(\\|\\nabla \\eta^n\\| + \\left\\|\\bar{\\delta}_t\\eta^n\\right\\| \\right) \\frac{\\|\\xi^n\\|_A+\\|\\xi^{n-1}\\|_A}{2}\\Big).\n \\end{multline*} \nChoose a suitable $\\ell$ such that $\\|\\xi^\\ell \\|_A = \\max\\limits_{0\\leq n \\leq m}\\left\\|\\xi^n \\right\\|_A$, we have for $j\\le m,$ \n\\begin{multline}\\label{eq7.13}\n \\|\\xi^\\ell\\|_A \\leq C \\tau \\sum\\limits_{n=1}^\\ell \\|\\xi^n\\|_A + \\left\\|\\xi^0 \\right\\|_A\n + C \\tau \\sum\\limits_{n=1}^\\ell \\|\\delta_t^2 \\eta^n\\| \n \\\\\n + C \\tau \\sum\\limits_{n=1}^\\ell\\Big( \\left\\|u'(t_n)-\\bar{\\delta}_tu^n\\right\\| + \\sum\\limits_{j=1}^{4}\\|R_j^n\\| + \\left(\\|\\nabla \\eta^n\\| +\\left\\|\\bar{\\delta}_t\\eta^n\\right\\| \\right)\\Big).\n \\end{multline}\nNext, we will estimate the terms of the right-hand side of \\eqref{eq7.13}. First, we get\n\\begin{align}\n & \\tau \\sum\\limits_{n=1}^{m} \\left\\|\\delta_t^{2}\\eta^n \\right\\|\\leq\n C \\tau \\sum\\limits_{n=1}^{m} \\|\\eta''_1(\\kappa_n)\\|\\leq Ch \\|u''\\|_{L^{\\infty}(H^1)}, \\;\\; \\kappa_n\\in [t_{n-1}, t_{n+1}], \\label{eq7.14} \\\\\n & \\tau \\sum\\limits_{n=1}^{m} \\left\\|u'(t_n)-\\bar{\\delta}_tu^n\\right\\| \\leq \\frac{\\tau^2}{2}\\int_0^{t_{m+1}}\\|u'''(s)\\|ds,\\nonumber\\\\\n & \\tau \\sum\\limits_{n=1}^{m} \\left(\\|\\nabla \\eta^n\\| +\\left\\|\\bar{\\delta}_t\\eta^n\\right\\| \\right) \\leq Ch\\left(\\|u\\|_{L^{\\infty}(H^2)}+\\|u'\\|_{L^{\\infty}(H^1)}\\right). \\nonumber\n\\end{align}\nThen we discuss the terms $R_j^n$ with $1\\leq j\\leq 4$. First, we bound the term on $R_1^n$ in \\eqref{qq1}. We apply the Taylor expansion with integral remainder to arrive at\n\\begin{equation*}\n \\begin{split}\n & u''(t_n) - \\delta_t^{2}u^n = \\frac{-1}{6\\tau^2} \\left[ \\int_{t_n}^{t_{n+1}}(t_{n+1}-t)^3u^{(4)}(t)dt + \\int_{t_{n-1}}^{t_{n}}(t-t_{n-1})^3u^{(4)}(t)dt\\right], \\\\\n& n\\geq 2, \\quad u''(t_1) - \\delta_t^{2}u^1 = \\frac{-1}{2\\tau^2} \\left[ \\int_{t_1}^{t_{2}}(t_{2}-t)^2 u'''(t)dt + \\int_{0}^{t_{1}}t^2u'''(t)dt\\right],\n \\end{split}\n \\end{equation*}\nwhich gives\n\\begin{equation*} \\begin{split}\n \\tau \\sum\\limits_{n=1}^{m} \\|R_1^n\\| \\leq \\tau \\int_0^{2\\tau} \\|u'''(t)\\|dt + \\tau^2 \\int_{\\tau}^{t_{m+1}} \\|u^{(4)}(t)\\|dt.\n \\end{split}\n \\end{equation*}\nTo estimate $R_2^n$ in \\eqref{qq1}, we use the identity \n\\begin{equation*}\n \\begin{split}\n u'(t_n)-\\Bar{\\delta}_tu^n &= \\frac{-1}{4\\tau} \\left[ \\int_{t_n}^{t_{n+1}}(t_{n+1}-t)^2u'''(t)dt + \\int_{t_{n-1}}^{t_{n}}(t-t_{n-1})^2u'''(t)dt\\right],\n \\end{split}\n \\end{equation*}\n and obtain \n\\begin{equation*} \\begin{split}\n \\tau \\sum\\limits_{n=1}^{m} \\left\\|R_2^n \\right\\|\\leq\n \\frac{g_1}{2} \\tau^2 \\int_0^{t_{m+1} }\\|u'''(t)\\|dt.\n \\end{split}\n \\end{equation*}\n However, to estimate $R_3^n$ in \\eqref{qq1}, we use\n \\begin{equation*}\n \\begin{split}\n \\Delta u(t_n) - \\Delta\\widetilde{u}^n & = \\frac{-1}{2} \\Bigg[ \\int_{t_{n}}^{t_{n+1}} (t_{n+1}-t) \\Delta u''(t)dt + \\int_{t_{n-1}}^{t_{n}} (t-t_{n-1}) \\Delta u''(t) dt \\Bigg],\n \\end{split}\n \\end{equation*} \n and get \n\\begin{equation*} \\begin{split}\n \\tau \\sum\\limits_{n=1}^{m} \\|R_3^n\\| \\leq \\tau^2 \\int_0^{t_{m+1} }\\|\\Delta u''(t)\\|dt.\n \\end{split}\n \\end{equation*}\nTo estimate $R_4^n$ in \\eqref{qq4},\nwe follow the procedure in \\cite[52]{McLean} with $\\hat{\\mu}_j = \\int_{t_j}^{t_{j+1}}K(s)ds$ to obtain \n\\begin{equation*}\n \\begin{split}\n \\tau \\sum\\limits_{n=1}^{m} \\|R_4^n\\|\n & \\leq\n \\hat{\\mu}_0 \\tau \\int_0^{\\tau} \\|\\Delta u''(t)\\|dt + \\tau^2 \\sum\\limits_{p=2}^{m}\\left(\\sum\\limits_{n=p}^{m} \\hat{\\mu}_{n-p} \\right) \\int_{t_{p-1}}^{t_p} \\|\\Delta u'''(t)\\|dt \\\\\n & + \\tau \\sum\\limits_{p=1}^{m} \\Delta\\left(u'(t_p)-\\bar{\\delta}_tu^p \\right) \\left( \\sum\\limits_{n=p}^{m} \\widetilde{\\kappa}_{np} \\right).\n \\end{split}\n \\end{equation*}\nThus, since $\\sum\\limits_{n=p}^{m} \\hat{\\mu}_{n-p} \\leq \\int_0^{t_m}K(t)dt \\leq C$ and since $\\sum\\limits_{n=p}^{m} \\widetilde{\\kappa}_{np} \\leq C\\tau \\sum\\limits_{n=p}^{m} \\leq C$ (which is due to \\eqref{coeff}), \n\\begin{equation*} \\begin{split}\n \\tau \\sum\\limits_{n=1}^{m} \\|R_4^n\\|\n & \\leq\n C \\tau \\int_0^{\\tau} \\|\\Delta u''(t)\\|dt + C\\tau^2 \\int_{\\tau}^{t_{m+1}} \\|\\Delta u'''(t)\\|dt. \n \\end{split}\n \\end{equation*}\nFinally, to estimate the first term of the right-hand side of \\eqref{eq7.13}, i.e., $\\|\\xi^0\\|_A$. we subtract \\eqref{ModelE} from \\eqref{eq2.16} and use \\eqref{qq5} to get\n\\begin{equation*} \\begin{split}\n & \\delta_t \\xi^1 = \\delta_t \\eta^1 + (u_{1h}-u_1) + \\frac{\\tau}{2}(u_{2h}-u_2) - \\mathcal{R}^0, \\\\\n & \\nabla \\xi^1 = \\tau \\left[ \\delta_t \\nabla\\eta^1 + (\\nabla u_{1h}-\\nabla u_1) + \\frac{\\tau}{2}(\\nabla u_{2h}-\\nabla u_2) - \\nabla\\mathcal{R}^0 \\right],\n \\end{split}\n\\end{equation*}\nwhich, together with \\eqref{eq6.3}, leads to\n\\begin{equation}\\label{eq7.22}\n \\begin{split}\n \\left\\|\\xi^0 \\right\\|_A & \\leq Ch \\left( \\|u'\\|_{L^{\\infty}(H^1) } + \\|u_1\\|_1 +\\tau \\|u_0\\|_2 \\right) + C\\tau \\int_0^{\\tau} \\| u'''(t)\\|dt.\n \\end{split} \n \\end{equation}\nWe invoke \\eqref{eq7.14}--\\eqref{eq7.22} in \\eqref{eq7.13} and use $\\|\\nabla \\eta(t_m)\\|\\leq Ch\\max\\limits_{0\\leq t \\leq t_m}\\|u(t)\\|_2$ and the discrete Gr\\\"{o}nwall's lemma to obtain \\eqref{eq2.20}. This completes the proof.\n\\end{proof}\n\n\n\\subsection{Discussion on temporal accuracy}\nSince the solution of \\eqref{eq1.1}-\\eqref{eq1.3} may exhibit initial singularity and is in general smooth away from the initial time under certain regularity assumptions on the given data, we focus the attention near the initial time and thus consider \\eqref{eq1.1}-\\eqref{eq1.3} with $q(t)= q(0):=q_0$ for simplicity. Hence, to consider the temporal solution regularity, one could apply the eigenpairs of $-\\Delta$ to decompose the linear version of \\eqref{eq1.1}-\\eqref{eq1.3} as ordinary differential equations of the following form for some $\\lambda>0$: for $t>0$, \n\\begin{equation}\\label{eq7.36}\n u'' + q_0 u' +\\lambda u -\\lambda \\int_0^{t}\\beta(t-s) u(s)ds = f(t), ~~{\\rm with}~~ u(0) = u_0~{\\rm and}~u'(0) = u_1.\n \\end{equation}\nSince \n\\[ u'(t)= u_1 + \\int_0^t u''(s) ds~{\\rm and}~ u(t)= u_0 + tu_1 + \\int_0^t (t-s)u''(s) ds, \\quad t>0,\\]\n\\begin{equation*}\n \\begin{split}\n u''(t) = f(t) & - q_0 \\left[u_1 + \\int_0^t u''(s) ds\\right] - \\lambda \\left[u_0 + tu_1 + \\int_0^t (t-s)u''(s) ds\\right] \\\\\n & + \\lambda \\int_0^{t}\\beta(t-s) \\left[u_0 + su_1 + \\int_0^s (s-z)u''(z) dz \\right]ds, \\quad t> 0.\n \\end{split}\n \\end{equation*}\nThus, for the case (i) in (\\ref{eq1.5}), we have the asymptotic behaviour of $u''$ and $u'''$ as follows \n\\begin{align*}\n u''(t)& \\simeq f(t) - q_0 u_1 - \\lambda u_0 + \\frac{\\lambda u_0}{\\Gamma(1+\\alpha)}t^{\\alpha} + O(t^{\\alpha}), \\quad t\\rightarrow 0^{+},\\\\\n u'''(t)& \\simeq f'(t) - q_0\\, u''(0) - \\lambda u_1 + \\frac{\\lambda u_0}{\\Gamma(\\alpha)}t^{\\alpha-1} + O(t^{\\alpha-1}), \\quad t\\rightarrow 0^{+},\n \\end{align*}\nwhich implies $|u''(t)| \\leq C$ and $|u'''(t)| \\leq C(1 + t^{\\alpha-1})$. Similarly we could get $|u^{(4)}(t)| \\leq C(1 + t^{\\alpha-2})$. For the case (ii) in (\\ref{eq1.5}), where there is no singularity in the kernel, the solutions could be smooth under smooth data. Thus, it is reasonable to assume that the solutions to \\eqref{eq1.1}-\\eqref{eq1.3} satisfy\n\\begin{equation}\\label{regul}\n \\begin{split}\n &t^{\\alpha-1}\\|\\Delta u''(t)\\| +\\|\\Delta u'''(t)\\| \\leq C(1+t^{\\alpha-1}), \\quad \\|u^{(4)}(t)\\| \\leq C(1+t^{\\alpha-2})\\text{ for case } (i), \\\\\n & \\|\\Delta u''(t)\\| + \\|\\Delta u'''(t)\\| + \\|u^{(4)}(t)\\| \\leq C\\text{ for case } (ii). \n \\end{split}\n \\end{equation}\nThen the last three terms on the right-hand side of \\eqref{eq2.20} could be further bounded as \n\\begin{equation*}\n \\begin{split}\n & \\int_0^{2\\tau} \\|u'''(t)\\|dt + \\tau \\int_{\\tau}^{t_{m+1}} \\|u^{(4)}(t)\\|dt \\leq C\\tau^\\alpha, \\\\\n & \\int_0^{t_{m+1} }\\Big(\\|u'''(t)\\| + \\|\\Delta u''(t)\\|\\Big)dt \\leq C, \\\\\n & \\int_0^{\\tau} \\|\\Delta u''(t)\\|dt + C\\tau \\int_{\\tau}^{t_{m+1}} \\|\\Delta u'''(t)\\|dt \\leq C\\tau,\n \\end{split}\n \\end{equation*}\nwhich results in $O(\\tau^{1+\\alpha})$ temporal accuracy with $\\alpha=1$ or $\\alpha=\\frac{1}{2}$.\n\n\n\\begin{remark}\\label{rem2.1}\nMotivated by the above analysis, Theorem \\ref{theorem2.5} indicates the $O(h+\\tau^{1+\\alpha})$ accuracy of the $H^1$-norm error with $\\alpha=1$ or $\\alpha=\\frac{1}{2}$. \n\\end{remark}\n\n\n\n\n\n\\section{Numerical experiment} \\label{sec5}\n\n\\subsection{One-dimensional case}\n Let $\\Omega=(0,1)$, $u_0(x)=\\sin(\\pi x)$, $u_1(x)=\\sin(2\\pi x)$ and $f(x,t)=t^{\\alpha}e^{-\\sigma t} \\cos(\\gamma t) \\sin(\\pi x)$. Let $G(z)=\\sqrt{1+z}$ with $\\mu_1=\\mu_2=1$ in \\eqref{eq1.2} and $h=\\frac{1}{M}$ for some $M>0$. Since the exact solution is unknown, to illustrate numerically the achieved $H^1$-norm convergence rates in Theorem \\ref{theorem2.5}, we define \n\\begin{align*}\n {E_t(M,N)}& =\\sqrt{h\\sum\\limits_{j=1}^{M-1}\\left|V_j^{N+1}-V_j^{2N+1} \\right|^2}, ~ {E_s(M,N)}=\\sqrt{h\\sum\\limits_{j=1}^{M-1}\\left|V_j^{N+1}-V_{2j}^{N+1} \\right|^2},~V_j^n = \\frac{U_j^n-U_{j-1}^n}{h}\\,.\\end{align*}\nHere $U_j^n$ approximates $u(x_j,t_n)$ and hence, $V_j^n$ approximates $\\nabla u (x_j,t_n)$,\n$E_t$ represents the difference between gradients of numerical solutions at time $T$ computed under the time step sizes $\\tau=T/N$ and $\\tau=T/(2N)$, and $E_s$ could be interpreted similarly.\nThe convergence rates in time and space are accordingly defined as\n\\begin{align*}\n CR_t =\\log_{2}\\left(\\frac{E_t(M,N)}{E_t(M,2N)}\\right), \\quad CR_s=\\log_{2}\\left(\\frac{E_s(M,N)}{E_s(2M,N)}\\right).\n\\end{align*}\n\n\nWe fix $T=1$ to evaluate the $H^1$ errors and convergence rates of the fully-discrete Galerkin scheme in Table \\ref{tab1}, which demonstrate the $O(\\tau^{1+\\alpha} + h)$ accuracy as predicted in Theorem \\ref{theorem2.5}. \n\n\n\n\n\\begin{table}[H]\n \\center \\footnotesize\n \\caption{ $H^1$ errors and time-space convergence rates under $T=1$.} \\label{tab1}\n \\vskip 2mm\n \\begin{tabular}{cccccccccccc}\n \\toprule\n & & & \\multicolumn{2}{c}{$N=32$, $\\gamma=3\\sqrt{3}$, $\\sigma=3$} & &\\multicolumn{2}{c}{$M=32$, $\\gamma=1.0$, $\\sigma=2.0$}\\\\\n \\cmidrule{4-5} \\cmidrule{7-8}\n $\\alpha=0.5$ &$M$ & & {$E_s$} & {$CR_s$} & $N$ & {$E_t$} & {$CR_t$} \\\\\n \\midrule\n & $32$ & & $2.4232 \\times 10^{-2}$ & * & $128$ & $2.3145 \\times 10^{-3}$ & * \\\\\n & $64$ & & $1.2798 \\times 10^{-2}$ & 0.92 & $256$ & $7.2638 \\times 10^{-4}$ & 1.67 \\\\\n & $256$ & & $6.5733 \\times 10^{-3}$ & 0.96 & $512$ & $2.3469 \\times 10^{-4}$ & 1.63 \\\\\n & $512$ & & $3.3306 \\times 10^{-3}$ & 0.98 & $1024$ & $7.7536 \\times 10^{-5}$ & 1.60 \\\\\n \\text{Predict} & & & & 1.00 & & & 1.50 \\\\\n \\midrule \n & & & \\multicolumn{2}{c}{$N=32$, $\\gamma=2$, $\\sigma=2$} & &\\multicolumn{2}{c}{$M=32$, $\\gamma=0.5$, $\\sigma=1.1$}\\\\\n \\cmidrule{4-5} \\cmidrule{7-8}\n $\\alpha=1.0$ &$M$ & & {$E_s$} & {$CR_s$} & $N$ & {$E_t$} & {$CR_t$} \\\\\n \\midrule\n & $16$ & & $3.6308 \\times 10^{-2}$ & * & $16$ & $1.2054 \\times 10^{-1}$ & * \\\\\n & $32$ & & $1.8470 \\times 10^{-2}$ & 0.98 & $32$ & $2.8221 \\times 10^{-2}$ & 2.09 \\\\\n & $64$ & & $9.5854 \\times 10^{-3}$ & 0.95 & $64$ & $6.8974 \\times 10^{-3}$ & 2.03 \\\\\n & $128$ & & $4.9102 \\times 10^{-3}$ & 0.97 & $128$ & $1.7148 \\times 10^{-3}$ & 2.01 \\\\\n \\text{Predict} & & & & 1.00 & & & 2.00 \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\nNext, we test the possible energy dissipation of the proposed model. It is shown in \\cite[Page 498]{Cavalcanti1} that the Euler-Bernoulli viscoelastic model, which coincides with the form of model (\\ref{eq1.1}) with $-\\Delta$ replaced by $\\Delta^2$, admits the energy dissipation with the corresponding energy \n\\begin{equation}\\label{mh1}\n\\hat E(t):=\\frac{1}{2} \\|u'(t)\\|^2 + \\frac{1}{2} \\|\\Delta u(t)\\|^2\n\\end{equation}\nMotivated by this definition, we define the following energy for problem \\eqref{eq1.1} with $f=0$ by \n\\begin{equation}\\label{energy1}\n \\begin{split}\n \\widetilde{E}(t) = \\frac{1}{2} \\|u'(t)\\|^2 + \\frac{1}{2} \\|\\nabla u(t)\\|^2, \\quad t\\geq 0,\n \\end{split}\n\\end{equation}\nand we numerically evaluate this energy as follows \n\\begin{equation*}\n \\begin{split}\n \\widetilde{E}^n = \\frac{1}{2} \\|\\bar{\\delta}_tU^n\\|^2 + \\frac{1}{2} \\|\\nabla U^n\\|^2, \\quad n\\geq 1, \\quad \\widetilde{E}^0 = \\frac{1}{2} \\|u_1\\|^2 + \\frac{1}{2} \\|\\nabla u_0\\|^2.\n \\end{split}\n\\end{equation*}\nNumerical results are presented in Figure \\ref{fig1}, which indicate the dissipative property of the energy for $\\alpha=0.5$ or $1$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=3.8in]{Graph1.eps}\n\\caption{Plots of the energy under $f=0$, $\\sigma=3$, $\\gamma=3\\sqrt{3}$, and $N=M=32$.}\n\\label{fig1}\n\\end{figure}\n\n\n\\subsection{Two-dimensional case} \nHere, let $\\Omega=(0,1)\\times (0,1)$, $u_0(x,y)=\\sin(\\pi x)\\sin(\\pi y)$, $u_1(x,y)=\\sin(2\\pi x)\\sin(2\\pi y)$, and $f(x,y,t)=0$. Let $G(z)=\\sqrt{1+z}$ with $\\mu_1=\\mu_2=1$ in \\eqref{eq1.2} and $h_x=h_y=h=\\frac{1}{M}$ for some $M>0$. In two-dimensional case, to illustrate numerically the achieved $H^1$-norm convergence rates in Theorem \\ref{theorem2.5}, we denote\n\\begin{align*}\n {E_t(M,N)}& =h\\,\\sqrt{\\sum\\limits_{i=1}^{M-1}\\sum\\limits_{j=1}^{M-1}\\left|W_{i,j}^{N+1}-W_{i,j}^{2(N+1)} \\right|^2}, \\quad {E_s(M,N)}=h\\,\\sqrt{\\sum\\limits_{i=1}^{M-1}\\sum\\limits_{j=1}^{M-1}\\left|W_{i,j}^{N+1}-W_{2i,2j}^{N+1} \\right|^2}, \\\\\n W_{i,j}^n & = \\sqrt{\\left(\\frac{U_{i,j}^n-U_{i-1,j}^n}{h}\\right)^2+\\left(\\frac{U_{i,j}^n-U_{i,j-1}^n}{h}\\right)^2}.\n\\end{align*}\n\nIn Table \\ref{tab3-1}, we test the $H^1$ errors and convergence orders of the fully discrete Galerkin scheme for the two-dimensional case. We observe that the numerical results are consistent with our theoretical analysis (see Remark \\ref{rem2.1}).\n\n\n\\begin{table}\n \\center \\footnotesize\n \\caption{ $H^1$ errors and time-space convergence rates under $T=1/2$.} \n \\label{tab3-1}\n \\vskip 2mm\n \\begin{tabular}{cccccccccccc}\n \\toprule\n & & & \\multicolumn{2}{c}{$N=16$, $\\gamma=3\\sqrt{3}$, $\\sigma=3$} & &\\multicolumn{2}{c}{$M=64$, $\\gamma=0.5$, $\\sigma=1.5$}\\\\\n \\cmidrule{4-5} \\cmidrule{7-8}\n $\\alpha=0.5$ &$M$ & & {$E_s$} & {$CR_s$} & $N$ & {$E_t$} & {$CR_t$} \\\\\n \\midrule\n & $64$ & & $6.9769 \\times 10^{-3}$ & * & $64$ & $4.2265 \\times 10^{-3}$ & * \\\\\n & $128$ & & $3.3304 \\times 10^{-3}$ & 1.07 & $128$ & $1.3363 \\times 10^{-3}$ & 1.66 \\\\\n & $256$ & & $1.6259 \\times 10^{-3}$ & 1.03 & $256$ & $4.3567 \\times 10^{-4}$ & 1.62 \\\\\n & $512$ & & $8.0321 \\times 10^{-4}$ & 1.02 & $512$ & $1.4597 \\times 10^{-4}$ & 1.58 \\\\\n \\text{Predict} & & & & 1.00 & & & 1.50 \\\\\n \\midrule \n & & & \\multicolumn{2}{c}{$N=16$, $\\gamma=2$, $\\sigma=2$} & &\\multicolumn{2}{c}{$M=64$, $\\gamma=0.5$, $\\sigma=1.1$}\\\\\n \\cmidrule{4-5} \\cmidrule{7-8}\n $\\alpha=1.0$ &$M$ & & {$E_s$} & {$CR_s$} & $N$ & {$E_t$} & {$CR_t$} \\\\\n \\midrule\n & $64$ & & $8.9036 \\times 10^{-3}$ & * & $32$ & $9.9870 \\times 10^{-3}$ & * \\\\\n & $128$ & & $4.3094 \\times 10^{-3}$ & 1.05 & $64$ & $2.4006 \\times 10^{-3}$ & 2.06 \\\\\n & $256$ & & $2.1189 \\times 10^{-3}$ & 1.02 & $128$ & $5.1903 \\times 10^{-4}$ & 2.21 \\\\\n & $512$ & & $1.0505 \\times 10^{-3}$ & 1.01 & $256$ & $1.2184 \\times 10^{-4}$ & 2.09 \\\\\n \\text{Predict} & & & & 1.00 & & & 2.00 \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\section{Concluding remarks} \\label{sec6}\n\nThis work performs numerical analysis for a hyperbolic integrodifferential equation. The developed techniques in this work overcome the difficulties caused by the non-positive variable-sign kernel and nonlinear-nonlocal damping, which could also be extended to investigate other related problems such as the Euler-Bernoulli viscoelastic problem in \\cite{Cavalcanti1} that coincides with the form of model (\\ref{eq1.1}) with $-\\Delta$ replaced by $\\Delta^2$, or (\\ref{eq1.1}) with a different damping coefficient \\cite{Xu3}\n \\begin{equation*}\n q(t) = G\\left( \\int_0^t \\int_{\\Omega}\\mu_1| u(x,s)|^2+\\mu_2|\\nabla u(x,s)|^2dxds \\right), \\;\\; \\mu_1, \\mu_2 \\geq 0, \\;\\; \\mu_1^2+ \\mu_2^2 \\neq 0.\n \\end{equation*}\n In particular, it is shown in \\cite[Theorem 2.1]{Cavalcanti1} that if the kernel function $\\beta(t)$ satisfies the assumption ($\\mathbf{A1}$) and\n \\begin{align}\n &\\qquad\\qquad \\beta(0)>0, \\quad 1-\\int_{0}^{\\infty}\\beta(s)ds = \\ell >0, \\label{ass1} \\\\\n & -c_1\\beta(t) \\leq \\beta'(t) \\leq -c_2 \\beta(t), \\quad 0\\leq \\beta''(t) \\leq c_3 \\beta(t) \\quad \\forall t\\geq 0, \\nonumber \n \\end{align}\n where $c_1,c_2,c_3$ are positive constants, the energy (\\ref{mh1}) decays exponentially in time. However, since the kernel $\\beta(t)$ in this work has a variable sign and initial singularity, it does not satisfy the conditions in \\eqref{ass1} such that it is difficult to follow the derivations in \\cite{Cavalcanti1} to derive the energy decay. Nevertheless, the numerical experiment suggests the energy decay as shown in Figure \\ref{fig1}, which motivates us to perform a further investigation for this issue.\n\n\nFinally, in the current work we only prove the error estimate in the $H^1$ norm. Due to the existence of the gradient term in $q(t)$, which may limit the improvement of the convergence order in deriving the $L^2$ error, it is not straightforward to obtain the optimal $L^2$ error estimate in the current circumstance. We will investigate this interesting question in the near future.\n\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{99}\n\n\n\\small\n\n \\bibitem{Alabau-Boussouira} F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342--1372.\n\n \\bibitem {Allegretto} W. Allegretto and Y. Lin, Numerical solutions for a class of differential equations in linear viscoelasticity, Calcolo, 30 (1993), 69--88.\n\n \\bibitem {Appleby} J. A. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677--1694.\n\n\n \\bibitem {Baker1} K. Baker and L. Banjai, Numerical analysis of a wave equation for lossy media obeying a frequency power law, IMA J. Numer. Anal., 42 (2022), 2083--2117.\n\n \\bibitem {Baker2} K. Baker, L. Banjai and M. Ptashnyk, Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping, Math. Comp., 93 (2024), 2711--2743.\n\n \\bibitem {Cavalcanti1} M. M. Cavalcanti, V. N. Domingos Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differ. Integral Equ., 17 (2004), 495--510.\n\n\n \\bibitem {Cannarsa1} P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equ., 250 (2011), 4289--4335.\n \n \\bibitem {Cannarsa} P. Cannarsa and D. Sforza, A stability result for a class of nonlinear integrodifferential equations with $L^1$ kernels, Appl. Math., 35 (2008), 395--430.\n\n \\bibitem {Cannon} J. R. Cannon, Y. Lin and C. Xie, Galerkin Methods and $L^2$-error estimates for hyperbolic integrodifferential equations, Calcolo, 26 (1989), 197--207.\n\n\n\\bibitem{Emm} E. Emmrich and M. Thalhammer, A class of integrodifferential equations incorporating nonlinear and nonlocal damping with applications in nonlinear elastodynamics: Existence via time discretization, Nonlinearity, 24 (2011), 2523--2546.\n\n\n \\bibitem {Fairweather} G. Fairweather, Spline collocation methods for a class of hyperbolic partial integrodifferential equations, SIAM J. Numer. Anal., 31 (1994), 444--460.\n\n \\bibitem {Georgiev} V. Georgiev, B. Rubino and R. Sampalmieri, Global existence for elastic waves with memory, Arch. Rational Mech. Anal., 176 (2005), 303--330.\n\n\n \\bibitem {Karaa} S. Karaa, A. K. Pani and S. Yadav, A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integrodifferential equations, Appl. Numer. Math., 96 (2015), 1--23.\n\n \\bibitem {Larsson} S. Larsson and F. Saedpanah, The continuous Galerkin method for an integrodifferential equation modeling dynamic fractional order viscoelasticity, IMA J. Numer. Anal., 30 (2010), 964--986.\n\n \\bibitem {Lebon} G. Lebon, C. Perez-Garcia and J. Casas-Vazquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068--5075.\n\n \n \\bibitem {McLean} W. McLean and V. Thom\\'{e}e, Numerical solution of an evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B, 35 (1993), 23--70.\n\n \\bibitem {McLean1} W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105 (2007), 481--510. \n\n \\bibitem {Messaoudi1} S. A. Messaoudi and B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277--287.\n\n \\bibitem {Messaoudi2} S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nach., 260 (2003), 58--66.\n \n \\bibitem {Mustapha} K. Mustapha and H. Mustapha, A second-order accurate numerical method for a semilinear integrodifferential equation with a weakly singular kernel, IMA J. Numer. Anal., 30 (2010), 555--578.\n\n\n \\bibitem {Renardy} M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematics Problems in Viscoelasticity, in: Pitman Monographs Pure Appl. Math., vol. 35, Longman Sci. Tech., Harlow, Essex, 1988.\n\n \\bibitem {Saedpanah} F. Saedpanah, Continuous Galerkin finite element methods for hyperbolic integro-differetial equations, IMA J. Numer. Anal., 35 (2015), 885--908.\n\n \\bibitem {Thomee} V. Thom\\'{e}e, Galerkin finite element methods for parabolic problems, Springer, 2007.\n\n \\bibitem {Xu2} D. Xu, Decay properties for the numerical solutions of a partial differential equation with memory, J. Sci. Comput., 62 (2015), 146--178.\n\n \\bibitem {Xu3} D. Xu, Analytical and numerical solutions of a class of nonlinear integrodifferential equations with L1 kernels, Nonlinear Anal.-Real, 51 (2020), 103002.\n\n \\bibitem {Xu4} D. Xu, Application of the Crank–Nicolson time integrator to viscoelastic wave equations with boundary feedback damping, IMA J. Numer. Anal., 42 (2022), 487--514.\n \n \\bibitem {Xu5} D. Xu, Finite element methods of the two nonlinear integrodifferential equations, Appl. Math. Comput., 58 (1993), 241--273.\n\n \\bibitem {Yanik} E. G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integrodifferential equations, Nonlinear Anal., 12 (1988), 785--809.\n\n \\bibitem {Zhao} M. Zhao and W. Qiu, Finite element approximation and analysis of damped viscoelastic hyperbolic integrodifferential equations with $L^1$ kernel, Appl. Math. Lett., 151 (2024), 108993.\n\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":83,"cells":{"id":{"kind":"string","value":"2412.07452v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07452v1"},"title":{"kind":"string","value":"On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature"},"full_text":{"kind":"string","value":"\\documentclass[12pt,twoside]{article} \t\\usepackage{geometry} \t\t\\geometry{left=3.5cm,right=3.5cm,top=3.5cm,bottom=3.5cm} \t\t\n\\usepackage{graphicx,subcaption}\t\t\t\t\t\n\\usepackage{amssymb}\n\\usepackage{indentfirst}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{bm}\n\\usepackage{changepage}\n\\usepackage{lineno}\n\\usepackage{setspace}\n\\usepackage{booktabs,multirow}\n\\usepackage{authblk}\n\\usepackage{graphicx}\n\\usepackage{float}\n\\usepackage[flushleft]{threeparttable} \\usepackage{tikz-cd}\n\\usepackage{cite}\n\\usepackage{url}\n\\usepackage{lipsum}\n\\usepackage[marginal]{footmisc}\\usepackage{fancyhdr}\n\\usepackage[colorlinks = true,\n linkcolor = blue,\n urlcolor = blue,\n citecolor = blue,\n anchorcolor = blue]{hyperref}\n\\renewenvironment{abstract}{ \\setlength\\textwidth{3in}\n \\begin{quote}\n}{ \\end{quote}}\\renewcommand{\\thefootnote}{}\n\n\n\\numberwithin{equation}{section}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{claim}{Claim}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\newcommand\\keywords[1]{\\textbf{ Keywords:}#1}\n\n\n\\title{On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature}\n\\author{Yuxing $\\text{Deng}^*$ and~~Yuehan~Hao }\n\n\n\n\n\n\n\\date{}\t\t\t\t\t\t\t\n\n\n\\begin{document}\n\n\t\\maketitle\n \\pagestyle{fancy} \\fancyhf{} \\fancyhead[OC]{\\small FUNDAMENTAL GROUP OF STEADY GRADIENT RICCI SOLITONS} \\fancyhead[EC]{\\small YUXING DEND~AND~YUEHAN~HAO} \\renewcommand{\\headrulewidth}{0pt} \\fancyfoot[c]{\\small \\thepage}\n \\footnote{ \\keywords{ Ricci flow, Ricci soliton, fundamental group}\\\\\n $^{*}$Supported by National Key R$\\&$D Program of China 2022YFA1007600.\n \n \n }\n\t\\begin{abstract}\n ABSTRACT. In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we show that an $n$-dimensional complete $\\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be diffeomorphic to $\\mathbb{R}^n$. \n \n \\end{abstract}\n \n\t\n\t\n\t\n\t\n\t\\section{Introduction} \t\\subsection{Introduction}\n\t\n\n\t\n\t A \\textbf{Ricci soliton} is a quadruple $(M ,g,X,\\lambda)$ consisting of a smooth manifold $M $, a Riemannian metric $g$, a smooth vector field $X$ and a real constant $\\lambda$, if these variables satisfy the equation\n\t\\begin{equation}\n\t\t\\label{ricci soliton definition}\n\t\t{\\rm Ric}+\\mathcal{L}_Xg=\\frac{\\lambda}{2}g\n\t\\end{equation}\n\ton $M$, where ${\\rm Ric}$ denotes the Ricci tensor of $g$, and $\\mathcal{L}$ denotes the Lie derivative. \n\tFor $\\lambda =0$ the Ricci soliton is \\textbf{steady}, for $\\lambda >0$ it is \\textbf{shrinking} and for $\\lambda <0$ \\textbf{expanding}.\n\tUp to scaling, we can normalize $\\lambda = 0,1,$ or $-1$, respectively.\n\t\n\n\t\n\t\n\tIf the vector field $X= \\nabla f$ for some smooth function $f$ on $M $, then the Ricci soliton is called a \\textbf{gradient Ricci soliton}.\n\tFor such a soliton, equation (\\ref{ricci soliton definition}) simplifies to \n\t\\begin{equation}\n\t\t\\label{ric+downtri f=}\n\t\t{\\rm Ric} +\\nabla^2f=\\frac{\\lambda}{2} g,\n\t\\end{equation}\n\tsince $\\mathcal{L}_{\\nabla f} g = 2 \\nabla^2 f$. Here $\\nabla^2$ denotes the Hessian of $f$. \n\n\t\n\t\n\t\n\t\n\tIn this paper, We will use the notation $(M,g,f, \\lambda)$ to denote a gradient Ricci soliton. If we can determine the \\textbf{scale} $\\lambda$ and the \\textbf{potential function} $f$ from the context, then the underlying manifold $(M,g)$ will be often referred to as the Ricci soliton. In addition to certain metric $g$ of $M$, we simplify the notation to $M$.\n \n Since a Ricci soliton that arises as the blow-up limit of a compact Ricci flow is $\\kappa$-noncollpased. We are interested in $\\kappa$-noncollapsed Ricci solitons. The notion of $\\kappa$-noncollapsing is defined as follows. \n \\begin{definition}[Noncollapsed manifolds]\n\t\t\\label{noncollapsed}\n\t\tLet $\\kappa, \\rho>0 $. We say that a Riemannian manifold $\\left(\\mathcal{M}, g\\right)$ is $\\mathbf{\\kappa }$\\textbf{-noncollapsed on the scale} $\\rho$ if for any $ x \\in \\mathcal{M}$ and $ r \\in(0, \\rho)$ satisfying $R \\leq r^{-2} $ in $B_{r}(x) $, we have\n\t\t\\begin{equation}\n\t\t\t{\\rm Vol } B_{r}(x) \\geq \\kappa r^{n},\n\t\t\\end{equation}\n\t\twhere $B_r(x)$ is the geodesic ball with the radius $r$ based at $x$.\n\t\t\n\t\tIf $\\left(\\mathcal{M} , g\\right) $ is $\\kappa $-noncollapsed on the scale $\\rho$ for all $ \\rho \\in(0, \\infty) $, we say that $ \\left(\\mathcal{M} , g\\right) $ is $\\kappa $\\textbf{-noncollapsed on all scales}.\n\t\\end{definition}\n\t\n\t\n\n Ricci solitons play an important role in the singularity analysis of the Ricci flow. Ricci flows generated by shrinking and steady Ricci solitons are special ancient solutions to the Ricci flow. $\\kappa$-solutions are $\\kappa$-noncollapsed ancient solutions. The classification of $\\kappa$-solutions with positive curvature may rely on the classification of shrinking and steady gradient Ricci with positive curvature, especially the rotational symmetry of positively curved steady gradient Ricci solitons. For recent progress on the classification of $\\kappa$-solutions to the Ricci flow, see \\cite{Cho-Li-3023,Li-2020-arXiv,Brendle-2020,Brendle-Daskalopoupos-Sesum-2021,Brendle_singularity_models_2022,Brendle_Daskalopoulos_Naff_Sesum_2023,Brendle_Naff_2023,Deng-Zhu-PAMS-2020,Li-Zhang-2022,cao-xie-2024-arxiv}. In dimension 3, the uniqueness of $\\kappa$-noncollapsed and non-flat steady gradient Ricci solitons was claimed by Perelman in \\cite{perelman_entropy_2002}. Perelman's claim was confirmed by Brendle in \\cite{brendle_rotational_2013}. There are also many works on the higher dimensional $\\kappa$-noncollapsed steady gradient Ricci solitons, such as \\cite{Deng-Zhu-2018,Cao_chen_2012,brendle_rotational_2014,Deng-Zhu-Mathann-2020,Deng-Zhu-SCM-2020,Deng-Zhu-JEMS-2020,Bamler-Chow-Deng-Ma-2022,Chow-Deng-Ma-2022,chan_dimension_2023,zhao-zhu-arxiv-2022,zhao-zhu-arxiv-2024,ma-Mahmoudian-Sesum-arxiv-2023}. Many new examples of steady gradient Ricci solitons have also been found, see \\cite{Lai-JDG-2024,appleton_family_2022,conlon-Deruelle-arxiv-2020,buttsworth-2021-arxiv,Buzano-Dancer-Wang-2015,wink-2023} and references therein. Due to the steady Ricci solitons constructed by Yi Lai \\cite{Lai-JDG-2024}, 4-dimensional $\\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature is not unique. How to classify these Ricci solitons is still an open problem. Even the topology of 4-dimensional $\\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature has not been classified yet.\n\n It is well-known that the topology of manifolds with nonnegative sectional curvature has been studied by Cheeger and Gromoll in \\cite{Cheeger-Gromoll-1972}. A manifold with nonnegative sectional curvature must be a normal bundle over a soul\\cite{Cheeger-Gromoll-1972,sarafutdinov-1979}. For the expanding case, an expanding gradient Ricci soliton with nonnegative Ricci curvature must be diffeomorphic to the Euclidean space, see Lemma 5.5 in \\cite{Cao-Catino-Chen-Mantegazza-Mazzieri-2014} or Lemma 2.2 in \\cite{Chen-Deruelle-2015}. For the shrinking case, the topology of a shrinking gradient Ricci soliton with nonnegative sectional curvature is still unknown. It is open whether a 4-dimensional compact shrinking gradient Ricci soliton with positive sectional curvature is diffeomorphic to a quotient of $\\mathbb{S}^4$ or $\\mathbb{CP}^2$. There are also many nontrivial noncompact shrinking gradient Ricci solitons with nonnegative sectional curvature and finite fundamental groups, such as $(\\mathbb{S}^2\\times \\mathbb{R})/\\mathbb{Z}_2$. In this paper, we study the topology of steady gradient Ricci solitons with nonnegative sectional curvature. We get the following result on the fundamental group of steady gradient Ricci solitons.\n\n \\begin{theorem}\n \t \\label{MAIN}\n\t\tSuppose that $(M,g,f)$ is an $n$-dimensional complete steady gradient Ricci soliton with nonnegative sectional curvature. Then the fundamental group $\\pi_1(M)$ of $M$ is either trivial or infinite. \n\n Moreover, if $\\pi_1(M)$ is trivial, then $M$ is diffeomorphic to $\\mathbb{R}^n$.\n\t\\end{theorem}\n\n Bamler\\cite{bamler_fundamental_2021} has proved that any $\\kappa$-noncollapsed ancient soliton to the Ricci flow must have a finite fundamental group.\n\t\\begin{theorem}[Theorem 1.1. \\cite{bamler_fundamental_2021}]\n\t\t\\label{bamler_noncollapsed finite fundamental group_theorem}\n\t\tLet $(M,(g_t)_{t \\le 0})$ be a $\\kappa$-noncollapsed ancient Ricci flow that has complete time-slices and bounded curvature on compact time-intervals. Then the fundamental group $\\pi_1(M)$ is finite.\n\t\\end{theorem}\n\n As an application of Bamler's finiteness theorem and Theorem \\ref{MAIN}. We have\n\n \\begin{theorem}\\label{MAIN-2}\n\t\tAn $n$-dimensional $\\kappa$-noncollapsed complete steady gradient Ricci soliton with nonnegative sectional curvature is diffeomorphic to $\\mathbb{R}^n$.\n\t\\end{theorem}\n\n By Theorem \\ref{MAIN-2}, we have the following refinement of Corollary 1.2 by Cao and Xie in \\cite{cao-xie-2024-arxiv}.\n \\begin{corollary}\n Let $(M^4,g,f)$ be a $4$-dimensional complete noncompact, non-flat, $\\kappa$-noncollapsed steady gradient Ricci soliton with nonnegative isotropic curvature. Then, either\n \n\t\t\t(i) $(M^4,g,f)$ has positive isotropic curvature and positive Ricci curvature, or \n \n\t\t\t(ii) $(M^4,g,f)$ is isometric to the product $N^3\\times\\mathbb{R}$ of the 3D Bryant soliton with the real line. \n\t\t\n \\end{corollary}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\t\n\t\n\t\\subsection{Outline}\n\tWe now outline the main steps involved in the proof of the main result. \n\t\n\n\n\n\n \n\tIn Section 2, we briefly review group actions\n and the geometry of the universal Riemannian covering \n of manifolds. \n Particularly, each compact subgroup of the affine group acting on affine spaces \n has a fixed point in this space\\cite{wolf_spaces_2011}.\n\tMoreover, \n covering automorphism groups, \n consisting of fixed-point-free isometries, \n are closely associated with the fundamental groups\n \\cite{lee_introduction_2011, lee_introduction_2012,petersen_riemannian_2006}. \n\n\n\n\n\n \n\tIn Section 3, \n\twe prove that the isometry groups of universal covers based on gradient Ricci solitons with nonnegative sectional curvature preserve the splitting property argued in \\cite{guan_rigidity_nodate}. \n\n\n\n\n \n\tIn Section 4, in addition to positive Ricci curvature, \n we argue that such solitons are simply connected. \n Furthermore, these solitons do not have \n non-trivial finite quotient manifolds.\n\n\n\n\n\n\n\n \n \n In Section 5, we present the proof of Theorem \\ref{MAIN}. \n Suppose that $M$ is a steady gradient Ricci soliton with nonnegative sectional curvature.\n Applying the splitting theorem in \\cite{guan_rigidity_nodate}, \n there is a universal Riemannian covering map \n $\\pi :N^k \\times \\mathbb{R}^{n-k} \\rightarrow M$, \n where the Ricci curvature on $N^k$ is strictly positive. \n Then we analyze the behaviour of components of a covering automorphism \n acting on $N^k$ and $\\mathbb{R}^{n-k}$ separately. \n We shall show that if its covering automorphism group is not trivial, \n then it must be infinite. \n\n\n\n\n\n\n\n\n \n\t\\section{Preliminaries}\n\t\n\tIn this section, we recall some results concerning the group actions on manifolds and Riemannian universal coverings. \n\n \\subsection{Group actions on manifolds}\n \\begin{definition}\t\tIf $G$ is a group and $M$ is a set, a \\textbf{left action of $\\mathbf{G}$ on $\\mathbf{M}$} is a map $G \\times M \\rightarrow M$, often written as $(g,x) \\mapsto g \\cdot x$, which satisfies\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] $g_1\\cdot( g_2 \\cdot x )= (g_1g_2) \\cdot x$ for all $g_1, g_2 \\in G$ and $x \\in M$; and \n\t\t\t\\item[(ii)] $e \\cdot x =x$ for all $x \\in M$. \n\t\t\\end{itemize}\n\t\\end{definition}\n\tFor any $x \\in M$, the set of images of $x$ under all elements of the group \n\t$G \\cdot x = \\{g\\cdot x : g \\in G\\} \\subseteq M$ is called the \\textbf{orbit} of $x$.\n\tA \\textbf{topological group} is a group $G$ with a topology such that the multiplication $ G \\times G \\rightarrow G$, $(x,y) \\mapsto xy$ and the inverse $ G \\rightarrow G$, $x \\mapsto x^{-1}$ are continuous.\n\tA \\textbf{Lie group} is a smooth manifold equipped with a group structure such that \n\tthe multiplication and inversion maps are both smooth. \n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\tThe following terminology regarding group actions are standard, see\\cite{lee_introduction_2011,lee_introduction_2012,lee_introduction_2018}.\n\t\\begin{definition}\n\t\t\\label{group terminology}\n\t\tSuppose that there is a left action of a group $G$ on a set $M$. \n\t\t\\begin{itemize}\n\t\t\t\\item The action is said to be \\textbf{free} if the only element of $G$ that fixes any point in $M$ is the identity; \n\t\t\tthat is, \n\t\t\t$g \\cdot x \\not= x$ for $g \\in G $ and $x \\in M $ except $g = e$,\n\t\t\twhere $e$ is the identity element of $G$.\n\t\t\t\n\t\t\t\\item If $M$ is a smooth manifold, $G$ is a Lie group, and \n\t\t\tthe defining map $G \\times M \\rightarrow M$ is smooth, \n\t\t\tthen the action is said to be a \\textbf{smooth action}.\n\t\t\t\\item A continuous left action of a Lie group $G$ on a smooth manifold $M$ is said to be a \\textbf{proper action} if \n\t\t\tthe map $G \\times M \\rightarrow M \\times M$ given by \n\t\t\t$(g,x) \\mapsto (g\\cdot x, x)$ is a proper map; \n\t\t\tthat is, the pre-image of each compact set is compact.\n\t\t\t\\item If in addition $M$ is equipped with a Riemannian metric, the action is said to be an \\textbf{isometric action} if the map $x \\mapsto \\varphi \\cdot x$ is an isometry for each $\\varphi \\in G$.\n\t\t\\end{itemize}\n\t\\end{definition}\n\t\n\t\n\tWe consider the Lie groups with the properties above due to the following theorem.\n \\begin{theorem}[Theorem 21.10. \\cite{lee_introduction_2012}]\\label{theorem-manifold/action is a manifold}\n Suppose $G$ is a Lie group acting smoothly, freely and properly on a smooth manifold $M$.\n Then the orbit space $M/G$ is a topological manifold of dimension equal to ${\\rm dim}M- {\\rm dim}G$ and has a unique smooth structure with the property that the quotient map $\\pi: M \\rightarrow M/G$ is a smooth submersion.\n \\end{theorem}\n\t The following lemma gives two alternative characterizations of proper actions. \n\t\\begin{lemma}[Proposition 21.5. \\cite{lee_introduction_2012}]\n\t\t\\label{proper_equivalence}\n\t\tLet $M$ be a smooth manifold, and let $G$ be a Lie group acting continuously on $M$. The following are equivalent:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] The action is proper.\n\t\t\t\\item[(b)] If $\\{x_i\\}$ is a sequence in $M$ \n\t\t\tand $\\{g_i\\}$ is a sequence in $G$ \n\t\t\tsuch that both $\\{x_i\\}$ and $\\{ g_i \\cdot x_i \\}$ converge, \n\t\t\tthen a subsequence of $\\{g_i\\}$ converges.\n\t\t\t\\item[(c)] For every compact subset $K \\subseteq M$, the set $G_K=\\{g \\in G: (g \\cdot K) \\cap K \\not= \\emptyset \\}$ is compact.\n\t\t\\end{itemize}\n\t\\end{lemma}\n \n By Lemma \\ref{proper_equivalence}, it is easy to check that if $G$ acts properly on $M$, then a finite subgroup of $G$ also acts properly on $M$. Hence, we have the following corollary.\n\n \\begin{corollary}\n \\label{corollary-finite subgroup inherited smooth proper free}\n Suppose a Lie group $G$ acts smoothly, freely and properly on a smooth manifold $M$. If $H$ is a finite subgroup of $G$, then $H$ also acts smoothly, freely and properly on $M$.\n \\end{corollary}\n \\begin{proof}\n By definition, it is easy to check that $H$ acts freely and smoothly on $M$. Since $H$ is finite, any subset of $H$ is also finite. For any compact set $K \\subseteq M$, $H_K\\subseteq H$ is finite and therefore is compact. By Lemma \\ref{proper_equivalence} (c), $H$ also acts properly.\n \\end{proof}\n\n \tIn the end of this subsection, we recall from \\cite{wolf_spaces_2011} the characterization of compact group actions on affine spaces.\n\t\\begin{lemma}[Lemma 3.1.2 \\cite{wolf_spaces_2011}]\n\t\t\\label{constant_compact affine subgroup has a fixed point_lemma}\n\t\tLet $A(n)$ be the affine group acting on affine spaces $A^n$, and let $B$ be a compact subgroup of $A(n)$. \n\t\tThen $B$ has a fixed point on $A^n$.\n\t\\end{lemma}\n \n \\subsection{Riemannian covering}\n Given a Riemannian covering map $\\pi: (\\tilde{M}, \\tilde{g}) \\rightarrow (M,g)$. There is an important Lie group which acts on $\\tilde{M}$ and is closely related to $\\pi$. \n\t\n\t\n\t\n\t\n \n \\begin{definition}[Covering automorphism groups of Riemannian covering maps]\n\t\t\\label{Covering Automorphism Groups}\n\t\tLet $(\\tilde{M}, \\tilde{g})$ and $(M, g )$ be Riemannian manifolds and $\\pi: \\tilde{M} \\rightarrow M$ be a Riemannian covering map. A \\textbf{covering automorphism} (or \\textbf{deck transformation}) of $\\pi$ is an isomorphism $\\varphi: \\tilde{M} \\rightarrow \\tilde{M}$ such that $\\pi \\circ \\varphi = \\pi$; \n\t\tthe set of all covering automorphisms is a group acting on $\\tilde{M}$ on the left, under composition of isomorphisms, \n\t\tcalled the \\textbf{covering automorphism group} (or \\textbf{deck transformation group}) and denoted by ${\\rm Aut}_{\\pi}(\\tilde{M})$.\n\t\\end{definition}\n\t\n\t\n\tLet $(M,g)$ be a Riemannian manifold. An isometry from $(M,g)$ to itself is called an \\textbf{isometry} of $(M,g)$. \n\tThe set of all isometries of $(M,g)$ is a group under compositions, called the \\textbf{isometry group } of $(M,g)$, denoted by ${\\rm Iso}(M,g)$. We may also denote ${\\rm Iso}(M,g)$ by ${\\rm Iso}(M)$ for convenience if it will not cause misunderstanding. Given a Riemannian covering map $\\pi: (\\tilde{M},\\tilde{g}) \\rightarrow (M,g)$ with $\\tilde{g}=\\pi^{\\ast}g$. Obviously, the covering automorphism group ${\\rm Aut}_{\\pi}(\\tilde{M})$ is a subgroup of the isometry group ${\\rm Iso}(\\tilde{M},\\tilde{g})$. In general, a subgroup of ${\\rm Iso}(\\tilde{M},\\tilde{g})$ may not act smoothly, freely and properly on $\\tilde{M}$. However, ${\\rm Aut}_{\\pi}(\\tilde{M})$ does.\n\t\n\n\t\n\t\n\t\n\t\\begin{proposition}[\\cite{lee_introduction_2012,petersen_riemannian_2006}]\n\t\t\\label{Aut:smooth+free+proper+isometrical proposition}\n\t\tLet $\\pi: (\\tilde{M}, \\tilde{g}) \\rightarrow (M , g )$ be a Riemannian covering map over a Riemannian manifold $M$. ${\\rm Aut}_{\\pi}(\\tilde{M})$ is a discrete Lie group acting smoothly, freely, properly and isometrically on $\\tilde{M}$.\n\t\\end{proposition}\n\n ${\\rm Aut}_{\\pi}(\\tilde{M})$ is closely related to the fundamental groups of $\\tilde{M}$ and $M$. See the covering automorphism group structure theorem below. \n\t\n\t\n\t\\begin{theorem}[Theorem 12.7. \\cite{lee_introduction_2011}]\n\t\t\\label{Covering Automorphism Group Structure Theorem}\n\t\tSuppose that $\\pi : \\tilde{M} \\rightarrow M$ is a covering map, \n\t\t$\\tilde{x} \\in \\tilde{M}$, \n\t\tand $x= \\pi(\\tilde{x})$. \n\t\t Then, we have:\n\t\t\\begin{equation}\\label{formula-structure theorem}\n\t\t\t{\\rm Aut}_{\\pi}(\\tilde{M}) \\cong \\frac{ {\\rm N}_{\\pi_1(M,x)}(\\pi_*\\pi_1(\\tilde{M},\\tilde{x})) }{ \\pi_*\\pi_1(\\tilde{M},\\tilde{x})},\n\t\t\\end{equation} \n\t\twhere $\\pi_* : \\pi_1(\\tilde{M} , \\tilde{x}) \\rightarrow \\pi_1(M, x)$ denotes the induced homomorphism by $\\pi$. \n\t\\end{theorem}\n\t\n\t\n\t${\\rm N}_{\\pi_1(M,x)}(\\pi_*\\pi_1(\\tilde{M},\\tilde{x}))$ in (\\ref{formula-structure theorem}) is the normalizer of $\\pi_*\\pi_1(\\tilde{M},\\tilde{x})$ in $\\pi_1(M,x)$. If \n$\\pi_1(\\tilde{M},\\tilde{x})$ is trivial, then ${\\rm N}_{\\pi_1(M,x)}(\\pi_*\\pi_1(\\tilde{M},\\tilde{x}))=\\pi_1(M,x)$. Consequently,\n \n \\begin{theorem}[Corollary 12.9. \\cite{lee_introduction_2011}]\n \\label{theorem-structure theorem on universal covering map}\n\t\tSuppose that $\\pi : \\tilde{M} \\rightarrow M$ is a universal covering map, \n\t\t$\\tilde{x} \\in \\tilde{M}$, \n\t\tand $x= \\pi(\\tilde{x})$. \n\t\t Then, we have:\n\t\t\\begin{equation}\n\t\t\t{\\rm Aut}_{\\pi}(\\tilde{M}) \\cong \\pi_1(M,x).\\notag\n\t\t\\end{equation} \n\t\t \n\t\\end{theorem}\n\n\tAs a result of Proposition \\ref{Aut:smooth+free+proper+isometrical proposition} and Theorem \\ref{theorem-structure theorem on universal covering map}, the following holds.\n\t\n\t\n\t\\begin{proposition}[Corollary 2.33. \\cite{lee_introduction_2018}]\\label{propositon-quotients isometric to universal cover module deck group}\n\t\tSuppose $(M,g)$ and $(\\tilde{M}, \\tilde{g})$ are Riemannian manifolds, and $\\pi : \\tilde{M} \\rightarrow M$ is a universal Riemannian covering map. Then $M$ is isometric to $\\tilde{M}/{\\rm Aut}_{\\pi}(\\tilde{M})$.\n\t\\end{proposition}\n\n\n\n\t\n\t\n\n\t\\section{The isometry group of the universal cover}\n\n Given a steady gradient Ricci soliton $(M,g,f)$. Suppose $(\\tilde{M},\\tilde{g})$ is the universal cover of $(M,g)$. By Theorem \\ref{theorem-structure theorem on universal covering map},\n \\begin{align}\n \\pi_1(M,x) \\cong {\\rm Aut}_{\\pi}(\\tilde{M})\\subseteq {\\rm Iso}(\\tilde{M},\\tilde{g})\\notag\n \\end{align}\n So, to figure out the structure of $\\pi_1(M,x)$, we need to study ${\\rm Iso}(\\tilde{M},\\tilde{g})$.\n\tIn \\cite{guan_rigidity_nodate}, Guan-Lv-Xu prove a splitting theorem for gradient Ricci solitons with nonnegative sectional curvature.\n\t\\begin{theorem}[Theorem 1.1. \\cite{guan_rigidity_nodate}]\n \\label{split theorem_universal}\n\t\tLet $(M^n ,g ,f)$ be a gradient Ricci soliton. If $g$ has nonnegative sectional curvature, then the rank of the Ricci curvature is constant. Thus, either the Ricci curvature is strictly positive or the universal covering $(\\tilde{M}, \\tilde{g}) = (N^k,h) \\times \\mathbb{R}^{n-k}$ splits isometrically and $(N^k, h)$ has strictly positive Ricci curvature.\n\t\\end{theorem}\n\n \n\t We need to study the relations of ${\\rm Iso}(N,g)$, ${\\rm Iso}(\\mathbb{R}^k)$ and ${\\rm Iso}(\\tilde{M} , \\tilde{g})$. \n\tWe first recall the following notion of holonomy group.\n\t\n\t\\begin{definition}[Riemannian holonomy groups\\cite{joyce_riemannian_nodate}]\n\t\t\\label{holonomy_group}\n\t\tLet $(M,g)$ be a Riemannian manifold of dimension $n$, \n\t\t$TM$ be the tangent bundle over $M$ and \n\t\t$\\nabla$ be the Levi-Civita connection of $g$ on $TM$. \n\t\tFix a point $x \\in M$. \n\t\tGiven a loop $\\gamma$ based at $x$ \n\t\t(i.e. a piecewise smooth path $\\gamma: [0,1]\\rightarrow M$ with $\\gamma(0 ) =\\gamma(1) =x$), \n\t\twe let $P_{\\gamma}$ denote the parallel transportation of $\\nabla$ along $\\gamma$ and $T_xM$ denote the fiber over $x$. \n\t\t\n\t\tThen, $P_{\\gamma} : T_xM \\rightarrow T_xM$ is an invertible linear transformation, hence an element of $GL(T_xM) \\cong GL(n, \\mathbb{R})$. The \\textbf{Riemannian holonomy group of $\\mathbf{(M,g)}$ based at $\\mathbf{x}$} is defined by \n\t\t\\begin{equation}\n\t\t\t{\\rm Hol}_x(g) := \\{ P_{\\gamma} : \\gamma\\text{ is a loop based at }x \\} \\subseteq GL(T_xM).\n\t\t\\end{equation}\n\t\t\n\t\\end{definition}\n\t\n\tThe definition of canonical decompositions of the tangent space $T_xM$\\cite{kobayashi_foundations_1996} is given as follows.\t\\begin{definition}[Canonical decompositions]\n\t\tWith the condition of Definition \\ref{holonomy_group}, let $T^{(0)}_x$ be the set of elements in $T_xM$ which are left fixed by ${\\rm Hol}_x(g)$. \n\t\tIt is the maximal linear subspace of $T_xM$ on which ${\\rm Hol}_x(g)$ acts trivially. \n\t\tLet $T_x'$ be the orthonormal complement of $T^{(0)}_x$ in $T_xM$. \n\t\tIt is invariant by ${\\rm Hol}_x(g)$ and can be decomposed into a direct sum \n\t\t$T_x' = \\sum_{i=1}^{k} T^{(i)}_x$ of \n\t\tmutually orthogonal, invariant and irreducible subspaces. \n\t\tWe shall call $T_xM= \\sum_{i=0}^{k} T_x^{(i)}$ \n\t\ta \\textbf{canonical decomposition} (or \\textbf{de Rham decomposition}) of $T_xM$.\n\t\\end{definition}\n\t\n\tLet $T^{(i)}$ be the involutive distribution on $M$ obtained by parallel transformation of $T_x^{(i)}$ for each \n$i=1,2,\\cdots,k$. Theorem 5.4 of Chapter IV in \\cite{kobayashi_foundations_1996} states that the maximal integral manifold $M_0$ of $T^{(0)}$ through a fixed point is locally Euclidean. Moreover, if $M$ is simply connected, then a canonical decomposition $T_xM= \\sum_{i=0}^{k} T_x^{(i)}$ is unique up to an order. Subsequently, the de Rham decomposition theorem shows that if $M$ is simply connected and complete, then $M$ is isometric to the direct product of the maximal integral manifolds of the parallel distributions of $T^{(i)},i=1,2,\\cdots,k$, respectively, see \\cite{de_rham_sur_1952,kobayashi_foundations_1996}. \n\n\n\t\\begin{lemma}[de Rham]\n\t\tLet $(M,g)$ be a simply connected and complete Riemannian manifold.\n\t\tThen, up to isometries and permutations of factors, there exists a unique decomposition as a direct product\n\t\t\\begin{equation}\n\t\t\t(M,g) = \\prod_{i=0}^{k}(M_i,g_i),\n\t\t\\end{equation}\n\t\twhere $(M_0, g_0)$ is a Euclidean space (possibly a point) and \n\t\tall $(M_i, g_i)$, $i=1,\\cdots,k,$ are simply connected, complete and irreducible Riemannian manifolds.\n\t\t\\label{sdecomposition}\n\t\\end{lemma}\n\nGiven the de Rham decomposition of $(M,g)$, Hano has proved the following relation between the isometry group of $M$ and those of $M_i$.\n\n\\begin{theorem}[Theorem 1 \\cite{Hano-1955}]\\label{theorem-decomposition of isometry group-identity components}\n Let $(M,g)$ be a simply connected complete Riemannian manifold and $(M,g)=(M_1,g_1)\\times(M_2,g_2)\\times\\cdots\\times(M_k,g_k)$ be the de Rham decomposition of $(M,g)$.\n\tThen,\n \\begin{align}\\label{decomposition of isometry group}\n {\\rm Iso}_{0}(M,g)\\cong {\\rm Iso}_{0}(M_1,g_1)\\times {\\rm Iso}_{0}(M_2,g_2)\\times\\cdots\\times {\\rm Iso}_{0}(M_k,g_k),\n \\end{align}\nwhere ${\\rm Iso}_{0}(M,g)$ and ${\\rm Iso}_{0}(M_i,g_i)$ are the connected components of the identity in ${\\rm Iso}(M,g)$ and ${\\rm Iso}(M_i,g_i)$ respectively.\n \\end{theorem}\n\n In general, ${\\rm Iso}_0(M,g)$ and ${\\rm Iso}_0(M_i,g_i)$ in (\\ref{decomposition of isometry group}) can not be replaced by ${\\rm Iso}(M,g)$ and ${\\rm Iso}(M_i,g_i)$. For example, given any irreducible Riemannian manifold $(M,g)$, ${\\rm Iso}(M,g)\\times {\\rm Iso}(M,g)$ is a proper subgroup of ${\\rm Iso}(M\\times M,g\\oplus g)$. In view of Theorem \\ref{split theorem_universal}, we can obtain the following decomposition result due to the difference between the curvatures on $N^k$ and $\\mathbb{R}^{n-k}$.\n\t\n\t\\begin{proposition}\n\t\tSuppose that $N$ is a simply connected complete Riemannian manifold with positive Ricci curvature. Then, we have ${\\rm Iso} (N \\times \\mathbb{R}^n) ={\\rm Iso} ( N ) \\times {\\rm Iso} ( \\mathbb{R}^n)$ for $n\\in \\mathbb{N}$.\n\t\t\\label{proposition-isometry separately N R^n}\n\t\\end{proposition}\n\t\\begin{proof}\n\t\tLet $\\varphi: N \\times \\mathbb{R}^n \\rightarrow N \\times \\mathbb{R}^n $ be \n\t\tan isometry of $N \\times \\mathbb{R}^n $. \n\t\tFix a point $ (x, y) \\in N \\times \\mathbb{R}^n $ and \n\t\tconsider the differential of $\\varphi$: \n\t\t\\begin{equation}\n\t\t\td \\varphi_{(x, y)}: \n\t\t\tT_{(x, y)}(N \\times \\mathbb{R}^n ) \n\t\t\t\\cong \n\t\t\tT_{x} N \\oplus T_{y} \\mathbb{R}^n \n\t\t\t\\rightarrow \n\t\t\tT_{\\left(x^{\\prime}, y^{\\prime}\\right)}(N \\times \\mathbb{R}^n ) \n\t\t\t\\cong\n\t\t\tT_{x^{\\prime}} N \\oplus T_{y^{\\prime}} \\mathbb{R}^n,\n\t\t\\end{equation}\n\t\twhere $\\left(x^{\\prime}, y^{\\prime}\\right)=\\varphi(x, y) $.\n\t\t\n\t\t\\begin{claim}\n\t\t\t$d\\varphi$ sends $T_{x} N $ to $T_{x'} N $ and $T_{y} \\mathbb{R}^n $ to $T_{y'} \\mathbb{R}^n $.\n\t\t\t\\label{isometrysep}\n\t\t\\end{claim}\n\t\t\n\t\t\\begin{proof}[Proof of claim \\ref{isometrysep}]\n\t\t\tRecall from Proposition 2.5 of Chapter IV in \\cite{kobayashi_foundations_1996} that if $\\varphi$ is an isometry of $N \\times \\mathbb{R}^n$, \n\t\t\tthen the differential of $\\varphi$ commutes with parallel transports.\n\t\t\tMore precisely, if $\\gamma $ is a loop based at $(x,y)$ in $N \\times \\mathbb{R}^n$, then the following diagram is commutative:\n\t\t\t\\begin{equation}\n\t\t\t\t\\begin{tikzcd}\n\t\t\t\t\tT_{(x,y)}(N \\times \\mathbb{R}^n) \\arrow[d,\"d\\varphi\"] \\arrow[r,\"\\gamma\"] & T_{(x,y)}(N \\times \\mathbb{R}^n) \\arrow[d,\"d\\varphi\"]\\\\\n\t\t\t\t\tT_{(x',y')}(N \\times \\mathbb{R}^n) \\arrow[r,\"\\gamma'\"] & T_{(x',y')}(N \\times \\mathbb{R}^n), \n\t\t\t\t\\end{tikzcd}\n\t\t\t\\end{equation}\n\t\t\twhere $\\gamma' = \\varphi(\\gamma)$. Regarding this, $d\\varphi$ sends a canonical decomposition of $T_{(x,y)}( N \\times \\mathbb{R}^n)$ to a canonical decomposition of $T_{\\varphi(x,y)}( N \\times \\mathbb{R}^n)$. \n\t\t\t\n Note that sectional curvatures are invariant under isometries. Since $N$ has positive Ricci curvature and lemma \\ref{sdecomposition}, $d\\varphi$ sends $T_{x} N $ to $T_{x'} N $ and $T_{y} \\mathbb{R}^n $ to $T_{y'} \\mathbb{R}^n $ separately.\n\t\t\\end{proof}\n\t\tAs an isometry maps geodesics to geodesics, $\\varphi$ sends totally geodesic submanifolds $N \\times\\{y\\} $ to $ N \\times\\left\\{y^{\\prime}\\right\\} $ and $\\{x\\} \\times \\mathbb{R}^n$ to $\\left\\{x^{\\prime}\\right\\} \\times \\mathbb{R}^n $. Suppose $\\pi_{N}$ and $\\pi_{\\mathbb{R}^n}$ are the projection from $N \\times \\mathbb{R}^n$ onto $N$ and $\\mathbb{R}^n$, respectively. \n\t\tWe define $\\varphi_1 : N \\rightarrow N$ by $\\varphi_1(u)= \\pi_N \\circ \\varphi(u,y)$ and $\\varphi_2 : \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ by $\\varphi_2(v)= \\pi_{\\mathbb{R}^n} \\circ \\varphi(x,v)$, where $u \\in N$ and $v \\in \\mathbb{R}^n$. \n\n Considering that $\\varphi_{1} \\in {\\rm Iso} (N)$ and $\\varphi_{2} \\in {\\rm Iso} (\\mathbb{R}^n) $, \n $ \\left(\\varphi_{1}, \\varphi_{2}\\right)$ is a product isometry of $N \\times \\mathbb{R}^n $. \n\t\tIsometries $\\varphi$ and $\\left(\\varphi_{1}, \\varphi_{2}\\right)$ satisfy $\\varphi(x,y) = \\left(\\varphi_{1}, \\varphi_{2}\\right)(x,y)$ and $d\\varphi_{(x,y)} = d\\left(\\varphi_{1}, \\varphi_{2}\\right)_{(x,y)}$. Hence they coincide globally and we complete the proof.\n\t\\end{proof}\n \n\n\n\n\t\\section{Steady gradient Ricci solitons with nonnegative sectional curvature and positive Ricci curvature} \n In this section, we will study steady gradient Ricci solitons with nonnegative sectional curvature and positive Ricci curvature. If the scalar curvature attains its maximum and the Ricci curvature is positive, then $M^n$ is diffeomorphic to $\\mathbb{R}^n$\\cite{Cao_chen_2012}. On a steady gradient Ricci soliton, the scalar curvature attains its maximum if and only if \n the potential $f$ has a critical point $x_0$, i.e., $|\\nabla f|(x_0)=0$. It is still unknown whether there exists a critical point on $M$, even if we assume the sectional curvature is nonnegative. Given a steady gradient Ricci soliton $(M^n,g,f)$, denote the level set of a function $f$ on $M$, by $M_s :=\\{ x\\in M: f(x) =s\\}$. When $M_s$ is compact, we show the existence of the critical point in subsection \\ref{subsection-compact}. When $M_s$ is noncompact, we will show that the level set is diffeomorphic to $\\mathbb{R}^{n-1}$ in subsection \\ref{subsection-noncompact}. We will also show that a steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature has only trivial quotients in subsection \\ref{subsection-quotient}.\n \n\t\\subsection{Compact level sets of \\textit{f}}\\label{subsection-compact}\n\n\n \n\tCounting the ends of a noncompact manifold is an important problem in the study of noncompact manifolds. For the definition of ends, one may refer to\\cite{chow_ricci_2023}. In \\cite{gromoll_complete_1969}, Gromoll and Meyer have proved that a complete open manifold $M$ of positive Ricci curvature has only one end. Munteanu and Wang generalize Gromoll and Meyer's results on steady gradient Ricci solitons. They prove that a steady gradient Ricci soliton is either connected at infinity or isometric to $N\\times \\mathbb{R}$, where $N$ is a compact Ricci flat manifold (See Theorem 4.2 in \\cite{Munteanu_wang_2011}). This result gives a strong restriction of the topology of steady gradient Ricci solitons.\n\t\\begin{lemma}\\label{lemma-existence of critical point}\n\t\tSuppose $(M,g,f)$ is an $n$-dimensional complete steady gradient Ricci soliton. Assume that $f$ has a compact level set $M_s$ for some $s \\in f(M)$. \n\t\tThen, there exists a point $x \\in M$, such that $|\\nabla f|(x) = 0$. \n\t\\end{lemma}\n\t\\begin{proof}\nWe prove this by contradiction. Suppose $|\\nabla f|(x)\\neq0$, $\\forall~x\\in M$. Then, $f$ must be an open map. Hence, $f(M)$ is a connected and open subset of $\\mathbb{R}$. We may assume that $f(M)=(a,b)$, where $-\\infty\\le a<b\\le +\\infty$. Note that $s\\in (a,b)$ and $|\\nabla f|$ is not vanishing everywhere. Then, $M$ is diffeomorphic to $(a,b)\\times M_s$. Hence, $M$ has two ends when $M_s$ is compact. It contradicts Theorem 4.2 in \\cite{Munteanu_wang_2011}.\n\t\\end{proof} \n\n \n\n \n\nBy Lemma \\ref{lemma-existence of critical point}, we can show that $M$ is diffeomorphic to the Euclidean space when the level set is compact.\n\\begin{theorem}\\label{theorem-level set compact}\n\t\tSuppose $(M,g,f)$ is an $n$-dimensional complete steady gradient Ricci soliton with positive Ricci curvature. \n\t\tIf $M_s$ is compact for some $s \\in f(M)$,\n\t\tthen, $M$ is diffeomorphic to $\\mathbb{R}^n$.\n\t\\end{theorem}\n \n\\begin{proof}\nHamilton\\cite{hamilton_formation_1995} has proved that a steady gradient Ricci soliton satisfies \n\t\\begin{equation}\\label{identity-hamilton}\n\t\tR+|\\nabla f|^2 = C,\n\t\\end{equation}\n\twhere $C$ is a constant. Here $R$ denotes the scalar curvature. \n By lemma \\ref{lemma-existence of critical point}, $f$ has a critical point $x_0$. So, the scalar curvature attains its maximum at point $x_0$ by (\\ref{identity-hamilton}). By Proposition 2.3 in \\cite{Cao_chen_2012}, $M$ is diffeomorphic to $\\mathbb{R}^n$.\n\\end{proof} \n \n\t\n\t\n\t\\subsection{Noncompact level sets of \\textit{f}}\\label{subsection-noncompact}\n\tAccording to the above, if an $n$-dimensional complete steady gradient Ricci soliton $(M,g,f)$ with positive Ricci curvature has no critical point, then all level sets of $f$ are noncompact.\n\t\n\t\\begin{theorem}\\label{theorem-noncompact level set}\n\t\tSuppose $(M,g,f)$ is an $n$-dimensional complete steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. If $M_s$ is noncompact for some $s \\in f(M)$,\n\t\tthen $M$ is diffeomorphic to $\\mathbb{R}^n$.\n\t\\end{theorem}\n\t\\begin{proof}\nIf there is a point $x_0 \\in M$ such that $|\\nabla f|(x_0)=0$, then we can show that $M$ is diffeomorphic to $\\mathbb{R}^n$ by the same argument as in the proof of Theorem \\ref{theorem-level set compact}. \n\nNow, we assume that $|\\nabla f|$ is not vanishing everywhere. Note that $M_s$ is a smooth manifold due to the regular value theorem for any $s \\in f(M)$. Then, we follow the computation in the Lemma 2.1 of \\cite{chan_dimension_2023}. The second fundamental form ${\\rm II}$ of $M_s$ is determined by \n\t\t\\begin{equation}\n\t\t\t{\\rm II}(X, Y ) := <\\nabla_X {\\rm n},Y> = \\frac{1 }{|\\nabla f |}{\\rm Ric}(X, Y ) \n\t\t\t\\label{II>0}\n\t\t\\end{equation} \n\t\tfor any $X, Y$ tangent to $M_s$, where ${\\rm n}= -\\frac{\\nabla f}{|\\nabla f|}$. Then ${\\rm II}>0$.\n\t\tMoreover, \n\t\t\\begin{equation}\n\t\t\t{\\rm sec}_{M_s} (X, Y ) = {\\rm sec}_{M}(X, Y ) + {\\rm II}(X, X){\\rm II}(Y, Y )-({\\rm II}(X, Y ))^2 > 0, \n\t\t\\end{equation}\n\t\twhere ${\\rm sec}$ denotes sectional curvature, and $X,Y$ are unit vectors tangent to $M_s$ with $X \\perp Y$.\n\t\tMore precisely, $M_s$ is an $(n-1)$-manifold with positive sectional curvature. Since Lemma \\ref{lemma-existence of critical point} implies that all level sets of $f$ are noncompact, $M_s$ is diffeomorphic to an Euclidean space. Hence, we obtain that $M$ is diffeomorphic to $\\mathbb{R}^n$ followed by the same argument as in the proof of Lemma \\ref{lemma-existence of critical point}. \n\t\\end{proof}\n\t As a corollary of Theorem \\ref{theorem-level set compact} and Theorem \\ref{theorem-noncompact level set}, we obtain the following main result of this section.\n\t \\begin{corollary}\\label{corollary-Ricci positive case is simply connected}\n\t \tSuppose $(M,g,f)$ is an $n$-dimensional complete steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. Then, $M$ is diffeomorphic to $\\mathbb{R}^n$.\n\t \t\n\t \\end{corollary}\n\n \\begin{remark}\n For $n=4$, Theorem \\ref{theorem-noncompact level set} and Corollary \\ref{corollary-Ricci positive case is simply connected} still hold if we only assume $(M,g,f)$ has positive Ricci curvature. Note that any $3$-dimensional noncompact manifold with positive Ricci curvature is diffeomorphic to $\\mathbb{R}^3$(cf.\\cite{Schoen-Yau-1982,Liu-2013} ). So, the argument in Theorem \\ref{theorem-noncompact level set} and Corollary \\ref{corollary-Ricci positive case is simply connected} remains true.\n \\end{remark}\n\n \\subsection{Quotients of steady gradient Ricci solitons}\\label{subsection-quotient}\n\n To prove Theorem \\ref{MAIN}, we still need to study the quotients of steady gradient Ricci soltions with nonnegative sectional curvature and positive Ricci curvature.\n\n \\begin{theorem}\\label{theorem-symmetry of f}\nLet $(M,g,f)$ be a complete gradient Ricci soliton. Suppose $G$ is a finite subgroup of ${\\rm Iso}(M,g)$. Then, one of the following holds:\n\n (1) $(M,g)$ locally splits off a line;\n \n (2) $f(x)=f(\\phi(x))$ for all $x\\in M$ and $\\phi\\in G$.\n \\end{theorem}\n\n \\begin{proof}\nLet $\\phi\\in G$ and $V_1,V_2\\in T_xM$ for some $x\\in M$. Assume $y=\\phi(x)$. By the Ricci soliton equation and $\\phi\\in {\\rm Iso}(M,g)$, we have \n\\begin{align}\n-\\nabla_{V_1}\\nabla_{V_2}f|_x=&{\\rm Ric}(V_1,V_2)|_x-\\frac{\\lambda}{2}g(V_1,V_2)|_x\\notag\\\\\n=&{\\rm Ric}_{\\phi^{\\ast}g}(V_1,V_2)|_x-\\frac{\\lambda}{2}g_{\\phi^{\\ast}g}(V_1,V_2)|_x\\notag\\\\\n=&{\\rm Ric}(\\phi_{\\ast}(V_1),\\phi_{\\ast}(V_2))|_y-\\frac{\\lambda}{2}g(\\phi_{\\ast}(V_1),\\phi_{\\ast}(V_2))|_y\\notag\\\\\n=&-\\nabla_{\\phi_{\\ast}(V_1)}\\nabla_{\\phi_{\\ast}(V_2)}f|_y\\notag\\\\\n=&-\\bar{\\nabla}_{V_1}\\bar{\\nabla}_{V_2}(f\\circ\\phi)|_x\\notag\\\\\n=&-\\nabla_{V_1}\\nabla_{V_2}(f\\circ\\phi)|_x\\label{Parallel-vector-field-1}\n\\end{align}\nwhere $\\bar{\\nabla}$ is the gradient with respect to the metric $\\phi^{\\ast}g$. By $(\\ref{Parallel-vector-field-1})$, we have \n\\begin{align}\n\\nabla(\\nabla f-\\nabla (f\\circ \\phi))\\equiv0.\\notag\n\\end{align}\nLet $V=\\nabla f-\\nabla (f\\circ \\phi)$. Then, $V$ is a parallel vector field by the equation above. If $V$ is nonzero, then $(M,g)$ locally splits off a line. If $V\\equiv0$, then there is a constant $c$ such that \n\\begin{align}\nf(x)-f(\\phi(x))=c,~\\forall~x\\in M.\\label{Parallel-vector-field-2}\n\\end{align}\nSince $\\phi\\in G$ and $G$ is a finite group, then there is a positive integer $k$ such that $\\phi^{k}=Id$. Hence, for any fixed point $x\\in M$,\n\\begin{align}\nkc=\\sum_{i=0}^{k-1}[f(\\phi^{i}(x))-f(\\phi^{i+1}(x))]=f(x)-f(\\phi^{k}(x))=0.\\notag\n\\end{align}\nIt follows that \n\\begin{align}\n c=0.\\label{Parallel-vector-field-3}\n\\end{align}\nBy (\\ref{Parallel-vector-field-2}) and (\\ref{Parallel-vector-field-3}), we get \n\\begin{align}\n f(x)=f(\\phi(x)),~\\forall~x\\in M.\n\\end{align}\nWe complete the proof.\n\n \\end{proof}\n \n \n\\begin{remark}\nIn Theorem \\ref{theorem-symmetry of f}, we assume that $G$ is a finite subgroup of ${\\rm Iso}(M,g)$. If the isomorphism $\\phi$ is related to a killing vector field, one may refer to \\cite{petersen-gradient-2009}.\n\\end{remark}\n\n\n\n As an application of Theorem \\ref{theorem-symmetry of f} and Corollary \\ref{corollary-Ricci positive case is simply connected}, we show that steady gradient Ricci solitons with nonnegative sectional curvature and positive Ricci curvature have no finite quotients.\n\n \\begin{corollary}\n \\label{corollary-sec>=0+Ric>0+finite =>simplyconnected}\n Let $(M,g,f)$ be a complete steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. Suppose $(N,h)$ is a finite quotient of $(M,g)$. Then, $N$ is simply connected, i.e., $(N,h)$ is isometric to $(M,g)$.\n \\end{corollary}\n\n \\begin{proof}\n By Corollary \\ref{corollary-Ricci positive case is simply connected}, $M$ is simply connected. Hence, $M$ is a universal cover of $N$. Let $\\pi: M\\rightarrow N$ be the covering map. By Proposition \\ref{propositon-quotients isometric to universal cover module deck group}, $N$ is isometric to $M/{\\rm Aut}_{\\pi}(M)$. Note that ${\\rm Aut}_{\\pi}(M)$ is a finite group by assumption. Recall that $(M,g,f)$ can not locally split off a line by Theorem \\ref{split theorem_universal}. Therefore, for any $\\phi\\in {\\rm Aut}_{\\pi}(M)$, we have $f=f\\circ\\phi$ on $M$ by Theorem \\ref{theorem-symmetry of f}. Hence, we can define a smooth function $\\bar{f}$ on $N$ such that $f=\\pi\\circ \\bar{f}$. Since $\\pi$ is a local isomorphism, one can check that $(N,h,\\bar{f})$ is a steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. By Corollary \\ref{corollary-Ricci positive case is simply connected}, $N$ is simply connected. Hence, we complete the proof.\n \\end{proof}\n\t\n\t\\section{The Proof of Theorem \\ref{MAIN}} \n\tNow we prove the main theorem. \n\t\\begin{proof}[Proof of Theorem \\ref{MAIN}]\n It suffices to consider the case that $\\pi_1(M)$ is finite. \n\tLet $(\\tilde{M} , \\tilde{g},\\tilde{f})$ be the universal cover of $(M,g,f)$ and $\\pi : (\\tilde{M} , \\tilde{g}) \\rightarrow (M,g)$ be the covering map. \n\tBy Theorem \\ref{split theorem_universal}, \n\t\teither $(M,g,f)$ has strictly positive Ricci curvature\n\t\tor the universal cover $(\\tilde{M}, \\tilde{g}) = (N ,h ) \\times \\mathbb{R}^k$ \n splits isometrically, where $(N , h)$ has strictly positive Ricci curvature and $k \\ge 1$. Due to Lemma 2.1 in \\cite{petersen-gradient-2009}, \n $f$ can also split so that $N$ and $\\mathbb{R}^k$ are both steady gradient Ricci solitons.\n\tIf $M$ has strictly positive Ricci curvature, \n then $\\pi_1(M)$ is trivial due to \n Corollary \\ref{corollary-Ricci positive case is simply connected}. \n Now. it remains to consider the case $k\\ge 1$. \n\n \n By Proposition \\ref{proposition-isometry separately N R^n}, \n \\begin{align}\n {\\rm Aut}_{\\pi} (\\tilde{M})\\subseteq {\\rm Iso} (N \\times \\mathbb{R}^k) = {\\rm Iso} ( N ) \\times {\\rm Iso} ( \\mathbb{R}^k)\\notag\n \\end{align}\n For any $\\phi \\in {\\rm Aut}_{\\pi} (\\tilde{M})$, we can assume that $\\phi = (\\phi_1 ,\\phi_2)$, where $\\phi_1 \\in {\\rm Iso}(N)$ and $\\phi_2 \\in {\\rm Iso}(\\mathbb{R}^k)$. \n By Theorem \\ref{theorem-structure theorem on universal covering map}, the fundamental group $\\pi_1(M) \\cong {\\rm Aut}_{\\pi} (\\tilde{M})$. Since $\\pi_1(M)$ is finite, we see that ${\\rm Aut}_{\\pi} (\\tilde{M})$ is also finite. It follows that $\\phi$, $\\phi_1$ and $\\phi_2$ all have finite orders.\n\n Let $G_{\\phi}=\\{\\phi^i|i\\in\\mathbb{N}\\}$. Then, $G_{\\phi}$ is a finite subgroup of ${\\rm Aut}_{\\pi} (\\tilde{M})$. By Corollary \\ref{corollary-finite subgroup inherited smooth proper free} and Proposition \\ref{Aut:smooth+free+proper+isometrical proposition}, $G_{\\phi}$ acts smoothly, freely, properly and isometrically on $\\tilde{M}$. Similarly, let $G_{\\phi_2}=\\{\\phi_2^i|i\\in\\mathbb{N}\\}$. Since $\\phi_2$ has a finite order, $G_{\\phi_2}$ is a finite subgroup of ${\\rm Iso}(\\mathbb{R}^k)$. So, $G_{\\phi_2}$ is also compact. By Lemma \\ref{constant_compact affine subgroup has a fixed point_lemma}, \n $G_{\\phi_2}$ has at least one fixed point. Therefore, $\\phi_2^i$ has at least one fixed point for all $i\\in\\mathbb{N}$. We also define $G_{\\phi_1}=\\{\\phi_1^i|i\\in\\mathbb{N}\\}$. Then, $G_{\\phi_1}$ is a finite subgroup of ${\\rm Iso}(N)$. \n\n \n\n Let the order of $\\phi$ be $r$. We want to show that $r=1$. Suppose $r\\ge 2$. For $0< i<r$, $\\phi^i$ has no fixed point on $N\\times \\mathbb{R}^k$ since $G_{\\phi}$ is a free action. Note that $\\phi^i=(\\phi^i_1,\\phi_2^i)$ and $\\phi_2^i$ has at least a fixed point on $\\mathbb{R}^k$. So, $\\phi^i_1$ has no fixed point on $N$ for all $0<i<r$. Let the order of $\\phi_1$ be $r_1$. Then, $r_1|r$ and therefore $r_1\\le r$. Note that $\\phi_1^i$ has no fixed point and cannot be the identity map for $0<i<r$. Hence, $r=r_1$. It follows that $\\phi^i_1$ has no fixed point on $N$ for all $0<i<r_1$.\n Then, $G_{\\phi_1}$ is a free action on $N$ by definition. By Corollary \\ref{corollary-finite subgroup inherited smooth proper free}, $G_{\\phi_1}$ acts freely and properly on $N$. By Theorem \\ref{theorem-manifold/action is a manifold}, $N/G_{\\phi_1}$ is a quotient of $N$. However, $G_{\\phi_1}$ is trivial by Corollary \\ref{corollary-sec>=0+Ric>0+finite =>simplyconnected}. It contradicts the fact that $\\phi_1^i$ has no fixed point for $0<i<r$. Hence, $r=1$ and $\\phi$ is the identity map. Since $\\phi$ is an arbitrary isomorphism in ${\\rm Aut}_{\\pi} (\\tilde{M})$, we obtain that $\\pi_1(M)\\cong {\\rm Aut}_{\\pi} (\\tilde{M})=\\{{\\rm Id}_{\\tilde{M}}\\}$.\n \n\t\n\nNow we are left to show that $M$ is diffeomorphic to $\\mathbb{R}^n$. We note that $(N,h)$ is a steady gradient Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. Hence, $N$ is diffeomorphic to $\\mathbb{R}^{n-k}$. Then, $\\tilde{M}$ is diffeomorphic to $\\mathbb{R}^n$. Since $\\pi_1(M)$ is trivial, $M$ is diffeomorphic to $\\tilde{M}$ and therefore is also diffeomorphic to $\\mathbb{R}^n$.\n\n \n\n \n Hence we complete the proof.\n\t\\end{proof}\n\n \n\n\n\n\\begin{spacing}{1.0}\n\n\\bibliographystyle{plain}\n\t\\bibliography{simplyconnected_Reference.bib}\n\t\n\t\n \n\\end{spacing}\n\t\n\t\n\t{\\footnotesize YUXING DENG, SCHOOL OF MATHEMATICS AND STATISTICS, BEIJING INSTITUTE OF TECHNOLOGY, BEIJING,100081, CHINA, 6120180026@BIT.EDU.CN}\n\t\n\t{\\footnotesize YUEHAN HAO, SCHOOL OF MATHEMATICS AND STATISTICS, BEIJING INSTITUTE OF TECHNOLOGY, BEIJING,100081, CHINA, 3120221441@BIT.EDU.CN}\n\t\n\t\n\t\n\t\n\t\n\\end{document}\n\n\n"}}},{"rowIdx":84,"cells":{"id":{"kind":"string","value":"2412.07510v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07510v1"},"title":{"kind":"string","value":"Roman domination number of zero-divisor graphs over commutative rings"},"full_text":{"kind":"string","value":"\\begin{filecontents*}{example.eps}\n\ngsave\nnewpath\n 20 20 moveto\n 20 220 lineto\n 220 220 lineto\n 220 20 lineto\nclosepath\n2 setlinewidth\ngsave\n .4 setgray fill\ngrestore\nstroke\ngrestore\n\\end{filecontents*}\n\\RequirePackage{fix-cm}\n\n\\documentclass[11pt]{svjour3}\n\\usepackage{amssymb}\n\\usepackage{amsmath, mathtools}\n\\usepackage[11pt]{extsizes}\n\\smartqed \\usepackage{booktabs,caption}\n\n\\usepackage{tikz}\n\\usepackage{calc}\n\\usetikzlibrary{decorations.markings}\n\\tikzstyle{vertex}=[circle, draw, inner sep=0pt, minimum size=1pt]\n\\newcommand{\\vertex}{\\node[vertex]}\n\\newcounter{Angle}\n\n\\usepackage{graphicx}\n\\usepackage{pdflscape}\n\\usepackage{geometry}\n \\geometry{\n a4paper,\n total={210mm,297mm},\n left=30mm,\n right=30mm,\n top=25mm,\n bottom=25mm,\n }\n\n\\numberwithin{equation}{section}\n\n\\usepackage{amsfonts}\n\\usepackage{xcolor}\n\\newcommand{\\overbar}[1]{\\mkern 1.5mu\\overline{\\mkern-1.5mu#1\\mkern-1.5mu}\\mkern 1.5mu}\n\\journalname{Indian J. Pure Appl. Math.}\n\\begin{document}\n\n\\title{Roman domination number of zero-divisor graphs over commutative rings}\n\n\\titlerunning{Roman domination number of zero-divisor graphs over commutative rings} \n\\author{Ravindra Kumar$^1$\\and Om Prakash$^{*1}$ }\n\\authorrunning{R. Kumar and O. Prakash}\n\n\n\\institute{\\at $^1$\n Department of Mathematics\\\\\n Indian Institute of Technology Patna, Patna 801 106, India \\\\\n \\email{om@iitp.ac.in(*corresponding author), ravindra.pma15@iitp.ac.in }\n }\n\\date{Received: date / Accepted: date}\n\\maketitle\n\\begin{abstract}\n For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \\rightarrow \\{0, 1, 2\\}$ satisfies the property that if $f(v) = 0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)= 2$. The weight of a Roman dominating function $f$ is the value $f(V)= \\Sigma_{u \\in V} f(u)$, and the minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$, denoted by $\\gamma_R(G)$. The main focus of this paper is to study the Roman domination number of zero-divisor graph $\\Gamma(R)$ and find the bounds of the Roman domination number of $T(\\Gamma(R))$.\n\n\\keywords{Commutative ring \\and Roman domination number \\and Total graph \\and Zero divisor graph.}\n\\subclass{13M99 \\and 05C25}\n\\end{abstract}\n\n\\section{Introduction}\nLet $R$ be a commutative ring with unity and $Z(R)$ be the set of zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\\Gamma(R)$, is a graph with set of vertices $Z(R)- \\{0\\}$ such that there is an edge (undirected) between the vertices $x, y \\in V(\\Gamma(R))$ if and only if $xy = 0$. It is noted that $\\Gamma(R)$ is an empty graph if and only if $R$ is an integral domain.\\\\\n\\indent The concept of the zero-divisor graph was introduced by Beck in \\cite{beck} in 1988. Later, Anderson and Livingston \\cite{ander} redefined Beck's definition in 1999 and established several fundamental results on $\\Gamma(R)$. Consequently, in the last four decades, plenty of works have been reported by several researchers, a few are \\cite{akbari1,akbari,ander,beck,kumar,kumar1}. Further, in $2002$, Redmond \\cite{redmond} extended the study of zero-divisor graph for noncommutative rings. He defined an undirected zero-divisor graph $\\Gamma(R)$ of a noncommutative ring $R$ with set of vertices $Z(R)^* = Z(R) \\setminus \\{0\\}$ and for distinct vertices $a$ and $b$, there is an edge between them if and only if either $ab= 0$ or $ba= 0$.\n\n\\par On the other hand, the concept of the Roman domination was motivated by the defence strategies used to defend the Roman empire during the reign of Emperor Constantine the great $274-337$ AD. There were mainly eight region from Asia minor to Britain of Roman empire at the time of Constantine. To defend all the region by the four groups of legions, he imposed the certain rules. He ordered that for all cities of the Roman empire, at most two group of legions should be stationed under following conditions.\n\n\\begin{itemize}\n\t\\item {A region is securable if a group of legion can be moved to it in a single step from an adjacent region.}\n\t\n\t\\item {At least two group of legions must occupy a region before a group of legion can move out of it (i.e., at least one group of legion must remain behind).}\n\\end{itemize}\n\nBased on the above conditions of the Roman Empire, presently we have the mathematical concept of Roman domination. It is initially defined and discussed by Stewart \\cite{stewart} in $1999$, and later by ReVelle and Rosing \\cite{revelle} in $2000$. The proper definition of Roman domination was introduced by Cockayne et al. \\cite{cockayne} in 2004. After that several works have been reported on various aspects of Roman domination in the graph, including generalizations such as weak Roman domination \\cite{henning}, double Roman domination \\cite{ahangar, beeler}.\n\n\\par A Roman dominating function on a graph $G= (V, E)$ is a function $f : V \\rightarrow \\{0, 1, 2\\}$ with the property that every vertex $u \\in V$ for which $f(u) = 0$ is adjacent to at least one vertex $v \\in V$ for which $f(v)= 2$. The weight of a Roman dominating function is the value $f(V)= \\Sigma_{u \\in V} f(u)$. The Roman domination number of a graph $G$, denoted by $\\gamma_R(G)$, is the minimum weight of an Roman dominating function on a graph $G$. Further, let $G= (V, E)$ be a graph with $f : V \\rightarrow \\{0, 1, 2\\},$ a function and $V_0, V_1, V_2$ be the ordered partition of $V$ induced by $f$, where $V_i= \\{v \\in V \\vert ~ f(v)= i \\}$ and $\\lvert V_i \\rvert = n_i$, for $i = 0, 1, 2$. It is noted that there exists a one-one correspondence between the function $f : V \\rightarrow \\{0, 1, 2\\}$ and the ordered partition $V_0, V_1, V_2$ of $V$. Therefore, it can be represented as $f = (V_0, V_1, V_2)$. A function $f = (V_0, V_1, V_2)$ is a Roman dominating function (RDF) if the set $V_2$ dominates the set $V_0,$ i.e., $V_0 \\subseteq N[V_2]$. A function $f = (V_0, V_1, V_2)$ is said to be a $\\gamma_R$-function if it is an RDF and $f(V)= \\gamma_R(G)$.\n\n\\par Now, we recall some definitions and notations that will be used throughout this paper. Let $G = (V, E)$ be a graph of order $n$. The open neighbourhood of any vertex $v \\in V$ is the set $N(v) = \\{u \\in V \\vert uv \\in E\\}$ and closed neighbourhood is the\nset $N[V] = N(v) \\bigcup \\{v\\}$. The open neighbourhood of a subset $S$ of $V$ is $N(S) = \\bigcup_{v \\in S}N(v)$ and the closed neighbourhood is $N[S] = N(S) \\bigcup S$. A set $S \\subseteq V$ is called a dominating set if every vertex of $V$ is either in $S$ or adjacent to at least one vertex in $S$. The domination number $\\gamma(G)$ of a graph $G$ is the minimum cardinality among the dominating sets of $G$. A graph $G$ of order $n$ is said to be complete if every vertex in $G$ is adjacent to every other vertex in $G$ and it is denoted by $K_n$. A graph is said to be regular or $k$-regular if all its vertices have the same degree $k$. Also, a graph $G=(V, E)$ is called a bipartite graph if its vertex set $V$ can be partitioned into two subsets $V_1$ and $V_2$ such that each edge of $G$ has one end vertex in $V_1$ and another end vertex in $V_2$. It is denoted by $K_{m,n}$ where $m$ and $n$ are the numbers of vertices in $V_1$ and $V_2$, respectively. A complete bipartite graph of the form $K_{1,n}$ is called a star graph. For more basic definitions and results on graph theory, we may refer \\cite{bala}.\n\n\\par Section $2$ contains some basic results on Roman domination graph. In section $3$, we present Roman domination number of a zero-divisor graph $\\Gamma(R)$ for $R = R_1 \\times R_2$ for different diameters of $R_1$ and $R_2$ and later we generalized it for $R= R_1 \\times R_2 \\times ... \\times R_n$. In section $4$, we present lower and upper bounds for the Roman domination number of $T(\\Gamma(R))$. Section 5 concludes the work.\n\n\n\n\n\n\\section{Basic Results}\nWe start this section with several classes of graphs with well-known Roman domination numbers and their straightforward calculations.\n\\par It is easy to see that for a complete graph $K_n$, $\\gamma_R(K_n) = 2$. Let $G$ be a complete $r-$ partite graph $(r\\geq 2)$ with partite set $V_1, V_2,...,V_r$ such that $\\lvert V_i \\rvert > 2$ for $1\\leq i\\leq r$. Then $\\gamma_R(G) = 4$. If $\\lvert V_i \\rvert = 2$ for some $i$, then $\\gamma_R(G) = 3$ because one vertex of that set assigned $2$ and another vertex is assigned $1$. If $\\lvert V_i \\rvert = 1$ for some $i$, then $\\gamma_R(G) = 2$. Hence, we can say that Roman domination number of any star graph is $2$ and bistar graph is $3$.\\\\\n\n\\begin{example}\n\tConsider a ring $R = \\mathbb{Z}_{25}$. The graph of $\\Gamma(\\mathbb{Z}_{25})$ is shown in figure $1$.\n\t\n\t\\[\\begin{tikzpicture}\n\t\n\t\n\t\n\t\\vertex (A) at (-1.5,1.5) [label=left:${5}$]{};\n\t\\vertex (B) at (0.5,1.5) [label=right:${10}$]{};\n\t\\vertex (C) at (0.5,-0.5) [label=right:${15}$]{};\n\t\\vertex (D) at (-1.5,-0.5) [label=left:${20}$]{};\n\t\\path\n\t(A) edge (B)\n\t(B) edge (C)\n\t(C) edge (D)\n\t(D) edge (A)\n\t(A) edge (C)\n\t(B) edge (D)\n\t;\n\t\n\t\\end{tikzpicture}\\]\n\t\\hspace{6.6cm} \\textbf{Figure 1} \\\\\t\n\t\n\\end{example}\n\n\nIn this case, the graph $\\Gamma(\\mathbb{Z}_{25})$ is a complete graph of $4$ vertices i.e.$K_4$. Now, we define a function $g: V(\\Gamma(\\mathbb{Z}_{25})) \\longrightarrow \\{0, 1, 2\\}$ in a way such that $g(5) = 0$, $g(10) = 0$, $g(15) = 0$ and $g(20) = 2$. Clearly, by the definition, $g$ is an RDF with weight $g(V) = \\sum_{u\\in V} f(u) =2$. Since, this weight is minimum, so $\\gamma_R(\\Gamma(\\mathbb{Z}_{25})) = 2$ or, $\\gamma_R(K_n) = 2$.\n\n\n\n\n\\par Moreover, some results on Roman domination number given by Cockayne et al. in \\cite{cockayne} are given below.\n\n\\begin{proposition}\n\tFor the classes of paths $P_n$ and cycles $C_n$,\n\t\\par \t $\\gamma_R(P_n) = \\gamma_R(C_n) = \\lceil \\frac{2n}{3} \\rceil$.\\\\\n\t\n\\end{proposition}\n\nAlso, they have proposed a relation between domination number and Roman domination number of a graph as follows.\n\n\\begin{proposition}\n\tFor any graph $G$,\\\\\n\t\\par\t$\\gamma(G) \\leq \\gamma_R(G) \\leq 2\\gamma(G)$.\n\t\n\\end{proposition}\n\n\\begin{proposition}\n\tFor any graph $G$ of order $n$, $\\gamma(G) = \\gamma_R(G)$ if and only if $G = \\overline{K_n}$.\n\t\n\\end{proposition}\n\n\n\n\\section{Main Results}\n\n\\begin{theorem}\n\tLet $S$ be a finite principal ideal local ring. Then $\\gamma_R(\\Gamma(S)) = 2$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tLet $M$ be a maximal ideal of the finite principal ideal local ring $S$. Suppose $a \\in S$ such that $M = <a>$, then $M= aS$. Let the set of unit elements of $S$ be $U= \\{ u_1, u_2,..., u_m \\}$. Since $S$ finite, there exists a positive integer $n$ such that $a^n = 0$ and $a^{n-1} \\neq 0$. Then the element of $\\Gamma(S)$ is of the form $u_i a^j$ where $i \\leq m, ~ j \\leq n$. Then $M = \\{ u_i a^j : i \\leq m, ~ j \\leq n\\}$. Since, $a^{n-1}$ is adjacent to all vertex of $M$. So, we define Roman dominating function $f = (V_0, V_1, V_2)$ such that $V_0 = M\\backslash \\{a^{n-1}\\}, ~ V_1= \\phi$ and $V_2= a^{n-1}$. Hence, every element $x$ of $V_0$ for which $f(x)= 0$ is adjacent to element of $V_2$. Thus, the Roman dominating number $\\gamma_R(\\Gamma(S)) = \\sum_{u\\in M}f(u)= \\sum_{u_0\\in V_0}f(u_0)+ \\sum_{u_1\\in V_1}f(u_1)+ \\sum_{u_2\\in V_2}f(u_2)= 0+0+2= 2$.\n\\end{proof}\n\n\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= diam(\\Gamma (R_2)) = 0$ and $\\lvert R_1 \\rvert \\geq 5 ~\\& ~\\lvert R_2 \\rvert \\geq 5$. Then $\\gamma_R(\\Gamma(R)) = 4$.\n\\end{theorem}\n\n\\begin{proof}\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= diam(\\Gamma (R_2)) = 0$. Then we have three cases.\n\t\n\t\n\t\\textbf{Case 1:} $Z(R_1)= \\{0, a\\}$ and $Z(R_2)= \\{0, b\\}$ and let $Reg(R_1)= \\{x_1, x_2,..., x_n\\}$ and $Reg(R_2)= \\{y_1, y_2,..., y_m\\}$. Now, we are going to construct a graph for this case.\n\t\n\t\n\t\\[\\begin{tikzpicture}\n\t\n\t\n\t\n\t\\vertex (A) at (-3,2) [label=below:${(x_i, b)}$]{};\n\t\\vertex (B) at (0,2) [label=above:${(0,b)}$]{};\n\t\\vertex (C) at (3,2) [label=right:${(x_i,0)}$]{};\n\t\\vertex (D) at (0,0.4) [label=right:${(a,0)}$]{};\n\t\\vertex (E) at (-3,-0.5) [label=below:${(a,b)}$]{};\n\t\\vertex (F) at (3,-0.5) [label=below:${(0,y_i)}$]{};\n\t\\vertex (G) at (0,-1.2) [label=below:${(a,y_i)}$]{};\n\t\\path\n\t(A) edge (B)\n\t(B) edge (C)\n\t(B) edge (D)\n\t(B) edge (E)\n\t(C) edge (F)\n\t(D) edge (E)\n\t(D) edge (G)\n\t(D) edge (F)\n\t;\n\t\n\t\\end{tikzpicture}\\]\n\t\\hspace{6.6cm} \\textbf{Figure 2} \\\\\n\t\n\tAlso, we define a function $g : V(\\Gamma(R)) \\longrightarrow \\{0,1,2\\}$ by\n\t\n\t\\[ g(x, y) = \\left\\{\n\t\\begin{array}{ll}\n\t2 & if~ (x, y) = (0, b) ~and ~(x,y)= (a,0) \\\\\n\t0 & otherwise \\\\\n\t\\end{array}\n\t\\right. \\]\n\t\n\tHere, it is easily seen that $g$ is a Roman dominating function such that $g(v)= 2+2=4$. Hence, $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\t\n\t\\textbf{Case 2:} Suppose $R_1$ is an integral domain and $Z(R_2)= \\{0,b\\}$, then we have the following induced subgraph.\n\t\n\t\n\t\n\t\\[\\begin{tikzpicture}\n\t\n\t\n\t\n\t\\vertex (A) at (-1.5,1.5) [label=left:${(x_i, b)}$]{};\n\t\\vertex (B) at (0.5,1.5) [label=right:${(0,b)}$]{};\n\t\\vertex (C) at (0.5,-0.5) [label=right:${(x_i,0)}$]{};\n\t\\vertex (D) at (-1.5,-0.5) [label=left:${(0,y_i)}$]{};\n\t\\path\n\t(A) edge (B)\n\t(B) edge (C)\n\t(C) edge (D)\n\t;\n\t\n\t\\end{tikzpicture}\\]\n\t\\hspace{6.6cm} \\textbf{Figure 3} \\\\\n\t\n\tAgain, we define a function $g$ as follows:\n\t\n\t\\[ g(x, y) = \\left\\{\n\t\\begin{array}{ll}\n\t2 & if~ (x, y) = (0, b) ~and ~(x,y)= (x_i,0)~ for~ a~ fixed~ i\\\\\n\t0 & otherwise \\\\\n\t\\end{array}\n\t\\right. \\]\n\t\n\tClearly, $g$ is a RDF with $g(v)= 2+2=4$. Therefore, $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\t\n\t\\textbf{Case 3:} Now, we suppose $R_1$ and $R_2$ are integral domains. In this case, $\\Gamma(R)$ is a complete bipartite graph and $\\lvert R_1 \\rvert \\geq 5 ~\\& ~\\lvert R_2 \\rvert \\geq 5$. Therefore, $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\t\n\t\n\\end{proof}\n\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= 0$ and $diam(\\Gamma (R_2)) = 1$. Then $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tSince $diam(\\Gamma (R_1))= 0$ and $diam(\\Gamma (R_2)) = 1$. Then we have two cases for the ring $R_1$.\\\\\n\t\n\t\\textbf{Case 1:} Let $Z(R_1)= \\{0,a\\}$. Then $Reg(R_1)= \\{x_1,x_2,...,x_n\\}$, $Reg(R_2)= \\{y_1,y_2,...,y_m\\}$. Suppose $Z(R_2)= \\{0,z_1,z_2,...,z_k\\}$ such that $z_i z_j = 0$ for all $i,j \\leq k$. Now, we are going to construct a graph for this condition.\n\t\n\t\\[\\begin{tikzpicture}\n\t\n\t\n\t\n\t\\vertex (A) at (1,4) [label=above:${(a, z_j)}$]{};\n\t\\vertex (B) at (-1,2) [label=above:${(0,z_j)}$]{};\n\t\\vertex (C) at (3,2) [label=above:${(a,0)}$]{};\n\t\\vertex (D) at (0,0) [label=left:${(x_i,0)}$]{};\n\t\\vertex (E) at (2,0) [label=right:${(0,y_j)}$]{};\n\t\\vertex (F) at (5,2) [label=below:${(a,y_i)}$]{};\n\t\\vertex (G) at (-3,2) [label=below:${(x_i,z_j)}$]{};\n\t\\path\n\t(A) edge (B)\n\t(A) edge (C)\n\t(B) edge (G)\n\t(B) edge (D)\n\t(B) edge (C)\n\t(C) edge (F)\n\t(C) edge (E)\n\t(D) edge (E)\n\t;\n\t\n\t\\end{tikzpicture}\\]\n\t\\hspace{6.6cm} \\textbf{Figure 4} \\\\\n\t\n\tAlso, we define a function $g$ as follows:\n\t\n\t\\[ g(x, y) = \\left\\{\n\t\\begin{array}{ll}\n\t2 & if~ (x, y) = (a, 0) ~and ~(x,y)= (0,z_j) for~ j=1 \\\\\n\t0 & otherwise \\\\\n\t\\end{array}\n\t\\right. \\]\n\t\n\tIt has been easily seen that $g$ is an RDF. Therefore, $g(v)= 2+2= 4$ and hence $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\t\\textbf{Case 2:} Let $R_1$ be an integral domain. Then we have an induced subgraph given in fig $4$.\n\t\n\t\\[\\begin{tikzpicture}\n\t\n\t\n\t\n\t\\vertex (A) at (-1,0) [label=left:${(x_i, z_j)}$]{};\n\t\\vertex (B) at (0,0) [label=right:${(0,z_j)}$]{};\n\t\\vertex (C) at (0,1) [label=right:${(x_i,0)}$]{};\n\t\\vertex (D) at (0,2) [label=left:${(0,y_j)}$]{};\n\t\\path\n\t(A) edge (B)\n\t(B) edge (C)\n\t(C) edge (D)\n\t;\n\t\n\t\\end{tikzpicture}\\]\n\t\\hspace{6.6cm} \\textbf{Figure 5} \\\\\n\t\n\tAgain, we define a function $g$ as follows.\n\t\n\t\\[ g(x, y) = \\left\\{\n\t\\begin{array}{ll}\n\t2 & if~ (x, y) = (x_i, 0) ~and ~(x,y)= (0,z_j) for~ i=j=1 \\\\\n\t0 & otherwise \\\\\n\t\\end{array}\n\t\\right. \\]\n\t\n\tIt can be easily verify that $g$ is an RDF. Then $g(v)= 2+2= 4$ and hence $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{proof}\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= diam(\\Gamma (R_2)) = 1$. Then $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tThe proof is the same as the proof of the Theorem $3.3$.\n\t\n\\end{proof}\n\n\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= 0$ and $diam(\\Gamma (R_2)) = 2$. Then $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tLet $R= R_1 \\times R_2$ be a ring and $R_2$ be a finite local ring generated by $x$, say, $Z(R_2)= xR_2$ with $x^l=0$ and $x^{l-1}\\neq 0$. Now, we have two cases. \\\\\n\t\n\t\\textbf{Case 1:} Suppose $Z(R_1)= \\{0,a\\}$, $Reg(R_1)= \\{u_1,u_2,...,u_n\\}$, $Reg(R_2)= \\{v_1,v_2,...,v_m\\}$ and $Z(R_2)= \\{0,v_1x,v_2x,...,v_mx^{l-1}\\}$ such that two vertices $v_ix^j$ and $v_sx^r$ of $\\Gamma (R)$ are adjacent if and only if $j+r \\geq l$. Now, we define the RDF $g$ on $V(\\Gamma(R))$ as follows. For any one value of $m$, $g(0, v_mx^{l-1}) = 2$ and $g(a,0)=2$ and for the remaining vertices $x,y$, let $g(x,y)=0$. It is easily seen that $g$ is an RDF and $g(v)=2+2=4$. \\\\\n\t\n\t\\textbf{Case 2:} Let $R_1$ be an integral domain. Then $\\Gamma(R)$ is an induced subgraph after deleting the vertices $(a,0), (a,v_j), (a, v_ix^j)$ for each $i~\\& ~j$ from $case 1$. Now, defining RDF $g$ as $g(u_i,0)=2$ for any one of $i's,$ say, $i=1$ and $g(0, v_mx^{l-1})=2$ for $m=1$ and for the remaining vertices $(x,y)$, let $g(x,y)=0$. Then $g(v)=2+2=4$. Hence, in both cases, $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{proof}\n\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2$ be a ring such that $diam(\\Gamma (R_1))= 1~ or ~ 2$ and $diam(\\Gamma (R_2)) = 2$. Then $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tThe proof is the same as given in Theorem $3.5$.\n\\end{proof}\n\n\\begin{remark}\n\tLet $R$ be a finite commutative ring with unity. If $R$ is a product of two local rings with diameters less than equal to $2$. Then Roman domination number is $4$.\n\t\n\\end{remark}\n\n\nLet $G$ and $H$ be a graph. We define the Cartesian product of $G$ and $H$ to be the graph $G \\Box H$ such that the vertex set of $G \\Box H$ is $V(G) \\times V(H)$, i.e., $\\{(x,y)\\vert x\\in G, y\\in H\\}$. Also, two vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G \\Box H$ if and only if one of the following is true:\n\\begin{itemize}\n\t\\item $x_1 = x_2$ and $y_1$ is adjacent to $y_2$ in $H$, or\n\t\n\t\\item $y_1 = y_2$ and $x_1$ is adjacent to $x_2$ in $G$.\n\t\n\\end{itemize}\n\n\\begin{proposition}\n\tLet $R_1$ and $R_2$ be two rings such that $\\lvert \\Gamma(R_1) \\rvert = m$ and $\\lvert \\Gamma(R_2) \\rvert = n$ and having $\\Delta (\\Gamma(R_1))= r_1$, $\\Delta (\\Gamma(R_2))= r_2$. Then $\\gamma_R(\\Gamma(R_1) \\Box \\Gamma(R_2)) \\leq mn-r_1-r_2+1$.\n\t\n\\end{proposition}\n\n\\begin{proof}\n\tSuppose $R_1$ and $R_2$ be two rings and $\\Delta (\\Gamma(R_1))= r_1$, $\\Delta (\\Gamma(R_2))= r_2$ with $\\lvert \\Gamma(R_1) \\rvert = m$ and $\\lvert \\Gamma(R_2) \\rvert = n$. Now, we know from the definition of Cartesian product of two graphs, $V(\\Gamma(R_1) \\Box \\Gamma(R_2)) = mn$. Therefore, there exists a vertex $v$ in $\\Gamma(R_1) \\Box \\Gamma(R_2)$ such that $deg(v)= r_1+r_2$. If $V_2$= \\{v\\}, $V_1= V- N[v]$ and $V_0= V-V_1-V_2$, then $f= (V_0, V_1, V_2)$ is a Roman dominating function with $f(V)= \\lvert V_1 \\rvert + 2\\lvert V_2 \\rvert = mn-(r_1+r_2+1)+2 = mn-r_1-r_2+1$. Hence, the weight of the function $f$ is $mn-r_1-r_2+1$ and $\\gamma_R(\\Gamma(R_1) \\Box \\Gamma(R_2)) \\leq mn-r_1-r_2+1$.\n\t\n\\end{proof}\n\n\\begin{corollary}\n\tSuppose that total number of non-zero zero-divisor in a ring $R_1$ is $1$, say $\\lvert Z(R_1)^*\\rvert = 1$ and $\\lvert Z(R_2) \\rvert \\geq 2$, then $\\gamma_R(\\Gamma(R_1) \\Box \\Gamma(R_2)) = \\gamma_R(\\Gamma(R_2))$, since $\\Gamma(R_1) \\Box \\Gamma(R_2) \\cong \\Gamma(R_2)$.\n\t\n\\end{corollary}\n\nNow, we give some examples.\n\n\n\\begin{example}\n\t\n\tAny graph $G$ has a Roman domination number equal to $2$, then a vertex of graph $G$ is adjacent to every other vertex of $G$. In paper \\cite[Theorem 2.5]{ander}, it is proved that for a commutative ring $R$, there is a vertex of $\\Gamma(R)$ which is adjacent to every other vertex if and only if either $R \\equiv \\mathbb{Z}_2 \\times A$ where $A$ is an integral domain, or $Z(R)$ is an annihilator ideal (and hence is a prime).\n\t\n\\end{example}\n\n\n\\begin{example}\n\t\n\t\\textbf{(a)}\tIn \\cite{akbari}. it is proved that for any finite ring $R$, if $\\Gamma(R)$ is a regular graph of degree $m$, then $\\Gamma(R)$ is a complete graph $K_m$ or a complete bipartite graph $K_{m,m}$. In this case, $\\gamma_R(\\Gamma(R)) = 2~or~ 4$, provided $m\\geq 3$. \\\\\n\t\n\t\\textbf{(b)} In \\cite[Theorem 9]{akbari}, let $R$ be a finite principal ideal ring. If $\\Gamma(R)$ is a Hamiltonian graph, then it is either a complete graph or complete bipartite graph. Thus $\\gamma_R(\\Gamma(R)) = 2~or~ 4$.\\\\\n\t\n\t\\textbf{(c)} In \\cite[Theorem 8]{akbari} , let $R$ be a finite decomposable ring. If $\\Gamma(R)$ is a Hamiltonian graph, then $\\Gamma(R) \\equiv K_{n,n}$ for some natural number $n$. Consequently, $\\gamma_R(\\Gamma(R)) = 4$.\n\t\n\\end{example}\n\n\\begin{corollary}\n\t\n\tIn \\cite[Corollary 1]{akbari}, the graph $\\Gamma(\\mathbb{Z}_n)$ is a Hamiltonian graph if and only if $n= p^2$ where $p$ is a prime greater than $3$ and in this case, $\\Gamma(\\mathbb{Z}_n) \\equiv K_{p-1}$. Thus, Roman domination number of $\\Gamma(R)$ is $2$.\n\t\n\\end{corollary}\n\n\n\\begin{theorem}\n\t\n\tLet $R= R_1 \\times R_2 \\times ...\\times R_n$, for a fixed integer $n\\geq 3$ and $R_i$ be an integral domain for each $i= 1,2,...,n$. Then $\\gamma_R(\\Gamma(R)) = 2n$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tLet $R= R_1 \\times R_2 \\times ...\\times R_n$ be a ring and each $R_i$ be an integral domain for $i= 1,2,...,n$. Then the set $S= \\{(1,0,0,...,0), (0,1,0,...,0),...,(0,0,0,...,1)\\}$ upto $n$ terms is a dominating set and no subset $T$ of $R_1 \\times R_2 \\times ...\\times R_n$ with cardinality less than $n$ can be a dominating set. Now, define a $\\gamma_R$- function $g$ in such a way that $g(1,0,0,...,0)= g(0,1,0,...,0)= g(0,0,1,...,0)= ...= g(0,0,0,...,1)= 2$ and $g(u)= 0$ for rest of the vertices of $\\Gamma(R)$ where $u$ is a vertex of $n$-tuples of $R_1 \\times R_2 \\times ...\\times R_n$. Therefore, $g(V)= 2+2+...+2$ upto $n$ terms, it follows that $g(V)= 2n$ and hence $\\gamma_R(\\Gamma(R)) = 2n$.\n\t\n\\end{proof}\n\nIt is known from \\cite[Theorem 8.7]{atiyah} that a finite Artinian ring $R$ is a product of local rings. Based on this result, we have the following:\n\n\n\\begin{theorem}\n\t\n\tLet $R$ be a finite commutative Artinian ring and $R= R_1 \\times R_2 \\times ...\\times R_n$ where each $R_i$ is a local ring for $i= 1,2,...,n$. Then $\\gamma_R(\\Gamma(R)) = 2n$ for $n\\geq 3$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tThe proof of this Theorem is the same as above.\n\t\n\\end{proof}\n\n\n\n\\begin{theorem}\n\t\n\tLet $n= p_1^{m_1} p_2^{m_2}...p_k^{m_k}$ for any fixed integer $k\\geq 1$ and for distinct primes $p_1,..., p_k$ and positive integers $m_1,..., m_k$. Then the following results hold.\n\t\n\\end{theorem}\n\n\\noindent \\textbf{(a)} $\\gamma_R(\\Gamma(\\mathbb{Z}_n)) = 2k,$ if $n= p_1^{m_1}... p_k^{m_k}$ and $ k\\geq 3$.\\\\\n\n\\noindent \\textbf{(b)} $\\gamma_R(\\Gamma(\\mathbb{Z}_n)) = 4,$ if $n= p_1^{m_1} p_2^{m_2}$ where either $m_1 \\geq 2~ or ~ m_2\\geq 2~ \\text{or}~ m_1= m_2=1$ and $p_1, p_2 \\geq 3$.\\\\\n\n\\noindent \\textbf{(c)} $\\gamma_R(\\Gamma(\\mathbb{Z}_n)) = 2,$ if $n= p_1^{m_1}$ and $m_1 \\geq 2$ or $n= p_1 p_2$ where either $p_1 = 2, p_2 \\geq 3 ~or~ p_1 \\geq 3, p_2 = 2$.\n\n\\begin{proof}\n\tIt is known that $\\mathbb{Z}_n \\cong \\mathbb{Z}_{p_1}^{m_1} \\times ... \\times \\mathbb{Z}_{p_k}^{m_k}$ where $n= p_1^{m_1}...p_k^{m_k}$.\n\tTo prove part (a), from \\cite[Proposition 1]{cockayne}, $\\gamma(G)\\leq \\gamma_R(G) \\leq 2\\gamma(G)$. We suppose that $k \\geq 3$ and $p_1, p_2,...,p_k$ are distinct primes and $m_1, m_2,..., m_k$ are positive integers. Now, we construct a set $T= \\{(p_1^{m_1-1},0,0,...,0), (0, p_2^{m_2-1},0,...,0),.$ $..,(0,0,0,...,p_k^{m_k-1})\\}$ and define a function $f= (V_0, V_1, V_2)$ such that $V_0 = V(\\Gamma(\\mathbb{Z}_n)) \\setminus T$, $V_1= \\phi$, $V_2 = T$. Then the function $f$ is a Roman dominating function with $f(V)= 2\\lvert T\\rvert = 2k$ and hence $\\gamma_R(\\Gamma(\\mathbb{Z}_n)) = 2k$. \\\\\n\tProof of (b) is same as part (a). To prove (c), it is noted that $p_1^{m_1-1}$ is a vertex adjacent to all vertices in $\\Gamma(\\mathbb{Z}_n)$ and whenever $n= p_1p_2$, where $p_1 = 2, p_2 \\geq 3 ~or~ p_1 \\geq 3, p_2 = 2,$ then $\\Gamma(\\mathbb{Z}_n)$ is a star graph. Thus $\\gamma_R(\\Gamma(\\mathbb{Z}_n)) = 2$.\n\t\n\\end{proof}\n\n\\begin{theorem}\n\t\n\tIf $R$ is a commutative ring with unity that contains at least one prime ideal, then $\\gamma_R(\\Gamma(R)) \\leq 2(\\lvert p\\rvert -1)$ where $p$ is a prime ideal.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tSuppose $R$ is a commutative ring with unity that contains at least one prime ideal $p$, then every edge in $\\Gamma(R)$ has at least one end vertex in $p$. Let $S= p\\setminus \\{0\\}$. Then $S$ is a dominating set of $\\Gamma(R)$. Since every edge has at least one vertex in $S$, we define a function $g= (V_0, V_1, V_2)$ such that $V_0= V(\\Gamma(R))\\setminus S$, $V_1= \\phi$, $V_2= S$. Also, $p$ is a prime ideal whose cardinality is minimal among the cardinalities of all prime ideals of $R$. Hence, we can conclude $g(V) \\leq 2\\lvert S\\rvert = 2(\\lvert p\\rvert -1)$. Thus, $\\gamma_R(\\Gamma(R)) \\leq 2(\\lvert p\\rvert -1)$.\n\t\n\\end{proof}\n\n\n\n\\section{Roman domination number of total graph}\n\nLet $R$ be a commutative ring with unity and $Z(R)$ be the set of zero-divisors of $R$. The total graph of a ring $R$, denoted by $T(\\Gamma(R))$, is the undirected graph with vertex set $R$ and two vertices $x, y \\in R$ are adjacent if and only if $x+y \\in Z(R)$. This concept was introduced and studied by Anderson and Badawi \\cite{anderson}. Here, $Z(\\Gamma(R))$ and $Reg(\\Gamma(R))$ are disjoint induced subgraphs of $T(\\Gamma(R))$. It is observed that if $Z(R)$ is an ideal with $\\lvert Z(R) \\rvert = \\beta$ and $2\\in Z(R)$, then $deg(v) = \\beta -1$ for every $v \\in V(T(\\Gamma(R)))$ and if $2 \\notin Z(R)$, then $deg(v) = \\beta -1$ for each $v \\in Z(R)$ and $deg(v) = \\beta$ for every $v \\notin Z(R)$. Hence, there is no vertex in $T(\\Gamma(R))$ which has degree $\\lvert R \\rvert - 1$. In this section, we obtain bounds for Roman domination number of $T(\\Gamma(R))$.\\\\\n\nTo obtain the bound for the Roman domination number of $T(\\Gamma(R))$, we first state the following result of Anderson and Badawi \\cite{anderson}.\n\n\n\\begin{lemma}[\\cite{anderson}, Theorem 2.1]\n\tLet $R$ be a commutative ring such that $Z(R)$ is an ideal of $R$. Then $Z(\\Gamma(R))$ is a complete (induced) subgraph of $T(\\Gamma(R))$ and $Z(\\Gamma(R))$ is disjoint from $Reg(\\Gamma(R))$.\n\t\n\\end{lemma}\n\n\\begin{theorem}\n\tLet $R$ be a commutative ring (not necessarily finite) with identity and $Z(R)$ be its ideal with $\\lvert R/Z(R) \\rvert = \\alpha$. Then $3 \\leq \\gamma_R(T(\\Gamma(R))) \\leq 2\\alpha$.\n\t\n\\end{theorem}\n\n\\begin{proof}\n\tIt is noted that $T(\\Gamma(R))$ has no vertex of degree $\\lvert R \\rvert -1$, therefore, $\\gamma_R(T(\\Gamma(R))) \\geq 3$. Suppose $\\lvert Z(R) \\rvert = \\beta$ and $Z(R)$ is an ideal of $R$. Then, by Theorem $2.2$ of \\cite{anderson}, there are two cases arise: $(i)$ If $2 \\in Z(R)$, then each coset $x+Z(R)$ is a complete graph $K_{\\beta}$ and $(ii)$ if $2 \\notin Z(R)$, then coset $(x+Z(R)) \\bigcup (-x+Z(R))$ is a complete bipartite graph $K_{\\beta, \\beta}$. Hence,\n\t\n\t\\[\n\tT(\\Gamma(R))=\n\t\\begin{cases}\n\tK_{\\beta}\\cup \\underbracket{K_{\\beta} \\cup K_{\\beta} \\cup ... \\cup K_{\\beta}}_{(\\alpha - 1) copies},& \\text{if } 2 \\in Z(R)\\\\\n\tK_{\\beta}\\cup \\underbracket{K_{\\beta, \\beta} \\cup K_{\\beta, \\beta} \\cup ... \\cup K_{\\beta, \\beta}}_{(\\frac{\\alpha - 1}{2}) copies},& \\text{if } 2 \\notin Z(R)\n\t\\end{cases}\n\t\\]\n\t\n\tNote that each of the component in case $(i)$ has Roman domination number $2$ as the definition of complete graph. Therefore, $\\gamma_R(T(\\Gamma(R)))= 2\\alpha$. In case $(ii)$, each complete bipartite graph has maximum Roman domination number $4$. Hence, $\\gamma_R(T(\\Gamma(R))) \\leq 2+ (\\frac{\\alpha -1}{2}) 4 = 2\\alpha$. Thus, in both cases, $3 \\leq \\gamma_R(T(\\Gamma(R))) \\leq 2\\alpha$.\n\t\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe main objective of this paper was to study the Roman domination number of zero-divisor graph of a commutative ring $R$. Here, we have first calculated the Roman domination number of $\\Gamma(R)$ where $R = R_1 \\times R_2$, for different diameters of $\\Gamma(R_1)$ and $\\Gamma(R_2)$ and then we generalized it for $R= R_1 \\times R_2 \\times ... \\times R_n$. At the end of this paper, we have discussed the upper and lower bounds of Roman domination number of total graph $T(\\Gamma(R))$.\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{20}\n\t\n\t\\bibitem{ahangar}\n\tAhangar, HA., Chellali, M., Sheikholeslami, SM.: On the double Roman domination in graphs. Discrete Appl. Math. \\textbf{232}, 1-7 (2017).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{akbari1}\n\tAkbari, S., Maimani, HR., Yassemi, S.: When a zero-divisor graph is planer or a complete r-partite graph. J. Algebra. \\textbf{270}, 169-180 (2003).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{akbari}\n\tAkbari, S., Mohammadian, A.: On the zero-divisor graph of a commutative ring. J. Algebra. \\textbf{274}, 847-855 (2004).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{anderson}\n\tAnderson, DF., Badawi, A.: The total graph of a commutative ring. J. Algebra. \\textbf{320}, 2706-2719 (2008).\n\t\n\t\\bibitem{ander}\n\tAnderson, DF., Livingston, PS.: The zero-divisor graph of a commutative ring, J. Algebra. \\textbf{217}, 434-447 (1999).\n\t\\vspace{.2 cm}\n\t\n\t\\bibitem{atiyah}\n\tAtiyah, MF., MacDonald, IG.: Introduction to commutative algebra. Reading, MA: Addison-Wesley (1969).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{bala}\n\tBalakrishnan, R., Ranganathan, K.: A Textbook of Graph Theory, (2nd edition), Springer New York Heidelberg Dordrecht London (2012).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{beck}\n\tBeck, I.: Coloring of commutative ring. J. Algebra. \\textbf{116}, 208-226 (1988).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{beeler}\n\tBeeler, RA., Haynes, TW., Hedetniemi, ST.: Double Roman domination, Discrete Appl. Math. \\textbf{211}, 23-29 (2016).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{cockayne}\n\tCockayne, EJ., Dreyer, Jr PA., Hedetniemi, SM., Hedetniemi, ST.: Roman domination in graphs. Discrete Math. \\textbf{278}, 11-22 (2004).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{henning}\n\tHenning, MA., Hedetniemi, ST.: Defending the roman empire- a new strategy. Discrete Math. \\textbf{266}, 239-251 (2003).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{kumar}\n\tKumar, R., Prakash, O.: Divisor graph of the complement of $\\Gamma(R)$. Asian-Eur. J. Math. \\textbf{12}(4), DOI: 10.1142/S1793557119500578 (2019).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{kumar1}\n\tKumar, R., Prakash, O.: Pancyclic zero divisor graph over the ring $\\mathbb{Z}_n[i]$. Discrete Math. Algorithms Appl. DOI: 10.1142/S1793830922500495 (2022).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{redmond}\n\tRedmond, SP.: The zero-divisor graph of a non-commutative ring. Int. J. Commutative Ring. \\textbf{1}, 203-211 (2002).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{revelle}\n\tReVelle, CS., Rosing, KE.: Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly. \\textbf{107}(7), 585-594 (2000).\n\t\\vspace{.3 cm}\n\t\n\t\\bibitem{stewart}\n\tStewart, I.: Defend the Roman empire!. Sci. Amer. \\textbf{281}(6), 136-139 (1999).\n\t\\vspace{.3 cm}\t\n\t\n\t\n\n\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n"}}},{"rowIdx":85,"cells":{"id":{"kind":"string","value":"2412.07483v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07483v1"},"title":{"kind":"string","value":"Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map"},"full_text":{"kind":"string","value":"\\documentclass[a4paper,11pt]{amsart}\n\n\\usepackage[a4paper,top=3cm,bottom=3cm,\n left=3cm,right=3cm,marginparwidth=60pt]{geometry}\n\n\\usepackage{anysize} \n\n\\marginsize{1.3in}{1.3in}{1in}{1in}\n\\usepackage{comment}\n\\usepackage{xcolor}\n\\usepackage{amsmath}\n\\usepackage{mathtools}\n\\usepackage[all]{xy}\n\\usepackage[utf8]{inputenc}\n\\usepackage{varioref}\n\\usepackage[normalem]{ulem}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{bbm}\n\\usepackage{esint}\n\\usepackage{graphicx}\n\\usepackage{tikz}\n\\usepackage{empheq}\n\\usepackage{enumitem}\n\\usepackage{tikz-cd}\n\\usepackage[font=small,labelfont=bf]{caption}\n\\usepackage{subcaption}\n\\usetikzlibrary{matrix,arrows,decorations.pathmorphing}\n\\usepackage{mathrsfs}\n\\usepackage[hypertexnames=false,backref=page,pdftex,\n \tpdfpagemode=UseNone,\n \tbreaklinks=true,\n \textension=pdf,\n \tcolorlinks=true,\n \tlinkcolor=blue,\n \tcitecolor=red,\n \turlcolor=blue,\n ]{hyperref}\n\\renewcommand*{\\backref}[1]{{-- cited on p.~#1}}\n\\usepackage{cleveref}\n\n\\usepackage{soul} \n\n\n\n\\def\\franco#1{{\\color{cyan}(#1)}}\n\\definecolor{brickred}{rgb}{0.8, 0.25, 0.33}\n\\def\\luca#1{{\\color{brickred}#1}}\n\n\n\\usepackage[textsize=small]{todonotes}\n\\newcommand\\Luca[1]{\\todo[color=yellow!40]{#1}}\n\\newcommand\\Lucaline[1]{\\todo[inline,color=yellow!40]{#1}}\n\n\n\\newcommand{\\scrG}{{\\mathscr G}}\n\\def\\Mon{\\operatorname{Mon}}\n\\def\\sing{\\operatorname{sing}}\n\\def\\Ram{\\operatorname{Ram}}\n\\def\\Branch{\\operatorname{Branch}}\n\\def\\II{\\operatorname{II}}\n\\renewcommand{\\P}{{\\mathbb P}}\n\\newcommand{\\bbR}{{\\mathbb R}}\n\n\\newcommand{\\sO}{\\mathcal O}\n\n\\newcommand{\\wh}[1]{{\\widehat{#1}}}\n\\newcommand{\\wt}[1]{{\\widetilde{#1}}}\n\n\\newcommand{\\Bl}{\\operatorname{Bl}}\n\\newcommand{\\Gr}{\\operatorname{Gr}}\n\\newcommand{\\Hilb}{\\operatorname{Hilb}}\n\\newcommand{\\Sym}{\\operatorname{Sym}}\n\n\\newcommand{\\x}{\\times}\n\n\n\\newcommand{\\ord}{{\\mathrm{ord}}}\n\\newcommand{\\et}{{\\mathrm{et}}}\n\\newcommand{\\flatt}{{\\mathrm{flat}}}\n\n\n\n\\theoremstyle{plain}\n\\newtheorem{satz}[subsection]{Satz}\n\\newtheorem{theorem}[subsection]{Theorem}\n\\newtheorem{definition}[subsection]{Definition}\n\\newtheorem{question}[subsection]{Question}\n\\newtheorem{lemma}[subsection]{Lemma}\n\\newtheorem{corollary}[subsection]{Corollary}\n\\newtheorem{assumption}[subsection]{Assumption}\n\\newtheorem{set}[subsection]{Setting}\n\\newtheorem{bigthm}{Theorem}\n\\renewcommand{\\thebigthm}{\\Alph{bigthm}} \n\\newtheorem{proposition}[subsection]{Proposition}\n\n\n\n\n\n\\theoremstyle{remark}\n\\newtheorem{example}[subsection]{Example}\n\\newtheorem{remark}[subsection]{Remark}\n\n\n\n\n\\title[Monodromy of the Voisin map]{Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map}\n\n\\author[]{Franco Giovenzana}\n\\address[F. Giovenzana]{Laboratoire de Math\\'ematiques d’Orsay\\\\ Universit\\'e Paris-Saclay\\\\Rue Michel Magat, B\\^at. 307, 91405\\\\ Orsay, France}\n\\email{franco.giovenzana@universite-paris-saclay.fr}\n\n\n\\author[Franco and Luca Giovenzana]{Luca Giovenzana}\n\\address[L. Giovenzana]{Department of Pure Mathematics\\\\ University of Sheffield\\\\ Hicks Building, Hounsfield Road\\\\ Sheffield, S3 7RH\\\\ UK}\n\\email{l.giovenzana@sheffield.ac.uk}\n\n\n\n\n\n\n\n\n\\usepackage{framed}\n\n\\begin{document}\n\\thispagestyle{empty}\n\n\\begin{abstract}\nFor a cubic fourfold \\( Y \\) with associated Fano variety of lines \\( F \\), we establish several properties of the finite-degree 16 self-rational map \\( \\psi \\colon F \\dashrightarrow F \\) introduced by Voisin. We begin by analyzing the singularities of the nodal quintic with 16 nodes associated with a general line under the specialization to a line in the branch locus of \\( \\psi \\).\nThis approach reveals that the ramification of the natural resolution of indeterminacy of \\( \\psi \\) is simple. The main part of the paper focuses on the intriguing interplay between \\( \\psi \\) and the fixed locus of the antisymplectic involution on the LLSvS variety \\( Z \\), examined via the degree 6 Voisin map \\( F \\times F \\dashrightarrow Z \\). As an application, we show that the monodromy of \\( \\psi \\) is maximal.\n\\end{abstract}\n\n\n\\makeatletter\n\\@namedef{subjclassname@2020}{\n\t\\textup{2020} Mathematics Subject Classification}\n\\makeatother\n\n\\subjclass[2020]{32J27 (primary), 32S15 (secondary).}\n\\keywords{Irreducible symplectic varieties, cubic fourfolds}\n\n\\maketitle\n\n\\setlength{\\parindent}{1em}\n\\setcounter{tocdepth}{1}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nLet $Y$ be a smooth cubic fourfold and $F$ be its (Fano) variety of lines. The rich geometry of these varieties has attracted great attention from the mathematical community for several reasons. Notably, the Fano variety \n$F$ is one of the earliest examples of a locally complete family of projective hyperkähler manifolds, and the binome $Y-F$ between a Fano variety and a hyperk\\\"ahler manifold has inspired numerous other constructions.\n\nOne distinguishing feature of $F$ is the degree 16 self-rational Voisin map \n$\\psi\\colon F\\dashrightarrow F$. Remarkably, \n$\\psi$ is the only known finite, non-birational self-map defined on a locally complete family of projective hyperkähler manifolds. Finite maps such as $\\psi$ are particularly subtle to study as, unlike birational maps, they cannot be detected through their action on the second cohomology group and lack a comprehensive classification result as Hodge Torelli theorem that would allow precise control over their behavior. Since its construction \\cite{voisin-map-F}, this map has continued to attract significant interest in the mathematical community, as demonstrated by the numerous works on the subject\\cite{Amerik,GK-invariants, GK-monodromy, GK-lines}.\n\nIn this paper, we focus on the monodromy of the Voisin map $\\psi$. The monodromy group is a discrete invariant that encodes the symmetries and intrinsic geometry of finite maps. Determining the monodromy group of branched coverings is a classical problem, originating with Jordan in the 1870s. This topic was revitalized by Harris, who provided a modern framework by proving that the monodromy and Galois groups coincide, and developed tools to establish when the monodromy group is maximal. Significant progress was made by Vakil, who introduced innovative techniques to study monodromy in Schubert problems on Grassmannians \\cite{vakil}.\n\n\nBuilding on these developments, we take a step further by investigating linear spaces on a cubic hypersurface of dimension 4. Our main result shows that the monodromy group of the Voisin map $\\psi$ is maximal, meaning that it is the full symmetric group on 16 elements. \nTo achieve this, we leverage another Voisin map involving twisted cubics on the cubic fourfold, which unveils a fascinating connection with the fixed locus of the natural anti-symplectic involution on the LLSvS variety. \n\\bigskip\n\n\nWe now introduce the necessary notation and recall some useful results in order to state our main theorems.\nLines on a cubic fourfold fall into two cases: For a general line $L$ the linear space\n\\[\n\\Lambda_\\ell := \\bigcap_{p\\in L} T_p Y\n\\]\nis 2-dimensional and in this case the line is said to be of \\textit{type I}. For special lines, called of \\textit{type II}, the dimension of $\\Lambda_\\ell$ is 3.\nFor the general line we have $\\Lambda_\\ell \\cap Y = 2\\ell + r$ for a line $r\\in F$, and one sets $\\psi(\\ell) := r$. This map has been studied in detail in \\cite{Amerik}, its indeterminacy locus consists of the lines of type $\\II$, which for a general cubic fourfold form a smooth surface $S_{\\II}$. Blowing up $S_{\\II}$ provides a resolution of the indeterminacy\n\\[\n\\xymatrix{\n&\\wh F\\ar[ld]\\ar[rd]^{\\wh \\psi}\\\\\nF\\ar@{-->}[rr]^{\\psi} && F.\n}\n\\]\nGounelas and Kouvidakis recently computed that the restricion of $\\wh \\psi$ to the exceptional divisor $E$ of the blow-up morphism, which coincides with the ramification divisor of $\\wh\\psi$, is birational onto the image and they posed the question of whether the ramification of $\\wh\\psi$ is simple \\cite[Theorem~B, Remark~3.13]{GK-monodromy}.\n\nIn Section \\ref{sec:ramification} we study the ramification of $\\wh\\psi$ by consider the projection of $Y$ from a general line $R$ in $Y$. Its resolution is a conic bundle over $\\mathbb P^3$ with discriminant locus a quintic surface $S_r$ with exactly 16 nodes. These nodes correspond to the preimage of $r$ under $\\wh\\psi$.\nNodal quintic surfaces have been classically studied by Beauville and Catanese \\cite{Beauville-nodal, Catanese} and more recently studied in \\cite{HUY-nodal-quintics,7auth, catanese-new}.\nOur first attention is to study the singularity of $S_r$ for a special line $r$, which lies in the branch locus of $\\wh\\psi$.\n\\begin{theorem}[see Theorem~\\ref{thm:A3-singularity-quintic}]\nLet $r\\in F$ be a general line in the branch divisor of $\\wh\\psi$. Then the quintic\nsurface $S_r$ has one singularity of type $A_3$\ncorresponding to the unique line $L$ of type $\\II$ over $r$. \n\\end{theorem}\n \n As a result of our analysis we affirmatively answer Gounelas and Kouvidakis' question:\n\\begin{corollary}[{see Corollary~\\ref{cor:ram-simple}}]\n The ramification at the general point of the ramification locus of $\\wh\\psi$ is\nsimple.\n\\end{corollary}\n\nIn Section~\\ref{sec:variety-P} we shift our focus to the geometry of twisted cubics. With any smooth cubic fourfold $Y$ not containing a plane Lehn, Lehn, Sorger, and van Straten associated an eight-dimensional hyperkähler variety $Z$ parametrizing families of twisted cubics and their flat degenerations. The variety $Z$ is equipped with a natural antisymplectic involution $\\tau$ \\cite{Lehn-oberwolfach}.\nIts fixed locus is a smooth Lagrangian submanifold with 2 connected components: one is isomorphic to the cubic fourfold $Y$, the other one, $W$, is of general type \\cite{FMSOG-II} and remains somewhat mysterious. Using a the degree 6 rational map $\\varphi\\colon F\\times F \\dashrightarrow Z$, constructed by Voisin \\cite{Voisin-map-varphi}, we offer an alternative description of $W$.\nWe define the variety $P$ as the closure in $F\\times F$ of \n\\[\n\\{ (\\ell_1,\\ell_2)\\in F\\times F : \\ell_i \\text{ are of type I,\\quad $\\ell_1\\not = \\ell_2$,\\quad and }\\quad \\psi(\\ell_1)=\\psi(\\ell_2) \\},\n\\]\nwhich is birational to an irreducible component of the self-product of $\\wh F$ over $F$.\n\\begin{theorem}[see~Theorem~\\ref{thm: P->W}]\n The variety $P$ is mapped onto $W$ under the Voisin map $\\varphi\\colon F\\times F \\dashrightarrow Z$.\n\\end{theorem}\n\nIn Section~\\ref{sec:monodromy} after revising the basic notions of monodromy we tackle the study of $\\wh \\psi$. Despite extensive study and numerous works concerning $\\wh\\psi$, for example about its entropy \\cite{Amerik}, many of its properties remain elusive. Through an investigation of the restriction of the map $\\varphi$ to the variety $P$ we prove the following\n\n\\begin{theorem}[see Theorem~\\ref{thm:monodromy-maximal}]\n The monodromy group of $\\psi$ is the entire symmetric group on 16 elements.\n\\end{theorem}\n\n\\subsection*{Acknowledgments} This project began long time ago, and over the years we benefited from conversations with many people. It is our pleasure to thank everybody who expressed interest and shared their point of view, especially Enrico Fatighenti, Frank Gounelas, Christian Lehn, Emanuele Macrì, Giovanni Mongardi, Alan Thompson and Yilong Zhang.\n\nFranco Giovenzana was funded by Deutsche Forschungsgemeinschaft (DFG, German\nResearch Foundation) Projektnummer\n509501007, and partially supported by the European Research Council (ERC) under\nthe European Union’s Horizon 2020 research and innovation programme (ERC-2020-\nSyG-854361- HyperK). All authors are members of INdAM GNSAGA.\n\n\\section{The ramification of the Voisin map is simple}\\label{sec:ramification}\nIn this section we recall basic facts abour lines on cubic fourfolds and various properties of the Voisin map on $F$. Then we move on to prove that the ramification of the Voisin map is simple.\n\n\nRecall that the Gauss map associates to any point of the smooth cubic fourfold $Y\\subset \\P(V)$ its projective tangent space: \n\\[\n\\mathscr G\\colon Y \\to \\P(V^\\vee),\\ P\\mapsto T_P Y \\simeq \\P^4.\n\\]\nClemens and Griffiths distinguished lines on cubic hypersurfaces into two types, we recall here the definition for cubic fourfolds \\cite{griffiths-clemens}.\n\\begin{proposition}\nGiven a line $L$ on a smooth cubic fourfold $Y$, either the following equivalent conditions hold:\n\\begin{enumerate}\n \\item $N_{L|Y} \\simeq \\sO_L^{\\oplus 2}\\oplus \\sO_L(1)$,\n \\item $\\mathscr G|_L\\colon L\\to \\mathscr G (L)$ is 1:1,\n \\item $\\mathscr G (L)$ is a smooth conic in $\\P(V^\\vee)$,\n \\item $\\bigcap_{P\\in L} T_P Y$ is isomorphic to $\\P^2$;\n\\end{enumerate}\nor the following equivalent conditions hold:\n\\begin{enumerate}\n \\item[(5)] $N_{L|Y} \\simeq \\sO_L(-1)\\oplus \\sO_L(1)^{\\oplus 2}$,\n \\item[(6)] $\\mathscr G|_L\\colon L\\to \\mathscr G (L)$ is 2:1,\n \\item[(7)] $\\mathscr G (L)$ is a line in $\\P(V^\\vee)$,\n \\item[(8)] $\\bigcap_{P\\in L} T_P Y$ is isomorphic to $\\P^3$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{definition}[{\\cite[Definition~6.6]{griffiths-clemens}}]\nGiven a line $L$, we say that $L$ is a line of type I if the equivalent conditions $(1)-(4)$ hold, whereas if $(5)-(8)$ hold we say that $L$ is a line of type II. We set $\\Lambda_L:= \\cap_{P\\in L} T_P Y$.\n\\end{definition}\nWe record here these elementary observations for future reference.\n\\begin{remark}\nIn case $\\ell\\in F$ is of type $\\II$, then the line $\\mathscr G(L) \\subset \\P(V^\\vee)$ is the projective dual of $ \\Lambda_L \\subset \\P(V)$. In case of a line of type I, $\\mathscr G(L)$ spans a $\\P^2$ in $\\P(V^\\vee)$ which is dual to $\\Lambda_L$.\n\\end{remark}\n\n\\begin{remark}\\cite[Remark~2.2.2]{HuyBookCubics}\\label{huy-derivatives}\n Let $Y = V(F)\\subset \\P^5$ be a smooth cubic fourfold, let $L$ be a line in $Y$, then $L$ is of type II if and only if the partial derivatives $\\partial_0 F|_L,...,\\partial_6 F|_L$ span a vector space of dimension 2 in $H^0(L,\\sO_{\\mathbb P^5}(2))$.\n\\end{remark}\nVoisin observed that if $L$ is a line of type I, then $\\Lambda_L\\cap Y$ is a cubic curve, which, as it contains $L$ with multiplicity 2, consists of $L$ and a residual line $R$. As the general line is of type I, one defines the rational map\n\\begin{align*}\n\\psi\\colon F \\dashrightarrow F, \\quad\n\\ell \\mapsto r.\n\\end{align*}\nThis map has been studied in \\cite{voisin-map-F} and \\cite{Amerik}, where it is proven to be finite of degree 16. Lines of type II form a surface $S_{\\II}$ in $F$, and blowing up $F$ in this surface resolves the indeterminacy of $\\psi$:\n\n\\[\n\\xymatrix{\n&\\wh F\\ar[ld]\\ar[rd]^{\\wh \\psi}\\\\\nF\\ar@{-->}[rr]^{\\psi} && F.\n}\n\\]\n\n\nThe blow up $\\wh F$ can be identified as the closure of the graph \\cite[Lemma 4.1]{GK-lines},\\cite[Remark 2.2.19]{HuyBookCubics}.\n\\begin{align*}\nF \\dashrightarrow \\Gr(3,6), \\ \\ell \\mapsto \\Lambda_\\ell.\n\\end{align*}\nElements in the exceptional locus are then just pairs $(\\ell,\\Xi)$, where $L$ is a line of type II and $\\Xi$ is a projective plane such that\n$L \\subset \\Xi \\subset \\Lambda_\\ell$. For such 2-dimensional spaces $\\Xi$ we have that $\\Xi \\cap Y = 2\\ell + r$ and we shall say that $r$ is \\textit{residual} to $\\ell$.\n\nGiven a general line $r\\in F$, we consider the diagram\n\\[\\xymatrix{\n&\\wt Y:=\\Bl_R Y\\ar[ld]^p\\ar[rd]^{\\wt \\pi}\\\\\nY \\ar@{-->}[rr]^{\\pi_R} && \\P^3\n}\n\\]\nwhere $\\pi_R$ is the projection from $R$ and $\\wt Y$ is the blow-up of $Y$ in $R$. The morphism $\\wt \\pi$ is a conic bundle with discriminant a quintic surface $S_r$, whose singular locus consists of 16 nodes. The 16 nodes correspond to the preimage of $R$ under $\\wh\\psi$. Indeed, if $p_i$ denote the nodes of $S_R$ for $i=1,..,16$, then $L_i:=p(\\wt \\pi^{-1}(p_i))$ are the 16 lines for which $\\psi(L_i) = R$ (see \\cite[\\S~6.4.5]{HuyBookCubics} for an account on the various results in the literature about this surface).\n\nLet $E$ be the exceptional divisor of the blow-up morphism $\\wh F \\to F$. As $F$ has trivial canonical bundle, the divisor $E$ coincide with the ramification locus of the map $\\wh \\psi$.\n\n\\begin{theorem}[Gounelas-Kouvidakis,\\cite{GK-lines}]\\label{GK-RamBirational}\n The restriction $\\wh \\psi|_E \\colon E \\to F$ is generically 1-to-1 onto the image.\n\\end{theorem}\n\nIn other words, over the general point $r$ in the branch divisor of $\\wh \\psi$, there is exactly one point of ramification, i.e. there exists exactly one line $L$ of type II with residual $R$, meaning that $\\wh\\psi (\\ell, \\langle L,R \\rangle) = r$.\n\\begin{theorem}\\label{thm:A3-singularity-quintic}\n Let $r\\in F$ be a general line in the branch divisor of $\\wh \\psi$. Then the quintic surface $S_r$ has exactly 14 singularities of type $A_1$ and exactly 1 singularity of type $A_3$ corresponding to the unique line $L$ of type $\\II$ with residual line $R$.\n\\end{theorem}\n\\begin{proof}\n Let $r$ be a general line in the branch locus of $\\wh\\psi$ and let $\\ell$ be the unique line of type $\\II$ with residual $r$. We may assume that $R$ is of type I and that is given as the zero set $V(x_2,x_3,x_4,x_5)$. Following Clemens and Griffiths \\cite[equation~6.9]{griffiths-clemens} we write the equation of the cubic fourfold as\n \\begin{equation}\n x_4x_0^2 + x_5x_0x_1 + x_3x_1^2 + x_0Q_0 + x_1Q_1 + P = 0,\n \\end{equation}\n where $Q_0,Q_1$ and $P$ are homogeneous polynomials in the variables $x_2,x_3,x_4,x_5$ of degree 2, 2, and 3 respectively. \n The quintic surface $S_r\\subset V(x_0,x_1) \\simeq \\mathbb P^3$ is given by the determinant of the matrix\n \\[\n M:=\\begin{pmatrix}\nP & Q_0 & Q_1\\\\\nQ_0 & x_4 & x_5 \\\\\nQ_1 & x_5 & x_3\n\\end{pmatrix}.\n\\]\n\nWe assume that the line $L$ is given by $V(x_1,x_2,x_4,x_5)$\\footnote{Notice that with this assumption the line $L$ is not contained in the plane tangent to $R$ is $\\Lambda_r = V(x_3,x_4,x_5)$, would have been in contradiction with $R$ being residual to $L$.},\nthen $p := (1:0:0:0:0:0)$ is the intersection of $L$ and $R$, and $q:=(0:0:0:1:0:0)$ is the point of intersection with $\\mathbb P^3$.\n\nWe translate these assumptions in conditions on the coefficients of $Q_0,\\ Q_1$, and $P$.\nFor this, we introduce the notation $Q_0 = \\sum a_I\\underline{x}^I$, $Q_1 = \\sum b_I\\underline x^I$ and $P = \\sum c_I\\underline x^I$.\n\nThe cubic fourfold $Y$ contains the line $L$ if $F(x_0,0,0,x_3,0,0,0)$ is the zero form in $\\mathbb C[x_0,x_3]$, hence we get the conditon\n\\begin{align}\\label{eq: cond1}\na_{33} = c_{333} = 0.\n\\end{align}\n\nThe fact that the line $R$ is residual to $L$, i.e. $\\langle L, R\\rangle \\cap Y = 2L+R$, translates into the further condition\n\\begin{align}\\label{eq: cond2}\nb_{33} = 0.\n\\end{align}\n\nBy Remark~\\ref{huy-derivatives} the line $L$ is of type II can be rephrased in terms of the derivatives $\\partial_i F|_L$, and we get the additional condition\n\\begin{align}\\label{eq: cond3}\n a_{23}c_{335} - a_{35}c_{233} = 0.\n\\end{align}\n\nTo conclude the proof we check with the aid of Macaulay2 that the surface defined by det$(M)$ for polynomials $Q_0,\\ Q_1,$ and $P$ with coefficients satisfying conditions~\\eqref{eq: cond1},~\\eqref{eq: cond2}, and \\eqref{eq: cond3}\nhas a singularity of type $A_3$ at the point $q$. See ancillary file. \n\\end{proof}\n\n\\begin{corollary}\\label{cor:ram-simple}\n The ramification at the general point of the ramification locus of $\\wh\\psi$ is simple.\n\\end{corollary}\nThis answers the natural question raised in \\cite[Remark~3.13]{GK-monodromy}.\n\\begin{proof}\n Let $b$ be a general point in the branch divisor of $\\psi$. Then by Theorem \\ref{GK-RamBirational} the preimage $\\wh\\psi^{-1}(b)$ consists of a finite number of points $p_1,\\ldots p_k$ where only $p_1$ is of ramification, while $\\wh\\psi$ is \\'etale at the points $p_2,\\ldots p_k$. All the points $p_i$ correspond to singularities of the quintic surface $S_b$: in particular $p_2,\\ldots p_k$ are singularities of type $A_1$, in contrast $p_1$ is a singular point of type $A_3$ by Theorem~\\ref{thm:A3-singularity-quintic}. \n The surface $S_b$ is specialization of the surface $S_r$ for the general line $r\\in F$, which has exactly 16 nodes. Thus we deduce that 2 points among these 16 specialize to the singularity of type $A_3$, whereas the remaining 14 specialize to the $p_2,..,p_k$, deducing that $S_b$ has 14 singularities of type $A_1$ apart from the $A_3$ singularity.\n\\end{proof}\nLet $r$ be any line on the cubic fourfold, we denote by $F_r$ the locus of lines meeting $r$\n\\[\nF_r :=\\lbrace \\ell \\in F: L\\cap R \\not = \\emptyset \\rbrace.\n\\]\nIf $r$ is general, then $F_r$ is a smooth surface \\cite[\\S 3~ Lemma~1]{voisin-torelli}. As an application of our analysis we determine the singularities of $F_r$ for a general point $r$ in the branch locus of $\\wh\\psi$.\n\n\\begin{corollary}\nLet $r\\in F$ be a general line in the branch locus of $\\wh \\psi$, then the surface $F_r$ has one singularity of type $A_1$ at the point $\\ell$ corresponding to the unique line $L$ of type $\\II$ with residual line $R$.\n\\end{corollary}\n\\begin{proof}\n Let $\\pi_R\\colon Y \\dashrightarrow \\P^3$ be the projection from $R$. Then there is a map \n \\[f\\colon F_r \\dashrightarrow S_r,\\quad \\ell \\mapsto \\pi_R(L).\\]\n The map clearly extends on the point $r$ by mapping $r$ to $\\pi_r(\\Lambda_r)$, when the line $r$ is of type $I$. The map $f$ is 2-to-1 and can be regarded as the quotient of an involution, which, in case $r$ is general, it has 16 fixed points mapped to the 16 singular points of $S_r$. If $r$ is as in the hypothesis, then $S_r$ is has 14 singularities of type $A_1$ and one singularity of type $A_3$ corresponding to $\\ell$. All the nodes get resolved in the 2-to-1 cover $F_r$, while the $A_3$ singularity gives an $A_1$ singularity, hence the claim.\n\\end{proof}\n\n\\section{On a lagrangian submanifold of $Z$ and singular cubic surfaces}\\label{sec:variety-P}\n\nGiven a smooth cubic fourfold $Y$, we denote by $Z=Z(Y)$ the associated LLSvS variety. It carries the action of an antisymplectic involution which has been studied in \\cite{FMSOG-II} whose fixed locus consists of two connected components, one isomorphic to the cubic fourfold, which we hence denote $Y$, and a second one which we denote $W$. Goal of this section is to study the less understood variety $W$. In order to do that we introduce a subvariety $P$ of $F\\times F$ which is related to $W$ via the Voisin map $\\varphi: F\\times F\\dashrightarrow Z$. A key tool in our strategy is to reduce statements to the study of cubic surfaces $S$ which are given by intersecting the cubic $Y$ with a projective space $\\mathbb P^3$ and consider lines and twisted cubics on $S$.\n\n\\subsection{The LLSvS variety $Z$}\nGiven a smooth cubic fourfold $Y$ not containing any plane, Lehn, Lehn, Sorger and van Straten constructed in \\cite{LLSvS} a hyperk\\\"ahler variety $Z$ of dimension 8, which parametrizes families of twisted cubics on $Y$ and their flat degenerations.\n\nAs any twisted cubic has $3t+1$ as Hilbert polynomial, the starting point is to consider the compactification $\\Hilb^{gtc}(Y)$ of the space of twisted cubics inside the Hilbert scheme $\\Hilb^{3t+1}(Y)$. Curves parametrized by this variety are called \\textit{generalised twisted cubics}. The variety $\\Hilb^{gtc}(Y)$ is smooth, of dimension 10 and admits a morphism to the Grassmannian $G:=\\Gr(4,6)$, which maps any generalised twisted cubic to its linear span. \nThis morphism factors through a $\\P^2$-fibration onto a variety $Z'$ which has a generically finite morphism to the Grassmannian.\nThe hyperk\\\"ahler projective manifold $Z$ is obtained by $Z'$ by contracting the divisor parametrising linear systems of curves which are not arithmetically Cohen-Macaulay. It comes with the natural polarisation $\\mathcal{L}$ induced by the map to the Grassmannian $G$. It has dimension 8 and the image of the divisor under the contraction is a Lagrangian subvariety isomorphic to the cubic fourfold $Y$.\n\n\\begin{remark}\nThe construction above of the variety $Z'$ is performed relatively over the Grassmannian $G$, so that can be carried out for any cubic surface $S$, thus we shall and will use the variety $Z'(S)$ for any cubic surface $S$. It clearly comes equipped with the $\\P^2$-fibration $\\mathrm{Hilb}^{gtc}(S)\\to Z'(S)$.\n\\end{remark}\n\n\nThe hyperk\\\"ahler manifold $Z$ has a biregular antisymplectic involution $\\tau\\colon Z \\to Z$ as pointed out in \\cite{Lehn-oberwolfach}. It can be defined as follows: Let $C\\subset Y$ be a generalised twisted cubic and let $Q$ be any quadric (surface) in the linear span $\\langle C \\rangle$, which contains $C$. The intersection $Q\\cap Y$ is a sextic which contains $C$, the residual cubic is another generalised twisted cubic $C'$.\nAntisymplectic involutions of K3$^{[n]}$-type manifolds and this particular example have been studied in \\cite{antisymplectic, FMSOG-II}.\n\n\\begin{theorem}[{\\cite[Theorem 1.4]{FMSOG-II}}]\nThe fixed locus of the antisymplectic involution has two connected components, one of which is the lagrangian subvariety $Y$. The other component $W$ is of general type, in particular $\\omega_{W} = 3 \\mathcal{L}|_{W}$, where $\\mathcal L$ is the primitive polarization on $Z$.\n\\end{theorem}\n\n\n\nVoisin constructed a degree 6 rational map $\\varphi\\colon F\\times F \\dashrightarrow Z$ in this way \\cite{Voisin-map-varphi}. \nGiven a general pair of skew lines $(\\ell_1, \\ell_2)$ and a point $p\\in L_1$, then the union of $\\ell_1$ and the quadric, which is residual to $\\ell_1$ in the intersection $\\langle L_1,\\ p\\rangle$, is a generalized twisted cubic. The indeterminacy locus, the resolution of it and the branch divisor of its resolution have been studied in \\cite{Muratore, chen, mio}.\n \nThe involution $\\tau$ fits in the following commutative diagram:\n\\[\\xymatrix{\nF\\x F \\ar[d]^\\sigma \\ar@{-->}[r]^{\\varphi} & Z\\ar[d]^\\tau\\\\\nF\\x F \\ar@{-->}[r]^{\\varphi} & Z,\n}\n\\]\nwhere $\\sigma\\colon (\\ell_1,\\ell_2)\\mapsto (\\ell_2,\\ell_1)$ switches the factors.\n\n\\subsection{A variety dominating $W$}\n\nWe denote by $P$ the closure in $F\\times F$ of \n\\[\n\\{ (\\ell_1,\\ell_2)\\in F\\times F : \\ell_i \\text{ are of type I, $\\ell_1\\not = \\ell_2$ and } \\psi(\\ell_1)=\\psi(\\ell_2) \\}.\n\\]\nIt turns out to have a central r\\^ole in our study: The following intriguing lemma is the starting point which allows us to relate properties of $\\psi$ and $\\phi$. \n\\begin{lemma}\\label{lem: fundamental}\n Let $(\\ell_1, \\ell_2)\\in P$ be a general pair of lines, satisfying the conditions\n\\begin{enumerate}\n \\item $\\ell_1$ and $\\ell_2$ are skew,\n \\item $\\ell_1$ and $\\ell_2$ are both of type I, \n \\item and $\\psi(\\ell_1) = \\psi(\\ell_2)$.\n\\end{enumerate}\n\n Then the linear span of $L_1$ and $L_2$ intersects $Y$ in a Cayley cubic surface.\n\\end{lemma}\n\\begin{proof}\nWe claim that the linear span $\\Lambda:=\\langle L_1, L_2\\rangle$ intersects $Y$ in a cubic surface with two singular points on $L_1$. We denote by $r$ the residual line $\\psi(\\ell_1)$, then $R$ and $L_1$ are coplanar and we consider the line $T$ in $\\P(V^*)$ of hyperplanes which contain $\\Lambda$. \n\n\nAs both $T$ and $\\scrG (L_1)$ lie in the 2 dimensional linear space $\\langle L_1, R \\rangle^* \\subset \\P(V^*)$, the line $T$ intersects the smooth conic $\\scrG (L_1)$ in two points, which correspond to the image under the Gauss map of two singular points of $S:=\\Lambda\\cap Y$ lying on the line $L_1$.\n\nRepeating the argument for $L_2$ we find two singular points of $S$ on $L_2$ and we deduce that the cubic surface $S$ is Cayley by \\cite[Corollary~9.2.3]{CAG}.\n\\end{proof}\nFollowing the construction of the Voisin map, one gets the following\n\\begin{lemma}\\label{fundamental lemma}\nLet $L_1$ and $L_2$ be two lines that satisfy the general conditions:\n\\begin{enumerate}\\label{general-conditions}\n \\item $L_1$ and $L_2$ are of type I,\n \\item $L_1$ and $L_2$ are skew.\n\\end{enumerate}\nIf $\\psi(\\ell_1)= \\psi(\\ell_2)=:r$ then $\\varphi(\\ell_1, \\ell_2) = \\varphi(\\ell_2, \\ell_1)$.\n\\end{lemma}\n\nWe report the following theorem which is a key result for our study, but we defer its proof to later, because it relies on some observations conerning Cayley cubics.\n\\begin{theorem}\\label{thm: P->W}\n The Voisin map $\\varphi\\colon F\\times F \\dashrightarrow Z$ restricts to a dominant rational map $\\varphi|_P\\colon P\\dashrightarrow W$. \n \n The regular map $W\\to G$ is birational onto the image, which consists of the projective spaces that intersect $Y$ in a Cayley cubic or a degeneration of it.\n\\end{theorem}\n\nIn the following we analyze the maps $P\\dashrightarrow W \\to G$ in greater detail. In order to do so, we describe singular cubic surfaces with focussing on lines on them.\n\n\n\\subsection{Cayley cubics and their degenerations}\nThe stratification of the space of cubic surfaces given by the type of singularities has been studied by Bruce in \\cite{bruce}. Here we are interested only in cubic surfaces with exactly 4 singular points of type $A_1$, called Cayley cubics, and their degenerations. From his analysis degenerations of Cayley cubics can be summed up by the following diagram:\n\\[\n4A_1\\ \\succeq\\ 2A_1A_3\\ \\succeq\\ A_1A_5\\ \\succeq\\ \\wt E_6\\ \\succeq X_6\\ \\succeq X_7\\ \\succeq X_8\\ \\succeq\\ X_9\n\\]\nwhere $\\wt E_6$ denotes the cubics with one simple elliptic singularity and $X_i$ are non-normal cubic surfaces.\n\nIn the following we focus on lines and twisted cubic on singular cubic surfaces to study the maps $\\varphi$ and $\\psi$. We notice that the map $\\psi$ is in general not defined on $Z(S)$ for a singular cubic surface $S$ because a line $L$ need not have a plane tangent at every point. However, if $\\pi\\subset \\langle S\\rangle$ is tangent at every point of $L\\subset S$ with $\\pi \\cap S = 2L + R$ for a line $R$, we say that the line $R$ is \\textit{residual}, and that $L$ is \\textit{triple} if $R=L$.\nGeneralised twisted cubics on singular cubic surfaces have already been carefully analysed in \\cite[\\S~2]{LLSvS}, which we closely follow.\nEvery cubic surface $S$ with at most double rational points has a minimal resolution $\\wt S\\to S$ with $\\wt S$ isomorphic to the blow-up of $\\mathbb P^2$ in six points.\nThe orthogonal $\\Gamma:=K_{\\wt S}^{\\perp}\\subset H^2(\\wt S, \\mathbb Z)$ is isomorphic to the negative definite root lattice $E_6$. We denote by $R_0$ the roots in $\\Gamma$ which are effective, they correspond to irreducible components of exceptional curves for the resolution. We also denote by $W(R_0)$ the subgroup of the Weyl group of $E_6$ generated by reflections in the effective roots $R_0$.\nThe moduli space of generalised twisted cubics on $S$ with reduced structure is then given by a union of projective planes \\cite[Theorem~2.1]{LLSvS}:\n\\begin{align*}\n\\mathrm{Hilb}^{gtc}(S)_{red} = (R/W(R_0)) \\times \\P^2.\n\\end{align*}\nIn particular, linear systems of generalised twisted cubics on $S$ are in bijection with the orbits of $R$ under the action of $W(R_0)$. These correspond to the set $Z'(S):=a^{-1}(E)$ where $E\\in G$ is the linear span $\\langle S \\rangle$. Along this bijection families of arithmetically Cohen-Macaulay (aCM) curves correspond to orbits containing an effective root.\nBy the definition of $\\tau$ it is clear that it restricts to an action on twisted cubics contained on a cuibic surface, and indeed admits a nice description in this setting: As observed in \\cite{Lehn-oberwolfach} it is indeed given by the association $\\alpha \\mapsto -\\alpha$ for the roots $\\alpha \\in R$.\n\nWe defer to Appendix~\\ref{appendix:cubic-surfaces} details about these singular surfaces, and recall here only the main facts we need to analyse the the Voisin maps.\n\n\nWe also introduce a piece of notation. We denote by $G_{4A1}$ the closure in the Grassmanian $G$ of \n\\[\n\\{ E\\in G: E\\cap Y \\mbox{ is a singular cubic surface of type }4A_1 \\}.\n\\]\n Analogously, we define the loci $G_{2A_1A_3}$, $G_{A_1A_5}$, $G_{\\wt E_6}$, and for the non-normal surfaces $G_{X_6}$, $G_{X_7}$, $G_{X_8}$, $G_{X_9}$.\n\\bigskip\n\n\n\\subsubsection*{Singular cubic surfaces of type $4A_1$}\nAny Cayley cubic, i.e. a surface with exactly 4 $A_1$ singular points, contains exactly 9 lines whose schematic representation resembles a tetrahedron with vertices at the singular points. We name \\textit{edges} the 6 of them connecting 2 singular points. In particular, if $P_1, P_2, P_3, P_4$ are the singular points, we denote $E_{ij}$ the line containing $P_i$ and $P_j$. Each of the other 3 lines intersect exactly 2 edges, we denote $J_{ij,kl}$ the line joining the edges $E_{ij}$ and $E_{kl}$ for $\\{i,j,k,l\\} = \\{1,2,3,4\\}$. See Figure~\\ref{fig:4A1}.\n\n\n\\begin{figure}[h!]\n \\centering\n \\captionsetup[subfigure]{justification=centering}\n \\begin{subfigure}[t]{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\n\\draw (313,38.47) -- (450.5,262.04) -- (175.5,262.04) -- cycle ;\n\\draw (313,38.47) -- (317.03,178.12) ;\n\\draw (317.03,178.12) -- (175.5,262.04) ;\n\\draw (317.03,178.12) -- (450.5,262.04) ;\n\\draw [draw opacity=0] (269.15,197.77) .. controls (268.26,193.46) and (267.8,189.03) .. (267.8,184.5) .. controls (267.8,155.17) and (287.19,129.69) .. (315.66,116.85) -- (357.65,184.5) -- cycle ; \\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (269.15,197.77) .. controls (268.26,193.46) and (267.8,189.03) .. (267.8,184.5) .. controls (267.8,155.17) and (287.19,129.69) .. (315.66,116.85) ; \n\\draw [draw opacity=0] (356.5,262) .. controls (356.5,262) and (356.5,262) .. (356.5,262) .. controls (317.84,262) and (285.01,240.33) .. (273.19,210.21) -- (356.5,185.5) -- cycle ; \\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (356.5,262) .. controls (356.5,262) and (356.5,262) .. (356.5,262) .. controls (317.84,262) and (285.01,240.33) .. (273.19,210.21) ; \n\\draw [draw opacity=0] (361.11,117.19) .. controls (371.99,142.82) and (368.74,171.71) .. (353.78,193.75) -- (287.72,147.9) -- cycle ; \\draw [color={rgb, 255:red, 5; green, 26; blue, 245 } ,draw opacity=1 ] (361.11,117.19) .. controls (371.99,142.82) and (368.74,171.71) .. (353.78,193.75) ; \n\\draw [draw opacity=0] (347.39,202.4) .. controls (342.83,207.39) and (337.52,211.81) .. (331.48,215.48) .. controls (307.03,230.36) and (277.09,229.15) .. (253.14,214.97) -- (289.6,146.64) -- cycle ; \\draw [color={rgb, 255:red, 5; green, 26; blue, 245 } ,draw opacity=1 ] (347.39,202.4) .. controls (342.83,207.39) and (337.52,211.81) .. (331.48,215.48) .. controls (307.03,230.36) and (277.09,229.15) .. (253.14,214.97) ; \n\\draw [draw opacity=0] (322.73,130.27) .. controls (327,131.57) and (331.13,133.39) .. (335.04,135.74) .. controls (357.76,149.43) and (367.69,177.83) .. (363.55,208.66) -- (285.92,217.34) -- cycle ; \\draw [color={rgb, 255:red, 126; green, 211; blue, 33 } ,draw opacity=1 ] (322.73,130.27) .. controls (327,131.57) and (331.13,133.39) .. (335.04,135.74) .. controls (357.76,149.43) and (367.69,177.83) .. (363.55,208.66) ; \n\\draw [draw opacity=0] (238.81,158.65) .. controls (259.05,137.86) and (285.3,127.04) .. (309.5,129.02) -- (288.16,218.74) -- cycle ; \\draw [color={rgb, 255:red, 126; green, 211; blue, 33 } ,draw opacity=1 ] (238.81,158.65) .. controls (259.05,137.86) and (285.3,127.04) .. (309.5,129.02) ; \n\n\\draw (308.75,266.88) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{12}};\n\\draw (208.67,135.62) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{02}};\n\\draw (395.13,128.85) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{01}};\n\\draw (291.27,89.62) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{03}};\n\\draw (358.87,220.02) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{13}};\n\\draw (221.47,233.42) node [anchor=north west][inner sep=0.75pt] [align=left] {E\\textsubscript{23}};\n\\draw (272,158.2) node [anchor=north west][inner sep=0.75pt] [font=\\footnotesize,xslant=0] {$J_{03,12}$};\n\\draw (298.4,206.6) node [anchor=north west][inner sep=0.75pt] [font=\\footnotesize] {$J_{01,23}$};\n\\draw (330.4,154) node [anchor=north west][inner sep=0.75pt] [font=\\footnotesize] {$J_{02,13}$};\n\\draw (314,15.8) node [anchor=north west][inner sep=0.75pt] {$P_{0}$};\n\\draw (459.2,257.4) node [anchor=north west][inner sep=0.75pt] {$P_{1}$};\n\\draw (147.6,256.6) node [anchor=north west][inner sep=0.75pt] {$P_{2}$};\n\\end{tikzpicture}\n\\caption{Lines on a singular cubic surface of type $4A_1$}\n \\label{fig:4A1}\n \\end{subfigure} \\hfill\n \\begin{subfigure}[t]{0.45\\textwidth}\n \\centering\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\n\\draw (279,63) -- (356,204) ;\n\\draw (212,206) -- (279,63) ;\n\\draw (212,206) -- (356,204) ;\n\\draw [draw opacity=0][line width=1.5] (260.19,24.83) .. controls (270.77,40.35) and (280.91,66.22) .. (287.21,96.41) .. controls (294.55,131.62) and (294.76,163.54) .. (288.93,181.22) -- (257.84,102.54) -- cycle ; \\draw [color={rgb, 255:red, 10; green, 15; blue, 219 } ,draw opacity=1 ][line width=1.5] (260.19,24.83) .. controls (270.77,40.35) and (280.91,66.22) .. (287.21,96.41) .. controls (294.55,131.62) and (294.76,163.54) .. (288.93,181.22) ; \n\\draw [draw opacity=0][line width=1.5] (273.85,127.49) .. controls (287.42,135.81) and (300.95,164.16) .. (306.8,199.29) .. controls (309.84,217.54) and (310.32,234.74) .. (308.66,248.77) -- (277.21,204.21) -- cycle ; \\draw [color={rgb, 255:red, 9; green, 9; blue, 207 } ,draw opacity=1 ][line width=1.5] (273.85,127.49) .. controls (287.42,135.81) and (300.95,164.16) .. (306.8,199.29) .. controls (309.84,217.54) and (310.32,234.74) .. (308.66,248.77) ; \n\n\\draw (272,14) node [anchor=north west][inner sep=0.75pt] [align=left] {L\\textsubscript{1}};\n\\draw (314,237) node [anchor=north west][inner sep=0.75pt] [align=left] {L\\textsubscript{2}};\n\\draw (244,211.4) node [anchor=north west][inner sep=0.75pt] {$R_{03}$};\n\\draw (208,125) node [anchor=north west][inner sep=0.75pt] [align=left] {R\\textsubscript{02}};\n\\draw (325,119) node [anchor=north west][inner sep=0.75pt] [align=left] {R\\textsubscript{23}};\n\\draw (288,50) node [anchor=north west][inner sep=0.75pt] [align=left] {P\\textsubscript{2}};\n\\draw (187,194) node [anchor=north west][inner sep=0.75pt] [align=left] {P\\textsubscript{0}};\n\\draw (360,197) node [anchor=north west][inner sep=0.75pt] [align=left] {P\\textsubscript{3}};\n\\draw (277,147) node [anchor=north west][inner sep=0.75pt] [align=left] {P\\textsubscript{1}};\n\\end{tikzpicture}\n\\caption{Lines on a $2A_1A_3$ surface}\n \\label{fig:2A1A3}\n \\end{subfigure}\n \\caption{}\n \\label{fig:figures}\n\\end{figure}\n\n\n\nWe denote by $K$ the linear system of generalised twisted cubics containing the reducible elements given by the lines through a singular point\n \\[\n K = [E_{ij}\\cup E_{ik} \\cup E_{il}] = [E_{ji}\\cup E_{jk} \\cup E_{jl}] \\in Z'_{aCM}(S).\n \\]\n\\begin{lemma}\\label{lem: opposite edges}\n Opposite edges map to $K$ under $\\varphi$, that is:\n \\begin{align*}\n \\varphi(E_{ij}, E_{kl}) = K\\quad \\text{ for indices such that } \\{i,j,k,l\\}=\\{1,2,3,4\\}.\n \\end{align*}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem: 4A_1 LLSvS}\nLet $S$ be a Cayley cubic. Then\n\\begin{align*}\n \\# Z'_{aCM}(S)=13\\quad \\mbox{ and }\\quad \\#Z'(S)^{\\tau} = 1.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n The first equality follows from \\cite[Table~1]{LLSvS}. The second one can be deduced by the description of $\\tau$ in terms of roots.\n\\end{proof}\n\\medskip\n\n\\begin{proof}(of Theorem~\\ref{thm: P->W})\n By Lemma~\\ref{lem: fundamental} the general point $(\\ell_1,\\ell_2)$ in $P$ corresponds to opposite edges on a Cayley cubic surface. By Lemma~\\ref{lem: opposite edges} its image $\\varphi((\\ell_1, \\ell_2))$ is an aCM twisted cubic fixed by the involution on $Z$, i.e. a point in $W$. In particular, $\\varphi$ restricts to a dominant map $\\varphi: P\\dashrightarrow W$. Finally, the morphism $W\\to G$ is 1:1 because by Lemma~\\ref{lem: 4A_1 LLSvS} given a projective space $\\Lambda \\simeq \\mathbb P^3$ which intersects $Y$ in a Cayley cubic $S$, $K$ is the unique linear system of $aCM$ generalised twisted cubics in $S$ which is fixed by $\\tau$.\n\\end{proof}\n\n\\subsubsection*{Singular cubic surfaces of type $2A_1A_3$}\n\nLet $S$ be a cubic surface with a singular point $P_2$ of type $A_3$, and 2 singular points $P_0,P_3$ of type $A_1$. It contains exactly 5 lines as depicted in Figure~\\ref{fig:2A1A3}: the line $R_{01}$ through $P_0,P_1$; the line $R_{02}$ through $P_0,P_2$; the line $R_{03}$ through $P_{03}$; the line $L_1$ containing the only singular point $P_2$, and the line $L_2$ contained in the smooth locus.\n\n\n\\begin{proposition}\\label{prop:2A1A3-residual}\nLet $S$ be a cubic surface of type $2A_1A_3$. Then:\n \\begin{align*}\n L_1\\mbox{ is residual to }R_{23},\\qquad\n L_1 \\mbox{ is residual to }R_{02} ,\\\\ \n L_2\\mbox{ is residual to }R_{03},\\qquad\n L_2\\mbox{ is residual to } L_1.\n \\end{align*}\n \nMoreover, there is no plane in $\\langle S\\rangle\\simeq \\mathbb P^3$ tangent to $S$ at every point of $L_2$.\n In particular, there are no triple lines on a surface with $2A_1A_3$ singularities.\n\\end{proposition}\n\n\n\\begin{remark}\n If $S$ is a cubic surface of type $2A_1A_3$ given by intersecting a smooth cubic fourfold $Y$ with a $\\mathbb P^3$, then one can conclude there are no triple lines on S.\n Indeed, if all the lines are of type I, Proposition~\\ref{prop:2A1A3-residual} can be restated as:\n \\begin{align*}\n \\psi(R_{23})=\\psi(R_{02}) = L_1, \\text{ and } \\psi(R_{03})= \\psi(L_1) = L_2.\n \\end{align*}\n If a line, e.g. $R_{23}$, is of type II one can conclude that $\\wh\\psi(R_{23}, \\pi)=L_1$ where $\\pi=\\langle R_{23}, L_1\\rangle$.\n\\end{remark}\n\n\\begin{lemma}\nLet $S$ be a cubic surface of type $2A_1A_3$. Then\n\\begin{align*}\n \\# Z'_{aCM}(S)=5\\quad \\mbox{ and }\\quad \\#Z_{aCM}'(S)^{\\tau} = 1.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n The first equality follows from \\cite[Table~1]{LLSvS}. The second one can be deduced by the description of $\\tau$ in terms of roots.\n\\end{proof}\n\n\n\\subsubsection*{Singular cubic surfaces of type $A_1A_5$}\\label{ssec:A1A5}\n\nLet $S$ be a cubic surface with an $A_1$ and an $A_5$ singularity. It contains exactly two lines, say $L_1$ passing through both singular points, and $L_2$ which intersects the singular locus in the $A_5$ point.\n\\begin{proposition}\\label{prop: lines-A1A5}\n Let $S$ be a cubic surface of type $A_1A_5$. Then\n \\begin{align*}\n L_2 \\text{ is triple, and } L_2 \\text{ is residual to } L_1.\n \\end{align*}\n\\end{proposition}\n\n\\begin{lemma}\\label{lem:gtc-on-A1A5}\nLet $S$ be a cubic surface of type $A_1A_5$. Then\n\\begin{align*}\n \\# Z'_{aCM}(S)= \\#Z_{aCM}'(S)^{\\tau} = 1.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\n Clear from \\cite[Table~1]{LLSvS}.\n\\end{proof}\n\n\\subsection{Simple elliptic cubic surfaces}\nAny cubic surface $S$ with a simple elliptic singularity is a cone over a planar elliptic curve $E$. Let $v$ be the vertex. The Fano variety $F(S)$ of $S$ can be naturally identified with the embedded elliptic curve $E$, indeed to any point $P$ of $E$ corresponds the line through $P$ and $v$.\nThe scheme of generalised twisted cubics is isomorphic to $\\Sym^3(E)$ and the summation map to $E$ is a $\\P^2$-bundle giving the fibration $\\Hilb^{gtc}(S)\\to Z'(S)$ \\cite[Proposition~2.7]{LLSvS}. A generalised twisted cubic corresponds to the union of 3 lines, which join the vertex with 3 points on $E$. If these are collinear, then they are the support of a nCM curve with an embedded point at the vertex. Via the isomorphism $F(S)\\simeq E$, the residual line can be expressed in terms of the group law of $E$, and the analogue of $P$ for lines on the surface $S$\n\\begin{align*}\n P(S):= \\{(p_1,p_2)\\in F(S)\\times F(S) \\mid \\mbox{ $p_1$ and $p_2$ have the same residual}\\}\n\\end{align*}\nadmits a nice description:\n\n\n\\begin{proposition}\n Let $S$ be a simple elliptic cubic surface. Then\n \\begin{itemize}\n \\item the residual to the line $p\\in E\\simeq F(S)$ is given by $-2p$;\n\n \\item $P(S) =\\{(p,p+\\xi) \\mid \\xi\\in E[2]\\}$;\n \\item the triple lines on $S$ are given by the 3-torsion points $E[3]\\subset E$. In particular, every simple elliptic cubic surface has 9 triple lines.\n \\end{itemize}\n\\end{proposition}\n\n\\begin{proposition}\\label{cor: A1A5-E6}\n Let $Y$ be a general cubic fourfold, and let $\\ell_1$ and $\\ell_2$ be general lines of type $I$ such that $\\psi(\\ell_1) = \\psi(\\ell_2) = \\ell_2$. Then $\\langle E_{\\ell_1}, E_{\\ell_2} \\rangle$ is a 3-dimensional space which cuts $Y$ in a cubic surface of type either $A_1A_5$, or $\\wt E_6$.\n\\end{proposition}\n\\begin{proof}\n By Lemma \\ref{lem: fundamental} a general point of $P$ consists of a pair of lines whose linear span intersects $Y$ in a Cayley cubic. Thus, if $\\ell_1$ and $\\ell_2$ are as in the hypothesis, then the 3 dimensional linear space $\\langle E_{\\ell_1},E_{\\ell_2}\\rangle$ intersects $Y$ in a cubic surface $S$ which is a degeneration of a Cayley cubic and contains $\\ell_2$ as a triple line on the surface. By the analysis of lines on cubic surfaces, $S$ is either of type $A_1A_5$, $\\wt E_6$ or non-normal.\n As by assumption the lines $\\ell_1$ and $\\ell_2$ are not of type $\\II,$ we can exclude the non-normal cases.\n\\end{proof}\n\nWe explore now some connection of our study of singular cubic surfaces with results of Gounelas and Kouvidakis \\cite{GK-lines}. As a consequence, we deduce the existence of a surface of type $A_1A_5$, which is a fundamental little ingredient for the main theorem.\n\nLet $Y$ be a smooth cubic fourfold. The natural map $\\gamma\\colon \\mathbb L \\to Y$ from the universal family of lines to the cubic fourfold is a fibration in (2,3)-complete intersection curves. Given a point $y\\in Y$, its fiber $\\gamma^{-1}(y)$ consists of the lines trough the point $y$. Gounelas and Kouvidakis studied the $g^1_3$ of these curves coming from the rulings of the corresponding quadric and their relation with the Voisin map.\n\n \\begin{corollary}\\label{cor:existence A1A5}\n Let $Y$ be a general cubic fourfold. Then there exists a $\\Lambda \\in G$ such that $\\Lambda\\cap Y$ is a cubic surface of type $A_1A_5$.\n\\end{corollary}\n\\begin{proof}\nGiven a pair of lines $\\ell_1$ and $\\ell_2$ both of type I, with same residual and such that one is triple, then these are naturally contained in the cubic surface $\\langle E_{\\ell_1}, E_{\\ell_2} \\rangle\\cap Y$, which is either of type $A_1A_5$ or $\\wt E_6$. To discern the 2 cases, we reason as follows. A triple line through the point $y\\in Y$ corresponds to a ramification point of the $g^1_3$ of the curves $C_y$ \\cite[Lemma 3.10]{GK-lines}. As the general triple ramification point of one of the possibly two $g^1_3$ of the curve $C_y$ occurs when $C_y$ is smooth, we deduce that the general $\\langle E_{\\ell_1}, E_{\\ell_2} \\rangle$ cuts $Y$ in a cubic surface of type $A_1A_5$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:lineSecTyp}\n Let $Y$ be a general cubic fourfold.\n\\begin{enumerate}\n \\item \n For any $\\ell\\in S_{\\II}$ the intersection $E_\\ell\\cap Y$ is a non-normal singular cubic surface.\n \\item If $\\ell\\in V\\cap S_{\\II}$, then $E_\\ell\\cap Y$ is of type $X_7$ or worse.\n \\item If $\\ell\\in (V\\cap S_{\\II})_{\\sing}$ is a node, then $E_\\ell\\cap Y$ is of type $X_8$ or worse. In particular, there are 3780 projective spaces of dimension 3 intersecting $Y$ in a singular surface of type $X_8$.\n \\item For any $\\ell\\in S_{\\II}$ the surface $E_\\ell\\cap Y$ is not of type $X_9$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n Given a line $L$ of type $\\II$, the cubic surface $E_\\ell \\cap Y$ is singular along the entire $L$.\n By the study of Gounelas and Kouvidakis a line as in item (2) corresponds to a triple type two line, and one as in item (3) corresponds to a type two line with two triple tangent 2-planes \\cite[comments right above Remark~4.20]{GK-lines}.\n The distinction in cases comes from an elementary analysis of lines on non-normal cubic surfaces (see Appendix~\\ref{appendix:cubic-surfaces}). The second statement in item (3) follows from \\cite[Theorem~B]{GK-lines}.\n\n For the last item, let $L$ be a line of type $\\II$ such that $E_L\\cap Y$ is a singular cubic surface of type $X_9$, that is a cone over a cuspidal cubic curve. Then the lines passing trough the vertex of the cone form a curve, which contains a cuspidal cubic. This is in contradiction with \n \\cite[Proposition 3.5]{GK-lines}.\n\\end{proof}\n\n\\section{The monodromy group of $\\psi$ is maximal}\\label{sec:monodromy}\nIn this section we prove that the monodromy group of $\\wh\\psi$ is maximal. In order to do so, we use that the ramification of $\\wh\\psi$ is simple, as proven in Section~\\ref{sec:ramification} and the results of Section~\\ref{sec:variety-P} about the variety $P$.\n\nGiven a generically finite dominant morphism $f\\colon X\\to Y$ of degree $d$ between irreducible varieties (necessarily of the same dimension), we obtain a degree d extension of function fields $k(X)/k(Y)$, and taking the Galois closure $K/k(Y)$ of this extension, we denote by $\\mathrm{Gal}_f = \\mathrm{Gal}(K/k(Y))$. This agrees with the usual monodromy group $\\mathrm{Mon}_f$ (see \\cite{Har,sottile-yahl}), which is defined as the image in $S_d$ of the group of deck transformations of the unramified (i.e., topological) cover $X\\setminus f^{-1}(\\mathrm{Branch}(f)) \\to U$, where $U := Y \\setminus\\mathrm{Branch}(f)$.\nRecall the following classical results for $f\\colon X\\to Y$ and $U$ as above.\n\\begin{proposition}[{\\cite[p.698]{Har}}]\\label{prop:transposition}\n If there exists a fibre of $f$ with a unique point of ramification index two and all other points unramified, then $\\mathrm{Mon}_f \\subset S_d$ contains a transposition.\n\\end{proposition}\n\nLet $X^{[s]}_U$ be the complement of the big diagonal in the fibre product of $X_U$ $s$-times with itself over $U$. In other words, the fiber of the natural morphism $X^{[s]}_U \\to U$ consists\nof $s$ distinct points in the fibre of $f$.\n\\begin{lemma}[{\\cite[ Proposition 2]{sottile-yahl}}] \\label{lem:2-transitive}\n$X^{(s)}_U$ is irreducible if and only if $\\mathrm{Mon}_f$ is an s-transitive subgroup of $S_d$.\n\\end{lemma}\nThanks to Corollary \\ref{cor:ram-simple} we can readily apply Proposition \\ref{prop:transposition} and deduce that the monodromy group $\\mathrm{Mon}_{\\wh\\psi}$ contains a transposition.\nIn order to apply Lemma \\ref{lem:2-transitive} we are led to study the irreducibility of $\\wh F^{(2)}_U$ which is strictly related to the variety $P$, we introduced in the previous section. The rest of the section is then devoted to prove the following which is the crucial ingredient to conclude the maximality of monodromy.\n\\begin{proposition}\\label{prop:irreducible}\n The variety $P$ is irreducible.\n\\end{proposition}\nWe postpone the proof of it and illustrate right away how to get the maximality of monodromy.\n\\begin{theorem}\\label{thm:monodromy-maximal}\nThe monodromy of $\\wh\\psi$ is maximal, i.e. $\\Mon_{\\wh\\psi} = S_{16}$. \n\\end{theorem}\n\\begin{proof}\n From Corollary \\ref{cor:ram-simple} and Proposition \\ref{prop:transposition} $\\Mon_{\\wh\\psi}$ contains a transposition and thanks to Proposition \\ref{prop:irreducible} and \\ref{lem:2-transitive} the action of $\\Mon_{\\wh \\psi}$ is $2$-transitive. Thus $\\Mon_{\\wh\\psi}$ contains all transpositions: Indeed up to renumbering we can assume the transposition $(1,2)\\in \\Mon_{\\wh \\psi}$. By 2-transitivity there exists $\\sigma\\in\\Mon_{\\wh\\psi}$ such that $\\sigma(1)=a$ and $\\sigma(2)=b$ for any $a$ and $b$ and we have\n\\[\n\\sigma \\circ (1,2) \\circ \\sigma^{-1} = (a,b).\n\\]\nHence $\\Mon_\\wh\\psi$ contains any transposition and coincides with the entire symmetric group of 16 elements.\n\\end{proof}\n\nIn order to prove the irreducibility of $P$ we take advantage of the map $\\varphi \\colon P \\dashrightarrow W$ and use its various properties we have showed in Section \\ref{sec:variety-P}, to compute the ramification of $\\varphi$ at some special points. Clearly there is an open subset $U_{et}\\subset W$ for which the base change $P_{U_{et}} \\to U_{et}$ is étale of degree 6 and proper, but as we are interested in the ramification of $\\varphi$ we need to study a slightly bigger open subset, thus we provide the following technical proposition.\n\\begin{proposition}\\label{prop:flat-open}\nThere is an open $U_{\\mathrm{flat}}\\subset W$ such that\n\\begin{enumerate}\n \\item The pullback $\\varphi\\colon P_{U_\\flatt} \\to U_\\flatt$ is a flat finite morphism.\n \\item The open set $U_\\flatt$ contains a point in $w \\in W$ such that the 3-dimensional linear space $a(w)$ intersects $Y$ in a cubic surface of type $A_1A_5$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $P \\xleftarrow{\\pi}\\wh P \\xrightarrow{\\wh\\varphi} W$ be any resolution of indeterminacy of $\\varphi$ and $\\wh P \\xrightarrow{\\varphi_1} P_{\\mathrm{Stein}} \\xrightarrow{\\varphi_2} W$ be its Stein factorization. \nWe denote by $E_\\pi$ and $E_1$ the exceptional loci of $\\pi$ and $\\varphi_1$ respectively. \\[\n \\wh{P}\\setminus \\wh \\varphi^{-1}\\left(\\wh \\varphi \\left( E_1\\cup E_\\pi \\right)\\right) \\to Y \\setminus \\wh \\varphi (E_1 \\cup E_\\pi) =:U\n \\]\n is proper, as base change of the proper map $\\wh \\varphi$, and with finite fibers, as we removed the exceptional divisor $E_1$). Thus it is a finite morphism. Having removed $E_\\pi$ the source space is actually a subset of $P$.\n Hence, we have proven that the restriction of $\\varphi$ to $P_U \\to U$ is a finite morphism.\n\n Moreover, the further restriction of $\\varphi$ to a regular locus of $P_U$ is a finite morphism of non singular varieties, hence flat by \\cite[Chapter~4, Remark~3.11]{Liu}. This open set satisfies then point (1) of the claim.\\medskip\n \n We now construct explicitly such an open set, for which we show point (2) of the claim. In order to do so, we consider the map $\\varphi$ and the auxiliary map $\\varphi_G$\n\\[\n\\begin{tikzcd}[row sep=tiny]\n P \\ar[r,dashed, \"\\varphi\"]\\ar[rr, dashed, bend left, \"\\varphi_G\"] & W\\ar[r,\"a\"] & G.\n\\end{tikzcd}\n\\]\nTheir graph closures are going to serve as an explicit resolution of indeterminacy.\n \\begin{align*}\n\\wh P:=\\mathrm{closure}\\{((\\ell_1,\\ell_2),w)\\in P\\times W \\mid \\varphi(\\ell_1,\\ell_2) = w \\};\\\\\n\\wh P_G:=\\mathrm{closure}\\{((\\ell_1,\\ell_2),g)\\in P\\times G \\mid \\varphi_G(\\ell_1,\\ell_2) = g \\}. \n\\end{align*}\nAs $\\varphi_G = a\\circ \\varphi$, by continuity we have a natural map $\\wh P \\to \\wh P_G$ given by\n\\begin{align}\\label{map-P--PG}\n\\wh P \\longrightarrow \\wh P_G, \\quad \\left( \\ell_1, \\ell_2, w \\right) \\longmapsto \\left( \\ell_1, \\ell_2, a(w) \\right).\n\\end{align}\n\nAs before, we factor the resolution of indeterminacy $\\wh \\varphi \\colon \\wh P \\to W$ in $\\wh\\varphi = \\varphi_2\\circ \\varphi_1$ using Stein and we denote by $E_1$ the exceptional divisor of $\\varphi_1$ and $E_\\pi$ the exceptional divisor of $\\pi\\colon \\wh P \\to P$.\n\n\nBy Corollary~\\ref{cor:existence A1A5} there exists a point $w\\in W$ such that $a(w)\\in G$ is a point that intersects $Y$ in a surface $S$ with singularities $A_1A_5$. By Proposition~\\ref{prop: lines-A1A5} $S$ contains exactly 2 lines $\\ell_1$ and $\\ell_2$, that by generality are of type $I$, such that $\\psi(\\ell_1) = \\ell_2 = \\psi (\\ell_2)$. \nBy the very definition of $\\wh P_G$, its points consist of a pair of lines and a 3-dimensional linear subspace, which contains the 2 lines, hence \n\\[\n\\wh\\varphi_G^{-1} \\big( a(w) \\big) = \\left\\lbrace \\Big( \\big( \\ell_1,\\ell_2 \\big), a(w) \\Big),\\ \\Big( \\big( \\ell_2,\\ell_1 \\big), a(w) \\Big) \\right\\rbrace.\n\\]\nNow, as there is just one linear system of arithmetically Cohen-Macaulay generalized twisted cubics on a cubic surface of type $A_1A_5$ by Proposition~\\ref{lem:gtc-on-A1A5} we get\n\\[\n\\wh\\varphi^{-1} \\big( w \\big) = \\left\\lbrace \\Big( \\big( \\ell_1,\\ell_2 \\big), w \\Big),\\ \\Big( \\big( \\ell_2,\\ell_1 \\big), w \\Big) \\right\\rbrace.\n\\]\nAs $\\wh\\varphi^{-1}(w)$ is finite, we have $w\\not \\in \\wh\\varphi (E_1)$.\n\nNotice that $\\varphi_G|_P:P\\dashrightarrow G$ is defined by the association $(\\ell_1,\\ell_2)\\mapsto \\langle L_1,L_2\\rangle = \\langle L_1,L_2, R\\rangle$ where $r = \\psi(\\ell_1) = \\psi(\\ell_2)$. Moreover, if both $\\ell_1$ and $\\ell_2$ are of type I, then we have $\\varphi_G((\\ell_1,\\ell_2)) = \\langle E_{\\ell_1}, E_{\\ell_2}\\rangle$, hence pairs of distinct lines of type I in $P$ are not in the indeterminacy locus of $\\varphi$.\nSince $Z_{aCM}(S)$ is a singleton for a cubic surface $S$ of type $A_1A_5$ by Lemma~\\ref{lem:gtc-on-A1A5}, the map $\\varphi$ is defined at the point $(\\ell_1,\\ell_2)$, thus we conclude that $w\\not \\in \\wh\\varphi(E_\\pi)$.\n\nNow we are left to prove flatness of $\\varphi$ at the points $(\\ell_1, \\ell_2)$ and $(\\ell_2, \\ell_1)$ in $P$. To do so we observe that $(\\ell_1, \\ell_2)$ is a regular point of $P$ as it follows from the lemma below.\n\\end{proof}\n\\begin{lemma}\\label{lem:singular-locus-P}\n The singular locus of $\\wh F \\times_{\\wh\\psi} \\wh F$ is contained in $E \\times_{\\wh\\psi} E$.\n\\end{lemma}\n\\begin{proof}\n Let $(f_1,f_2)\\in \\wh F \\times_{\\wh\\psi} \\wh F$ be a point with $\\wh \\psi(f_1) = \\wh \\psi(f_2) = r$. Then its tangent space is the fiber product over the diagram\n \\[\n\\begin{tikzcd}[row sep=tiny]\n T_{f_1} \\wh F \\ar[rd, \"d\\wh\\psi\"] && T_{f_2} \\wh F \\ar[ld,\"d\\wh\\psi\", swap]\\\\\n & T_r F\n\\end{tikzcd}\n\\]\nwhere $T_{f_1} \\wh F$ and $T_{f_2} \\wh F$ have dimension 4 as $\\wh F$ is a non-singular fourfold.\nIf one among $f_1$ and $f_2$ is not in the ramification of $\\wh \\psi$, which coincide with the exceptional divisor $E$, then $d\\wh\\psi$ is an isomorphism at that point and the fiber product is a vector space of dimension 4, hence the claim.\n\\end{proof}\n\n\\begin{proof}(of Proposition~\\ref{prop:irreducible})\nFirst we are going to prove that $P$ modulo the switching involution $\\sigma$ is irreducible. In order to do so, we start with a point $q\\in U_{\\mathrm{flat}}\\cap W_{A_1A_5}$ which exists by Proposition~\\ref{prop:flat-open}. As exactly 2 lines lie on a surface with singularities $A_1A_5$ (see Proposition~\\ref{prop: lines-A1A5}), the preimage $\\varphi^{-1}(q)$ consists of two points, say:\n\\[\n(\\ell_1,\\ell_2) \\quad \\text{and} \\quad (\\ell_2, \\ell_1).\n\\]\nUsing the involution $\\sigma$ on $F\\times F$ switching the factors, which clearly restrict to an involution of $P$, we see that $P$ at $(\\ell_1, \\ell_2)$ is locally isomorphic to $P$ at $\\sigma((\\ell_1,\\ell_2)) = (\\ell_2, \\ell_1)$.\nAs $\\varphi$ is flat, proper and of degree 6 at the point $q$, we conclude that the schematic preimage $\\varphi^{-1}(q)$ is a finite scheme with multiplicity 3 both at $(\\ell_1,\\ell_2)$ and $(\\ell_2,\\ell_1)$.\nWe pick now a point $p\\in U_{\\mathrm{et}}$ close enough to $q$ and focus on the preimages $\\varphi^{-1}(p) = \\{p_1,\\cdots, p_6\\}$: If we let $p$ tend to $q$, then 3 points out of the 6 in $\\varphi^{-1}(p)$, say $p_1, p_2 $ and $p_3$, tend to $(\\ell_1,\\ell_2)$ and the other 3 points, i.e. $p_4,p_5$ and $p_6$, tend to $(\\ell_2,\\ell_1)$. Moreover, $\\sigma$ maps the first triple to the second one, i.e. $\\sigma(\\{p_1,p_2,p_3\\}) = \\{p_4,p_5,p_6\\}$, thus $p_1,p_2$ and $p_3$ lie in the same connected component of $P$, in other words the quotient of $P$ by the switching involution is path-connected. By Lemma \\ref{lem:singular-locus-P} we deduce that it is irreducible too.\n\n\nHence, it is now sufficient to show that for some point $p_1\\in P_{U_{et}}$, the points $p_1$ and $\\sigma(p_1)$ lie in the same connected component of $P_{U_{et}}$. To do so, we consider one transposition $\\tau$ in $\\Mon_\\psi$, whose existence is granted by Proposition~\\ref{prop:transposition} and Corollary~\\ref{cor:ram-simple}. If we consider it as an element of deck transformations, this means that there exists a point $r\\in F\\setminus \\Branch(\\wh \\psi)$, and a loop $\\gamma$ centered at $r$ such that $\\gamma$ acts on the fiber of $r$ exchanging two points, say $\\ell_1\\mapsto \\ell_2$ and $\\ell_2\\mapsto \\ell_1$, and as identity on the remaining other points of the fiber. \nWe now consider the action under deck transformation of $\\gamma$ for the natural map $P \\dashrightarrow F$ defined by $\\psi$. A moment's thought tells us that under the action of $\\gamma$ the point $p=(\\ell_1,\\ell_2)\\in P$, with $\\psi(\\ell_1)=\\psi(\\ell_2)=r$, is mapped to $(\\ell_2,\\ell_1)$.\nThus $P$ is irreducible as we claimed.\n\\end{proof}\n\n\n\\section{Appendix}\\label{appendix:cubic-surfaces}\n\n\nIn this appendix we study singular cubic surfaces that are degenerations of a cubic surface with exactly 4 singular points of type $A_1$. These surfaces are called \\textit{Cayley}.\n\n\nThe stratification of the space of cubic surfaces given by the type of singularities has been studied by Bruce in \\cite{bruce}. From his analysis Cayley cubics can degenerate to a singular surface of type $2A_1A_3$, which in turn degenerates to a singular surface $A_1A_5$, which in turn degenerates to surfaces with simple elliptic singularities and then to non-normal ones. Here is the diagram of degenerations:\n\\[\n4A_1 \\succeq 2A_1A_3 \\succeq A_1A_5 \\succeq \\wt E_6 \\succeq X_6 \\succeq X_7 \\succeq X_8 \\succeq X_9\n\\]\nwhere $\\wt E_6$ denotes the cubics with one simple elliptic singularity and $X_i$ are non-normal cubic surfaces. We discuss them in details in the subsequent subsections, focussing on lines and generalised twisted cubics and the restrictions of the Voisin maps $\\varphi$, and $\\psi$ to a cubic surface.\n\n \n\n\\subsection*{Singular cubic surfaces of type $4A_1$}\n\nUnder a suitable change of coordinate every cubic surface with 4 singular points of type $A_1$ can be expressed as zero locus of the following polynomial:\n\\begin{align*}\n F = t_0t_1t_2 + t_0t_1t_3 + t_0t_2t_3 + t_1t_2t_3\n\\end{align*}\nIts singular locus consists of the coordinates points\n\\begin{align*}\n p_0=(1:0:0:0),\\quad p_1=(0:1:0:0),\\quad p_2=(0:0:1:0),\\quad p_3=(0:0:0:1).\n\\end{align*}\n\nSuch a surface can be obtained as the anticanonical model of the blow-up of $\\mathbb P^2$ in special configuration of points. Indeed, let $\\widetilde S$ be the blow-up of $P^2$ in the six points given by the intersection of 4 general lines $L_0, L_1,...,L_3$. Then the anticanonical linear system of $\\widetilde S$ maps to a cubic surfaces $S$ and contracts the strict tranforms of $L_i$ to 4 singular $A_1$ points. The exceptional divisors map to lines in a tetrahedron-like disposition, we refer to them as edges, and denote $E_{ij}$ the line connecting the $i$-th and the $j$-th coordinate point. In particular, the edge $E_{ij}$ is cut by the equations $(t_k=t_l=0)$ where $\\{i,j,k,l\\}=\\{1,2,3,4\\}$. Besides the edges there are 3 more lines on $\\bar S$ Joining opposite edges; we denote by $J_{ij,kl}$ the line intersecting the edges $E_{ij}$ and $E_{kl}$, where $\\{i,j,k,l\\}=\\{1,2,3,4\\}$. These are images of the strict transforms of lines through the blown-up points corresponding to the edges it intersects. For example, the line $J_{01,23}$ is the zero locus $(t_0+t_1 = t_2+t_3=0)$.\n\nGeneralised twisted cubics on singular cubic surfaces have been studied in \\cite[\\S~2]{LLSvS}. Any generalised twisted cubic on an integral normal cubic surface moves in a 2-dimensional linear system and can be described as the image of the pull-back of a line in $\\mathbb P^2$ along the diagram of a determinantal representation of $S$. The determinantal representation for the Cayley cubic described above defines $K\\in Z'(S)$ which contains as reducible elements of the family, the union of all the edges passing through a fixed vertex of the tetrahedron:\n \\[\n K = [E_{ij}\\cup E_{ik} \\cup E_{il}] \\in Z'(S).\n \\]\n\n\\begin{lemma}\n Opposite edges map to $K$ under $\\varphi$, that is:\n \\begin{align*}\n \\varphi(E_{ij}, E_{kl}) = K\\quad \\text{ for indices such that } \\{i,j,k,l\\}=\\{1,2,3,4\\}.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n The proof is elementary and can be checked directly by the definition of the Voisin map $\\varphi$. For example to compute $\\varphi(E_{02}, E_{13})$, consider the plane $\\pi=(t_1=0)$ containing the line $E_{02}$ and the point $p_\\in E_{13}$. Its intersection with $S$ is given by $\\pi\\cap S=E_{02}\\cup E_{03}\\cup E_{23}$, hence $\\varphi(E_{02}, E_{13}) = \\left[E_{03}\\cup E_{13} \\cup E_{23}\\right]$ and the claim is shown.\n\n The other cases are completely similar.\n\\end{proof}\n\n\\subsection*{Singular cubic surfaces $2A_1A_3$}\\phantom.\n\nLet $S$ be a cubic surface with a singular point $P_2$ of type $A_3$, and 2 singular points $P_0,P_3$ of type $A_1$. It contains exactly 5 lines as depicted in Figure~\\ref{fig:2A1A3}: the line $R_{01}$ through $P_0,P_1$; the line $R_{02}$ through $P_0,P_2$; the line $R_{03}$ through $P_{03}$; the line $L_1$ containing the only singular point $P_2$, and the line $L_2$ contained in the smooth locus.\n\nSuch a surface is given in an opportune coordinate system by the equation\n \\[\n t_0t_2t_3 - t_1^2t_3-t_0t_1^2 = 0\n \\]\nIts singular locus consists of the point $P_2:=(0:0:1:0)$ of type $A_3$ and of the points $P_0:=(1:0:0:0)$, $P_3:=(0:0:0:1)$ of type $A_1$.\\\\\nThe lines are then described by the equations $R_{23}=\\{t_0=t_1=0\\},R_{02}=\\{t_1=t_3=0\\}$ and $R_{03}=\\{t_1=t_2=0\\}$, $L_1=(t_0=t_3=0)$, and $L_2=(t_2=t_0+t_3=0)$.\\\\\n\n\n\n\\begin{proposition}\nLet $S$ be a cubic surface of type $2A_1A_3$. Then:\n \\begin{align*}\n L_1\\mbox{ is residual to }R_{23},\\qquad\n L_1 \\mbox{ is residual to }R_{02} ,\\\\ \n L_2\\mbox{ is residual to }R_{03},\\qquad\n L_2\\mbox{ is residual to } L_1.\n \\end{align*}\nMoreover, there is no plane in $\\langle S\\rangle\\simeq \\mathbb P^3$ tangent to $S$ at every point of $L_2$.\n In particular, there are no triple lines on a surface with $2A_1A_3$ singularities.\n\\end{proposition}\n\n \\begin{proof}\n The proof can be checked through straightforward computations. For example, intersecting the surface $S$ with the plane $\\pi=(t_0=0)$, one gets:\n \\begin{align*}\n \\pi\\cap S &= (t_0=0)\\cap (t_0t_2t_3 - t_1^2t_3-t_0t_1^2 = 0) \\\\\n &= (t_0=0)\\cap(t_1^2t_3=0) = 2R_{23} + L_1,\n \\end{align*}\n which shows that $L_1$ is residual to $R_{23}.$\n \\end{proof}\n \n\n\n\\subsection*{Singular cubic surfaces $A_1A_5$}\n\nA singular cubic surface $S$ with exactly one $A_1$ and one $A_5$ singularities is given in an opportune of coordinate system by the equation\n \\[\n t_3(t_0t_2-t_1^2)+t_0^3 = 0\n \\]\n whose singular locus consists of the 2 points $P_1:=(0:0:0:1)$ of type $A_5$ and $P_2:=(0:0:1:0)$ of type $A_1$. There are exactly 2 lines on $S$ and they have equations $L_1=\\{t_0=t_1=0\\}$ and $L_2= \\{t_0=t_3=0\\}$.\n\n\n\\begin{proposition}\n Let $S$ be a cubic surface of type $A_1A_5$. Then\n \\begin{align*}\n L_2 \\text{ is triple, and } L_2 \\text{ is residual to } L_1.\n \\end{align*}\n\\end{proposition}\n\n\\begin{proof}\n Straightforward computation: The intersection\n \\[\n \\{t_3=0\\}\\cap S = 3L_2.\n \\]\n shows the first statement, while\n \\[\n \\{t_0=0\\}\\cap S = L_1+2L_2\n \\]\n shows the second one.\n\\end{proof}\n\n\n\n\\subsection*{The cubic surface $X_6$}\n\nIn a suitable system of coordinates, the surface $X_6$ is given by the equation\n\\begin{align*}\n t_0^2t_2 + t_1^2t_3 = 0.\n\\end{align*}\n\nIt is a non-normal surface whose singular locus consists of the line $L=(t_0=t_1=0)$.\nWe record only the following which is necessary for the proof of Corollary~\\ref{cor:lineSecTyp}\n\n\\begin{proposition}\n The line $L$ is not triple on $X_6$.\n\\end{proposition}\n\\begin{proof}\n The proof is an elementary computation:\n Each plane containing $L$ is of the form $\\pi=(at_0 + bt_1=0)$ with $(a:b)\\in \\mathbb P^1$.\n By considering the intersection with such a plane $\\pi$ one has:\n \\begin{align*}\n X_6 \\cap (-t_0 + bt_1 = 0) = (-t_0 + bt_1 = t_1^2(b^2t_2 + t_3) =0) = 2L + R\n \\end{align*}\n with $R= b^2t_2 + t_3 = bt_1 + t_0 = 0$ different from $L$, and\n \\begin{align*}\n X_6 \\cap (t_1=0) = (t_1 = t_0^2t_2=0) =2L + R \n \\end{align*}\n with $R= (t_1 = t_2 =0)$. This shows that $L$ is not triple.\n\\end{proof}\n\n\\subsection*{The cubic surface $X_7$}\n\nIn a suitable system of coordinates, the surface $X_7$ is given by the equation\n\\begin{align*}\n t_1^3 + t_2^3 + t_1t_2t_3.\n\\end{align*}\n\nIt is a non-normal surface isomorphic to the cone over a nodal planar cubic whose singular locus consists of the line $L=(t_1=t_2=0)$.\nWe record only the following which is necessary for the proof of Corollary~\\ref{cor:lineSecTyp}\n\n\\begin{proposition}\n There is a unique plane $\\pi$ making $L$ triple, i.e. such that $\\pi\\cap X_7 = 3L$.\n\\end{proposition}\n\n\\begin{proof}\n The proof is an elementary computation: Each plane containing $L$ is of the form $\\pi =(at_1+bt_2=0)$ with $(a:b)\\in \\mathbb P^1$.\n\n By considering the intersection with such a plane $\\pi$ one has:\n \\begin{align*}\n X_7 \\cap (t_0 - bt_1 = 0) = (t_0 - bt_1 = t_1^2(bt_2 + b^2t_3 + t_1) = 0) = 2L + R.\n \\end{align*}\n with $R = (t_0 - bt_1 = t_1 + bt_2 + b^2t_3=0)$.\n Also, one has:\n \\begin{align*}\n X_7 \\cap (t_0 = 0) = (t_0 = t_1^3=0) = 3L.\n \\end{align*}\n This shows the claim.\n\\end{proof}\n\n\\subsection*{The cubic surface $X_8$}\n\nIn a suitable system of coordinates, the surface $X_8$ is given by the equation\n\\begin{align*}\n t_0t_1t_2 + t_0^2t_3 + t_1^3 = 0.\n\\end{align*}\n\nIt is a non-normal surface whose singular locus consists of the line $L=(t_0=t_1=0)$.\nWe record only the following which is necessary for the proof of Corollary~\\ref{cor:lineSecTyp}\n\n\\begin{proposition}\n There are 2 planes $\\pi$ and $\\pi'$ making $L$ triple, i.e. such that $\\pi\\cap X_8 = \\pi'\\cap X_8 = 3L$.\n\\end{proposition}\n\\begin{proof}\n The proof is an elementary computation: Each plane containing $L$ is of the form $\\pi =(at_1+bt_2=0)$ with $(a:b)\\in \\mathbb P^1$.\n\n By considering the intersection with such a plane $\\pi$ one has:\n \\begin{align*}\n X_8 \\cap (t_1 - bt_2 = 0) = (t_1 - bt_2 = t_2^2(b^3t_2 + t_2 + bt_3) = 0) = 2L + R.\n \\end{align*}\n with $R = (t_0 - bt_1 = t_1 + bt_2 + b^2t_3=0)$.\n Also, one has:\n \\begin{align*}\n X_8 \\cap (t_2 = 0) = (t_1^3 = t_2=0) = 3L.\n \\end{align*}\n The planes $\\pi = (t_1=0)$ and $\\pi'= (t_2=0)$ are hence the only planes as in the claim.\n\\end{proof}\n\n\n\\bibliography{literatur}\n\\bibliographystyle{alpha}\n\n\\end{document}\n\n\n"}}},{"rowIdx":86,"cells":{"id":{"kind":"string","value":"2412.07561v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07561v1"},"title":{"kind":"string","value":"The $L_q$ Minkowski problem for $\\mathbf{p}$-harmonic measure"},"full_text":{"kind":"string","value":"\\documentclass[12pt]{article}\n\\usepackage{amsmath, graphicx, amsfonts,amssymb, calrsfs}\n\\usepackage{amsfonts,mathrsfs, color, amsthm}\n\n\n\\addtolength{\\topmargin}{-0.1\\textheight}\n\\addtolength{\\textheight}{0.2\\textheight}\n\\addtolength{\\textwidth}{0.2\\textwidth}\n\\setlength{\\oddsidemargin}{-0.08in}\n\\setlength{\\evensidemargin}{-0.08in} \\pretolerance=5000\n\n\\usepackage{accents}\n\\usepackage{enumitem}\n\\usepackage{indentfirst}\n\\usepackage{fancyhdr}\n\n\n\n\\def\\cK{\\mathcal{K}}\n\\def\\sphere{\\mathbb{S}^{n-1}}\n\\def\\N{\\mathbb{N}}\n\\def\\Rn{{\\mathbb R^n}}\n\\def\\R{\\mathbb{R}}\n\\def\\cH{\\mathcal{H}}\n\\def\\dV{\\,d \\widetilde{V}_K(u)}\n\n\n\\def\\theequation{\\thesection.\\arabic{equation}}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}{Lemma}[section]\n\\newtheorem{remark}{Remark}[section]\n\\newtheorem{proposition}{Proposition}[section]\n\\newtheorem{corollary}{Corollary}[section]\n\\newtheorem{example}{Example}[section]\n\\newtheorem{definition}{Definition}[section]\n\n\\def\\cC{\\mathcal{C}}\n\\def\\bpf{\\begin{proof}}\n\\def\\epf{\\end{proof}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\\def\\bt{\\begin{theorem}}\n\\def\\et{\\end{theorem}}\n\\def\\bl{\\begin{lemma}}\n\\def\\el{\\end{lemma}}\n\\def\\br{\\begin{remark}}\n\\def\\er{\\end{remark}}\n\\def\\bc{\\begin{corollary}}\n\\def\\ec{\\end{corollary}}\n\\def\\bd{\\begin{definition}}\n\\def\\ed{\\end{definition}}\n\\def\\bp{\\begin{proposition}}\n\\def\\ep{\\end{proposition}}\n\n\n\\numberwithin{equation}{section}\n\n\\begin{document}\n\\title{The $L_q$ Minkowski problem for $\\mathbf{p}$-harmonic measure\n\\footnote{Keywords: Minkowski problem, convex body, $\\mathbf{p}$-harmonic measure, $\\mathbf{p}$-Laplacian. These authors contributed equally: Hai Li, Longyu Wu. $^\\ddagger$Corresponding author: email: bczhu@snnu.edu.cn}}\n\n\\author{Hai Li$^\\dagger$, Longyu Wu$^\\dagger$, Baocheng Zhu$^{\\dagger,\\ \\ddagger}$}\n\\date{\\quad}\n\n\n\\maketitle\n\\begin{abstract}\nIn this paper, we consider an extremal problem associated with the solution to a boundary value problem.\nOur main focus is on establishing a variational formula for a functional related to the $\\mathbf{p}$-harmonic measure,\nfrom which a new measure is derived. This further motivates us to study the Minkowski problem for this new measure.\nAs a main result, we prove the existence of solutions to the $L_q$ Minkowski problem associated with the $\\mathbf{p}$-harmonic\nmeasure for $0<q<1$ and $1<\\mathbf{p}\\ne n+1$.\n\n\n\\vskip 2mm\n2020 Mathematics Subject Classification: 31B05, 35J25, 42B37, 52A20, 52A40.\n\\end{abstract}\n\n\\section{Introduction}\\label{sect:1}\n\nThe $L_q$ Minkowski problem is one of the most important contents in convex geometry.\nIt can be stated as: For any given $q\\in \\R$ and a finite nonzero Borel measure $\\mu$\non the unit sphere $\\sphere$ in $\\R^n$, whether there exists a convex body whose $L_q$\nsurface area measure is the given measure $\\mu$. When $q=1$, the $L_q$ Minkowski problem\nreduces to the classical one, which dates back to the early works\nby Minkowski and was developed further by Aleksandrov, Fenchel and Jessen.\nThe $L_q$ Minkowski problem for $q>1$ was first studied by Lutwak \\cite{L93}. Since then,\nthis problem has received significant attention, leading to remarkable progress (see e.g., \\cite{HS04, HZ05, LZ04, U03}).\nWhen $q<1$, the problem is more challenging (see e.g., \\cite{C06, CW06, DZ12, JZ16, LW13, Z15}).\nParticularly for $q=0$, it becomes the logarithmic Minkowski problem (see e.g.,\n\\cite{BZ13, CL22, LX24, S02, S03, TX23, Z14}). For more progress on the $L_q$ Minkowski problem,\nwe refer to \\cite{CL20, HX15, M24} and the references therein. It is well known that\nthe solutions to the $L_q$ Minkowski problem are key ingredients in the rapidly developing\n$L_q$ Brunn-Minkowski theory of convex bodies. For instance, they have played an important role\nin establishing affine Sobolev inequalities (see e.g., \\cite{CZ09, HS09, LZ02, Z99}).\n\nAlong with the rapid development of the Brunn-Minkowski theory, the Minkowski\nproblem has been greatly enriched. Examples include the Minkowski problem for\nthe dual curvature measure \\cite{HZ16, LW20}, the Gaussian\nsurface area measure \\cite{CZ23, FX23, HZ21}, the chord measure \\cite{GZ24, LZ24++, XZ23},\nand the Minkowski problem for unbounded closed convex sets \\cite{LZ24+, S18, S24, YZ23},\nas well as for log-concave functions \\cite{CK15, FY22, R22}. These problems are well-known\nfor their close relationships among convex geometry, integral geometry,\ndifferential geometry, and PDEs. Jerison systematically integrated the Brunn-Minkowski theory with\npotential theory and the regularity theory of fully nonlinear equations. In his earlier works \\cite{J89, J91},\nhe first studied the Minkowski problem for harmonic measure. Later, in another paper \\cite{J96},\nhe examined a similar problem for electrostatic capacity. Jerison's contributions sparked significant research\ninto Minkowski problems. A notable example of ongoing research is the study of the Minkowski problem\nfor $\\mathbf{p}$-capacity by Colesanti et al. \\cite{CZ15}.\nRecently, this problem has been extended to the $L_q$\ncase \\cite{ZX20}. In fact, such kind of Minkowski problem\nis closely related to a boundary value problem. More examples\nof Minkowski problems associated with the boundary value problems\ninclude those for capacity \\cite{AV22, HZ18, LH23, X20, XX19} and for\ntorsional rigidity \\cite{CF10, HZ23, LZ20}.\n\nLet $K$ be a bounded convex domain with boundary $\\partial K$ and $N$\nbe a neighborhood of $\\partial K$. In this paper, we consider the\nfollowing boundary value problem\n\\begin{equation}\\label{1.1}\n\\left\\{\n\\begin{aligned}\n&\\text{div}\\left({{\\left|{\\nabla u}\\right|}^{\\mathbf{p}-2}\n\\nabla u}\\right)=0&&\\text{in}\\ K\\cap N,\\\\\n&u>0&&\\text{in}\\ K,\\\\\n&u=0&&\\text{on}\\ \\partial K.\n\\end{aligned}\n\\right.\n\\end{equation}\nHere, $N$ is chosen so that the solution $u_K$ satisfies\n$\\left\\|u_K\\right\\|_{L^\\infty\\left(\\bar N\\cap K\\right)}\n+\\left\\|\\nabla u_K\\right\\|_{L^\\infty\\left(\\bar N\\cap K\\right)}<\\infty$\nand $\\left|{\\nabla u_K}\\right|\\ne0$ in $K\\cap N$,\nwhere ${\\left\\|\\cdot\\right\\|_{L^\\infty}}$ is the ${L^\\infty}$ norm, $\\nabla$ is\nthe gradient operator and $\\bar N$ is the closure of $N$.\nThroughout this paper, we assume that $\\partial N$ is of class $C^{\\infty}$.\nLet $W^{1,\\mathbf{p}}$ denote the usual Sobolev space with $1<\\mathbf{p}<\\infty$.\nFollowing Akman-Mukherjee \\cite{AM24}, the $\\mathbf{p}$-harmonic function\n$u_K\\in W^{1,\\mathbf{p}}\\left(K\\cap N\\right)$ can be used to define the measure\n$\\omega_\\mathbf{p}\n=\\left|\\nabla u_K\\right|^{\\mathbf{p}-1}\n\\mathcal{H}^{n-1}\\llcorner_{\\partial K}$.\nMoreover, the $\\mathbf{p}$-harmonic measure $\\mu_K$ is defined by\n$\\mu_K=(g_K)_*\\omega_\\mathbf{p}$, that is,\n\\begin{equation}\\label{1.2}\n\\mu_K\\left(E\\right)\n=\\int_{g_K^{-1}\\left(E\\right)}\n{\\left|\\nabla u_K\\right|}^{\\mathbf{p}-1}\nd{\\mathcal H}^{n-1}\n\\end{equation}\nfor any Borel set $E$ on the unit sphere $\\mathbb{S}^{n-1}$, where $g_K:\\partial K\\to\\mathbb{S}^{n-1}$\nis the Gauss map and $\\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.\n\nAccording to Akman-Mukherjee \\cite{AM24}, the definition \\eqref{1.2} is valid for any convex set,\nand the $\\mathbf{p}$-harmonic measure is of variation meaning. In fact, the $\\mathbf{p}$-harmonic measure\nhas been studied by Lewis et al. \\cite{L06, L13}, and Jerison's work \\cite{J91} on harmonic measure has been\nnontrivially extended to the $\\mathbf{p}$-harmonic measure setting by Akman-Mukherjee \\cite{AM24}.\nBy studying the discrete measure case and using the approximation arguments, Akman-Mukherjee \\cite{AM24}\ndemonstrated the solvability of the Minkowski problem for $\\mathbf{p}$-harmonic measure, provided that\nthe given measure is not concentrated on any great subsphere and its centroid is at the origin.\nRecently, smooth solutions have been established by using the Gauss curvature flow \\cite{LZ24}.\nDetailed discussions on the relationships among the Minkowski problem for $\\mathbf{p}$-harmonic measure,\nharmonic measure \\cite{J91}, and $\\mathbf{p}$-capacitary measure \\cite{CZ15} can be found on page 13 of \\cite{AM24}.\n\nIn this paper, we focus on the following problem concerning the $\\mathbf{p}$-harmonic measure,\nwhere $1<\\mathbf{p}<\\infty$, unless specified otherwise.\n\n\\vskip.2cm\n\n\\textbf{$L_q$ Minkowski problem for $\\mathbf{p}$-harmonic measure.}\n{\\it Let $q\\in\\mathbb{R}$ and $\\mu$ be a finite Borel measure on $\\mathbb{S}^{n-1}$.\nWhat are the necessary and sufficient conditions for $\\mu$ such that there exists\na convex body $\\Omega$ satisfying\n$\\mu=h_{\\Omega}^{1-q}\\mu_\\Omega$? Here $h_{\\Omega}$ is the support function of $\\Omega$.\n}\n\n\\vskip.2cm\n\nActually, the measure $h_{\\Omega}^{1-q}\\mu_\\Omega=\\mu_{\\Omega,q}$ in the above problem\ncan be derived from our new variational formula (see Theorem \\ref{th:3.1} below),\nand we call it the $L_q$ $\\mathbf{p}$-harmonic measure. As mentioned above,\nthe $L_1$ Minkowski problem for $\\mathbf{p}$-harmonic measure was recently studied\nby Akman-Mukherjee \\cite{AM24}. By studying an extremal problem for a functional\nrelated to the $\\mathbf{p}$-harmonic measure, we can obtain a solution to\nthe $L_q$ Minkowski problem for $\\mathbf{p}$-harmonic measure for $0<q<1$.\nThis can be stated as main result of this paper as follows.\n\n\\begin{theorem}\\label{th:1.1}\nLet $0<q<1$, $1<\\mathbf{p}\\ne n+1$, and $\\mu$ be a finite Borel measure on $\\mathbb{S}^{n-1}$. If $\\mu$ is not concentrated on any closed hemisphere,\nthere exists a convex body $\\Omega$ containing the origin\nin its interior so that $\\mu=c\\mu_{\\Omega,q}$,\nwhere $c$ is a positive explicit constant. In particular $c=1$,\nif $\\mathbf{p}\\ne n+1-q$.\n\\end{theorem}\n\nThis paper is organized as follows. In Section \\ref{sect:2}, we review some necessary notations\nand background on convex sets, $\\mathbf{p}$-harmonic functions and $\\mathbf{p}$-harmonic measures.\nIn Section \\ref{sect:3}, after establishing a variational formula associated with the\n$\\mathbf{p}$-harmonic measure, we further introduce the $L_q$ $\\mathbf{p}$-harmonic\nmeasure for $q\\in\\mathbb{R}$ and prove its weak convergence.\nIn Section \\ref{sect:4}, we complete the proof of Theorem \\ref{th:1.1}.\n\n\n\\section{Preliminaries}\\label{sect:2}\n\\subsection{Background for convex sets}\\label{subsect:2.1}\n\nIn this subsection, we collect the necessary background, notations and preliminaries.\nMore details on convex sets can be found in \\cite{G06, G07, S14}.\n\nLet $K\\subset \\mathbb{R}^{n}$ be a convex set with boundary $\\partial K$,\none can define the multi-valued Gauss map $g_K:\\partial K\\to\\mathbb{S}^{n-1}$ by\n\n\\begin{equation}\\label{2.1}\n{g_K}\\left(x\\right)\n=\\left\\{{\\xi\\in {\\mathbb{S}^{n-1}}:\n\\left\\langle{y-x,\\xi }\\right\\rangle< 0\\ \\text{for all}\\ y\\in K}\\right\\},\n\\end{equation}\ni.e., the set of all unit outward normal vectors at $x\\in \\partial K$,\nwhere $\\left\\langle{\\cdot, \\cdot}\\right\\rangle$ is the standard inner product\non $\\mathbb{R}^{n}$.\nThe set defined in \\eqref{2.1} is a singleton for\n$\\mathcal{H}^{n-1}$-a.e. $x\\in\\partial K$. For a measurable subset $E\\subset\\mathbb{S}^{n-1}$,\nlet $g_K^{-1}(E):=\\{{x\\in\\partial K:g_K(x)\\cap E\\ne\\emptyset}\\}$ be the inverse\nimage of $g_K$, and ${\\left(g_K\\right)_*}$ be the push forward of $g_K$ given by\n\\[\\left({{\\left(g_K\\right)}_*}\\mu\\right)\\left(E\\right)\n=\\mu\\left({g_K^{-1}\\left(E\\right)}\\right),\\]\nwhere $\\mu$ is a measure defined on any measurable subsets of $\\partial K$.\nIf $E$ is a Borel subset of $\\mathbb{S}^{n-1}$, $g_K^{-1}\\left(E\\right)$\nis $\\mathcal{H}^{n-1}$-measurable.\n\nFor a compact convex set $K\\subset\\mathbb{R}^{n}$ and nonzero $x\\in\\mathbb{R}^{n}$,\nthe support function of $K$ is defined by\n$h_K\\left(x\\right)\n=\\max\\limits_{y\\in K}\\left\\langle {x,y}\\right\\rangle$,\nand the support hyperplane of $K$ is given by\n\n$${H_K}(x)\n=\\left\\{{y\\in {\\mathbb{R}^n}:\\left\\langle{x,y} \\right\\rangle\n={h_K}(x)}\\right\\}.$$\nIf $K\\cap{H_K}\\left( x \\right)$ consists of only a single point for all $x$, then $K$ is strictly convex.\nIn particular, a convex and compact subset in $\\Rn$ with nonempty interior is called a convex body.\n\nA convex set $K$ is said to be of class $C_+^2$ (resp. $C_ +^{2,\\alpha }$\nfor $\\alpha\\in\\left({0,1}\\right]$) if $\\partial K$ is of class $C_+^2$\n(resp. $C_+^{2,\\alpha}$) and the Gauss map $g_K: \\partial K\\to\\mathbb{S}^{n-1}$ is\na diffeomorphism. For any convex set $K$ of class $C_+^{2}$, we have\n$K\\cap {H_K}\\left( {{g_K}\\left( x \\right)} \\right)\n=\\left\\{ x \\right\\}$, where $x\\in \\partial K$.\nMoreover, the support function\nis differentiable and\n\\[\\nabla {h_K}\\left( {{g_K}\\left( x \\right)} \\right) = x,\\]\nwhere $\\nabla $ is the gradient operator on $\\mathbb{R}^{n}$.\nFor $\\xi \\in \\mathbb{S}^{n-1}$, there exists an orthonormal basis\n$\\left\\{ {{e^1}, \\ldots ,{e^{n - 1}},\\xi } \\right\\}$ of $\\mathbb{R}^{n}$, where $\\left\\{ {e^i}\\right\\}$\nspans the tangent space ${T_\\xi }\\left(\\mathbb{S}^{n-1}\\right)$. Then, for any $x\\in \\mathbb{R}^{n}$,\nwe have the decomposition\n\\begin{equation}\\nonumber\nx=\\sum\\limits_{i=1}^{n-1}x^ie^i\n+\\left\\langle{x,\\xi}\\right\\rangle\\xi\\ \\ \\text{with}\\ \\\nx^i=\\left\\langle x,e^i\\right\\rangle.\n\\end{equation}\nLet $\\xi ={g_K}\\left( x \\right)$ for any $x\\in\\partial K$, then we have\n\\begin{equation}\\label{2.2}\n\\nabla {h_K}\\left(\\xi\\right)\n=\\sum\\limits_{i = 1}^{n - 1} {{\\bigtriangledown_i}{h_K}\\left( \\xi \\right){e^i}}\n+\\left\\langle {\\nabla {h_K}\\left( \\xi \\right),\\xi } \\right\\rangle \\xi,\n\\end{equation}\nwhere\n${\\bigtriangledown _i}{h_K}\\left(\\xi\\right)\n=\\left\\langle {\\nabla {h_K}\\left(\\xi\\right),{e^i}}\\right\\rangle$.\n\nLet $\\mathcal{A}_+^{2,\\alpha}$ be the set of all compact convex sets that are of class $C_+^{2,\\alpha}$.\nFor a sequence of compact convex sets $\\left\\{\\Omega_j\\right\\}_{j=0}^{\\infty}$, we say that $\\Omega_j$\nconverges to $\\Omega_0$ and denote it as $\\Omega_j\\to \\Omega_{0}$, if the Hausdorff distance $d_{\\mathcal H}\\left({\\partial \\Omega_j,\\partial\\Omega_0}\\right)$ between\n${\\Omega_j}$ and $\\Omega_{0}$ converges to $0$ as $j \\to\\infty$.\nAccording to Theorem 2.46 of \\cite{AM24}, for any compact convex set $\\Omega$ with Gaussian\ncurvature $\\kappa$, there exists a sequence\n$\\left\\{\\Omega_j \\right\\}_{j=1}^\\infty\\subset\\mathcal{A}_+^{2,\\alpha}$ with Gaussian curvature\n$\\kappa_{j}$ such that $\\Omega_{j}\\to \\Omega$, and for any continuous function $f$ defined on the unit sphere $\\mathbb{S}^{n-1}$,\n\\begin{equation}\\nonumber\n\\int_{\\mathbb{S}^{n-1}}\\frac{f\\left(\\xi\\right)}{\\kappa_j\\left({g_{\\Omega_j}^{-1}\\left(\\xi\\right)}\\right)}d\\xi\n\\to\n\\int_{\\mathbb{S}^{n-1}}\\frac{f\\left(\\xi\\right)}\n{\\kappa\\left({g_\\Omega^{-1}\\left(\\xi\\right)}\\right)}d\\xi,\n\\end{equation}\nas $j\\to\\infty$.\n\nLet $C\\left(E\\right)$ denote the set of all continuous functions\ndefined on subset $E\\subset\\mathbb{S}^{n-1}$ and let $C_{+}\\left(E\\right)\\subset C\\left(E\\right)$ denote\nthe set of all strictly positive functions. The Wulff shape\n$K_f$ associated with a nonnegative function $f\\in C\\left(E\\right)$\nis defined by\n\\begin{equation}\\nonumber\n{K_f}\n=\\left\\{{x\\in\\mathbb{R}^{n}:\\left\\langle {x,\\xi}\\right\\rangle\n\\le f\\left(u\\right)}\\\n\\text{for all}\\ \\xi\\in E\\right\\}.\n\\end{equation}\nLet $\\mathcal K_o^n$ be the set of convex bodies containing\nthe origin $o$ in their interiors.\nA well-known fact is that $K_f\\in\\mathcal K_o^n$ if $f\\in C_{+}\\left(\\mathbb{S}^{n-1}\\right)$, and $h_{K_f}=f$\nalmost everywhere with respect to the surface area measure of $K_f$.\nSchneider \\cite{S14} proved that if $\\{f_j\\}_{j=1}^\\infty\\subset C_{+}\\left(\\mathbb{S}^{n-1}\\right)$\nconverges to $f\\in C_{+}\\left(\\mathbb{S}^{n-1}\\right)$ uniformly as $j\\to\\infty$, then the sequence\n$\\{K_{f_j}\\}$ is also convergent in the sense of the Hausdorff metric, i.e.,\n\\begin{equation}\\label{2.3}\nK_{f_j}\\to K_f,\\ \\text{as}\\ j\\to\\infty.\n\\end{equation}\n\n\\subsection{The $\\mathbf{p}$-harmonic functions and\n$\\mathbf{p}$-harmonic measures}\\label{subsect:2.2}\n\nWe now review some properties of the $\\mathbf{p}$-harmonic function, which are also\nreferenced in \\cite{AM24} for more details.\n\nThe $\\mathbf{p}$-harmonic functions minimize the $\\mathbf{p}$-Dirichlet energy\n$\\int_{K}{\\left|{\\nabla u}\\right|}^\\mathbf{p}dx$\nand are weak solutions to the $\\mathbf{p}$-Laplacian equation\n$\\Delta_\\mathbf{p}u=\\text{div}\\left({{\\left|{\\nabla u}\\right|}^{\\mathbf{p}-2}\n\\nabla u}\\right)=0$ in a convex domain $K$.\nThe existence of a weak solution $u_K\\in W^{1,\\mathbf{p}}\\left(K\\right)$ to $\\Delta_\\mathbf{p}u=0$\nin $K$, with boundary condition $u=f$ on $\\partial K$, is known. The uniqueness of the weak solution\nfollows directly from the comparison principle, while the regularity theory presents more complex challenges.\nLet $K\\in \\mathcal{A}_+^{2,\\alpha}$ and $f\\in C^{1,\\alpha}\\left(\\partial K\\right)$,\nit follows from \\cite{L88} that\n$u_K\\in C^{1,\\beta}\\left(\\bar{K}\\right)$ for some $\\beta(n,\\mathbf{p},\\alpha)\\in(0,1)$.\nTolksdorf \\cite{T84} has proved that the weak solutions to $\\Delta_\\mathbf{p}u = 0$ in $K$\nare locally $C^{1,\\beta}$ for some $\\beta(n,\\mathbf{p})\\in \\left(0,1\\right)$.\nThis shows that for any compact subset $K^\\prime\\subset \\subset K$,\nthe weak solutions are continuously differentiable on $K^\\prime$ and their first derivatives are H\\\"older continuous.\nHence, the weak solution $u$ to \\eqref{1.1} belongs to $C^{1,\\beta}(\\bar K\\cap N)$.\nSince $\\left|{\\nabla u}\\right|\\ne0$ in $K\\cap N$, the $\\mathbf{p}$-Laplacian operator\nis uniformly elliptic in $K\\cap N$. It follows from the boundary Schauder estimates \\cite{GT01}\nthat the Hessian matrix $D^{2}u$ is well-defined on $\\partial K$.\nLet $u_{K_j}$ be the weak solution to \\eqref{1.1} for $K_j$. Then, by Proposition 3.65 of \\cite{AM24}, $\\nabla u_{K_j}\\to\\nabla u_K$ uniformly in $N$, if $K_j\\to K$.\n\nFor the $\\mathbf{p}$-harmonic function, we provide two important lemmas. The first one can be stated as follows.\n\n\\begin{lemma}\\label{lem:2.1}\nLet $K$ be a bounded convex domain containing the origin and $u$ be the solution to \\eqref{1.1},\nthere exists a constant $M>0$, independent of $K$, such that\n\\begin{equation}\\nonumber\n\\left|{\\nabla u}\\right|\\le M\\ \\mathrm{on}\\ \\partial K.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nBy Theorem 2.46 of \\cite{AM24}, for any convex domain $K$, there exists a sequence of convex domains $\\{K_j\\}\\subset\\mathcal{A}_+^{2,\\alpha}$\nthat converges to $K$ as $j\\to\\infty$. Thus, we only need to consider the case that $K\\in \\mathcal{A}_+^{2,\\alpha}$.\n\nLet $u$ be a solution to the boundary value problem\n\\begin{equation}\\label{2.4}\n\\left\\{\n\\begin{aligned}\n&\\text{div}\\left({{\\left|{\\nabla u}\\right|}^{\\mathbf{p}-2}\n\\nabla u}\\right)=0&&\\text{in}\\ K\\setminus \\bar\\Omega_0,\\\\\n&u>0&&\\text{in}\\ K,\\\\\n&u=0&&\\text{on}\\ \\partial K,\\\\\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\bar\\Omega_0:=K\\setminus N$. If $u=1$ in $\\bar\\Omega_0$,\nit follows from page 204 of \\cite{L77} that $u$ is a $\\mathbf{p}$-capacity function of\n$K\\setminus\\bar\\Omega_0$. By Theorem 2 of \\cite{CS03}, we conclude that\n$u\\in C^{\\infty}\\left(K \\setminus \\bar\\Omega_0\\right)\n\\cap C\\left(K \\setminus \\Omega_0\\right)$, $0<u<1$ in $K \\setminus\\bar\\Omega_0$\nand $K_s= \\left\\{ {x\\in K:u(x)\\ge s} \\right\\}$ is convex for $0\\le s\\le 1$.\n\nSince $\\left| {\\nabla u\\left( x \\right)} \\right| > 0$ in $K \\setminus \\bar \\Omega_0$,\nby Theorem 4 of \\cite{CS03}, we obtain\n\\begin{equation}\\label{2.5}\n- \\frac{{\\partial {h_{{K_s}}}\\left( {{{-\\nabla u\\left( x \\right)} \\mathord{\\left/\n{\\vphantom {{\\nabla u\\left( x \\right)} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right.\n\\kern-\\nulldelimiterspace} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right)}}{{\\partial s}}\n=\\frac{1}{{\\left| {\\nabla u\\left( x \\right)} \\right|}},\n\\end{equation}\nfor all $x\\in\\partial K_s$. By applying Proposition 1 of \\cite{CS03}, we further have\n\\[\\frac{{\\partial {h^2_{{K_s}}}\\left( {{{-\\nabla u\\left( x \\right)} \\mathord{\\left/\n{\\vphantom {{\\nabla u\\left( x \\right)} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right.\n\\kern-\\nulldelimiterspace} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right)}}{{\\partial s^{2}}} \\ge 0,\\]\nthus $\\frac{{\\partial {h_{{K_s}}}\\left( {{{-\\nabla u\\left( x \\right)} \\mathord{\\left/\n{\\vphantom {{\\nabla u\\left( x \\right)} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right.\n\\kern-\\nulldelimiterspace} {\\left| {\\nabla u\\left( x \\right)} \\right|}}} \\right)}}{{\\partial s}} $\nis non-decreasing for every fixed $x$. This, together with \\eqref{2.5}, shows that\n$\\left|{\\nabla u\\left(x\\right)}\\right|$\nattains its maximum on $\\partial\\bar\\Omega_0$.\nLet $B_r$ be a ball with radius $r$ included in $\\bar\\Omega_0$ and\ninternally tangent to $\\partial\\bar\\Omega_0$ at $x\\in\\partial\\bar\\Omega_0$,\nand let $v$ be a solution to the equation \\eqref{2.4} with $\\bar \\Omega_0$ replaced by $B_r$.\nAs $B_{r}\\subset \\bar \\Omega_0$, we have $K \\setminus \\bar \\Omega_0 \\subset K \\setminus B_{r}$,\nthus\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\Delta_{\\mathbf{p}}u= \\Delta _{\\mathbf{p}}v && \\text{in}\\ K \\setminus\\bar\\Omega_0,\\\\\n&u=v=0\\ &&\\text{on}\\ \\partial K,\\\\\n&v\\le u\\ &&\\text{on}\\ \\partial \\Omega_0.\\\\\n\\end{aligned}\n\\right.\n\\end{equation*}\nThen, by the comparison principle (cf. Theorem 2.1 of \\cite{G13}), $v\\le u$ on\n$K \\setminus\\bar\\Omega_0$. This, combined with $u(x)=v(x)$, implies that\n$\\left|{\\nabla u\\left(x\\right)}\\right|\\le\\left|{\\nabla v\\left(x\\right)} \\right|$\nfor $x\\in\\partial\\bar\\Omega_0$.\nThen, we can calculate the value of $\\left|{\\nabla v\\left(x\\right)}\\right|$ and obtain a positive constant $m$\ndepending on $r$ and $n$ such that\n\\begin{equation}\\label{2.6}\n\\left|{\\nabla u}\\right|\\le m\n\\end{equation}\nin $K\\setminus\\bar\\Omega_0$.\n\nMoreover, since $u\\in C^{1,\\beta}\\left({\\bar K\\cap N}\\right)$\nwith $\\beta=\\beta(n,\\mathbf{p},\\alpha)$, it follows that $\\nabla u$ is $\\beta$-H\\\"older continuous.\nThen, there exists a constant $\\Lambda>0$ such that\n$$\n\\left| {\\nabla u\\left( y \\right)-\\nabla u\\left(z\\right)}\\right|\\le \\Lambda{\\left|{y-z} \\right|^\\beta}\n$$\nfor $y,z\\in {\\bar K\\cap N}$. Thus, we have\n\\[\\left| {\\nabla u\\left(z \\right)} \\right| \\le \\Lambda {\\left| {y-z} \\right|^\\beta } + \\left| {\\nabla u\\left(y\\right)} \\right|\\]\nfor any $z\\in \\partial K$ and $y\\in K\\cap N$.\nThis, together with \\eqref{2.6} and the boundedness of ${\\bar K\\cap N}$, shows that\nthere exists a finite positive constant $M$, independent of $K$, such that\n$$|{\\nabla u\\left(z\\right)}|\\le M$$ for all $z\\in\\partial K$.\nThis completes the proof of Lemma \\ref{lem:2.1}.\n\\end{proof}\n\nThe second order covariant derivative of $h_K:\\mathbb{S}^{n-1}\\to\\mathbb{R}$ is locally given by\n$${\\bigtriangledown}^2{h_K}\n=\\sum\\limits_{{i,j= 1}}^{n-1}(\\bigtriangledown_{i,j}h_K)\ne^i\\otimes e^j,$$\nwhere $\\bigtriangledown_{i,j}h_K(x)=\\partial_{i,j}(h_K\\circ\\varphi^{-1})(\\varphi(x))$ with $U\\subset\\mathbb{S}^{n-1}$\nand $\\varphi:U\\to V\\subset\\mathbb{R}^{n-1}$ being a coordinate chart.\nLet $\\mathbb{I}$ be the unit matrix of order $(n-1)$ and $C[\\bigtriangledown^2h_K+h_K\\mathbb{I}]$ be the cofactor matrix of\n$\\left({{\\bigtriangledown^2}{h_K}+{h_K}{\\mathbb{I}}}\\right)$ with element\n${C_{i,j}}\\left[\\cdot\\right]\n=\\left\\langle{C\\left[\\cdot\\right]{e^j},{e^i}}\\right\\rangle$.\nThe following lemma directly follows from Lemma 3.44 of \\cite{AM24}.\n\n\\begin{lemma}\\label{lem:2.2}\nLet $\\left\\{{e^1,\\ldots,e^{n-1},\\xi}\\right\\}$ be an orthonormal basis of $\\mathbb{R}^{n}$,\nand let $u$ be the solution to \\eqref{1.1} for a convex domain $K$ that is of class\n$C_+^{2,\\alpha}$. Then we have\n\\begin{enumerate}[label=\\upshape(\\roman*)]\n\\item \t$\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\xi ,\\xi } \\right\\rangle\n= \\frac{1}{{\\mathbf{p}-1}}\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\left| {\\nabla u\\left( {\\nabla {h_K}\n\\left( \\xi \\right)} \\right)} \\right|{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]} \\right)$,\n\\item $\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right){e^i},\\xi } \\right\\rangle\n=-\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\n\\sum\\limits_{j = 1}^{n - 1} {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]} {\\bigtriangledown _j}\n\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right)$.\n\\end{enumerate}\n\\end{lemma}\n\nAt the end of this subsection, we review the weak convergence of the $\\mathbf{p}$-harmonic measure.\nLet $u\\in W^{1,\\mathbf{p}}\\left(K\\cap N\\right)$ be a $\\mathbf{p}$-harmonic function, a solution to \\eqref{1.1} in $K\\cap N$.\nFollowing Akman-Mukherjee \\cite{AM24}, one can define\nthe $\\mathbf{p}$-harmonic measure\n\\begin{equation}\\nonumber\n{\\mu_{\\Bar K}}\\left(E\\right)\n={\\mu_K}\\left(E\\right)\n=\\int_{g_K^{-1}\\left(E\\right)}{\\left|\n{\\nabla u\\left(x\\right)} \\right|}^{\\mathbf{p}-1}d{\\mathcal{H}}^{n-1}\\left(x\\right),\n\\end{equation}\nwhere $E\\subset\\mathbb{S}^{n-1}$ is a Borel subset.\nIf $K\\in \\mathcal{A}_+^{2,\\alpha }$, we have $\\nabla h_K\\left(\\xi\\right)=g_K^{-1}\\left(\\xi\\right)$,\nand we can use the transformation rule of\nthe Jacobian (cf. page 8 of \\cite{AM24}) to obtain\n\\begin{equation}\\label{2.7}\n(g_K)_*\\mathcal{H}^{n- 1}\\llcorner_{\\partial K}\n=|\\det\\left({\\bigtriangledown}^2h_K+h_K{\\mathbb{I}}\\right)|\n\\mathcal{H}^{n- 1}\\llcorner_{\\mathbb{S}^{n-1}}\n=\\frac{1}{\\left(\\kappa\\circ g_K^{-1}\\right)}\n\\mathcal{H}^{n- 1}\\llcorner_{\\mathbb{S}^{n-1}}.\n\\end{equation}\nTherefore,\n\\begin{equation}\\nonumber\n\\begin{split}\nd{\\mu _K} = {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left(\\xi\\right)} \\right)} \\right|^{\\mathbf{p}-1}}d{\\mathcal{H}^{n - 1}} \\llcorner_{\\partial K}={\\left| {\\nabla u\\left( \\nabla {h_{{K}}}\\left(\\xi\\right) \\right)}\n\\right|^{\\mathbf{p}-1}}\\det \\left( {{\\bigtriangledown ^2}{h_K} + {h_K}\\mathbb{I}} \\right)d\\xi.\n\\end{split}\n\\end{equation}\nFor a compact convex set $K$ and a sequence of compact convex sets $\\left\\{K_{j}\\right\\}$ with $K_{j}\\to K$ as $j\\to\\infty$,\nAkman-Mukherjee \\cite{AM24} proved that\n\\begin{equation}\\label{2.8}\n\\mathop{\\lim}\\limits_{j\\to\\infty }\n\\int_{\\mathbb{S}^{n-1}}{f\\left(\\xi\\right)}d\\mu_{K_j}\\left(\\xi\\right)\n=\\int_{\\mathbb{S}^{n-1}}{f\\left(\\xi\\right)} d{\\mu_K}\\left(\\xi\\right)\n\\end{equation}\nfor any $f \\in C\\left(\\mathbb{S}^{n-1}\\right)$. This shows that the $\\mathbf{p}$-harmonic measure is weakly convergent.\nMoreover, it can be checked that the centroid of the $\\mathbf{p}$-harmonic measure is at the origin.\n\n\\begin{lemma}\\label{lem:2.3}\nLet $K$ be a bounded convex domain, then for any $x_0\\in \\mathbb{R}^{n}$,\n$$\n\\int_{{\\mathbb{S}^{n - 1}}} {\\left\\langle {{x_0},\\xi } \\right\\rangle } d{\\mu_K}(\\xi)=0.\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $u_K$ be a weak solution to the $\\mathbf{p}$-Laplace equation\nin $K \\cap N$, or equivalently,\n\\begin{equation}\\label{2.9}\n\\int_{K\\cap N} {{{\\left|\\nabla u_K(x)\\right|}^{\\mathbf{p}-2}}\n\\left\\langle{\\nabla u_K(x),\\nabla\\phi(x)}\\right\\rangle}dx\n=0\n\\end{equation}\nfor any smooth function $\\phi$ defined in $K\\cap N$ with compact support. Consider the boundary value problem \\eqref{1.1} and\nlet $f$ be a function in $C^\\infty\\left(\\overline{K\\cap N}\\right)$ such that $f=u_K$ on $\\partial N\\cap K$ and $f=1$ on $\\partial K$.\nNotice that $$g_K(x)=-\\frac{\\nabla u_K(x)}{\\left|{\\nabla u_K(x)}\\right|},$$\nthen for any $x_0\\in \\mathbb{R}^{n}$, we have the following calculation:\n\\begin{equation*}\n\\begin{split}\n&\\int_{{\\mathbb{S}^{n - 1}}} {\\left\\langle {{x_0},\\xi } \\right\\rangle } d{\\mu_K}\\left( \\xi \\right)\\\\\n=&\\int_{{\\mathbb{S}^{n - 1}}} {\\left\\langle {{x_0},\\xi } \\right\\rangle } {\\left| {\\nabla {u_K}\n\\left( {g_K^{ - 1}\\left( \\xi \\right)} \\right)} \\right|^{{\\mathbf{p}} - 1}}d{S_K}\\left( \\xi \\right)\\\\\n=&\\int_{\\partial K} {{{\\left| {\\nabla {u_K}\\left( x \\right)} \\right|}^{{\\mathbf{p}} - 1}}\n\\left\\langle {{x_0}, g_K(x)} \\right\\rangle } d{\\mathcal{H}^{n - 1}}\\\\\n=&\\int_{\\partial K} {{{\\left| {\\nabla {u_K}\\left( x \\right)} \\right|}^{{\\mathbf{p}} - 2}}} \\left\\langle {\\nabla {u_K}\\left( x \\right),{g_K}\\left( x \\right)} \\right\\rangle\n\\left\\langle {{x_0}, g_K(x)} \\right\\rangle \\left( {u_K\\left( x \\right)-f\\left( x \\right) } \\right)d{\\mathcal{H}^{n - 1}}\\\\\n&+\\int_{\\partial N \\cap K} {{{\\left| {\\nabla {u_K}\\left( x \\right)} \\right|}^{{\\mathbf{p}} - 2}}} \\left\\langle {\\nabla {u_K}\\left( x \\right),{\\nu _{\\partial N \\cap K}}\\left( x \\right)} \\right\\rangle \\left\\langle {{x_0}, g_K(x)} \\right\\rangle \\left( {u_K\\left( x \\right)-f\\left( x \\right) } \\right)d{\\mathcal{H}^{n - 1}}\\\\\t\n=&\\int_{\\partial \\left( {K \\cap N} \\right)} {{{\\left| {\\nabla {u_K}\\left( x \\right)} \\right|}^{{\\mathbf{p}} - 2}}\\left\\langle {\\nabla {u_K}\\left( x \\right),{\\nu _{\\partial \\left( {K \\cap N} \\right)}}\\left( x \\right)} \\right\\rangle \\left\\langle {{x_0}, g_K(x)} \\right\\rangle \\left( {u_K\\left( x \\right) - f\\left( x \\right)} \\right)} d{\\mathcal{H}^{n - 1}}\\\\\n=&\\int_{K \\cap N} {\\text{div}\\left( {{{\\left| {\\nabla {u_K}\\left( x \\right)} \\right|}^{{\\mathbf{p}} - 2}}\\nabla {u_K}\\left( x \\right)\\left\\langle {{x_0}, g_K(x)} \\right\\rangle \\left( {u_K\\left( x \\right) - f\\left( x \\right)} \\right)} \\right)} dx\\\\\n=& 0,\\\\ \t\n\\end{split}\n\\end{equation*}\nwhere we have used the divergence theorem and \\eqref{2.9}.\nThis proves the desired property.\n\\end{proof}\n\n\\section{The variational formula associated with $\\mathbf{p}$-harmonic measure}\\label{sect:3}\n\nAssociated with the $\\mathbf{p}$-harmonic measure $\\mu_K$ of\na compact convex set $K\\subset\\R^n$, Akman-Mukherjee \\cite{AM24} introduced a continuous functional\n\\begin{equation}\\label{3.1}\n\\Gamma\\left(K\\right)\n=\\int_{\\mathbb{S}^{n-1}}h_K\\left(\\xi\\right) d{\\mu_K}\\left(\\xi\\right).\n\\end{equation}\nBy Lemma \\ref{lem:2.3}, it can be verified that the functional $\\Gamma(\\cdot)$ is translation invariant. That is, for any $x_0\\in \\mathbb{R}^{n}$,\n\\begin{equation}\\label{3.2}\n\\Gamma \\left({K+x_0}\\right)=\\Gamma\\left(K\\right).\n\\end{equation}\n\nIn the following part of this section, we will focus on calculating the variation of $\\Gamma\\left(K\\right)$ with respect to the $q$-sum for $q>0$\nand introduce the $L_q$ $\\mathbf{p}$-harmonic measure. To do so, we will briefly review the concept of the $q$-sum.\n\nLet $K$ and $L$ be two compact convex sets containing the origin.\nFor $q\\ge1$ and $t\\ge 0$, Firey's $q$-sum $K^t$ can be defined by\n$h_{K^t}^q=h_K^q+th_L^q$ on $\\mathbb{S}^{n-1}$.\nFollowing B\\\"or\\\"oczky et al. \\cite{BZ12}, the $q$-sum $K^t$ for $0<q<1$ can be\ndefined as the Wulff shape of the function\n$\\left(h_K^q+ th_L^q\\right)^{\\frac{1}{q}}$, that is\n\\begin{equation}\\label{3.3}\n{K^t}=\\left\\{{x\\in \\mathbb{R}^{n}:\\left\\langle {x,\\xi}\\right\\rangle\n\\le{\\left( {h_K^q\\left(\\xi\\right)\n+th_L^q\\left(\\xi\\right)} \\right)}^{\\frac{1}{q}}}\\\n\\text{for all}\\ \\xi\\in\\mathbb{S}^{n-1}\\right\\}.\n\\end{equation}\nIn this case, $h_{K^t}^q=h_K^q+th_L^q$ holds almost everywhere on\n$\\mathbb{S}^{n-1}$ with respect to the surface area measure $S_{K^t}$ of $K^t$. Thus, we have\n$S_{K^t}\\left(\\omega_t\\right)=0$,\nwhere\n\\[{\\omega_t}=\\left\\{\\xi\\in {\\mathbb{S}^{n- 1}}:h_{{K^t}}^q(\\xi)\\ne h_K^q\n(\\xi)+th_L^q(\\xi)\\right\\}.\\]\n\nLet $K,L\\in \\mathcal{A}_+^{2,\\alpha}$ and $q>0$. We take a small enough\n\\begin{equation}\\label{3.4}\n\\tau\n:=\\tau\\left(d_{\\mathcal H}\\left({\\partial K,\\partial N}\\right),\nd_{\\mathcal H}\\left({\\partial L,\\partial N}\\right),\n\\left\\|u\\right\\|_{W^{1,\\mathbf{p}}\\left(N\\right)}\\right)\n>0,\n\\end{equation}\nwhere $u$ is the solution to \\eqref{1.1},\nsuch that ${K^t}\\in\\mathcal{A}_+^{2,\\alpha}$, $\\partial K^{t}\\subset N$, and\n$K^{t}\\cap\\partial N=K\\cap\\partial N$ for all $\\left|t\\right|\\le\\tau$.\nWith this choice, we conclude that\n$g_{K^t}:\\partial K^t\\to\\mathbb{S}^{n-1}$ is a diffeomorphism. It follows that ${\\mathcal{H}^{n-1}}\\left({\\omega _t}\\right)=0$ and\n\\[\\int_{\\mathbb{S}^{n-1}}h_{K^t}^qd\\xi\n=\\int_{\\mathbb{S}^{n-1}}{(h_K^q+th_L^q)}d\\xi.\\]\n\nNext, we consider the $\\mathbf{p}$-harmonic measure corresponding\nto $u(\\cdot,t)\\in W^{1,\\mathbf{p}}(K^{t}\\cap N)$, which is a weak solution\nto the Dirichlet problem\n\\begin{equation}\\label{3.5}\n\\left\\{\n\\begin{aligned}\n&\\text{div}\\left({{\\left|{\\nabla u\\left(x,t\\right)}\\right|}^{\\mathbf{p}-2}\n\\nabla u\\left(x,t\\right)}\\right)=0&&x\\in K^t\\cap N,\\\\\n&u\\left(x,t\\right)=0&&x\\in\\partial K^t,\\\\\n&u\\left(x,t\\right) = u\\left(\\frac{x}{\\left(1+t\\right)^{\\frac{1}{q}}}\\right)&&x\\in\\partial N\\cap K^t,\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\left| t \\right|$ is small enough so that upon zero extension,\n$u\\left(x,t\\right) \\in {W^{1,\\mathbf{p}}}\\left( N \\right)$.\nBy defining\n\\begin{equation}\\label{3.6}\n\\mathcal{F}\\left[h_{K^t}\\right]\\left(\\xi\\right)\n:={\\left|{\\nabla u\\left({\\nabla{h_{K^t}}\n\\left(\\xi\\right),t}\\right)}\\right|^{\\mathbf{p}-1}}\\det\n\\left({{\\bigtriangledown^2}{h_{{K^t}}}+h_{K^t}\\mathbb I}\\right),\n\\end{equation}\nwe obtain\n\\begin{equation*}\nd{\\mu_{{K^t}}}\n={\\left|{\\nabla u\\left({\\nabla{h_{{K^t}}}\\left(\\xi\\right),t} \\right)}\\right|^{\\mathbf{p}-1}}\nd{\\cal H}^{n-1}{\\llcorner_{\\partial K^t}}\n=\\mathcal{F}\\left[h_{K^t}\\right]\\left(\\xi\\right)d\\xi,\n\\end{equation*}\nand\n\\begin{equation}\\label{3.7}\n\\Gamma\\left(K^t\\right)\n=\\int_{\\mathbb{S}^{n-1}}{{h_{K^t}}\\left(\\xi\\right)}\nd{\\mu_{K^t}}\\left(\\xi\\right)\n=\\int_{{\\mathbb{S}^{n-1}}}h_{K^t}\n\\left(\\xi\\right)\\mathcal{F}\\left[h_{K^t}\\right]\n\\left(\\xi\\right)d\\xi.\n\\end{equation}\n\n\\begin{lemma}\\label{lem:3.1}\nLet $1<\\mathbf{p}<\\infty$ and $q>0$, and let $\\mathcal{F}$ be given by \\eqref{3.6}. Then we have\n\\begin{equation}\\label{3.8}\n\\mathcal{F}\\left[ {\\left( {1 + t} \\right)^{\\frac{1}{q}}{h_K}} \\right]\\left( \\xi \\right)\n= {\\left( {1 + t} \\right)^{\\frac{n-\\mathbf{p}}{q}}}\\mathcal{F}\\left[ {{h_K}} \\right]\\left( \\xi \\right),\n\\end{equation}\nfor all $\\left| t \\right|\\le\\tau$. Here $\\tau$ is given in \\eqref{3.4}.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is similar to that of Lemma 3.12 in \\cite{AM24}. For completeness, we provide a proof as follows.\n\nWe first deal with the case that $0<q<1$. By setting $L=K$ in \\eqref{3.3},\nwe obtain that $K^{t}=\\lambda K$ is the Wulff shape\nof the support function $\\lambda h_{K}$, where $\\lambda=\\left(1+t\\right)^{\\frac{1}{q}}$.\nLet $u_{\\lambda}\\left(\\cdot\\right):=u\\left(\\cdot,\\lambda^{q}-1\\right)$ be the weak solution\nto the Dirichlet problem\n\\begin{equation}\\label{3.9}\n\\left\\{\n\\begin{aligned}\n&\\text{div}\\left( {{{\\left| {\\nabla u_{\\lambda}\\left( x \\right)} \\right|}^{\\mathbf{p}-2}}\n\\nabla u_{\\lambda}\\left( x \\right)} \\right) = 0&&x\\in\\lambda K\\cap N,\\\\\n&u_{\\lambda}\\left(x\\right)=0&&x\\in\\partial(\\lambda K),\\\\\n&u_{\\lambda}\\left(x\\right)= u\\left(\\frac{x}{\\lambda}\\right)&&x\\in\\partial N\\cap\\lambda K,\n\\end{aligned}\n\\right.\n\\end{equation}\nfor $\\left| {{\\lambda ^q} - 1} \\right| \\le \\tau $.\nThen we have\n\\begin{equation}\\label{3.10}\n\\begin{split}\n\\mathcal{F}\\left[ {{\\lambda h_{K}}} \\right]\\left( \\xi \\right)\n&={\\left| {\\nabla u_{\\lambda}\\left( {\\lambda \\nabla {h_{K}}\\left( \\xi \\right)} \\right)}\n\\right|^{\\mathbf{p}-1}}\\lambda^{n-1}\\det \\left( {{\\bigtriangledown ^2}{h_{K}}+{h_{K}}\\mathbb{I}} \\right)\\\\\n&= \\left({\\frac{\\left|{\\nabla {u_\\lambda }\\left( {\\lambda \\nabla {h_K}\\left( \\xi \\right)} \\right)}\\right|}{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}}\\right)^{\\mathbf{p} - 1}\n{\\lambda ^{n - 1}}\\mathcal{F} \\left[ {{ h_{K}}} \\right]\\left( \\xi \\right).\n\\end{split}\n\\end{equation}\nAs $u$ is the solution to \\eqref{1.1}, we have that $u\\left(\\frac{x}{\\lambda}\\right)$\nis also the solution to \\eqref{3.9} in $\\lambda K$. By the uniqueness of\nthe solution to \\eqref{3.9},\n$u_{\\lambda}\\left(x\\right)=u\\left(\\frac{x}{\\lambda}\\right)$\nin $\\lambda K$. It follows that\n$\\nabla {u_\\lambda }\\left( x \\right)\n= \\frac{1}{\\lambda }\\nabla u\\left( {\\frac{x}{\\lambda }} \\right)$, thus \\eqref{3.10} gives\n\\[\\mathcal{F}\\left[ {\\lambda {h_K}} \\right]\\left( \\xi \\right)\n= {\\lambda^{n-\\mathbf{p}}}\\mathcal{F}\\left[ {{h_K}} \\right]\\left( \\xi \\right)\\]\nfor $\\left| {{\\lambda ^q} - 1} \\right| \\le \\tau $.\nThis proves the case $0<q<1$.\n\nNote that the $q$-sum $K^t$ for $q\\ge1$ can also be given by \\eqref{3.3},\nand the argument for the case $q\\ge1$ follows along the same lines. Therefore, the remaining case of the proof is omitted.\n\\end{proof}\n\nWe define $\\dot u\\left(x\\right)\n={{{\\left. {\\frac{\\partial}{{\\partial t}}}\\right|}_{t= 0}}u\\left( {x,t} \\right)}$\nand present a differentiability lemma as follows.\n\n\\begin{lemma}\\label{lem:3.2}\nLet $1<\\mathbf{p}<\\infty$ and $q>0$, and let $K, L\\in \\mathcal{A}_+^{2,\\alpha}$\nbe two compact convex sets containing the origin. If $u\\left(\\cdot,t\\right)\\in W^{1,\\mathbf{p}}\\left(K^{t} \\cap N\\right)$ is the solution to \\eqref{3.5},\nthe following holds:\n\\begin{enumerate}[label=\\upshape (\\roman*)]\n\n\\item The map $t\\mapsto u\\left({x,t} \\right)$ is differentiable at $t=0$\nfor all $x\\in\\bar K\\cap N$, and $\\dot u\\in C^{2,\\beta}\\left(\\overline{K\\cap N}\\right)$ with $\\beta=\\beta(n,\\mathbf{p},\\alpha)$;\n\t\t\t\t\n\\item For $x\\in\\partial K$ and $q\\ge1$,\n$\\dot u(x)=\\left| {\\nabla u\\left(x\\right)}\\right|\n\\left({\\frac{1}{q}h_K^{1-q}\\left(g_K\\left(x\\right)\\right)h_L^q\n\\left( {{g_K}\\left(x\\right)}\\right)}\\right)$. If $0<q<1$, this equality\nholds almost everywhere with respect to $S_K$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nPart (i) comes from Proposition 3.20 of \\cite{AM24}. Here, we provide a brief proof of (ii) for the case $0<q<1$; the case $q\\ge1$ follows similarly.\n\nDefine $\\omega\\left(x,t\\right)=\\frac{u\\left(x,t\\right)-u\\left(x,0\\right)}{t}$\nfor $t\\neq0$. According to (3.23) in \\cite{AM24}, there exists a sequence $\\{t_k\\}$ such that\n$t_k\\to 0$ as $k\\to\\infty$, and the limit\n\\begin{equation*}\n\\lim\\limits_{k\\to\\infty }\\omega\\left(x,{t_k}\\right)\n=\\lim\\limits_{k\\to\\infty }\n\\frac{u\\left({x,{t_k}} \\right)-u\\left({x,0}\\right)}{t_k}\n=:\\omega\\left(x\\right)\n\\end{equation*}\nexists for all $x\\in K\\cap N$. Moreover, for $x\\in \\partial K$, there\nexists a sequence $\\left\\{x_j\\right\\}\\subset \\text{int}K$ such that $x_j\\to x$\nas $j\\to\\infty$, and\n\\begin{equation*}\n\\omega\\left(x\\right)\n=\\lim\\limits_{j\\to\\infty }\\omega\\left(x_j\\right)\n=\\lim\\limits_{j\\to\\infty }\\lim \\limits_{k\\to\\infty}\\omega\\left({x_j},{t_k}\\right)\n=\\lim\\limits_{k\\to\\infty }\\frac{u\\left(x,t_k\\right)-u\\left(x,0\\right)}{t_k},\n\\end{equation*}\nfor any $x\\in\\partial K$. Hence, the function $t\\to u\\left({\\cdot,t}\\right)$\nis differentiable at $t=0$ for all $x\\in\\bar K\\cap N$.\nIt follows from (3.26) and (3.27) of \\cite{AM24} that\n$\\dot u\\in C^{2,\\beta} \\left(\\overline{K\\cap N}\\right)$,\nand\n\\[\\left| {\\omega \\left( {{x_k},{t_k}} \\right)\n-\\omega\\left( {{x_k},0}\\right)}\\right|\\leqslant\\Lambda\\left|{x_k-x}\\right|\\]\nfor $\\Lambda>0$ and any $x_k\\in\\partial K^{t_k}$. Thus,\n\\begin{equation*}\n\\omega\\left(x\\right)\n=\\lim\\limits_{k\\to\\infty}\\omega\\left(x_k,t_k\\right)\n=\\lim\\limits_{k\\to\\infty}\\frac{u\\left(x_k,t_k\\right)-u\\left(x_k,0\\right)}{t_k}\n=\\lim\\limits_{k\\to\\infty}\\frac{u\\left(x\\right)-u\\left(x_k,0\\right)}{t_k}\n\\end{equation*}\nfor any $x\\in\\partial K$.\n\nFor $\\xi\\in\\mathbb{S}^{n-1}$, there exists $x\\in\\partial K$ and\n$x_k\\in\\partial K^{t_k}$ so that $x=\\nabla h_{K}\\left(\\xi\\right)$,\n$x_{k}=\\nabla h_{K^{t_k}}\\left(\\xi\\right)$.\nThen, we compute:\n\\begin{equation*}\n\\begin{split}\n\\nabla h_{K^{t_k}}\n&=\\nabla {\\left({h_K^q+t_kh_L^q}\\right)^{\\frac{1}{q}}}\\\\\n&= {\\left({h_K^q+t_kh_L^q}\\right)^{\\frac{{1-q}}{q}}}h_K^{q - 1}\\nabla {h_K}\n+t_k{\\left( {h_K^q + t_kh_L^q} \\right)^{\\frac{{1 - q}}{q}}}h_L^{q - 1}\\nabla {h_L}\\\\\n&={\\left({1+t_kh_L^qh_K^{-q}} \\right)^{\\frac{{1 - q}}{q}}}\\nabla {h_K}\n+t_k{\\left( {{{\\left( {h_L^qh_K^{-q}}\\right)}^{-1}}+t_k} \\right)^{\\frac{{1 - q}}{q}}}\\nabla {h_L}\\\\\n&=\\nabla {h_K}+\\left( {{{\\left( {1 +t_kh_L^qh_K^{ - q}} \\right)}^{\\frac{{1 - q}}{q}}} - 1} \\right)\\nabla {h_K}\n+ t_kh_L^{q - 1}h_K^{1 - q}{\\left( {1 + t_kh_L^qh_K^{ - q}} \\right)^{\\frac{{1 - q}}{q}}}\\nabla {h_L},\n\\end{split}\n\\end{equation*}\n$S_{K^{t_k}}$-almost everywhere. Taking the limit as $k\\to \\infty$, we obtain:\n\\begin{equation*}\n\\begin{split}\n\\mathop {\\lim }\\limits_{k \\to \\infty} \\frac{{{x_k} - x}}{t_k}\n&= \\mathop {\\lim }\\limits_{k \\to \\infty} \\frac{{\\left( {{{\\left( {1 + t_kh_L^qh_K^{ - q}} \\right)}^{\\frac{{1 - q}}{q}}} - 1} \\right)\\nabla {h_K} + t_kh_L^{q - 1}h_K^{1 - q}{{\\left( {1 +t_kh_L^qh_K^{ - q}} \\right)}^{\\frac{{1 - q}}{q}}}\\nabla {h_L}}}{t_k}\\\\\n&= \\frac{{1 - q}}{q}h_L^qh_K^{ - q}\\nabla {h_K} + h_L^{q - 1}h_K^{1 - q}\\nabla {h_L}\\\\\n&= \\nabla \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right),\n\\end{split}.\n\\end{equation*}\n$S_{K}$-almost everywhere. Thus,\n\\begin{equation*}\n\\begin{split}\n\\omega \\left( x \\right)\n=\\mathop {\\lim }\\limits_{k \\to \\infty } \\frac{{u\\left( x \\right) - u\\left( {{x_k},0} \\right)}}{{{t_k}}}\n=-\\left\\langle {\\nabla u\\left( x \\right),\\nabla \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right\\rangle,\n\\end{split}\n\\end{equation*}\n$S_{K}$-almost everywhere for all $x\\in \\partial K$.\nNotice that\n$\\xi=-\\frac{{\\nabla u\\left( x \\right)}}{{\\left| {\\nabla u\\left( x \\right)} \\right|}}$\nand\n\\[\\frac{1}{q}h_K^{1 - q}\\left( \\xi \\right)h_L^q\\left( \\xi \\right)\n=\\left\\langle {\\xi ,\\nabla \\left( {\\frac{1}{q}h_K^{1 - q}\n\\left(\\xi\\right)h_L^q\\left(\\xi\\right)} \\right)} \\right\\rangle,\\]\ndue to the Euler's homogeneous function theorem.\nWe can conclude that\n\\[\\omega \\left( x \\right) = \\left| {\\nabla u\\left( x \\right)} \\right|\\left( {\\frac{1}{q}h_K^{1 - q}\\left( {{g_K}\\left( x \\right)} \\right)h_L^q\\left( {{g_K}\\left( x \\right)} \\right)} \\right).\\]\nThis completes the proof of the second assertion for the case $0<q<1$.\n\\end{proof}\n\nIn the following, we prove two lemmas which are critical for establishing the variational formula of\n$\\Gamma\\left(K \\right)$ with respect to the $q$-sum. The first one can be stated as follows.\n\\begin{lemma}\\label{lem:3.3}\nLet $1<\\mathbf{p}<\\infty$, and let $K, L\\in \\mathcal{A}_+^{2,\\alpha}$\nbe two compact convex sets containing the origin.\nThen, for the Wulff shape $K^t$ with $\\left| t \\right|\\le\\tau$ (where $\\tau$ is given in \\eqref{3.4}),\nif $0<q<1$, we have\n\\begin{equation*}\n\\begin{split}\n{\\left.{\\frac{d}{{dt}}}\\right|_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left(\\xi\\right)\n=&\\sum\\limits_{i,j=1}^{n-1} {{\\bigtriangledown _j}\n\\left({{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K}+{h_K}{\\mathbb I}}\\right]\n{{\\left|{\\nabla u\\left({\\nabla {h_{{K}}}\\left(\\xi\\right)}\\right)}\\right|}^{\\mathbf{p}-1}}\n{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1-q}h_L^q} \\right)}\\right)}\\\\\n&-\\left( {\\mathbf{p}-1}\\right)\n{\\left| {\\nabla u\\left( {\\nabla {h_{{K}}}\\left(\\xi\\right)}\\right)}\\right|^{\\mathbf{p}-2}}\n\\det\\left( {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}}\\right)\n\\left\\langle{\\nabla\\dot u\\left({\\nabla{h_K}\\left(\\xi\\right)}\\right),\\xi}\\right\\rangle\n\\end{split}\n\\end{equation*}\n$S_K$-almost everywhere on $\\mathbb{S}^{n-1}$. If $q\\ge1$, this equality always holds on $\\mathbb{S}^{n-1}$.\n\\end{lemma}\n\n\\begin{proof}\t\nSince the proof for the case $q\\ge1$ is similar to that for the case $0<q<1$,\nwe will focus only on the latter.\n\nAccording to \\eqref{3.6}, we have the following calculation\n\\begin{equation}\\label{3.11}\n\\begin{split}\n&{\\left.{\\frac{d}{{dt}}}\\right|_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}}\\right]\\left(\\xi\\right)\\\\\n=&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\left( {{{\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left(\\xi\\right),t} \\right)} \\right|}^{\\mathbf{p}-1}}\\det \\left( {{\\bigtriangledown ^2}{h_{K^t}} + {h_{K^t}}{\\mathbb I}} \\right)} \\right)\\\\\n=&\\left( {\\mathbf{p}-1} \\right){\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-2}}{\\left. {\\det \\left( {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right)\\frac{d}{{dt}}} \\right|_{t = 0}}\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)} \\right| \\\\\n&+{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-1}}{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\det \\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb I}} \\right).\t\t\n\\end{split}\n\\end{equation}\nNotice that\n\\[\\int_{{\\mathbb{S}^{n-1}}} {\\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb{I}}} \\right)} d{S_{{K^t}}}\n= \\int_{{\\mathbb{S}^{n-1}}} \\left({\\bigtriangledown ^2}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}\n+ {{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}{\\mathbb{I}}\\right) d{S_{{K^t}}},\\]\nwe differentiate both sides with respect to $t$ at $t=0$ and obtain\n\\begin{equation*}\n\\begin{split}\n&\\int_{{\\mathbb{S}^{n-1}}}{{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{h_{{K^t}}}\n+ {h_{{K^t}}}{\\mathbb{I}}} \\right)} d{S_K}\n+\\int_{{\\mathbb{S}^{n-1}}} \\left({{\\bigtriangledown ^2}{h_{{K}}} + {h_{{K}}}{\\mathbb{I}}}\\right)\n{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}d{S_{{K^t}}}\\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\n\\left( {{\\bigtriangledown ^2}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}\n+ {{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}{\\mathbb{I}}} \\right)} d{S_K}\n+\\int_{{\\mathbb{S}^{n-1}}} \\left({{\\bigtriangledown ^2}{{ {h_K}}}\n+ { {h_K} }{\\mathbb{I}}}\\right) {\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}d{S_{{K^t}}}.\n\\end{split}\n\\end{equation*}\nThis implies that\n\\[\\int_{{\\mathbb{S}^{n-1}}} {{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb{I}}} \\right)} d{S_K}\n= \\int_{{\\mathbb{S}^{n-1}}} {{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}} + {{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}{\\mathbb{I}}} \\right)} d{S_K}.\\]\nTherefore,\n\\[{{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb{I}}} \\right) = {{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}} + {{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}{\\mathbb{I}}} \\right)}\\]\n$S_K$-almost everywhere. Hence,\n \\begin{equation}\\label{3.12}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\det \\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb I}} \\right)\\\\\n=&{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{h_{{K^t}}} + {h_{{K^t}}}{\\mathbb I}} \\right)} \\right)\\\\\n=&{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\left( {{\\bigtriangledown ^2}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}} + {{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}{\\mathbb I}} \\right)} \\right)\\\\\n=&{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {{\\bigtriangledown ^2}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right) + \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right){\\mathbb I}} \\right)} \\right).\n\\end{split}\n\\end{equation}\n$S_K$-almost everywhere.\n\nAs the unit outer normal $\\xi$ of $K^t$ satisfies the identity\n\\begin{equation*}\n\\xi = - \\frac{{\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)}}{{\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)} \\right|}},\n\\end{equation*}\nthen\n$\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)} \\right| = - \\left\\langle {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right),\\xi } \\right\\rangle$,\nand we have the following calculation\n\\begin{equation*}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)} \\right|\\\\\n=&-{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\left\\langle {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right),\\xi } \\right\\rangle \\\\\n=&-\\left( {\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right){{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\nabla {h_{{K^t}}}\\left( \\xi \\right),\\xi } \\right\\rangle + \\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle } \\right)\\\\\n=&-\\left( {\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\nabla \\left( {{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}{{\\left( {h_K^q + th_L^q} \\right)}^{\\frac{1}{q}}}} \\right),\\xi } \\right\\rangle + \\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle } \\right)\\\\\n=&-\\left( {\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\nabla \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right),\\xi } \\right\\rangle + \\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle } \\right)\\\\\n=&-\\left( {{J_1} + {J_2}} \\right),\n\\end{split}\n\\end{equation*}\n$S_K$-almost everywhere.\nSince\n$$\n\\nabla {h_K}\\left( \\xi \\right) = {h_K}\\left( \\xi \\right)\\xi + \\sum\\limits_{i = 1}^{n - 1} {{\\bigtriangledown _i}{h_K}\\left( \\xi \\right){e^i}}\n$$\nand\n$$\n\\nabla {h_L}\\left( \\xi \\right) = {h_L}\\left( \\xi \\right)\\xi + \\sum\\limits_{i = 1}^{n - 1} {{\\bigtriangledown _i}{h_L}\\left( \\xi \\right){e^i}},\n$$\nwe have\n\\begin{equation}\\label{3.13}\n\\begin{split}\n\\nabla \\left( {\\frac{1}{q}h_K^{1-q}(\\xi)h_L^q(\\xi)} \\right)\n=\\left( {\\frac{1}{q}h_K^{1-q}(\\xi)h_L^q(\\xi)} \\right)\\xi\n+\\sum\\limits_{i=1}^{n-1} {{\\bigtriangledown_i}\n\\left({\\frac{1}{q}h_K^{1-q}(\\xi)h_L^q(\\xi)}\\right){e^i}}.\n\\end{split}\n\\end{equation}\nThis, together with Lemma \\ref{lem:2.2}, yields that\n\\begin{equation*}\n\\begin{split}\n{J_1}=&\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\nabla\n\\left( {\\frac{1}{q}h_K^{1-q}h_L^q} \\right),\\xi } \\right\\rangle\\\\\n=&\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\xi ,\\xi } \\right\\rangle \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\n+\\sum\\limits_{i = 1}^{n - 1} {\\left\\langle {{D^2}u\\left( {\\nabla {h_K}\\left(\\xi\\right)} \\right){e^i},\\xi} \\right\\rangle {\\bigtriangledown _i}\n\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\\\\n=&\\frac{1}{{\\mathbf{p}-1}}\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\n\\left|{\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|\n{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K}+{h_K}{\\mathbb I}} \\right]} \\right)\n\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\\\\\n&-\\sum\\limits_{i = 1}^{n - 1} {\\kappa \\left( {\\nabla {h_K}\\left(\\xi\\right)} \\right)\\sum\\limits_{j = 1}^{n - 1} {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K}\n+ {h_K}{\\mathbb I}} \\right]} {\\bigtriangledown _j}\n\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\\\\n=&\\frac{1}{{\\mathbf{p}-1}}\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\n\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|\n{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K}+{h_K}{\\mathbb I}} \\right]} \\right)\n\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\\\\\n&-\\kappa \\left( {\\nabla {h_K}\\left(\\xi\\right)} \\right)\\sum\\limits_{i,j = 1}^{n - 1} {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]} {\\bigtriangledown _j}\n\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\n\\left( \\xi \\right)} \\right)} \\right|} \\right){\\bigtriangledown _i}\n\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right).\n\\end{split}\n\\end{equation*}\t\nThen, using $\\sum\\limits_{j=1}^{n-1} {{\\bigtriangledown _j}{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]}=0$ (cf. (4.3) of \\cite{CY76}), we have\n\\begin{equation*}\n\\begin{split}\n{J_1} =& \\frac{1}{{\\mathbf{p}-1}}\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]} \\right)\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\\\\\n&- \\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)}.\n\\end{split}\n\\end{equation*}\nHence,\n\\begin{equation}\\label{3.14}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\left| {\\nabla u\\left( {\\nabla {h_{{K^t}}}\\left( \\xi \\right),t} \\right)} \\right| \\\\\n=&\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right)} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\\\\n&-\\frac{1}{{\\mathbf{p}-1}}\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]} \\right)\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right) \\\\\n&-\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle,\n\\end{split}\n\\end{equation}\n$S_K$-almost everywhere.\n\nApplying \\eqref{2.7} and substituting both \\eqref{3.14} and \\eqref{3.12} into \\eqref{3.11},\nwe obtain that\n\\begin{equation*}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left( \\xi\\right)\\\\\n=&\\left( {\\mathbf{p}-1} \\right){\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-2}}\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right)} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)}\\\\\n&-{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-1}}{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]} \\right)\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right) \\\\\n&-\\left( {\\mathbf{p}-1} \\right)\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle \\\\\n&+{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-1}}{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {{\\bigtriangledown ^2}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right) + \\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right){\\mathbb I}} \\right)} \\right)\\\\\n=&\\left( {\\mathbf{p}-1} \\right){\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-2}}\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right)} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\\\\n&-\\left( {\\mathbf{p}-1} \\right)\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle \\\\\n&+ {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-1}}{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {{\\bigtriangledown ^2}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right)} \\right),\n\\end{split}\n\\end{equation*}\n$S_K$-almost everywhere. Since\n\\begin{equation*}\n\\begin{split}\n&\\sum\\limits_{i,j = 1}^{n - 1}{{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right)}\\\\\n=&\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}} \\right)} {\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\\\\\n&+\\sum\\limits_{i,j = 1}^{n - 1} {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}} {\\bigtriangledown _{j,i}}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)\\\\\n=&\\left( {\\mathbf{p}-1} \\right){\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-2}}\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]\\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|} \\right)} \\right){\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)}\\\\\n&+ {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|^{\\mathbf{p}-1}}{\\rm{Tr}}\\left( {C\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{\\bigtriangledown ^2}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right).\n\\end{split}\n\\end{equation*}\n Hence,\n\\begin{equation*}\n\\begin{split}\n{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)\n=& \\sum\\limits_{i,j = 1}^{n - 1}\n{{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right)} \\\\\n&-\\left( {\\mathbf{p}-1} \\right)\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle,\n\\end{split}\n\\end{equation*}\n$S_K$-almost everywhere.\n\\end{proof}\n\nLemmas \\ref{lem:3.2} and \\ref{lem:3.3} can be employed to prove the following result.\n\n\\begin{lemma}\\label{lem:3.4}\nLet $1<\\mathbf{p}<\\infty$ and $q>0$, and let $K, L\\in \\mathcal{A}_+^{2,\\alpha}$\nbe two compact convex sets containing the origin.\nThen, for the Wulff shape $K^t$ with $\\left| t \\right|\\le\\tau$ (where $\\tau$ is given in \\eqref{3.4}), we have\n\\begin{equation}\\label{3.15}\n\\int_{{\\mathbb{S}^{n-1}}} {{h_K}{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi = \\int_{{\\mathbb{S}^{n-1}}} {h_K^{1 - q}h_L^q{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F}\\left[ {{{\\left( {1 + t} \\right)}^{\\frac{1}{q}}}{h_K}} \\right]\\left( \\xi \\right)} d\\xi.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $K\\in\\mathcal{A}_+^{2,\\alpha}$, by Lemma \\ref{lem:3.3},\nwe have\n\\begin{equation}\\label{3.16}\n\\begin{split}\n&\\int_{{\\mathbb{S}^{n-1}}}{{h_K}{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi \\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {{h_K}\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb I}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right)} } d\\xi \\\\\n&-\\int_{{\\mathbb{S}^{n-1}}} {{h_K}\\left( {\\mathbf{p}-1} \\right)\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle } d\\xi \\\\\n=&I_1-I_2.\n\\end{split}\n\\end{equation}\nThen, by repeatedly applying Stokes's theorem for a compact manifold without boundary, we can calculate the term $I_1$ as follows.\n\\begin{equation}\\label{3.17}\n\\begin{split}\nI_1\n& = \\int_{{\\mathbb{S}^{n-1}}} {\\sum\\limits_{i,j = 1}^{n - 1} {{h_K}{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right)} \\right)} } d\\xi \\\\\n&=-\\int_{{\\mathbb{S}^{n-1}}} {\\sum\\limits_{i,j = 1}^{n - 1} {{C_{i,j}}\\left[ {{\\bigtriangledown^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K^{1 - q}h_L^q} \\right){\\bigtriangledown _j}} } {h_K}d\\xi\\\\\n&=\\int_{{\\mathbb{S}^{n-1}}} {\\sum\\limits_{i,j = 1}^{n - 1} {h_K^{1-q}h_L^q{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}{h_K}} \\right)} \\right)} } d\\xi.\n\\end{split}\n\\end{equation}\nBy using (ii) of Lemma \\ref{lem:3.2}, along with the formulas \\eqref{3.13} and \\eqref{2.2}, we can calculate\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\mathbf{p}-1}{I_2}\n=&\\int_{{\\mathbb{S}^{n-1}}} {{h_K}\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\dot u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right),\\xi } \\right\\rangle } d\\xi \\\\\n=&\\int_{\\partial K} {{{\\left| {\\nabla u} \\right|}^{\\mathbf{p}-2}}{h_K} \\circ {g_K}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u} \\right|\\left( {\\frac{1}{q}{{\\left( {{h_K} \\circ {g_K}} \\right)}^{1 - q}}{{\\left( {{h_L} \\circ {g_K}} \\right)}^q}} \\right)} \\right),{g_K}} \\right\\rangle } d{\\mathcal{H}^{n - 1}}\\\\\n=&\\int_{\\partial K} {{{\\left| {\\nabla u} \\right|}^{\\mathbf{p}-2}}{h_K} \\circ {g_K}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u} \\right|} \\right)\\left( {\\frac{1}{q}{{\\left( {{h_K} \\circ {g_K}} \\right)}^{1 - q}}{{\\left( {{h_L} \\circ {g_K}} \\right)}^q}} \\right),{g_K}} \\right\\rangle } d{\\mathcal{H}^{n - 1}}\\\\\n&+\\int_{\\partial K} {{{\\left| {\\nabla u} \\right|}^{\\mathbf{p}-2}}{h_K} \\circ {g_K}\\left| {\\nabla u} \\right|\\frac{1}{q}{{\\left( {{h_K} \\circ {g_K}} \\right)}^{1 - q}}{{\\left( {{h_L} \\circ {g_K}} \\right)}^q}} d{\\mathcal{H}^{n - 1}}\\\\\n=&\\int_{\\partial K} {{{\\left| {\\nabla u} \\right|}^{\\mathbf{p}-2}}{{\\left( {{h_K} \\circ {g_K}} \\right)}^{1 - q}}{{\\left( {{h_L} \\circ {g_K}} \\right)}^q}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u} \\right|} \\right)\\frac{1}{q}{h_K} \\circ {g_K},{g_K}} \\right\\rangle } d{\\mathcal{H}^{n - 1}}\\\\\n&+\\int_{\\partial K} {{{\\left| {\\nabla u} \\right|}^{\\mathbf{p}-2}}{{\\left( {{h_K} \\circ {g_K}} \\right)}^{1 - q}}{{\\left( {{h_L} \\circ {g_K}} \\right)}^q}\\left\\langle {\\left| {\\nabla u} \\right|\\nabla \\left( {\\frac{1}{q}{h_K} \\circ {g_K}} \\right),{g_K}} \\right\\rangle } d{\\mathcal{H}^{n - 1}}\\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {h_K^{1 - q}h_L^q\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|\\left( {\\frac{1}{q}{h_K}} \\right)} \\right),\\xi } \\right\\rangle } d\\xi.\n\\end{split}\n\\end{equation*}\nThis, together with \\eqref{3.17} and \\eqref{3.16}, yields that\n\\begin{equation}\\label{3.18}\n\\begin{split}\n&\\int_{{\\mathbb{S}^{n-1}}}{{h_K}{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F}\\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi\\\\\n=& \\int_{{\\mathbb{S}^{n-1}}} {h_K^{1 - q}h_L^q\\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}{h_K}} \\right)} \\right)} d\\xi } \\\\\n&- \\left( {\\mathbf{p}-1} \\right)\\int_{{\\mathbb{S}^{n-1}}} {h_K^{1 - q}h_L^q\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|\\left( {\\frac{1}{q}{h_K}} \\right)} \\right),\\xi } \\right\\rangle } d\\xi.\n\\end{split}\n\\end{equation}\n\nOn the other hand, by Lemma \\ref{lem:3.2} and Lemma \\ref{lem:3.3} with $L=K$, we have\n\\begin{equation*}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}{\\cal F}\\left[ {{{\\left( {1 + t} \\right)}^{\\frac{1}{q}}}{h_K}} \\right]\\left( \\xi \\right)\\\\\n=& \\sum\\limits_{i,j = 1}^{n - 1} {{\\bigtriangledown _j}\\left( {{C_{i,j}}\\left[ {{\\bigtriangledown ^2}{h_K} + {h_K}{\\mathbb{I}}} \\right]{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-1}}{\\bigtriangledown _i}\\left( {\\frac{1}{q}h_K} \\right)} \\right)} \\\\\n&-\\left( {\\mathbf{p}-1} \\right)\\frac{{{{\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|}^{\\mathbf{p}-2}}}}{{\\kappa \\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)}}\\left\\langle {\\nabla \\left( {\\left| {\\nabla u\\left( {\\nabla {h_K}\\left( \\xi \\right)} \\right)} \\right|\\left( {\\frac{1}{q}{h_K}} \\right)} \\right),\\xi } \\right\\rangle,\n\\end{split}\n\\end{equation*}\nfor $q\\ge1$. Note that the above equality holds almost everywhere with respect to\n$S_K$ if $0<q<1$, then by substituting it into \\eqref{3.18}, we can obtain \\eqref{3.15}.\n\\end{proof}\n\nNow, the main result of this section can be stated as follows.\n\n\\begin{theorem}\\label{th:3.1}\nLet $1<\\mathbf{p}<\\infty$, $q>0$, $K\\in\\mathcal K_o^n$ and $L\\subset \\mathbb{R}^{n}$ be a compact convex set containing the origin.\nThen, for the Wulff shape $K^t$ with $\\left| t \\right|\\le\\tau$ (where $\\tau$ is given in \\eqref{3.4}), we have\n\\begin{equation}\\label{3.19}\n\\begin{split}\n{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\Gamma \\left( {{K^t}} \\right) = \\frac{{n-\\mathbf{p} + 1}}{q}\\int_{{\\mathbb{S}^{n-1}}}\n{h_L^q\\left( \\xi \\right)h_K^{1 - q}\\left( \\xi \\right)} d\\mu_{K}\\left(\\xi\\right) .\n\\end{split}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nLet $K\\in\\mathcal K_o^n$ and $L\\subset \\mathbb{R}^{n}$ be a compact convex set containing the origin. We first prove the case that $K, L\\in \\mathcal{A}_+^{2,\\alpha}$.\nThen, by formula \\eqref{3.7} and Lemmas \\ref{lem:3.4} and \\ref{lem:3.1}, we have\n\\begin{equation*}\n\\begin{split}\n&{\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\Gamma \\left( {{K^t}} \\right) \\\\\n=& {\\left. {\\frac{d}{{dt}}} \\right|_{t = 0}}\\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {h_K^q\\left( \\xi \\right) + th_L^q\\left( \\xi \\right)} \\right)}^{\\frac{1}{q}}}\\mathcal{F} \\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi \\\\\n=& \\int_{{\\mathbb{S}^{n-1}}} {{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}{{\\left( {h_K^q\\left( \\xi \\right) + th_L^q\\left( \\xi \\right)} \\right)}^{\\frac{1}{q}}}\\mathcal{F} \\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi + \\int_{{\\mathbb{S}^{n-1}}} {{h_K}\\left( \\xi \\right){{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F} \\left[ {{h_{{K^t}}}} \\right]\\left( \\xi \\right)} d\\xi \\\\\n=& \\frac{1}{q}\\int_{{\\mathbb{S}^{n-1}}}\n{h_L^qh_K^{1-q}\\mathcal{F} \\left[ {{h_K}} \\right]\\left( \\xi \\right)} d\\xi\n+\\int_{{\\mathbb{S}^{n-1}}} {h_L^q h_K^{1 - q}{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\mathcal{F} \\left[ {{{\\left( {1 + t} \\right)}^{\\frac{1}{q}}}{h_K}} \\right]\\left( \\xi \\right)} d\\xi \\\\\n=&\\frac{1}{q}\\int_{{\\mathbb{S}^{n-1}}}\n{h_L^qh_K^{1-q}\\mathcal{F} \\left[ {{h_K}} \\right]\\left( \\xi \\right)} d\\xi\n+\\frac{{n-\\mathbf{p}}}{q}\\int_{{\\mathbb{S}^{n-1}}}\n{h_L^qh_K^{1-q}\\mathcal{F}\\left[ {{h_K}} \\right]\\left( \\xi \\right)} d\\xi \\\\\n=&\\frac{{n-\\mathbf{p} + 1}}{q}\\int_{{\\mathbb{S}^{n-1}}} {h_L^qh_K^{1 - q}\\mathcal{F} \\left[ {{h_K}} \\right]\\left( \\xi \\right)}d\\xi \\\\\n=&\\frac{{n-\\mathbf{p} + 1}}{q}\\int_{{\\mathbb{S}^{n-1}}}\n{h_L^qh_K^{1-q}}d\\mu_{K}.\n\\end{split}\n\\end{equation*}\nThis proves \\eqref{3.19} for the case\nthat $K, L\\in \\mathcal{A}_+^{2,\\alpha}$.\n\nFor $K\\in\\mathcal K_o^n$ and a compact convex set $L \\subset \\mathbb{R}^{n}$ containing the origin, we can respectively choose two sequences\n$\\{K_j\\}_{j=1}^\\infty$ and $\\{L_j\\}_{j=1}^\\infty$ in $\\mathcal{A}_+^{2,\\alpha}$, such that $K_j\\to K$ and $L_j\\to L$ as $j\\to\\infty$. It follows that $h_{K_j}\\to h_K$ and $h_{L_j}\\to h_L$\nuniformly. Then, by \\eqref{2.3}, the continuity of the functional $\\Gamma$ on compact convex sets and the weak convergence \\eqref{2.8}, we can verify the\ndesired \\eqref{3.19}.\n\\end{proof}\n\nIn view of the variational formula \\eqref{3.19}, one can generalize the $\\mathbf{p}$-harmonic measure\nand introduce the following $L_{q}~\\mathbf{p}$-harmonic measure.\n\n\\begin{definition}\\label{def:3.1}\nLet $q\\in\\mathbb{R}$, $1<\\mathbf{p}<\\infty$, and $K\\in\\mathcal K_o^n$.\nWe define the $L_q$ $\\mathbf{p}$-harmonic measure $\\mu _{K,q}$ for each Borel\n$E\\subset \\mathbb{S}^{n-1}$ as\n\\begin{equation}\\nonumber\n{\\mu _{K,q}}\\left(E\\right) = \\int_E {h_K^{1 - q}\\left( \\xi \\right)} d{\\mu _K}\\left(\\xi\\right).\n\\end{equation}\n\\end{definition}\n\nThe weak convergence of the $L_q$ $\\mathbf{p}$-harmonic measure is critical and can be stated as follows.\n\n\\begin{lemma}\\label{lem:3.5}\nLet $q\\in\\mathbb{R}$, $1<\\mathbf{p}<\\infty$, and $K\\in\\mathcal K_o^n$.\nThen for any sequence of convex bodies $\\{K_j\\}$ in $\\mathcal K_o^n$,\nif $K_{j}\\to K$ as $j\\to\\infty$, then $\\mu_{K_j,q}$ converges to $\\mu_{K,q}$ weakly, as $j\\to\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nIt follows from \\eqref{2.8} that the $\\mathbf{p}$-harmonic measure is convergent weakly. Then, by Definition \\ref{def:3.1} and $K_{j}\\to K$ as $j\\to\\infty$, for any function $f\\in C\\left(\\mathbb{S}^{n-1}\\right)$, we have\n\\begin{equation}\\nonumber\n\\lim\\limits_{j\\to\\infty}\\int_{\\mathbb{S}^{n-1}}fd\\mu_{K_j,q}\n=\\lim\\limits_{j\\to\\infty}\n\\int_{\\mathbb{S}^{n-1}}fh_{K_j}^{1-q}d\\mu_{K_j}\n=\\int_{\\mathbb{S}^{n-1}}fh_K^{1-q}d\\mu_K\n=\\int_{\\mathbb{S}^{n-1}}fd\\mu_{K,q}.\n\\end{equation}\nThus, the desired weak convergence follows.\n\\end{proof}\n\n\\section{The proof of Theorem \\ref{th:1.1}}\\label{sect:4}\n\nIn this section, we study the $L_q$ Minkowski problem associated with $\\mathbf{p}$-harmonic measure for $0<q<1$\nand $1<\\mathbf{p}\\ne n+1$. By introducing an appropriate functional\nand studying a related extremal problem as well as the existence of a solution,\nwe can finally prove Theorem \\ref{th:1.1} via the variation method.\nTo begin with, we prove the following lemma, which is critical for our later approximation argument.\n\n\\begin{lemma}\\label{lem:4.1}\nLet $0<q<1$. If $f:\\mathbb{S}^{n-1}\\to \\mathbb{R}$ is a positive, twice continuously\ndifferentiable function, there exists a convex body $L$ containing the origin in its interior and a constant $r>0$ such that\n$$f^q=h_L^q-h^q_{rB_2^n},$$\nwhere $B_2^n$ is the standard unit ball in $\\mathbb{R}^n$.\n\\end{lemma}\n\n\\begin{proof}\nWe extend the function $f$ to $\\mathbb{R}^{n}\\setminus\\left\\{o\\right\\}$\nby defining $F\\left(x\\right):=\\left|x\\right|f\\left(\\frac{x}{\\left|x\\right| }\\right)$\nand we define $G\\left(x\\right):=\\left|x\\right|$ for $x\\in\\mathbb{R}^n$. Then, we can verify that\nthe function $\\left(F^q+r^qG^q\\right)^{\\frac{1}{q}}$ is positively homogeneous of degree one,\nwhere $r>0$. According to Euler's homogeneous function theorem,\n\\[\\left\\langle {x,\\nabla{{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}}} \\right\\rangle\n= {\\left( {{F^q} + {r^q}{G^q}} \\right)^{\\frac{1}{q}}},\\]\nwe then take the first derivative with respect to each component $x_j$ of $x$ and obtain\n\\[\\sum\\limits_{i = 1}^n {\\left( {\\frac{{\\partial {x_i}}}{{\\partial {x_j}}}\n\\frac{{\\partial \\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)}}{{\\partial {x_i}}}\n+ {x_i}\\frac{{{\\partial ^2}\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)}}\n{{\\partial {x_i}\\partial {x_j}}}} \\right)}\n= \\frac{{\\partial \\left( {{{\\left( {{F^q}+{r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)}}{{\\partial {x_j}}},\\]\nwhere $j=1,\\ldots,n$. Thus, we have\n\\begin{equation}\\label{4.1}\n\\sum\\limits_{i = 1}^n {\\left( {{x_i}\\frac{{{\\partial ^2}\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)}}{{\\partial {x_i}\\partial {x_j}}}} \\right)}\n=0,\n\\end{equation}\nfor all $j=1,\\ldots,n$. Let $D^2_{x}\\left(\\left(F^q+r^qG^q\\right)^{\\frac{1}{q}}\\right)$\nbe the second differential of function ${F^q}+{r^q}{G^q}$ at $x$,\nthat is\n\\[D_x^2\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)\n= {\\left( {\\frac{{{\\partial ^2}\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)}}{{\\partial {x_i}\\partial {x_j}}}} \\right)_{ij}}.\\]\nIt follows from \\eqref{4.1} that\n\\begin{equation}\\label{4.2}\nxD^2_{x}\\left(\\left(F^q+r^qG^q\\right)^{\\frac{1}{q}}\\right)z^\\intercal=0,\n\\end{equation}\nwhere $z^\\intercal$ is the transpose of $z\\in\\mathbb{R}^{n}$.\n\nFor any two vectors $x,y\\in \\mathbb{S}^{n-1}$ with $x\\perp y$, we can verify\n\\[yD_x^2\\left( {{F^q}+{r^q}{G^q}}\\right)y^\\intercal\n=yD_x^2\\left( {{F^q}} \\right)y^\\intercal+q{r^q}.\\]\nSince the second differential $D_x^2\\left( {{F^q}} \\right)$ of\nfunction $F^q$ is continuous on $\\mathbb{S}^{n-1}$,\nand $yD_x^2\\left({{F^q}} \\right)y^\\intercal$ has a minimum, we can choose a suitable $r>0$ so that\n\\begin{equation}\\label{4.3}\nyD_x^2\\left( {{ {{F^q} + {r^q}{G^q}}}} \\right)y^\\intercal\n\\ge 0.\n\\end{equation}\n\nLet $x\\in \\mathbb{S}^{n-1}$. Then for any nonzero $z\\in\\mathbb{R}^{n}$, there exists $\\alpha_1,\\alpha_2\\in\\mathbb{R}$\nsuch that $z=\\alpha_1x+\\alpha_2x'$, where $x'\\perp x$ and $x'\\in \\mathbb{S}^{n-1}$. Since\n\\begin{equation*}\n\\begin{split}\n&D_x^2\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)\\\\\n=&\\frac{1}{q}\\left( {\\frac{1}{q} - 1} \\right){\\left( {{F^q} + {r^q}{G^q}} \\right)^{\\frac{1}{q}-2}}|{\\nabla}\\left( {{F^q}+{r^q}{G^q}} \\right)|^2\\mathrm{I}\n+\\frac{1}{q}{\\left( {{F^q} + {r^q}{G^q}} \\right)^{\\frac{1}{q} - 1}}D_x^2\\left( {{F^q} + {r^q}{G^q}} \\right),\n\\end{split}\n\\end{equation*}\nwhere $\\mathrm{I}$ is the unit matrix of order $n$.\nThis, together with \\eqref{4.2} and \\eqref{4.3}, shows that\n\\begin{equation}\\nonumber\nzD^2_{x}\\left(\\left(F^q+r^qG^q\\right)^{\\frac{1}{q}}\\right)z^\\intercal\n\\ge 0,\n\\end{equation}\nfor any nonzero $z\\in\\mathbb{R}^{n}$ and $x\\in\\mathbb{S}^{n-1}$.\nIt follows that the matrix\n$D_x^2\\left( {{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}} \\right)$ is positive semi-definite\nfor any nonzero $x\\in\\mathbb{R}^{n}$. Then, by Theorem 1.5.13 of \\cite{S14},\nwe can verify that the function ${{{\\left( {{F^q} + {r^q}{G^q}} \\right)}^{\\frac{1}{q}}}}$ is sublinear.\nThe existence of the convex body $L$ directly follows from Theorem 1.7.1 of \\cite{S14}.\n\\end{proof}\n\nLet $Q$ be a compact convex set, $\\mu$ be a finite Borel measure on $\\mathbb{S}^{n-1}$, and $0<q<1$.\nWe define the functional\n$\\Phi_Q:Q\\to\\mathbb{R}$ as follows:\n\\begin{equation}\\label{4.4}\n{\\Phi_Q}\\left(\\zeta\\right)=\\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {{h_Q}\\left(\\xi\\right)-\\left\\langle {\\zeta,\\xi}\\right\\rangle}\\right)}^{q}}}d\\mu\\left(\\xi\\right).\n\\end{equation}\nNext, we proceed to prove two necessary lemmas concerning the functional $\\Phi_Q$.\n\n\\begin{lemma}\\label{lem:4.2}\nLet $0<q<1$ and $Q$ be a compact convex set, there exists a unique\n$\\zeta\\left(Q\\right)\\in \\rm{int} Q$ such that\n\\[{\\Phi_Q}\\left( {\\zeta \\left( Q \\right)}\\right)\n=\\mathop {\\sup }\\limits_{\\zeta\\in Q} {\\Phi_Q}\n\\left(\\zeta\\right),\\]\nand for any $x_0\\in\\mathbb{R}^{n}$, we have ${\\zeta\\left(Q+x_0\\right)}={\\zeta\\left(Q\\right)}+x_0$.\n\\end{lemma}\n\n\\begin{proof}\nLet $0<\\lambda<1$ and $\\zeta_{1},\\zeta_{2}\\in Q$.\nFrom equality \\eqref{4.4} and the concavity of the function $s^q$ with $s\\ge0$ and $0<q<1$, we obtain that\n\\begin{equation*}\n\\begin{split}\n&\\lambda {\\Phi_Q}\\left( {{\\zeta_1}} \\right)\n+\\left( {1-\\lambda } \\right){\\Phi_Q}\\left( {{\\zeta _2}} \\right)\\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {\\lambda {{\\left( {{h_Q}\\left( \\xi \\right) - \\left\\langle {{\\zeta _1},\\xi } \\right\\rangle } \\right)}^q} + } \\left( {1 - \\lambda } \\right){\\left( {{h_Q}\\left( \\xi \\right) - \\left\\langle {{\\zeta _2},\\xi } \\right\\rangle } \\right)^q}d\\mu \\left( \\xi \\right)\\\\\n\\le&{\\int_{{\\mathbb{S}^{n-1}}} {\\left( {{h_Q}\\left( \\xi \\right) - \\left( {\\lambda \\left\\langle {{\\zeta _1},\\xi } \\right\\rangle + \\left( {1 - \\lambda } \\right)\\left\\langle {{\\zeta _2},\\xi } \\right\\rangle } \\right)} \\right)} ^q}d\\mu \\left( \\xi \\right)\\\\\n=&{\\Phi_Q}\\left( {\\lambda {\\zeta _1} + \\left( {1 - \\lambda } \\right){\\zeta _2}} \\right),\n\\end{split}\n\\end{equation*}\nwhere the equality holds if and only if\n$\\left\\langle{{\\zeta_1},\\xi}\\right\\rangle\n=\\left\\langle{{\\zeta_2},\\xi}\\right\\rangle$\nfor all ${\\xi\\in \\mathbb{S}^{n-1}}$, implying\n${{\\zeta _1} = {\\zeta _2}}$.\nTherefore, ${\\Phi_Q}$ is strictly concave on $Q$, it follows that there exists a unique point\n${\\zeta\\left( Q \\right)}\\in Q$ such that\n${\\Phi_Q}\\left({\\zeta\\left(Q\\right)}\\right)\n=\\sup\\limits_{\\zeta\\in Q}{\\Phi_Q}\\left(\\zeta\\right)$.\n\t\nNext, we prove $\\zeta\\left(Q\\right)\\in \\text{int}Q$. Suppose\nto the contrary that $\\zeta\\left(Q\\right)\\in \\partial Q$, and let $\\omega$ be the set of all unit outward normal vectors at $\\zeta\\left(Q\\right)$:\n\\[\\omega = \\left\\{ {\\left. {\\xi \\in {\\mathbb{S}^{n-1}}} \\right|{h_Q}\\left( \\xi \\right)\n=\\left\\langle {\\zeta \\left( Q \\right),\\xi } \\right\\rangle } \\right\\}.\\]\nTake $x_{0}\\in \\text{int}Q$ and define\n\\[{\\xi _0} := \\frac{{{x_0} - \\zeta \\left( Q \\right)}}{{\\left| {{x_0} - \\zeta \\left( Q \\right)} \\right|}}.\\]\nIt can be verified that ${\\left\\langle {{\\xi_0},\\xi} \\right\\rangle }<0$ for $\\xi \\in \\omega$.\nDefine\n\\begin{equation}\\nonumber\n\\omega_+\n:=\\left\\{\\left.{\\xi \\in{\\mathbb{S}^{n-1}}\\setminus\\omega}\\right|\\left\\langle{{\\xi _0},\\xi }\\right\\rangle \\ge 0\\right\\}\n\\ \\text{and}\\\n\\omega_-\n:=\\left\\{\\left.{\\xi \\in{\\mathbb{S}^{n-1}}\\setminus\\omega}\\right|\\left\\langle{{\\xi _0},\\xi }\\right\\rangle < 0\\right\\},\n\\end{equation}\nthen for $\\xi\\in\\omega_+$, there exists a $\\epsilon>0$ such that\n${{h_Q}\\left(\\xi\\right)-\\left\\langle{\\zeta\\left(Q \\right),\\xi} \\right\\rangle}\n\\ge\\epsilon$. Choose $0<\\delta<\\frac{\\epsilon}{2}$ small enough\nso that $\\zeta\\left(Q\\right)+\\delta \\xi_0\\in\\text{int}Q$, which further gives\n\\begin{equation}\\nonumber\n{h_Q}\\left(\\xi\\right)-\\left\\langle {\\zeta\\left(Q \\right)+\\delta{\\xi _0},\\xi}\\right\\rangle\n>\\frac{{\\epsilon}}{2},\n\\end{equation}\nfor $\\xi\\in\\omega_+$. These, together with \\eqref{4.4} and the Lagrange mean value theorem, imply that\n\n\\begin{equation}\\nonumber\n\\begin{split}\n&{\\Phi_Q}\\left({\\zeta\\left(Q\\right)+\\delta{\\xi_0}}\\right)- {\\Phi_Q}\\left({\\zeta\\left(Q\\right)}\\right)\\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {{h_Q}\\left( \\xi \\right)-\\left\\langle{\\zeta\\left(Q\\right)+\\delta{\\xi_0},\\xi } \\right\\rangle}\\right)}^q}}d\\mu \\left(\\xi\\right) - \\int_{{\\mathbb{S}^{n-1}}} {{{\\left({{h_Q}\\left(\\xi\\right)-\n\\left\\langle {\\zeta \\left(Q\\right),\\xi}\\right\\rangle } \\right)}^q}}d\\mu\\left(\\xi\\right)\\\\\n=&\\int_\\omega{{{\\left({-\\left\\langle {\\delta {\\xi_0},\\xi }\\right\\rangle }\\right)}^q}} d\\mu \\left(\\xi \\right)+\\int_{{\\mathbb{S}^{n-1}}\\setminus\\omega} {{{\\left( {{h_Q}\\left(\\xi\\right)-\\left\\langle {\\zeta \\left(Q \\right)\n+\\delta {\\xi_0},\\xi } \\right\\rangle } \\right)}^q} - {{\\left({{h_Q}\\left(\\xi\\right)-\\left\\langle {\\zeta\\left( Q \\right),\\xi}\\right\\rangle}\\right)}^q}} d\\mu \\left(\\xi\\right)\\\\\n\\ge&\\int_\\omega{{{\\left({-\\left\\langle {\\delta{\\xi_0},\\xi } \\right\\rangle}\\right)}^q}} d\\mu \\left(\\xi \\right)\n-\\int_{{\\omega_+}}{{{\\left({{h_Q}\\left(\\xi \\right)-\\left\\langle{\\zeta \\left(Q\\right),\\xi} \\right\\rangle}\\right)}^q}\n-{{\\left( {{h_Q}\\left(\\xi \\right)- \\left\\langle{\\zeta \\left( Q \\right)+\\delta {\\xi _0},\\xi } \\right\\rangle}\\right)}^q}}\nd\\mu\\left(\\xi \\right)\\\\\n>&\\int_\\omega{{{\\left({-\\left\\langle{\\delta {\\xi _0},\\xi } \\right\\rangle}\\right)}^q}}\nd\\mu \\left(\\xi \\right)-\\int_{{\\omega_+}}{q{{\\left({\\frac{\\epsilon}{2}} \\right)}^{q-1}}\\left\\langle{\\delta{\\xi_0},\\xi } \\right\\rangle }d\\mu\\left(\\xi\\right).\n\\end{split}\n\\end{equation}\nNotice that $\\lim\\limits_{\\delta \\to 0^{+}}{\\delta^{1-q}}=0$. Hence, there exists a small enough $\\delta_{0}>0$ such that\n${\\Phi _Q}\\left( {\\zeta \\left( Q \\right) + \\delta {\\xi _0}} \\right) > {\\Phi _Q}\\left( {\\zeta \\left( Q \\right)} \\right)$,\nwhich leads to a contradiction, as $\\zeta(Q)$ was chosen such that $\\Phi_Q\\left( \\zeta(Q) \\right) = \\sup\\limits_{\\zeta \\in Q} \\Phi_Q(\\zeta)$. Therefore, we conclude that $\\zeta\\left(Q\\right)\\in \\text{int}Q$.\n\nThus, for any $x_0\\in \\mathbb{R}^{n}$, we have\n\\begin{equation*}\n\\begin{split}\n{\\Phi _{Q + x_0}}\\left( {\\zeta \\left( {Q + x_0} \\right)} \\right)\n&=\\mathop {\\sup }\\limits_{\\zeta \\in Q + x_0} \\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {{h_{Q + x_0}}\n\\left( \\xi \\right) - \\left\\langle {\\zeta,\\xi } \\right\\rangle } \\right)}^q}} d\\mu \\left( \\xi \\right)\\\\\n&=\\mathop {\\sup }\\limits_{\\zeta \\in Q} \\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {{h_Q}\\left( \\xi \\right)\n- \\left\\langle {\\zeta ,\\xi } \\right\\rangle } \\right)}^q}} d\\mu \\left( \\xi \\right)\\\\\n&={\\Phi _Q}\\left( {\\zeta \\left( Q \\right)} \\right)\\\\\n&=\\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {{h_{Q + x_0}}\\left( \\xi \\right)\n-\\left\\langle {{\\zeta(Q)+ x_0},\\xi } \\right\\rangle } \\right)}^q}} d\\mu \\left( \\xi \\right)\\\\\n&={\\Phi _{Q + x_0}}\\left( {\\zeta(Q)+ x_0} \\right).\n\\end{split}\n\\end{equation*}\nTherefore, by the uniqueness of the extreme point $\\zeta\\left(Q+x_0\\right)$, we conclude that ${\\zeta\\left(Q+x_0\\right)}={\\zeta\\left(Q\\right)}+x_0$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:4.3}\nLet $0<q<1$, $\\mu$ be a finite Borel measure on $\\mathbb{S}^{n-1}$, and\n$\\left\\{Q_{j}\\right\\}^\\infty_{j=1}$ be a sequence of compact convex sets.\nIf $Q_j$ converges to a compact convex set $Q$ as $j\\to\\infty$, then we have\n$\\lim\\limits_{j\\to \\infty}\\zeta\\left(Q_{j}\\right)=\\zeta\\left(Q\\right)$ and $\\mathop {\\lim }\\limits_{j \\to \\infty }\\Phi_{Q_{j}}\\left(\\zeta \\left(Q_{j}\\right)\\right)=\\Phi_{Q}\\left(\\zeta \\left(Q\\right)\\right)$.\n\\end{lemma}\n\n\\begin{proof}\nSince the sequence $\\{\\zeta(Q_j)\\}$ is bounded, there exists a convergent subsequence\n(still denoted by $\\{\\zeta(Q_j)\\}$) that converges to some $\\zeta_{0}\\in Q$.\n\nNext, we prove that $\\zeta_{0}=\\zeta(Q)$. If otherwise, by using \\eqref{4.4}\nand Lemma \\ref{lem:4.2}, we have\n\\[\\mathop {\\lim }\\limits_{j \\to \\infty } {\\Phi _{{Q_j}}}\\left( {\\zeta \\left( {{Q_j}} \\right)} \\right)\n= {\\Phi _Q}\\left( {{\\zeta _0}} \\right)<{\\Phi _Q}\\left( {{\\zeta \\left(Q\\right)}}\\right)\n=\\mathop {\\lim }\\limits_{j \\to \\infty } {\\Phi _{{Q_j}}}\\left( {\\zeta \\left( {{Q}} \\right)} \\right).\\]\nOn the other hand, since ${\\zeta \\left( {{Q}} \\right)}\\in \\text{int}Q_{j}$ for sufficiently large $j$, it follows that\n${\\Phi _{{Q_j}}}\\left( {\\zeta \\left( {{Q_j}} \\right)} \\right)\n>{\\Phi _{{Q_j}}}\\left( {\\zeta \\left( Q \\right)} \\right)$\nfor sufficiently large $j$. This contradiction implies that\n$\\zeta_{0}=\\zeta\\left( {{Q}} \\right)$. Using \\eqref{4.4} again,\nwe can verify that\n$\\mathop {\\lim }\\limits_{j \\to \\infty }\\Phi_{Q_{j}}\\left(\\zeta \\left(Q_{j}\\right)\\right)\n=\\Phi_{Q}\\left(\\zeta \\left(Q\\right)\\right)$.\n\\end{proof}\n\nNow, we are able to prove the Theorem \\ref{th:1.1} as follows.\n\n\\begin{proof}[Proof of Theorem \\ref{th:1.1}]\nRecall that the Wulff shape $K_f$ associated with a function $f\\in C_+\\left(\\mathbb{S}^{n-1}\\right)$\nis given by\n\\begin{equation}\\nonumber\n{K_f}\n=\\left\\{{x\\in\\mathbb{R}^{n}:\\left\\langle {x,u}\\right\\rangle\n\\le f\\left(u\\right)}\\\n\\text{for all}\\ u\\in\\mathbb{S}^{n-1}\\right\\}.\n\\end{equation}\nThen for $0<q<1$, $f\\in C_+\\left(\\mathbb{S}^{n-1}\\right)$, and a finite Borel measure $\\mu$ on $\\mathbb{S}^{n-1}$, we introduce a functional $\\Phi_f:K_f\\to\\mathbb{R}$ by\n\\begin{equation}\\label{4.5}\n{\\Phi_f}\\left(\\zeta\\right)\n=\\int_{{\\mathbb{S}^{n-1}}} {{{\\left( {f\\left(\\xi\\right)-\\left\\langle {\\zeta,\\xi}\\right\\rangle}\\right)}^{q}}}d\\mu\\left(\\xi\\right),\n\\end{equation}\nfor $\\zeta\\in K_f$. We then construct the following minimization problem:\n\\begin{equation}\\label{4.6}\n\\mathop {\\inf }\\limits_{f \\in {C_+}\n\\left( {{\\mathbb{S}^{n-1}}} \\right)}\n\\left\\{ {\\mathop {\\sup }\\limits_{\\zeta \\in {K_f}} {\\Phi _{f}}\n\\left( \\zeta \\right):\\Gamma \\left( K_{f} \\right)= \\Gamma \\left( B_2^n \\right)} \\right\\}.\n\\end{equation}\n\nSince $h_{K_f}\\le f$ and $K_{h_{K_f}}=K_f\\in \\mathcal K_o^n$ for any\n$f\\in C_+\\left(\\mathbb{S}^{n-1}\\right)$, by \\eqref{4.4} and \\eqref{4.5}, we obtain that\n$$\\Phi_{K_f}\\left(\\zeta\\right)=\\Phi_{h_{K_f}}\\left(\\zeta\\right)\\le{\\Phi_f}\\left(\\zeta\\right),$$\nwhere $\\zeta\\in{K_f}$. It follows that\n$\\mathop{\\sup}\\limits_{\\zeta\\in{K_f}} {\\Phi_{K_f}}\\left(\\zeta\\right)\n\\le\\mathop{\\sup}\\limits_{\\zeta\\in{K_f}} {\\Phi_f}\\left(\\zeta\\right).$\nTherefore, we can search for the minimum for \\eqref{4.6} among the support functions\nof convex bodies that contain the origin in their interiors,\nand we can verify that $h_K$ is a solution to \\eqref{4.6} if and only if $K$\nis a solution to the problem\n\\begin{equation}\\label{4.7}\n\\mathop {\\inf }\\limits_{Q \\in \\mathcal K_o^n}\n\\left\\{ {\\mathop {\\sup }\\limits_{\\zeta \\in Q} {\\Phi_Q}\n\\left( \\zeta \\right):\\Gamma \\left( Q \\right)=\\Gamma \\left(B_2^n\\right)} \\right\\}.\n\\end{equation}\n\t\nLet $\\left\\{{Q_j}\\right\\}_{j = 1}^\\infty $ be a minimizing sequence for the problem \\eqref{4.7}.\nThat is, $\\Gamma\\left(Q_j \\right)=\\Gamma\\left(B_2^n \\right)$ and\n$$\\mathop {\\lim}\\limits_{j\\to\\infty} {\\Phi _{Q_j}}\n\\left( {\\zeta \\left( {Q_j} \\right)} \\right)\n= \\mathop {\\inf }\\limits_{Q \\in \\mathcal K_o^n}\n\\left\\{ {\\Phi_Q\\left(\\zeta \\left( {Q} \\right)\\right):\\Gamma \\left( Q \\right)\n= \\Gamma \\left( B_2^n \\right)} \\right\\}.$$\nAccording to Lemma \\ref{lem:4.2}, we can suitably translate each $Q_j$ to\nobtain a sequence $\\left\\{K_j\\right\\}_{j=1}^\\infty$\nin $\\mathcal K_o^n$\nsuch that $\\zeta\\left(K_j\\right)=o$\nand $\\Gamma \\left( {{K_j}} \\right)=\\Gamma \\left( B_2^n \\right)$ by \\eqref{3.2}.\nTherefore, $\\left\\{ {{K_j}} \\right\\}_{j = 1}^\\infty$ is also the minimizing sequence for the problem \\eqref{4.7}, and ${\\Phi _{K_j}}\\left(o\\right)$ converges to\n$$\\mathop {\\inf }\\limits_{Q \\in {\\mathcal{K}_o^n}}\n\\left\\{ {{\\Phi_Q}\\left( {\\zeta \\left( Q \\right)} \\right):\\Gamma \\left( Q \\right)\n= \\Gamma \\left( B_2^n \\right)} \\right\\},$$\nas $j\\to\\infty$.\t\n\t\nWe now prove that the sequence $\\left\\{{K_j} \\right\\}$ is uniformly bounded.\nTo do so, we let\n$R_j:=\\mathop {\\max }\\limits_{\\xi \\in {\\mathbb{S}^{n-1}}} {h_{{K_j}}}\\left(\\xi\\right)$\nand assume that the maximum can be achieved by some $\\xi_0\\in \\mathbb{S}^{n-1}$.\nThen, we have\n$$R_{j}{\\left\\langle {{\\xi _0},\\xi } \\right\\rangle _ + } \\le {h_{{K_j}}}\\left( \\xi \\right)$$\nfor all $j$ and $\\xi\\in\\mathbb{S}^{n-1}$, and hence\n\\begin{equation}\\label{4.8}\n\\int_{\\mathbb{S}^{n-1}}\n{{\\left({R_{j}{{\\left\\langle{{\\xi _0},\\xi} \\right\\rangle}_+}}\\right)}^q}d\\mu\n\\left(\\xi\\right)\n\\le\\int_{\\mathbb{S}^{n-1}}{{\\left( {{h_{{K_j}}}\\left( \\xi \\right)} \\right)}^q} d\\mu \\left( \\xi\\right)\n={\\Phi_{K_j}}\\left(o\\right).\n\\end{equation}\nOn the other hand, for sufficiently large $j$, we have\n\\begin{equation}\\nonumber\n{\\Phi_{K_j}}\\left(o\\right)\n\\le\\Phi_{B_2^n-\\zeta\\left(B_2^n\\right)}\\left(o\\right)\n=\\int_{\\mathbb{S}^{n-1}}{{\\left(1-\\left\\langle{\\zeta\\left(B_2^n\\right),\\xi} \\right\\rangle\\right)}^q} d\\mu \\left( \\xi\\right).\n\\end{equation}\nThis, together with \\eqref{4.8}, implies that $\\{R_j\\}$ is uniformly bounded,\nwhere we have used the fact that the measure $\\mu$ is finite and not concentrated on\nany closed hemisphere. Therefore, the boundedness of the sequence $\\left\\{{K_j}\\right\\}$\nfollows.\nBy the Blaschke selection theorem, there exists a subsequence (still denoted by\n$\\left\\{ {{K_j}} \\right\\}$) that converges to some compact convex set $\\Omega$ as\n$j \\to \\infty$. In the following, we prove that $\\dim (\\Omega)=n$.\nIf $\\dim (\\Omega)<n-1$, then\n$\\mathcal{H}^{n-1}\\left(\\Omega\\right)=0=\\mathcal{H}^{n-1}\\left(\\partial \\Omega\\right)$. It follows from definition \\eqref{3.1} and Lemma \\ref{lem:2.1} that $\\Gamma \\left( \\Omega \\right)=0$, which contradicts to the following\n\\begin{equation}\\label{4.9}\n\\Gamma\\left(\\Omega\\right)\n=\\lim\\limits_{j\\to\\infty}\\Gamma\\left(K_j \\right)\n=\\Gamma\\left(B_2^n\\right)\n>0.\n\\end{equation}\nIf $\\dim (\\Omega) = n - 1$, there are at least two half-spaces containing $\\Omega$ that\nshare a common boundary, and $\\Omega$ degenerates to a $1$-codimensional\nsubset of a hyperplane.\nBy Lemma \\ref{lem:2.1} again,\n\\begin{equation}\\nonumber\n\\left| {\\nabla u} \\right|\\le M,\n\\end{equation}\nthus we obtain that\t\n\\begin{equation}\\nonumber\n\\Gamma\\left(\\Omega\\right)\n=\\int_{\\mathbb{S}^{n-1}}{h_\\Omega}d{\\mu_\\Omega}\n\\le{M^{\\mathbf{p}-1}}\\int_{\\mathbb{S}^{n-1}} {{h_\\Omega}} d{S_\\Omega}\\left(\\xi\\right)\n=0,\n\\end{equation}\nwhich again contradicts to \\eqref{4.9}. Therefore, $\\text{dim}(\\Omega)=n$ and $\\Omega$ is indeed a convex body. By Lemma \\ref{lem:4.3}, we have ${\\zeta\\left(\\Omega\\right)=o}$ and\n\\begin{equation}\\label{4.10}\n{\\Phi _{{h_\\Omega}}}\\left( o \\right)\n=\\mathop {\\inf }\\limits_{f \\in {C^ + }\\left( {{\\mathbb{S}^{n-1}}} \\right)}\n\\left\\{ {\\mathop {\\sup }\\limits_{\\zeta\\in{K_f}} {\\Phi _f}\\left( \\zeta \\right):\n\\Gamma \\left( {{K_f}} \\right)\n= \\Gamma \\left( B_2^n \\right)} \\right\\}.\n\\end{equation}\n\nLet $\\Omega_1$ be a compact convex set containing the origin and $\\Omega^t$ be\nthe Wulff shape of ${\\left({h_\\Omega^q+t{h_{\\Omega_1}^q}}\\right)}^{\\frac{1}{q}}$\nfor a small enough $t$, where\n$$\\lambda\\left(t\\right)\n:={\\left( {\\frac{{\\Gamma \\left( B_2^n \\right)}}{\\Gamma \\left(\\Omega^t\\right)}}\\right)^{\\frac{1}{n-\\mathbf{p}+1}}}.$$\nHere, we have used the condition that $\\mathbf{p}\\neq n+1$.\t\nThen, by equalities \\eqref{3.1} and \\eqref{3.8}, we can verify that\n$\\Gamma\\left({\\lambda\\left(t\\right){\\Omega^t}}\\right)=\\Gamma\\left(B_2^n\\right)$.\nIn the following, we prove that\n$\\zeta\\left(t\\right):=\\zeta\\left({\\lambda\\left(t\\right)\\Omega^t}\\right)$\nis differentiable at $t=0$.\n\t\nLet $\\zeta=(\\zeta_1,\\zeta_2,\\ldots,\\zeta_n)$ and $F=(F_1,F_2,\\ldots,F_n)$ be a vector-value function\nfrom an open neighbourhood of the origin $\\left({0,0,0,\\ldots,0}\\right)$ in $\\mathbb{R}^{n+1}$ to $\\mathbb{R}^{n}$,\nwhere\n$${F_i}\\left( {t,{\\zeta _1},{\\zeta _2}, \\ldots ,{\\zeta _n}} \\right)\n=\\int_{{\\mathbb{S}^{n-1}}} {\\frac{{{\\xi _i}}}{{{{\\left( {\\lambda \\left( t \\right){h_{{\\Omega^t}}}\n\\left( \\xi \\right) - \\left( {{\\zeta _1}{\\xi _1}+{\\zeta _2}{\\xi _2}+ \\cdots +{\\zeta _n}{\\xi _n}} \\right)} \\right)}^{1 - q}}}}} d\\mu \\left( \\xi \\right)$$\nfor $i=1,2,\\ldots,n$.\nAs $\\zeta(t)$ is an extreme point of $\\Phi_{\\lambda(t)\\Omega^t}\\left(\\zeta\\right)$ for $\\zeta\\in\\lambda(t)\\Omega^t$, it follows that $F_i(t,\\zeta(t))=0$.\nThen, two functions both\n\\begin{equation}\\nonumber\n{\\left. {\\frac{{\\partial {F_i}}}{{\\partial t}}} \\right|_{\\left( {t,{\\zeta _1},{\\zeta _2}, \\ldots ,{\\zeta _n}} \\right)}} = \\int_{{\\mathbb{S}^{n-1}}} {\\frac{{\\left( {q - 1} \\right){\\xi _i}\\left( {\\lambda '\\left( t \\right){h_{\\Omega^t}}\\left( \\xi \\right)\n+\\lambda\\left(t\\right){h_{\\Omega^t}'}\\left( \\xi \\right)} \\right)}}{{{{\\left( {\\lambda \\left( t \\right){h_{{\\Omega^t}}}\\left( \\xi \\right) - \\left( {{\\zeta _1}{\\xi _1}+{\\zeta _2}{\\xi _2}+ \\cdots +{\\zeta _n}{\\xi_n}} \\right)} \\right)}^{2-q}}}}} d\\mu \\left( \\xi \\right)\n\\end{equation}\nand\n\\begin{equation}\\nonumber\n{\\left. {\\frac{{\\partial {F_i}}}{{\\partial {\\zeta _j}}}} \\right|_{\\left( {t,{\\zeta _1},{\\zeta _2}, \\ldots ,{\\zeta _n}} \\right)}} = \\int_{{\\mathbb{S}^{n-1}}} {\\frac{{\\left( {1 - q} \\right){\\xi _i}{\\xi _j}}}{{{{\\left( {\\lambda \\left( t \\right){h_{{\\Omega^t}}}\\left( \\xi \\right) - \\left( {{\\zeta _1}{\\xi _1}+{\\zeta _2}{\\xi _2}+ \\cdots +{\\zeta _n}{\\xi _n}} \\right)} \\right)}^{2-q}}}}} d\\mu \\left( \\xi \\right)\n\\end{equation}\nare all continuous on a small neighbourhood of $\\left(0,0,0,\\ldots,0\\right)$, and\n\\begin{equation}\\label{4.11}\n{\\left( {{{\\left. {\\frac{{\\partial F}}{{\\partial \\zeta }}} \\right|}_{\\left( {0,0,0, \\ldots ,0} \\right)}}} \\right)_{n \\times n}} = \\int_{{\\mathbb{S}^{n-1}}} {\\frac{{\\left( {1 - q} \\right)\\xi^\\intercal \\xi}}{{h_\\Omega^{2 - q}\\left( \\xi \\right)}}} d\\mu \\left( \\xi \\right),\n\\end{equation}\nwhere ${{\\xi ^\\intercal}\\xi }$ is an $(n\\times n)$ matrix.\n\t\nAs the measure $\\mu$ is not concentrated on any closed hemisphere, for any\nnonzero $x\\in\\mathbb{R}^{n}$, we have\n\\begin{equation}\\nonumber\n{x}{\\left( {{{\\left. {\\frac{{\\partial F}}{{\\partial \\zeta }}} \\right|}_{\\left( {0,0,0, \\ldots ,0} \\right)}}} \\right)_{n \\times n}}x^\\intercal\n= \\int_{{\\mathbb{S}^{n-1}}} {\\frac{{\\left( {1 - q} \\right){{\\left\\langle {x,\\xi } \\right\\rangle }^2}}}{{h_\\Omega^{2 - q}\\left( \\xi \\right)}}} d\\mu \\left( \\xi \\right)\n>0.\n\\end{equation}\nIt follows that the matrix in \\eqref{4.11} is positive definite. Then, by\n${F_i}\\left({0,0,0,\\ldots,0}\\right)=0$ and the continuity of ${\\partial {F_i}}\n\\mathord{\\left/{\\vphantom{\\partial{F_i} {\\partial{\\zeta _j}}}}\\right.\n\\kern-\\nulldelimiterspace} {\\partial {\\zeta _j}}$\non a neighbourhood of $\\left({0,0,0,\\ldots ,0}\\right)$, one can use the implicit\nfunction theorem to obtain that $\\zeta\\left( t \\right)$ is continuously differentiable\non a small neighbourhood of $\\left({0,0,0,\\ldots,0}\\right)$.\nHence, the derivative $\\zeta'\\left(0\\right)$ of $\\zeta \\left(t\\right)$ at $t=0$ exists.\n\t\nPut $\\Phi\\left(t\\right):\n=\\Phi_{\\lambda \\left( t \\right){{\\left(h_\\Omega^q + th_{\\Omega_1}^q\n\\right)}^{\\frac{1}{q}}}}\n\\left( {\\zeta\\left( t \\right)} \\right)$,\nthen \\eqref{4.10} shows that $\\Phi\\left(t\\right)$\nattains the minimal value at $t=0$.\nThus by \\eqref{4.10} and\n\\begin{equation}\\nonumber\n\\lambda'\\left( 0 \\right)\n= - \\frac{1}{\\left( {n-\\mathbf{p} + 1}\\right)\\Gamma \\left(B_2^n\\right)}\n{\\left.{\\frac{d}{dt}}\\right|_{t = 0}}\\Gamma\\left({\\Omega^t}\\right),\n\\end{equation}\nwe have the following calculation:\n\\begin{equation}\\label{4.12}\n\\begin{split}\n0=&{\\left.{\\frac{d}{dt}}\\right|_{t = 0}}\\Phi \\left( t \\right)\\\\\n=&{\\left.{\\frac{d}{dt}}\\right|_{t = 0}}\\int_{{\\mathbb{S}^{n-1}}}\n{{\\left({\\lambda\\left(t\\right){{\\left( {h_\\Omega^q + t{h_{\\Omega_1}^q}}\\right)}^{\\frac{1}{q}}}\n-\\left\\langle{\\zeta \\left( t \\right),\\xi } \\right\\rangle} \\right)}^{q}} d\\mu\\left(\\xi\\right)\\\\\n=&q\\int_{{\\mathbb{S}^{n-1}}} {h_\\Omega^{q - 1}\\left( {\\lambda '\\left( 0 \\right){h_\\Omega}\n+ \\frac{1}{q}h_\\Omega^{1 - q}h_{\\Omega_1}^q -\\left\\langle {\\zeta'\\left(0\\right),\\xi } \\right\\rangle } \\right)}\nd\\mu \\left( \\xi \\right)\\\\\n=&q\\int_{{\\mathbb{S}^{n-1}}} {h_\\Omega^{q - 1}\\left( { - \\frac{{{h_\\Omega}}}{{\\left( {n-\\mathbf{p} + 1} \\right)\n\\Gamma \\left( B_2^n \\right)}}{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\Gamma \\left({\\Omega^t} \\right)\n+\\frac{1}{q}h_\\Omega^{1 - q}h_{\\Omega_1}^q} \\right)} d\\mu \\left( \\xi\\right)\\\\\n&-q\\int_{{\\mathbb{S}^{n-1}}} {\\left\\langle {\\zeta'\\left(0\\right),h_\\Omega^{{q} - 1}\\xi } \\right\\rangle } d\\mu\n\\left( \\xi \\right)\\\\\n=&-\\frac{q}{{n-\\mathbf{p} + 1}}\\int_{{\\mathbb{S}^{n-1}}} {\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n\\right)}}\n{{\\left. {\\frac{d}{{dt}}}\\right|}_{t = 0}}\\Gamma \\left(\\Omega^t\\right)} d\\mu\\left( \\xi \\right)\n+\\int_{{\\mathbb{S}^{n-1}}} {h_{\\Omega_1}^q} d\\mu \\left( \\xi \\right)\\\\\n&-q\\left\\langle {\\zeta'\\left(0\\right),\\int_{{\\mathbb{S}^{n-1}}} {h_\\Omega^{{p}-1}\\xi }\nd\\mu \\left( \\xi \\right)}\\right\\rangle.\n\\end{split}\n\\end{equation}\nSince ${\\zeta\\left(\\Omega\\right)=o}$ is an extreme point of $\\Phi_\\Omega\\left(\\zeta\\right)$\nfor $\\zeta\\in \\Omega$, we have\n\\begin{equation}\\nonumber\n\\int_{{\\mathbb{S}^{n-1}}} h_\\Omega^{q-1}\\xi d\\mu(\\xi)=o.\n\\end{equation}\nThis, together with \\eqref{4.12} and Theorem \\ref{th:3.1}, gives that\n\\begin{equation}\\label{4.13}\n\\begin{split}\n\\int_{{\\mathbb{S}^{n-1}}} {{h_{\\Omega_1}^q}} d\\mu\n=&\\frac{q}{{n-\\mathbf{p} + 1}}\n\\int_{{\\mathbb{S}^{n-1}}} {\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n \\right)}}\n{{\\left. {\\frac{d}{{dt}}} \\right|}_{t = 0}}\\Gamma \\left( {\\Omega^t} \\right)} d\\mu \\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n \\right)}}\n\\int_{{\\mathbb{S}^{n-1}}} {{h_{\\Omega_1}^q}} d{\\mu _{\\Omega,q}}} d\\mu \\\\\n=&\\int_{{\\mathbb{S}^{n-1}}} {{h_{\\Omega_1}^q}\n\\int_{{\\mathbb{S}^{n-1}}} {\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n \\right)}}d\\mu }}\nd{\\mu_{\\Omega,q}}.\n\\end{split}\n\\end{equation}\n\t\t\nFor any $f\\in C_{+}\\left(\\mathbb{S}^{n-1} \\right)$, there exists a sequence of positive twice\ncontinuously differentiable functions $\\left\\{ {{f_j}} \\right\\}_{j = 1}^\\infty$ that converges to $f$.\nThen for each $f_{j}$, Lemma \\ref{lem:4.1} shows that there exists a convex body $L_j$ containing the origin in its interior and a constant $r_j>0$, such that $f_j^q=h^q_{L_j}-h^q_{r_jB_2^n}$.\nHence, by \\eqref{4.13}, we have\n\\begin{equation}\\label{4.14}\n\\int_{{\\mathbb{S}^{n-1}}} {{h_{L_j}^q}} d\\mu\n=\\int_{{\\mathbb{S}^{n-1}}} {{h_{L_j}^q}\\int_{{\\mathbb{S}^{n-1}}}\n{\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n \\right)}}d\\mu }} d{\\mu _{\\Omega,q}},\n\\end{equation}\nand similarly,\n\\begin{equation}\\label{4.15}\n\\int_{{\\mathbb{S}^{n-1}}} {{h_{r_jB_2^n}^q}} d\\mu\n=\\int_{{\\mathbb{S}^{n-1}}} {{h_{r_jB_2^n}^q}\\int_{{\\mathbb{S}^{n-1}}}\n{\\frac{{h_\\Omega^q}}{{\\Gamma \\left( B_2^n \\right)}}d\\mu }} d{\\mu _{\\Omega,q}}.\n\\end{equation}\nBy subtracting \\eqref{4.15} from \\eqref{4.14} and using the approximate argument,\nwe conclude\n\\begin{equation}\\nonumber\n\\int_{{\\mathbb{S}^{n-1}}} {{f^q}} d\\mu\n=c\\int_{{\\mathbb{S}^{n-1}}} {{f^q}} d{\\mu _{\\Omega,q}},\n\\end{equation}\nwhere\n$$\nc=\\int_{{\\mathbb{S}^{n-1}}} \\frac{{h_\\Omega^q}}{\\Gamma \\left( B_2^n \\right)}d\\mu .\n$$\nBy the Riesz representation theorem, we have\n$\\mu=c{\\mu_{\\Omega,q}}$.\nFurthermore, Lemma \\ref{lem:3.1} and Definition\n\\ref{def:3.1} imply that the $L_q$ $\\mathbf{p}$-harmonic measure is\npositively homogeneous of degree $(n-\\mathbf{p}+1-q)$,\nthen there exists a convex body $\\tilde{\\Omega}$\nso that $\\mu=\\mu_{\\tilde{\\Omega},q}$, if $\\mathbf{p}\\neq n+1-q$.\n\nWe have completed the proof of Theorem \\ref{th:1.1}.\n\\end{proof}\n\n\\vskip 5mm \\noindent {\\bf Acknowledgements} This paper is partially supported by the China Postdoctoral Science Foundation (No. 2024M761902), the National Natural Science Foundation of China (No. 12371060) and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSZ012). The third author also received support from the Mathematical Sciences Institute at the Australian National University.\n\n\\vskip 5mm\n\n\n\n\n\\begin{thebibliography}{99}\n\\addtolength{\\itemsep}{-1.5ex}\n\n\\bibitem{AV22}\nM. Akman, J. Gong, J. Hineman, J.L. Lewis and A. Vogel,\n{\\it The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity},\nMem. Amer. Math. Soc. {\\bf275} (2022), vi+115.\n\n\\bibitem{AM24}\nM. Akman and S. Mukherjee,\n{\\it On the Minkowski problem for $p$-harmonic measures},\narXiv:2310.10247 (2024), https://arxiv.org/abs/2310.10247.\n\n\\bibitem{BZ12}\nK. B\\\"or\\\"oczky, E. Lutwak, D. Yang and G. Zhang,\n{\\it The log-Brunn-Minkowski inequality},\nAdv. Math. {\\bf231} (2012), 1974-1997.\n\n\\bibitem{BZ13}\nK. B\\\"or\\\"oczky, E. Lutwak, D. Yang and G. Zhang,\n{\\it The logarithmic Minkowski problem},\nJ. Amer. Math. Soc. {\\bf26} (2013), 831-852.\n\n\\bibitem{CL22}\nS. Chen, Y. Feng and W. Liu,\n{\\it Uniqueness of solutions to the logarithmic Minkowski problem in $\\mathbb{R}^3$},\nAdv. Math. {\\bf411} (2022), Paper No. 108782.\n\n\\bibitem{CZ23}\nS. Chen, S. Hu, W. Liu and Y. Zhao,\n{\\it On the planar Gaussian-Minkowski problem},\nAdv. Math. {\\bf435} (2023), part A, Paper No. 109351.\n\n\\bibitem{CL20}\nS. Chen, Y. Huang, Q. Li and J. Liu,\n{\\it The $L_p$-Brunn-Minkowski inequality for $p<1$},\nAdv. Math. {\\bf368} (2020), Paper No. 107166.\n\n\\bibitem{C06}\nW. Chen,\n{\\it $L_p$ Minkowski problem with not necessarily positive data},\nAdv. Math. {\\bf201} (2006), 77-89.\n\n\\bibitem{CY76}\nS.Y. Cheng and S.T. Yau,\n{\\it On the regularity of the solution of the $n$-dimensional Minkowski problem},\nComm. Pure Appl. Math. {\\bf29} (1976), 495-516.\n\n\\bibitem{CW06}\nK.S. Chou and X.J. Wang,\n{\\it The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry},\nAdv. Math. {\\bf205} (2006), 33-83.\n\n\\bibitem{CZ09}\nA. Cianchi, E. Lutwak, D. Yang and G. Zhang,\n{\\it Affine Moser-Trudinger and Morrey-Sobolev inequalities},\nCalc. Var. PDE. {\\bf36} (2009), 419-436.\n\n\\bibitem{CF10}\nA. Colesanti and M. Fimiani,\n{\\it The Minkowski problem for torsional rigidity},\nIndiana Univ. Math. J. {\\bf59} (2010), 1013-1039.\n\n\\bibitem{CZ15}\nA. Colesanti, K. Nystr\\\"om, P. Salani, J. Xiao, D. Yang and G. Zhang,\n{\\it The Hadamard variational formula and the Minkowski problem for $p$-capacity},\nAdv. Math. {\\bf285} (2015), 1511-1588.\n\n\\bibitem{CS03}\nA. Colesanti and P. Salani,\n{\\it The Brunn-Minkowski inequality for $p$-capacity of convex bodies},\nMath. Ann. {\\bf327} (2003), 459-479.\n\n\\bibitem{CK15}\nD. Cordero-Erausquin and B. Klartag,\n{\\it Moment measures},\nJ. Funct. Anal. {\\bf268} (2015), 3834-3866.\n\n\\bibitem{DZ12}\nJ. Dou and M. Zhu,\n{\\it The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents},\nAdv. Math. {\\bf230} (2012), 1209-1221.\n\n\\bibitem{FY22}\nN. Fang, S. Xing and D. Ye,\n{\\it Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_p$ Minkowski problem},\nCalc. Var. PDE. {\\bf61} (2022), Paper No. 45.\n\n\\bibitem{FX23}\nY. Feng, S. Hu and L. Xu,\n{\\it On the $L_p$ Gaussian Minkowski problem},\nJ. Differ. Equ. {\\bf363} (2023), 350-390.\n\n\\bibitem{G13}\nE.I. Galakhov,\n{\\it Comparison principles for the $p$-Laplacian operator},\nJ. Math. Sci. (N.Y.) {\\bf202} (2014), 825-834.\n\n\\bibitem{G06}\nR.J. Gardner,\n{\\it Geometric Tomography},\nsecond ed., Cambridge Univ. Press, New York, 2006, xxii+492.\n\n\\bibitem{GT01}\nD. Gilbarg, N.S. Trudinger,\n{\\it Elliptic partial differential equations of second order},\nSpringer-Verlag, Berlin, 2001, xiv+517.\n\n\\bibitem{G07}\nP. Gruber,\n{\\it Convex and discrete geometry},\nSpringer, Berlin, 2007, xiv+578.\n\n\\bibitem{GZ24}\nL. Guo, D. Xi and Y. Zhao,\n{\\it The $L_p$ chord Minkowski problem in a critical interval},\nMath. Ann. {\\bf389} (2024), 3123-3162.\n\n\\bibitem{HS09}\nC. Haberl and F. Schuster,\n{\\it Asymmetric affine $L_p$ Sobolev inequalities},\nJ. Funct. Anal. {\\bf257} (2009), 641-658.\n\n\\bibitem{HZ18}\nH. Hong, D. Ye and N. Zhang,\n{\\it The $p$-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems},\nCalc. Var. PDE. {\\bf57} (2018), Paper No. 5.\n\n\\bibitem{HS04}\nC. Hu, X. Ma and C. Shen,\n{\\it On the Christoffel-Minkowski problem of Firey's $p$-sum},\nCalc. Var. PDE. {\\bf21} (2004), 137-155.\n\n\\bibitem{HZ23}\nZ. Hu and H. Li,\n{\\it On the existence of solutions to the Orlicz-Minkowski problem for torsional rigidity}, Arch. Math. {\\bf120} (2023), 543-555.\n\n\\bibitem{HX15}\nY. Huang, J. Liu and L. Xu,\n{\\it On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in $\\mathbb{R}^3$},\nAdv. Math. {\\bf281} (2015), 906-927.\n\n\\bibitem{HZ16}\nY. Huang, E. Lutwak, D. Yang and G. Zhang,\n{\\it Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems},\nActa Math. {\\bf216} (2016), 325-388.\n\n\\bibitem{HZ21}\nY. Huang, D. Xi and Y. Zhao,\n{\\it The Minkowski problem in Gaussian probability space},\nAdv. Math. {\\bf385} (2021), Paper No. 107769.\n\n\\bibitem{HZ05}\nD. Hug, E. Lutwak, D. Yang and G. Zhang,\n{\\it On the $L_p$ Minkowski problem for polytopes},\nDiscrete Comput. Geom. {\\bf33} (2005), 699-715.\n\n\\bibitem{J89}\nD. Jerison,\n{\\it Harmonic measure in convex domains},\nBull. Amer. Math. Soc. (N.S.) {\\bf21} (1989), 255-260.\n\n\\bibitem{J91}\nD. Jerison,\n{\\it Prescribing harmonic measure on convex domains},\nInvent. Math. {\\bf105} (1991), 375-400.\n\n\\bibitem{J96}\nD. Jerison,\n{\\it A Minkowski problem for electrostatic capacity},\nActa Math. {\\bf176} (1996), 1-47.\n\n\\bibitem{JZ16}\nH. Jian, J. Lu and G. Zhu,\n{\\it Mirror symmetric solutions to the centro-affine Minkowski problem},\nCalc. Var. PDE. {\\bf55} (2016), Paper No. 41.\n\n\\bibitem{L77}\nJ.L. Lewis,\n{\\it Capacitary functions in convex rings},\nArch. Ration. Mech. Anal. {\\bf66} (1977), 201-224.\n\n\\bibitem{L06}\nJ.L. Lewis,\n{\\it Note on $p$-harmonic measure},\nComput. Methods Funct. Theory {\\bf6} (2006), 109-144.\n\n\\bibitem{L13}\nJ.L. Lewis, K. Nystr\\\"om and A. Vogel,\n{\\it On the dimension of $p$-harmonic measure in space},\nJ. Eur. Math. Soc. {\\bf15} (2013), 2197-2256.\n\n\\bibitem{LZ24}\nC. Li and X. Zhao,\n{\\it Flow by Gauss curvature to the Minkowski problem of $p$-harmonic measure},\narXiv:2404.18757v2 (2024), https://arxiv.org/abs/2404.18757.\n\n\\bibitem{LH23}\nH. Li and Z. Hu,\n{\\it On the polar Orlicz Minkowski type problem for the general mixed $\\mathfrak{p}$-capacity},\nJ. Math. Anal. Appl. {\\bf522} (2023), Paper No. 126925.\n\n\\bibitem{LZ24+}\nN. Li, D. Ye and B. Zhu,\n{\\it The dual Minkowski problem for unbounded closed convex sets},\nMath. Ann. {\\bf388} (2024), 2001-2039.\n\n\\bibitem{LZ20}\nN. Li and B. Zhu,\n{\\it The Orlicz-Minkowski problem for torsional rigidity},\nJ. Differ. Equ. {\\bf269} (2020), 8549-8572.\n\n\\bibitem{LW20}\nQ. Li, W. Sheng and X.J. Wang,\n{\\it Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems},\nJ. Eur. Math. Soc. {\\bf22} (2020), 893-923.\n\n\\bibitem{L88}\nG. Lieberman,\n{\\it Boundary regularity for solutions of degenerate elliptic equations},\nNonlinear Anal. {\\bf12} (1988), 1203-1219.\n\n\\bibitem{LX24}\nY. Liu, X. Lu, Q. Sun and G. Xiong,\n{\\it The logarithmic Minkowski problem in $\\mathbb{R}^2$},\nPure Appl. Math. Q. {\\bf20} (2024), 869-902.\n\n\\bibitem{LW13}\nJ. Lu and X.J. Wang,\n{\\it Rotationally symmetric solutions to the $L_p$-Minkowski problem},\nJ. Differ. Equ. {\\bf254} (2013), 983-1005.\n\n\\bibitem{L93}\nE. Lutwak,\n{\\it The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem},\nJ. Differ. Geom. {\\bf38} (1993), 131-150.\n\n\\bibitem{LZ24++}\nE. Lutwak, D. Xi, D. Yang and G. Zhang,\n{\\it Chord measures in integral geometry and their Minkowski problems},\nComm. Pure Appl. Math. {\\bf77} (2024), 3277-3330.\n\n\\bibitem {LZ02}\nE. Lutwak, D. Yang and G. Zhang,\n{\\it Sharp affine $L_p$ Sobolev inequalities},\nJ. Differ. Geom. {\\bf62} (2002), 17-38.\n\n\\bibitem{LZ04}\nE. Lutwak, D. Yang and G. Zhang,\n{\\it On the $L_p$-Minkowski problem},\nTrans. Amer. Math. Soc. {\\bf356} (2004), 4359-4370.\n\n\\bibitem{M24}\nE. Milman,\n{\\it A sharp centro-affine isospectral inequality of Szeg\\\"o-Weinberger type and the $L^p$-Minkowski problem},\nJ. Differ. Geom. {\\bf127} (2024), 373-408.\n\n\\bibitem{R22}\nL. Rotem,\n{\\it A Riesz representation theorem for functionals on log-concave functions},\nJ. Funct. Anal. {\\bf282} (2022), Paper No. 109396.\n\n\\bibitem{S14}\nR. Schneider,\n{\\it Convex Bodies: The Brunn-Minkowski theory},\nCambridge Univ. Press, Cambridge, 2014, xxii+736.\n\n\\bibitem{S18}\nR. Schneider,\n{\\it A Brunn-Minkowski theory for coconvex sets of finite volume},\nAdv. Math. {\\bf332} (2018), 199-234.\n\n\\bibitem{S24}\nR. Schneider,\n{\\it A weighted Minkowski theorem for pseudo-cones},\nAdv. Math. {\\bf450} (2024), Paper No. 109760.\n\n\\bibitem{S02}\nA. Stancu,\n{\\it The discrete planar $L_0$-Minkowski problem},\nAdv. Math. {\\bf167} (2002), 160-174.\n\n\\bibitem{S03}\nA. Stancu,\n{\\it On the number of solutions to the discrete two dimensional $L_0$-Minkowski problem},\nAdv. Math. {\\bf180} (2003), 290-323.\n\n\\bibitem{U03}\nV. Umanskiy,\n{\\it On solvability of two-dimensional $L_p$-Minkowski problem},\nAdv. Math. {\\bf180} (2003), 176-186.\n\n\\bibitem{TX23}\nJ. Tao, G. Xiong and J. Xiong,\n{\\it The logarithmic Minkowski inequality for cylinders},\nProc. Amer. Math. Soc. {\\bf151} (2023), 2143-2154.\n\n\\bibitem{T84}\nP. Tolksdorf,\n{\\it Regularity for a more general class of quasilinear elliptic equations},\nJ. Differ. Equ. {\\bf51} (1984), 126-150.\n\n\\bibitem{XZ23}\nD. Xi, D. Yang, G. Zhang and Y. Zhao,\n{\\it The $L_p$ chord Minkowski problem},\nAdv. Nonlinear Stud. {\\bf23} (2023), Paper No. 20220041.\n\n\\bibitem{X20}\nJ. Xiao,\n{\\it Prescribing capacitary curvature measures on planar\nconvex domains},\nJ. Geom. Anal. {\\bf30} (2020), 2225-2240.\n\n\\bibitem{XX19}\nG. Xiong, J. Xiong and L. Xu,\n{\\it The $L_p$ capacitary Minkowski problem for polytopes},\nJ. Funct. Anal. {\\bf277} (2019), 3131-3155.\n\n\\bibitem{YZ23}\nJ. Yang, D. Ye and B. Zhu,\n{\\it On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets},\nInt. Math. Res. Not. {\\bf2023} (2023), 6252-6290.\n\n\\bibitem {Z99}\nG. Zhang,\n{\\it The affine Sobolev inequality},\nJ. Differ. Geom. {\\bf53} (1999), 183-202.\n\n\\bibitem{Z14}\nG. Zhu,\n{\\it The logarithmic Minkowski problem for polytopes},\nAdv. Math. {\\bf262} (2014), 909-931.\n\n\\bibitem{Z15}\nG. Zhu,\n{\\it The $L_p$ Minkowski problem for polytopes for $0<p<1$},\nJ. Funct. Anal. {\\bf269} (2015), 1070-1094.\n\n\\bibitem{ZX20}\nD. Zou and G. Xiong,\n{\\it The $L_p$ Minkowski problem for the electrostatic $\\mathfrak{p}$-capacity},\nJ. Differ. Geom. {\\bf116} (2020), 555-596.\n\n\n\\end{thebibliography}\n\\vskip 2mm\n\\noindent Hai Li, {\\small\\tt lihai121455@163.com}\\\\\n{Department of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China}\n\n\\vskip 2mm\n\\noindent Longyu Wu, {\\small\\tt wulongyu66@gmail.com}\\\\\n{Department of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China}\n\n\\vskip 2mm\n\\noindent Baocheng Zhu, {\\small\\tt zhubaocheng814@163.com}\\\\\n{Department of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China}\n\n\\end{document}\n\n\n\n\n"}}},{"rowIdx":87,"cells":{"id":{"kind":"string","value":"2412.07613v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07613v1"},"title":{"kind":"string","value":"Discontinuous Galerkin methods for the complete stochastic Euler equations"},"full_text":{"kind":"string","value":"\\documentclass[10pt]{amsart}\n\\usepackage{enumitem} \n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{comment}\n\\usepackage{hyperref}\n\\usepackage{mathrsfs} \n\\usepackage{breqn}\n\\usepackage{xcolor}\n\\usepackage{graphicx}\n\\usepackage{stmaryrd}\n\\usepackage{geometry}\n\\geometry{margin= 1in}\n\\let\\vec\\mathbf\n\n\n\\usepackage{subcaption}\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{ex}[thm]{Example}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{asp}[thm]{Assumption}\n\\newtheorem{con}[thm]{Condition}\n\\newtheorem{dfn}[thm]{Definition}\n\\newtheorem{rmk}[thm]{Remark}\n\\newtheorem{cnj}[thm]{Conjecture}\n\n\n\\newcommand{\\UU}{\\mathfrak{U}}\n\\newcommand{\\CC}{\\mathcal{C}}\n\\newcommand{\\BB}{\\mathcal{B}}\n\\newcommand{\\FF}{\\mathfrak{F}}\n\\newcommand{\\XX}{\\mathcal{X}}\n\\newcommand{\\LL}{\\mathcal{L}}\n\\newcommand{\\PP}{\\mathcal{P}}\n\\newcommand{\\Ss}{\\mathcal{S}}\n\\newcommand{\\OO}{\\mathcal{O}}\n\\newcommand{\\yy}{\\mathcal{Y}}\n\\newcommand{\\TT}{\\mathscr{T}}\n\\newcommand{\\MM}{\\mathcal{M}}\n\\newcommand{\\V}{\\mathcal{V}}\n\n\\newcommand{\\rr}{\\vec{r}_t}\n\n\n\\newcommand{\\T}{\\mathcal{O}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\G}{\\mathbb{G}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\F}{\\mathbb{F}}\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\p}{\\mathbb{P}}\n\\newcommand{\\I}{\\mathbb{I}}\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\B}{\\mathscr{B}}\n\\usepackage{authblk}\n\n\\numberwithin{equation}{section}\n\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\dx}{\\,\\mathrm{d}x}\n\\newcommand{\\dt}{\\,\\mathrm{d}t}\n\\newcommand{\\dxt}{\\,\\mathrm{d}x\\,\\mathrm{d}t}\n\\newcommand{\\ds}{\\,\\mathrm{d}\\sigma}\n\\newcommand{\\dxs}{\\,\\mathrm{d}x\\,\\mathrm{d} s}\n\\newcommand{\\dsx}{\\,\\mathrm{d}\\sigma\\,\\mathrm{d}x}\n\\newcommand{\\dz}{\\,\\mathrm{d}z}\n\\newcommand{\\bxi}{\\boldsymbol{\\xi}}\n\\newcommand{\\bfw}{\\mathbf w}\n\\newcommand{\\bfr}{\\mathbf r}\n\\newcommand{\\bfU}{\\mathbf U}\n\\newcommand{\\bfH}{\\mathbf H}\n\\newcommand{\\bfG}{\\mathbf G}\n\\newcommand{\\EE}{\\mathscr E}\n\\newcommand{\\bfh}{\\mathbf h}\n\\newcommand{\\ee}{\\mathrm e}\n\\newcommand{\\bn}{\\mathbf n}\n\\newcommand{\\VV}{\\mathcal V}\n\\newcommand{\\bU}{\\mathbf{U}}\n\\newcommand{\\bu}{\\mathbf u}\n\\newcommand{\\bh}{\\mathbf h}\n\\newcommand{\\bm}{\\mathbf m}\n\\newcommand{\\bV}{\\mathbf V}\n\\newcommand{\\diff}{\\mathrm{d}}\n\\newcommand{\\bx}{\\mathbf x}\n\\newcommand{\\bg}{\\mathbf{g}}\n\\newcommand{\\diag}{\\mathrm{diag}}\n\\newcommand{\\IdxM}{\\mat{\\mathbb I_4}}\n\\newcommand{\\bvu}{\\underline{\\bu}}\n\\newcommand{\\bvf}{\\underline{\\bf}}\n\\newcommand{\\bfnum}{\\mathbf f^{\\mathrm{num}}}\n\\newcommand{\\bvecfnum}{\\underline{{\\bf}_{e_m}^{\\mathrm{num}}}}\n\\newcommand{\\CV}{\\mathcal{V}_{t,x}}\n\\usepackage[foot]{amsaddr}\n\\newcommand{\\armean}[1]{\\{\\{#1\\}\\}}\n\\newcommand{\\logmean}[1]{\\{\\{#1\\}\\}_{\\log}}\n\\newcommand{\\jump}[1]{\\llbracket#1\\rrbracket}\n \\newcommand{\\mat}[1]{\\underline{\\underline{#1}}}\n \\newcommand{\\norm}[1]{\\|#1\\|}\n \n \\newcommand{\\PO}[1]{{\\color{blue}#1}}\n \n \n\n\\title{Discontinuous Galerkin methods for the complete stochastic Euler equations}\n\\author{Dominic Breit}\n\\author{Thamsanqa Castern Moyo}\n\\author{Philipp \\\"Offner}\n\\address{TU Clausthal, Institute of Mathematics, Erzstra\\ss e 1, 38678 Clausthal-Zellerfeld, Germany.}\n\\email{dominic.breit@tu-clausthal.de}\n\\email{thamsanqa.castern.moyo@tu-clausthal.de}\n\\email{mail@philippoeffner.de}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\n\nIn recent years, stochastic effects have become increasingly relevant for describing fluid behaviour, particularly in the context of turbulence. The most important model for inviscid fluids\nin computational fluid dynamics are the Euler equations of gas dynamics which we focus on in this paper. \nTo take stochastic effects into account, we incorporate a stochastic forcing term in the momentum equation of the Euler system. \nTo solve the extended system, we apply an entropy dissipative discontinuous Galerkin spectral element method including the Finite Volume setting, adjust it to the stochastic Euler equations and analyze its convergence properties.\nOur analysis is grounded in the concept of \\textit{dissipative martingale solutions}, as recently introduced by Moyo (J. Diff. Equ. 365, 408-464, 2023). Assuming no vacuum formation and bounded total energy, we proof that our scheme converges in law to a dissipative martingale solution. During the lifespan of a pathwise strong solution, we achieve convergence of at least order 1/2, measured by the expected $L^1$ norm of the relative energy. The results built a counterpart of corresponding results in the deterministic case.\nIn numerical simulations, we show the robustness of our scheme, visualise different stochastic realizations and analyze our theoretical findings. \n\n\n\n\n\n\n\\end{abstract}\n\\section{Introduction}\n\nThe Euler equations of gas dynamics are one of -if not the- most studied system for describing inviscid, compressible fluid flow. \nThey contain the three conservation laws of mass, momentum, and energy and serve as foundational tools for understanding and predicting fluid behavior in applications like aerodynamics, astrophysics, and meteorology. The system has been thoroughly investigated from an analytical and numerical perspective. Numerically, many different schemes have been developed to solve the Euler equations in a highly efficient way. The aim is always to obtain the most accurate result for real-world problems. From a modeling perspective, the inclusion of uncertainty and/or to describe \nturbulent effects, researchers have recently started to introduce stochastic forcing into the Euler equations (an other models as well) to describe more realistic scenarios. \\\\\nIn this paper, we follow this trend and study \nthe complete Euler equations describing the motion of a general inviscid, compressible and heat-dependent fluid subject to a stochastic perturbation in the momentum equation. In the variables density $\\varrho$, momentum $\\vec m$ and specific internal energy $e$, the considered system reads then as \n\\begin{align}\\label{Euler}\n\\begin{aligned}\n\\dd \\varrho + \\mathrm{div}_{x}(\\vec m)\\, \\dd t &=0 \\quad \\text{in}\\, Q,\\\\\n\\dd \\vec m+ \\mathrm{div}_x\\Big(\\frac{\\vec m \\otimes \\vec m}{\\varrho}\\Big)\\, \\dd t + \\nabla_x p\\,\\dd t&=\\varrho \\phi \\,\\dd W\\quad \\text{in}\\, Q,\\\\\n\\dd (\\varrho e) + \\mathrm{div}_x\\left(\\varrho e\\frac{\\vec m}{\\varrho}\\right)\\dd t &=-p\\,\\mathrm{div}\\Big(\\frac{\\vec m}{\\varrho}\\Big)\\dd t \\quad \\text{in}\\, Q,\\end{aligned}\n\\end{align}\nwhere $Q=(0,T)\\times\\mathcal O$ for some $T>0$ and $\\mathcal O\\subset\\R^n$, $n=1,2,3$, bounded.\nHere, $W$ denotes a (cylindrical) Wiener process on a filtered probability space $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ and $\\phi$ the noise coefficient (a Hilbert-Schmidt operator\\footnote{We refer to Section \\ref{sec:prob} for the precise probability framework. }). \nThe key idea behind adding the stochastic perturbation to the deterministic Euler system is to take physical, empirical or numerical uncertainties and thermodynamical fluctuations into account. Additionally, it is also commonly used to model turbulent effects, see \\cite{Bi,MiRo} and the references therein. \nHowever, it is well-known that the deterministic counterpart of \\eqref{Euler} is well-posed only locally in time and solutions can develop shocks \\cite{Sm}. Recently, also the precise behaviour of singularities could be described \\cite{BSV}. Due to the technique of convex integration, there has been, and continues to be, significant interest in constructing scenarios where the deterministic Euler equations yield multiple entropy solutions from the same initial data. See, for example, \\cite{ChDeO,DeSz} for the isentropic Euler system and \\cite{FKKM} for the complete Euler equations.\nThis is not only valid in the deterministic case but also if stochastic perturbations are added, cf. \\cite{BFHcv} for the isentropic Euler equations and in \\cite{ChFeFl} for the complete Euler equations.\nThe only chance to obtain globally defined solutions which obey the laws of thermodynamics in a stable way is the concept of measure-valued solutions. \n It was already introduced (in the deterministic case for incompressible fluids) in the 80's \\cite{DiP,DiMa} but attracted a lot of attention again recently due to the weak-strong uniqueness principle available for dissipative solutions \\cite{HoFeBr,Bren,FEJB,KrZa,Neu} and to demonstrate convergence of structure preserving methods \\cite{feireisl2020convergence,feireisl2021numerical} within this framework. Weak-strong uniqueness results were recently extended to the stochastic case by the second author in \\cite{Mo} where he showed the existence of a dissipative martingale solution to \\eqref{Euler}. Such a solution is weak in the probabilistic sense which means the probability space is not a priori given but becomes an integral part of the solution. \\\\\nMeanwhile from the numerical perspective, there has been a long history on solving the deterministic Euler equations. \nAn instrumental method in this context is the approach pioneered by Godunov \\cite{godunov1959finite}, which can be conceptualised as a finite volume (FV) technique that tackles exact Riemann problems at each cell interface. Godunov's method has served as the foundation for numerous extensions and has been employed as the groundwork for the development of higher-order methods, exemplified in works such as \\cite{harten1983upstream,roe1981approximate,toro1997riemann}. Despite its historical significance, Godunov's method continues to attract attention, with recent studies showcasing its convergence through dissipative weak (DW) solutions of the Euler equations \\cite{LuYu}. Additionally, novel error estimates for the multidimensional compressible Euler system have been introduced in \\cite{LuSh}. Notably, the results of \\cite{LuYu} build upon convergence outcomes for several first-order FV schemes, including the (local) Lax-Friedrichs method, as discussed in \\cite{feireisl2020convergence}. For a comprehensive overview of recent convergence results, we direct readers to \\cite{feireisl2021numerical} for further details.\nIn parallel with low-order methods, contemporary applications involve high-order methods for solving hyperbolic conservation laws, particularly the Euler equations. Among these methods, the Discontinuous Galerkin (DG) method stands out as one of the most favoured within the hyperbolic community. Originating from the work of Reed and Hill in 1973, initially devised for solving the hyperbolic neutron transport equation in nuclear reactors \\cite{ReHi}, DG has evolved over the years. Mathematical foundations were solidified in the 90s \\cite{CoKa, HeWa}, and more recent advancements have focused on ensuring structure-preserving (or property-preserving) properties, as pointed out in \\cite{KuHa, oeff2023} and references therein. In the context of the Euler equation, recent convergence results via dissipative weak solutions have been demonstrated for DG in \\cite{LuOe}. This marked a significant milestone as the first convergence result for a high-order method within the DW framework. Later one these results were extended to other structure-preserving Finite Element (FE) based schemes in \\cite{AbLu, KuLu}.\nAll convergence results using the DW framework are made on the assumption that the numerical solution $(\\varrho^h,\\vec m^h,\\mathcal E^h)$, usually formulated in terms of the total energy $\\mathcal E^h$ rather than the internal energy, satisfies for some $K>0$\n \\begin{align}\\label{ass:mainA}\n\\inf_{t\\in[0,T],x\\in\\T}\\varrho^h(t,x)\\geq \\frac{1}{K},\\quad \\sup_{t\\in[0,T],x\\in\\T}\\mathcal E^h(t,x)\\leq K,\n\\end{align}\nuniformly in $h$. Here $h$ is the corresponding grid parameter and the superscript denotes that the quantity is the numerical approximation, cf. \\cite{AbLu,feireisl2021numerical, KuLu, LuYu}. The physical interpretation of \\eqref{ass:mainA} is that the density is not reaching vacuum and we have no blow up in terms of energy (and so also the speed is bounded). \nIn the stochastic case\\footnote{Here, we refer that we consider stochastic forcing inside the equations not uncertain initial/boundary data or uncertainty inside the flux where other techniques like Monte-Carlo simulations, general Polynomial chaos or stochastic collocation methods are used and heavily considered \\cite{AbMi}.}, the only available result -up to the authors knowledge- is the paper \\cite{ChKo}, where a finite volume scheme is considered for the barotropic Euler equations. The authors from \\cite{ChKo} work under assumption \\eqref{ass:mainA}. Unfortunately, this is not realistic for stochastic PDEs. Due to the Gaussian character inherited from the driving Wiener process solutions are in general not bounded in probability, not even locally in time. This is even the case for linear problems with smooth data which are globally well-posed, see, for instance, \\cite{DP}. \\\\\nDue to this we replace \\eqref{ass:mainA} by the following assumption which is on the continuous level certainly satisfied by a strong solution (see Definition \\ref{def:compstrong}) and thus seems much more plausible. We suppose that there is a stopping time $\\mathfrak t$ with $\\p(\\mathfrak t>0)=1$ and a deterministic constant $K>0$ such that $\\p$-a.s.\n\\begin{align}\\label{ass:mainB}\n\\inf_{t\\in[0,\\mathfrak t],x\\in\\T}\\varrho^h(t,x)\\geq \\frac{1}{K},\\quad \\sup_{t\\in[0,\\mathfrak t],x\\in\\T}\\mathcal E^h(t,x)\\leq K,\n\\end{align}\nuniformly in $h$. Heuristically speaking, we assume that \\eqref{ass:mainB} is only fulfilled up to some time $\\mathfrak t$ which exists with probability $\\p(\\mathfrak t>0)=1$.\nLet us explain some heuristics behind this assumption. If the velocity gradient is bounded in space-time the standard maximum principle for the continuity equation \\eqref{Euler}$_1$ implies strict positive of the density. However such a bound on the velocity field can never be uniform in probability (as the stochastic forcing is not uniformly bounded). Hence, for any given deterministic time $T>0$, any positiv deterministic bound can be deceeded with a positive probability. So, out of all possible paths the solution can attain some of them will not satisfy \\eqref{ass:mainA}. One can consequently either neglect certain paths or adapt the time horizon. For any given path there is a (possibly small) time for which the movement of the noise is small. Thus \\eqref{ass:mainB} holds provided the solution is sufficiently smooth in space.\\\\\n Based on this assumption we are able to prove the consistency and convergence\\footnote{Convergence means in such context: Convergence in law up to a subsequence and it is based on the stochastic compactness method employing Jakubowski's extension of the Skorokhod representation theorem from \\cite{Jaku} (due to the defect measures, which appear in the nonlinear term when passing to the limit in the discrete solution, the classical version for Polish spaces does not apply). A main feature is that we have to include the stopping from \\eqref{ass:mainA} in the path space which creates several technical difficulties and will be explained in all details in the following chapters.} between a DG discretization (including the FV case) of \\eqref{Euler} and the dissipative martingale solution from \\cite{Mo}. The convergence result can be improved in the life-space of a strong solution to \\eqref{Euler} which exists locally in time up to a stopping time\n$\\mathfrak s$ (which we expect to be significantly smaller than $\\mathfrak t$ in \\eqref{ass:mainB}), cf. Definition \\ref{def:compstrong}, and is defined on a given stochastic basis. \nIn this case we obtain convergence of order $1/2$, where the error is measured as the expectation of the $L^1$ norm of the relative energy between discrete and continuous variables evaluated at the final time. \nThe result is based on the relative entropy of the discrete solution and provides a stochastic counterpart of the deterministic case. \nTherefore, the paper is organized as follows:\nIn Section \\ref{sec:cr}, we introduce the notation, the mathematical framework and the discontinuous Galerkin (DG) method under consideration. We give a focus on the stochastic part and how we are dealing with it.\n Then, in Section \\ref{se_concept} we repeat the definition of dissipative measure-valued martingale solutions from \\cite{Mo}, strong solutions in the probabilistic and PDE sense, and formulate the main results about consistency and convergence results. The chapter should give a detailed introduction inside the topic and provide the main results in this context. In Section \\ref{mvsproof}, the consistency results are proven in all technical details where in Section \\ref{se_Convergence} the focus is on the convergence result. In addition, in Section \\ref{sec_numerical} we show -up to the authors knowledge- for the first time numerical results using the extended Euler system \\eqref{Euler}. Furthermore, we verify our theoretical findings and discuss problems which may rise as well inside the numerical experiments.\nA conclusion and an outlook in Section \\ref{se_outlock} finishes this paper. \n\n\n\\section{Mathematical framework}\\label{Asec}\nIn this section, we introduce the stochastic Euler system \\eqref{Euler} and describe different formulations for it. On the continuous level (for sufficiently smooth solutions) they are equivalent, but from a numerical perspective one is more favorable than the other as already can be seen in the deterministic case in \\cite{abgrall2018,abgrall2010}. Next, we provide a brief introduction to the probability framework; for further details, see \\cite[Chapter 2]{FrBrHo}. Finally, we introduce the discontinuous Galerkin (DG) method under consideration the well-known DG spectral element method using summation-by-parts operators and a flux differencing approach. The method is well investigated and well-used applied for the deterministic Euler equations, cf. \\cite{chen2017,He,LuOe,renac_2019} and references therein. We adjust it to the stochastic Euler system.\n\\subsection{Constitutive relations}\\label{sec:cr}\nThe fluid model is described by means of three basic state variables: the mass density $\\varrho=\\varrho(t,x)$, the momentum $\\vec m = \\vec m(t,x)$, and the (absolute) temperature $\\vartheta=\\vartheta(t,x)$, where $t$ is the time, $x$ is the space variable (Eulerian coordinate system). The time evolution of the fluid flow is governed by the system of partial differential equations given by\n\\begin{align}\\label{EulerB}\n\\begin{aligned}\n\\dd \\varrho + \\mathrm{div}_{x}\\vec m\\, \\dd t &=0 \\quad \\text{in}\\, Q,\\\\\n\\dd \\vec m+ \\mathrm{div}_x\\Big(\\frac{\\vec m \\otimes \\vec m}{\\varrho}\\Big)\\, \\dd t + \\nabla_x p\\,\\dd t&=\\varrho \\phi \\,\\dd W\\quad \\text{in}\\, Q,\\\\\n\\dd e + \\mathrm{div}_x\\left(e\\frac{\\vec m}{\\varrho}\\right)\\dd t &=-p\\,\\mathrm{div}\\Big(\\frac{\\vec m}{\\varrho}\\Big)\\dd t \\quad \\text{in}\\, Q\n,\\end{aligned}\n\\end{align}\ndescribing: the balance of mass, momentum and internal energy, respectively. Here, $p(\\varrho,\\vartheta)$ denotes pressure and $e=e(\\varrho,\\vartheta)$ is the internal energy. \nIn the theoretical part, we focus on the two-dimensional Euler system \\eqref{EulerB}. However, the same arguments apply in both one and three dimensions\\footnote{In Section \\ref{sec_numerical}, we demonstrate also some results in one dimension for completeness.}. The total energy is the sum of the kinetic and internal energy, respectively, that is\n\\begin{align*}\n\\mathcal E=\\mathcal E(\\varrho,\\vartheta)=\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}+\\varrho e(\\varrho,\\vartheta).\n\\end{align*}\nApplying formerly It\\^{o}'s formula to the process $t\\mapsto \\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}$, the internal energy equation \nin \\eqref{EulerB} can equivalently be written in terms of the total energy as\\footnote{In the case of a cylindrical Wiener process, see \\eqref{Dwiener} below, one has $\\|\\phi\\|^2_{\\ell_2}:=\\sum_{k\\in\\N}|\\phi e_k|^2$. Note that this is a function depending on space.}\n\\begin{equation}\\label{eq_total}\n\\dd \\mathcal E + \\mathrm{div}_x\\left(\\left(\\mathcal E+p\\right)\\frac{\\vec m}{\\varrho}\\right)\\dd t =\\frac{1}{2}\\varrho\\|\\phi\\|_{\\ell_2}^2 \\,\\dd t+ \\phi\\cdot \\vec m\\,\\dd W.\n\\end{equation}\n For completeness, the system (\\ref{EulerB}) is supplemented by a set of constitutive relations characterising the physical principles of a compressible inviscid fluid. In particular, we assume that the pressure $p(\\varrho,\\vartheta)$ and the internal energy $e =e(\\varrho,\\vartheta)$ satisfy the caloric equation of state\n\\begin{equation}\\label{caloric}\n p=(\\gamma-1)\\varrho e, \n\\end{equation}\nwhere $\\gamma>1$ is the adiabatic constant. In addition, we suppose that the absolute temperature $\\vartheta$ satisfies the Boyle-Mariotte thermal equation of state:\n\\begin{equation}\\label{boyle}\n p =\\varrho\\vartheta \\quad \\mathrm{yielding}\\quad e= c_v\\vartheta, c_v =\\frac{1}{\\gamma-1}.\n\\end{equation}\nFinally, we suppose that the pressure $p=p(\\varrho, \\vartheta)$, the specific internal energy $e =e(\\varrho,\\vartheta)$, and the specific entropy $ s = s(\\varrho,\\vartheta)$ are interrelated through Gibbs' relation\n\n\\begin{equation}\\label{gibbs}\n \\vartheta D s (\\varrho,\\vartheta) = D e(\\varrho,\\vartheta)+ p(\\varrho,\\vartheta)D\\left(\\frac{1}{\\varrho}\\right).\n\\end{equation}\nIf $p,e,s$ satisfy (\\ref{gibbs}), in the context of \\textit{smooth} solutions to (\\ref{Euler}), the second law of thermodynamics is enforced through the entropy balance equation\n\n\\begin{equation}\\label{entrB}\n \\dd (\\varrho s (\\varrho,\\vartheta))+\\mathrm{div}_x( s(\\varrho,\\vartheta)\\vec m)\\, \\dd t =0, \n\\end{equation}\nwhere $s(\\varrho,\\vartheta)$ denotes the (specific) entropy and is of the form\n\\begin{equation}\\label{entropy}\n s(\\varrho,\\vartheta)=\\log(\\vartheta^{c_v})-\\log(\\varrho).\n\\end{equation}\nDenoting by $$S=S(\\varrho,\\vartheta)=\\varrho s(\\varrho,\\vartheta)=\\varrho\\log(\\vartheta^{c_v})-\\varrho\\log(\\varrho)$$ the total entropy, equation \\eqref{entropy} can also be written as\n\\begin{equation}\\label{EntrB}\n \\dd S (\\varrho,\\vartheta)+\\mathrm{div}_x\\Big(S(\\varrho,\\vartheta)\\frac{\\vec m}{\\varrho}\\Big)\\, \\dd t =0.\n\\end{equation}\nFor weak solutions, the equalities in (\\ref{entrB}) and (\\ref{EntrB}) no longer hold, the entropy balance is given as an inequality, for more details see \\cite{SBGD}. \nFinally, system \\eqref{EulerB} can be reformulated as\n\\begin{align}\n\\dd \\varrho + \\mathrm{div}_x\\vec m\\,\\dd t &=0\\qquad\\,\\,\\text{in $Q$},\\label{eq:aEuler}\\\\\n\\dd \\vec m + \\mathrm{div}_x\\left(\\frac{ \\vec m \\otimes \\vec m}{\\varrho}\\right)\\dd t+\\nabla_x p\\,\\dd t &= \\varrho\\phi\\dd W\\,\\,\n \\text{in $Q$,}\\label{eq:bEuler}\\\\\n\\dd S +\\mathrm{div}_x\\left(\\frac{S\\vec m}{\\varrho} \\right)\\,\\dd t &=0 \\quad\\qquad\\text{in $Q$.}\\label{eq:cEuler}\n\\end{align}\nFor physical relevant solutions, the problem is augmented by the total energy balance\n\\begin{equation}\\label{eq:tenergy}\n \\dd \\int_{\\T}\\mathcal E\\, \\dd x= \\dd \\int_{\\T}\\left [\\frac{1}{2}\\varrho|\\vec u|^2 +\\varrho e\\right]\\, \\dd x= \\int_{\\T}\\varrho\\phi \\cdot\\vec u\\,\\dd x \\dd W+ \\frac{1}{2}\\int_{\\T}\\|\\phi\\|_{\\ell_2}^2\\dx\\,\\dd t.\n\\end{equation}\nStrong solutions of the system (\\ref{Euler}) satisfy (\\ref{eq:tenergy}), but for weak solutions it has to be added in the definition. In fact, weak solutions to not posses sufficient spatial regularity to derive \\eqref{eq:tenergy} form the other equations. It needs to be derived on the approximate level and subsequently kept in the limit procedure.\n\\begin{ex}\nA particular instance is given by the two-dimensional standard Wiener process\n\\begin{align}\\label{eq:14.02}\n\\phi W=\\begin{pmatrix} 1\\\\0\\end{pmatrix}\\beta_1+\\begin{pmatrix} 0\\\\1\\end{pmatrix}\\beta_2\n\\end{align}\nwith two independent standard one-dimensional Wiener processes $ \\beta_1$ and $\\beta_2$, where $\\|\\phi\\|_{\\ell^2}^2=2$. We will apply this noise in our numerical section \\ref{sec_numerical} multiplied with some constant to handle the strength and so the influence of it. \n\\end{ex}\n\\subsection{Probability framework}\n\\label{sec:prob}\nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ be a complete stochastic basis with a probability measure $\\p$ on $(\\Omega,\\FF)$ and right-continuous filtration $(\\FF_t)_{t\\geq 0}$.\nThe filtration is a family of $\\sigma$-sub-algebras of $\\FF$ indexed by time $t\\geq0$, where each $\\FF_t$ represents the information available up to time $t$. In other words, it is the collections of events that are \\emph{known} at time $t$. \n Let $\\UU$ be a separable Hilbert space with orthonormal basis $(e_k)_{k\\in \\N}$ (a natural choice would be $\\UU=L^2(\\mathcal O)$). We denote by $ L_2(\\UU,L^2(\\T))$ the set of Hilbert-Schmidt operators from $\\UU$ to $L^2(\\T)$, i.e., the set of bounded linear mappings $\\Phi:\\UU\\rightarrow L^2(\\mathcal O)$ satisfying\n \\begin{equation}\\label{eq:HSa}\n \\sum_{k\\geq 1}\\|\\phi(e_k)\\|_{L^{2}(\\T)}^2<\\infty.\n\\end{equation}\n The stochastic process $W$ appearing in \\eqref{EulerB} is a cylindrical Wiener process $W =(W_t)_{t \\geq 0}$ in $\\UU$, meaning it is of the form\n\\begin{equation}\nW(s) = \\sum_{k\\in \\N}e_k \\beta_k(s),\n\\label{Dwiener}\n\\end{equation}\n\nwhere $(\\beta_k)_{k \\in \\N}$ is a sequence of independent real-valued one-dimensional Wiener processes relative to $(\\FF_t)_{t\\geq0}$. To identify the precise definition of the diffusion coefficient, set $\\UU =\\ell^2$ and consider $\\varrho \\in L^{1}(\\T), \\varrho > 0$. For a mapping $\\phi \\in L_2(\\UU,L^2(\\T))$, we set\n$\n\\phi_k=\\phi(e_k)\\in L^2(\\T)\n$, $k\\in\\N,$\nand suppose that\n\\begin{equation}\\label{eq:HS}\n \\sum_{k\\geq 1}\\|\\phi_k\\|_{L^{\\infty}(\\T)}^2= \\sum_{k\\geq 1}\\|\\phi(e_k)\\|_{L^{\\infty}(\\T)}^2<\\infty.\n\\end{equation}\nConsequently, since $\\phi_k$ is bounded we deduce \n\\begin{equation}\\label{fyB}\n\\|\\sqrt{\\varrho}\\phi_k\\|_{L_2(\\UU, L^2(\\T))}^2\\lesssim c(\\phi)( \\|\\varrho\\|_{L^{1}(\\T)}).\n\\end{equation}\nThe stochastic integral \n\\[\n\\int_{0}^{\\tau} \\varrho \\phi \\,\\dd W = \\sum_{k\\geq1}\\int_{0}^{\\tau}\\varrho\\phi(e_k)\\, \\dd \\beta_k,\n\\]\ntakes values in the Banach space $C([0,T];W^{-\\mathfrak k,2}(\\T))$ in the sense that\n\\begin{equation}\n \\int_{\\T}\\left(\\int_{0}^{\\tau}\\varrho\\phi \\,\\dd W\\cdot \\varphi \\right)\\,\\dd x =\\sum_{k\\geq1}\\int_{0}^{\\tau}\\left(\\int_{\\T}\\varrho\\phi(e_k)\\cdot \\varphi \\,\\dd x\\right)\\,\\dd \\beta_k, \\quad \\varphi \\in W^{\\mathfrak k,2}(\\T), \\mathfrak k>\\frac{n}{2}.\n\\end{equation}\nFinally, we define the auxiliary space $\\UU_0$ with $\\UU \\subset \\UU_0$ as \n\\begin{eqnarray}\n\\UU_0 :&=&\\Bigg\\{ e = \\sum_{k}\\alpha_k e_k:\\sum_{k}\\frac{\\alpha_{k}^{2}}{k^2} < \\infty \\Bigg\\}, \\nonumber\\\\\n\\| e\\|_{\\UU_0}^{2}:& =& \\sum_{k}^{\\infty}\\frac{\\alpha_{k}^{2}}{k^2}, \\ e=\\sum_{k}\\alpha_ke_k,\n\\label{auxiliary}\n\\end{eqnarray}\nso that the embedding $\\UU \\hookrightarrow \\UU_0$ is Hilbert-Schmidt, and the trajectories of $W$ belong $\\p$-a.s. to the class $C([0,T];\\UU_0)$ (see \\cite{Prato}).\n\n\\subsection{Discontinuous Galerkin schemes}\\label{sec:DG}\nAs mentioned in Subsection \\ref{sec:cr}, we focus on the two-dimensional setting. \nThe spatial domain $\\T \\subset \\R^2$ is then discretized with a mesh of tensor-product elements, e.g., a regular quadrilateral grid, denoted by $\\TT_h$. We denote the generic cell $K$ and the uniform mesh size with $h$. They are given by \n$$\\mathcal{K}:= [x_{i-1/2,j}, x_{i+1/2,j}] \\times [y_{i,j-1/2}, y_{i,j+1/2}]$$\nwith\n$h:= x_{i+1/2,j}-x_{i-1/2,j}=y_{i,j+1/2}-y_{i,j-1/2}$, for simplicity. \nExtensions to (unstructured) rectangular meshes with cell sizes $h_x\\neq h_y$ are straightforward. \nWith $\\partial \\mathcal{K}$ we denote the boundary of an element $\\mathcal{K}$ and by $\\EE$ the set of all interfaces of all cells $\\mathcal{K} \\in \\TT_h$ where $\\ee$ is one interface of $\\partial \\mathcal{K}$. Between two elements $\\mathcal{K}^-$ and $\\mathcal{K}^+$ we have a normal vector $\\bn$. It is given either by $\\bn=(n_x, 0)$ or by $\\bn=(0,n_y)$ depending on the interface. \nLet $\\mathcal{Q}^{\\mathfrak p}([-1,1]^2)$ be the space of all multivariate \npolynomials of degree at most $\\mathfrak p\\in\\N_0$ in each variable.On each element $\\mathcal{K} \\in \\TT_h$, we have a linear map $T_{\\mathcal{K}}:[-1,1]^2\\to \\mathcal{K}$ and $\\mathcal{Q}^{\\mathfrak p}(\\mathcal{K})$ is spanned by functions $\\psi \\circ T_{\\mathcal{K}}^{-1}$ with $\\psi\\in \\mathcal{Q}^{\\mathfrak p}([-1,1]^2)$.\nThe DG solution space $\\VV^h$ is given by\n \\begin{equation}\\label{eq:spaceA}\n \\VV^h = \\left\\{ v^h\\in L^1(\\T) \\Big| v^h|_{\\mathcal{K}} \\in \\mathcal{Q}^{\\mathfrak p}(\\mathcal{K}) \\text{ for all } \\mathcal{K}\\in \\TT_h \\right\\}.\n \\end{equation}\nApproximate solutions to the Euler equations \\eqref{Euler} live in the vector version of this \n space, that is $\\tilde{\\VV}^h= [\\VV^h]^4$. \n To describe an element of the finite dimensional space $\\VV^h$, we apply a nodal Gauss-Lobatto basis. We denote by $\\xi_i$ the Gauss-Lobatto points in the interval $[-1,1]$. Further, $L_i$ is the Lagrange polynomial which fulfils $L_i(\\xi_j)=\\delta_{i,j}$. Here, $\\delta_{i,j}$ is the Kronecker delta. Lagrange polynomials form a basis for $\\mathcal{Q}^{\\mathfrak p}([-1,1])$ in one dimension. We obtain a basis for\n $\\mathcal{Q}^{\\mathfrak p}([-1,1]^2)$ via the tensor product of the one-dimensional basis, i.e., $L_i(x)L_j(y)$.\n In each element, each component of the conservative variable vector of the Euler equations is approximated by a polynomial in the reference space \\eqref{eq:spaceA}. The nodal values (interpolation points) are our degrees of freedom (DOFs) where we have to calculate the time-dependent nodal coefficients for all components (density, momentum, energy) in the following.\n To make this point clear, our numerical solution is given by $\\bU^h=(\\varrho^h, \\bm^h,\\mathcal E^h)^\\top \\in \\tilde{\\VV}^h$ \n and each component is represented by a polynomial, e.g. in the reference element the approximated density is given by \n \\begin{equation}\\label{eq:approx}\n \\varrho^h(x,y,t) = \\sum_{i,j=0}^{\\mathfrak p} \\hat{\\varrho}^h(\\xi_i, \\nu_j, t) L_i(x) L_j(y)= \\sum_{i,j=0}^{\\mathfrak p} \\hat{\\varrho}_{i,j}(t) L_i(x) L_j(y)\n \\end{equation}\nwhere $ \\xi$ and $\\nu$ are our Gauss-Lobatto nodes in $x-$ and $y-$direction\nand \n$\\hat{\\varrho}_{i,j}(t)$ is the time-dependent coefficient of our polynomial presentation, i.e. $\\hat{\\varrho}_{i,j}(t)= \\varrho^h(\\xi_i, \\nu_j,t)$. \nAnalogous notation holds for the other components of $\\bU^h$. \\\\\nNext, we explain our DG discretization for the stochastic Euler system \\eqref{EulerB}. To avoid the non-conservative products inside the flux formulation which can yield to problems and needs special treatment \\cite{abgrall2018,abgrall2010, renac_2019}, we use equation \\eqref{eq_total} instead of the internal energy equation in \\eqref{EulerB}. Therefore, the third conservative variable is the total energy $\\mathcal E$ and we consider the system\n\n\\begin{align}\\label{EulerC}\n\\begin{aligned}\n\\dd \\varrho + \\mathrm{div}_{x}(\\vec m)\\, \\dd t &=0 \\quad \\text{in}\\, Q,\\\\\n\\dd \\vec m+ \\mathrm{div}_x\\Big(\\frac{\\vec m \\otimes \\vec m}{\\varrho}\\Big)\\, \\dd t + (\\gamma-1)\\nabla_x \\bigg(\\mathcal E-\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}\\bigg)\\,\\dd t&=\\varrho \\phi \\,\\dd W\\quad \\text{in}\\, Q,\\\\\n\\dd \\mathcal E + \\mathrm{div}_x\\left(\\left(\\mathcal E+(\\gamma-1)\\Big(\\mathcal E-\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}\\Big)\\right)\\frac{\\vec m}{\\varrho}\\right)\\dd t &=\\frac{1}{2}\\varrho\\|\\phi\\|_{\\ell_2}^2 \\,\\dd t+ \\phi\\cdot \\vec m\\,\\dd W\\quad \\text{in}\\, Q.\\end{aligned}\n\\end{align}\nOr in shorter terms\n\\begin{align}\\label{EulerC_short}\n\\dd\\bU+\\mathrm{div}(\\mathbf{f}(\\bU))\\dt=\\bh(\\bU)\\dt+\\bg(\\bU)\\,\\dd W,\n\\end{align} \nwhere\n\\begin{align*}\n\\mathbf{f}(\\bU)&=\\mathbf{f}\\begin{pmatrix}\\varrho\\\\\\vec m\\\\\\mathcal E\n\\end{pmatrix}=\\begin{pmatrix}\\vec m\\\\\\frac{\\vec m \\otimes \\vec m}{\\varrho}+ (\\gamma-1)\\bigg(\\mathcal E-\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}\\bigg)\\mathbb I_{2\\times 2}\\\\\\left(\\mathcal E+(\\gamma-1)\\Big(\\mathcal E-\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}\\Big)\\right)\\frac{\\vec m}{\\varrho}\\end{pmatrix},\\\\\n\\mathbf{h}(\\bU)&=\\mathbf{h}\\begin{pmatrix}\\varrho\\\\\\vec m\\\\\\mathcal E\n\\end{pmatrix}=\\varrho\\begin{pmatrix}0\\\\0\\\\\\tfrac{1}{2}\\|\\phi\\|_{\\ell_2}^2 \\end{pmatrix},\\quad\\mathbf{g}(\\bU)=\\mathbf{g}\\begin{pmatrix}\\varrho\\\\\\vec m\\\\\\mathcal E\n\\end{pmatrix}=\\begin{pmatrix}0\\\\\\varrho \\phi \\\\\\phi\\cdot \\vec m\\end{pmatrix}.\n\\end{align*}\nIn the case of the two-dimensional noise from \\eqref{eq:14.02} we have\n\\begin{align}\\label{eq:14.02b}\n\\mathbf{h}(\\bU)=&\\varrho\\begin{pmatrix}0\\\\0\\\\ 0 \\\\1\\end{pmatrix},\\quad\\mathbf{g}(\\bU)\\dd W=\\begin{pmatrix}0\\\\\\varrho\\, \\dd\\beta_1\\\\\\varrho\\,\\dd \\beta_2 \\\\m_1\\,\\dd\\beta_1+m_2\\,\\dd\\beta_2\\end{pmatrix}.\n\\end{align}\nIf the noise is truly infinite dimensional, it has to be approximated for practical purposes as outlined in Remark \\ref{rmk:noise} below.\n\n We start by defining the DG discretization for the flux term $\\mathrm{div}(\\mathbf{f}(\\bU))$ in the Euler equation \\eqref{EulerC}. \nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ be a complete stochastic basis with a probability measure $\\p$ on $(\\Omega,\\FF)$ and right-continuous filtration $(\\FF_t)_{t\\geq 0}$, $W$ an $(\\mathfrak F_t)$-adapted (cylindrical) Wiener process and $\\phi\\in L_2(\\mathfrak U;L^2(\\T))$.\n\n Multiplying by a test function $\\bV^h \\in \\tilde{\\VV}^h$ and integrate over the domain $\\T$ it reads as\n \\begin{align}\\label{eq:DG_strong_analytical}\n\\begin{aligned}\n \\dd\\int_{\\T} \\bU^h \\cdot \\bV^h \\diff \\bx& +\\sum_{K\\in \\TT_h} \\int_K\n (\\mathrm{div}\\,\\mathbf{f} (\\bU^h) )\\cdot \\bV^h \\diff \\bx\\dt \\\\=&- \\sum_{\\partial K^- \\in \\TT_h} \\int_{\\partial K^-} (\\bfnum(\\bU^{h,-}, \\bU^{h,+},\\bn^-) - \\mathbf{f}(\\bU^h) \\cdot \\bn^-) \\cdot \\bV^{h,-}\\,\\dd\\mathcal H^1\\dt\\\\\n&+\\int_{\\T}\\bh(\\bU^h) \\cdot \\bV^h \\diff \\bx\\,\\dd t+\\int_{\\T}\\bg(\\bU^h) \\cdot \\bV^h \\diff \\bx\\,\\dd W,\n\\end{aligned}\n \\end{align}\nwhere $\\mathbf{f}(\\bU^h)=(\\mathbf{f}_1,\\mathbf{f}_2)$ and for the numerical flux $\\bfnum$ we choose\nthe local Lax-Friedrichs flux given by \n\\begin{equation}\\label{eq:LLF}\n f^{\\mathrm num}(\\bU^{h,-}, \\bU^{h,+}, \\bn^-):= \\frac{1}{2} \\left(\\mathbf{f}(\\bU^{h,-}) + \\mathbf{f}(\\bU^{h,+}) \\right)\\cdot \\bn^- - \\frac{\\lambda}{2} \\left( \\mathbf{f}(\\bU^{h,+})-\\mathbf{f}(\\bU^{h,-}) \\right),\n\\end{equation}\nwhere $\\lambda \\geq \\lambda_{max}$ is an upper bound for the maximal wave speed. For the Euler equations \\eqref{EulerC}, it holds that \n$\\lambda_{max}=\\max\\{ |\\bu^h_{K^-}|+c_{K^-},|\\bu^h_{K^+}|+c_{K^+} \\}$ with $c=\\sqrt{\\gamma \\frac{p^h}{\\varrho^h} }$.\\\\\nNote that an inner product denoted by $\\cdot$ means that \\eqref{eq:DG_strong_analytical}\n has to be solved for each component of the Euler equation separately. \nLater all the calculations in our high-order DG method are performed in a reference element $I=[-1,1]^2$.\nTo evaluate the integrals, we proceed by choosing the Gauss-Lobatto nodes as collocated quadrature points. \nFinally, by this selection of a tensor-product basis and Gauss-Lobatto quadrature, the DG operators have a Kronecker-product structure as well. We denote by $\\mat{M}_1$ on $[-1,1]$ the one-dimensional mass matrix.\nIt has diagonal form with quadrature weight on the diagonal. Further, we obtain the one-dimensional differentiation matrix $\\mat{D}_{1}$ by evaluating the derivatives of the basis functions at the nodal points. We use the index here to clarify that we have the one-dimensional setting and working on the reference element $I$. \nTo clarify the setting, let $\\xi_j$ be the Gauss-Lobatto quadrature points $-1=\\xi_0<\\xi_1<\\cdots<\\xi_{\\mathfrak p}=1$ in $[-1,1]$\nwith corresponding quadrature weights \n $\\{\\omega_j\\}_{j=0}^{\\mathfrak p}$. The nodal Lagrangian basis is given by $L_j(\\xi_l)=\\delta_{jl}$ and we can define \nthe discrete inner $\\langle u,v\\rangle_{\\omega}:=\\sum_{j=0}^{\\mathfrak p} \\omega_j u(\\xi_j) v(\\xi_j)$ in one space-dimension. Then, the above described operators are given by \n \\begin{itemize}\n\\item Difference matrix $\\mat{D}_1$ with $\\mat{D}_{1,jl} =L_l'(\\xi_j)$ with $L_l'$ be the first derivative of the $l-$Lagrange polynomial,\n\\item Mass matrix $\\mat{M}_{1,jl}=\\langle L_j,L_l\\rangle_{\\omega} =\\omega_j \\delta_{jl}$, so that\n $\\mat{M}_1 = \\diag \\{\\omega_0, \\dots, \\omega_{\\mathfrak p}\\}$.\n\\end{itemize}\nLater, we need also the operators \n\\begin{itemize}\n\\item Stiffness matrix $\\mat{Q}_{1,jl}=\\langle L_j',L_l\\rangle_{\\omega} =\\langle L_j,L_l'\\rangle_{\\omega}$\n\\item Interface matrix $\\mat{B}_1=\\diag(-1,0,\\cdots,0,1)$\n\\end{itemize}\nUp to this point, these operators would work separately on every component of the conservative variable vector $\\bU^h$, i.e. \non the density $\\varrho^h$, momentum $\\bm^h$, and energy $\\mathcal{E}^h$. To extend these operators to the Euler system, a simple Kronecker product can be used. It is\n \\begin{equation*}\n \\mat{\\mathbf{M}}_1=\\mat{M}_1 \\otimes \\IdxM \\qquad \\mat{\\mathbf{D}}_1=\\mat{D}_1 \\otimes \\IdxM,\n \\end{equation*}\nwhere $ \\IdxM$ is the $4\\times4$-identy matrix.\\\\\nSimilar in two-dimension, we obtain the local mass matrix and differentiation matrix in the standard element $I$ through Kronecker products:\n $\\mat{\\mathbf{M}}_I= \\mat{\\mathbf{M}}_1\\otimes \\mat{\\mathbf{M}}_1,\n \\mat{\\mathbf{D}}_{1,I}= \\IdxM\\otimes \\mat{\\mathbf{D}}_1,\n \\mat{\\mathbf{D}}_{2,I}= \\mat{\\mathbf{D}}_1 \\otimes \\IdxM.$\nTo solve \\eqref{eq:DG_strong_analytical}, we have to evaluate the \ncell interface integrals. Due to the tensor structure ansatz, we evaluate at each interface the one-dimensional Gauss-Lobatto quadrature rule. Due to the Kronecker ansatz, we obtain the\ninterface operators \n$\n\\mat{\\mathbf{B}}_{\\ee_1}= \\mat{\\mathbf{M}}_1 \\otimes \\mat{\\mathbf{B}}_1 \\text{ and } \\mat{\\mathbf{B}}_{\\ee_2}= \\mat{\\mathbf{B}}_1 \\otimes \\mat{\\mathbf{M}}_1 \n$\ndepending on the considered cell interfaces. The defined operators fulfil the summation-by-parts property meaning that they mimic discretely integration-by-parts. \\\\\nIn \\eqref{eq:DG_strong_analytical}, we have to calculate the time-dependent coefficients of our polynomial representation \\\\\n$\\bU^h= (\\varrho^h, \\bm^h, \\mathcal{E}^h) \\in\\tilde{\\VV}^h.$ We have to distinguish in numbering between the interpolation (quadrature) points in $x$- and $y$- direction in \\eqref{eq:approx}. For simplicity, we are renumbering the points. We have $n_{\\mathfrak p}=({\\mathfrak p}+1)^2$ quadrature/interpolation points\\footnote{Alternatively, a multi-index can be used for the notation.} denoted by $\\boldsymbol{\\xi}_i$ in each element and $n_{b}=4{\\mathfrak p}$ at the interfaces. On each face we have ${\\mathfrak p}+1$ Gauss-Lobatto quadrature nodes but on the corners they intersect. The basis is given by $\\mathbf{L}_i$ with $i \\in \\{1,n_{\\mathfrak p}\\}$.\n We denote by $\\bvu$ the vector of coefficients (i.e., the nodal values) of $\\bU^h$ on $K$:\n\\begin{equation}\\label{eq:coefficients}\n\\bvu= \\left(\\varrho^h(\\boldsymbol{\\xi}_1), \\bm^h(\\boldsymbol{\\xi}_1), \\mathcal{E}^h(\\boldsymbol{\\xi}_1 );\n\\varrho^h(\\boldsymbol{\\xi}_2), \\bm^h(\\boldsymbol{\\xi}_2), \\mathcal{E}^h(\\boldsymbol{\\xi}_2 ); \\dots ;\\varrho^h(\\boldsymbol{\\xi}_{n_{\\mathfrak p}}), \\bm^h(\\boldsymbol{\\xi}_{n_{\\mathfrak p}}), \\mathcal{E}^h(\\boldsymbol{\\xi}_{n_{\\mathfrak p}}) \\right)^T.\n\\end{equation}\nLet $ \\underline{\\mathbf{f}}_m$ ($m=1,2$) denote the vector of values of $\\mathbf{f}_m(\\bU^h)$ evaluated at the nodal points. For each cell interface $\\ee \\in \\partial K \\subset \\EE$, we have to evaluate $\n\\mathbf{f}(\\bU^h)$ at the one-dimensional Gauss-Lobatto nodes on the cell interface $\\ee$ (where the trace of $\\bU^h$ is taken from inside $K$ dotted with the scaled normal vector $\\bn$ facing outwards from $\\ee$). We denote this by $\\mat{\\mathbf{R}}_{\\ee_m}\\underline{\\mathbf{f}}_{\\ee_m}$. Likewise $ \\underline{\\mathbf{f}}^{\\mathrm{num}}_{\\ee_m}$ denotes the nodal values of $\\bfnum(\\bU^{h,-}, \\bU^{h,+},\\bn^-)$.\nWith these operators, we can finally re-write the DG semidiscretization \\eqref{eq:DG_strong_analytical} on the reference element as follows\n\\begin{align}\\label{eq:DG_standard}\n\\begin{aligned}\n \\dd \\bvu &+ \\big(\\mat{\\mathbf{D}}_{1,I} \\underline{\\mathbf{f}}_1 +\\mat{\\mathbf{D}}_{2,I} \\underline{\\mathbf{f}}_2\\big)\\dt\\\\&\\qquad\\qquad= \\mat{\\mathbf{M}}_I^{-1} \\sum_{j \\in \\partial I} \\mat{\\mathbf{B}}_{j} \\left(\\mat{\\mathbf{R}}_{j}\\underline{\\mathbf{f}}_{j}-\\underline{\\mathbf{f}}^{\\mathrm{num}} \\right)\\dt+\\underline\\bh\\,\\dd t+\\underline\\bg\\,\\dd W,\n\\end{aligned}\n\\end{align}\nwhere $\\partial I$ denotes the cell interfaces of the reference element. \nEquation \\eqref{eq:DG_standard} describes the classical discontinuous Galerkin spectral element method (DGSEM) in two-space dimension.\nThe method is by construction not entropy conservative/ dissipative for the deterministic Euler equations \\cite{zbMATH07517718,gassner2016}. To obtain an high-order entropy dissipative DG method for the Euler equation, we apply the flux differencing approach \\cite{fischer_2013,gassner2016}. To this end we replace the volume flux $\\sum_{m=1}^2 \\mat{\\mathbf{D}}_{m,I} \\bvf_m $ in equation \\eqref{eq:DG_standard} above using consistent, symmetric two-point numerical fluxes. The resulting DG scheme in the reference element reads than \n \\begin{align}\\label{eq:DG_standard_2}\n\\begin{aligned}\n \\dd \\bvu &+ 2 \\left( \\mat{\\mathbf{D}}_{1,I} \\underline{\\mathbf{f}}^{\\mathrm{num}}_{1,Vol}(\\bvu, \\bvu)+\\mat{\\mathbf{D}}_{2,I} \\underline{\\mathbf{f}}^{\\mathrm{num}}_{2,Vol}(\\bvu, \\bvu) \\right)\\dt\\\\&\\qquad\\qquad= \\mat{\\mathbf{M}}_I^{-1} \\sum_{j \\in \\partial I} \\mat{\\mathbf{B}}_{j} \\left(\\mat{\\mathbf{R}}_{j}\\bvf_{j}-\\underline{\\mathbf{f}}^{\\mathrm{num}}_j \\right)\\dt+\\underline\\bh\\,\\dd t+\\underline\\bg\\,\\dd W,\n\\end{aligned}\n\\end{align}\nwhere $\\underline{\\mathbf{f}}^{\\mathrm{num}}_{Vol}$ denotes the numerical volume flux working on each degree of freedom and $\\underline{\\mathbf{f}}^{\\mathrm{num}}_j $ is the classical numerical flux at the interface. \nFor the numerical volume flux $\\mathbf{f}^{\\operatorname{num}}_{m,Vol}$ with $m=1,2$ , we select the consistent, symmetric and entropy conservative two-point flux from \\cite{ranocha2018}. It is defined for each component separately: \n \\begin{equation}\\label{eq:Ranocha_flux} \n \\begin{aligned}\n f^{\\mathrm num}_{\\varrho,1} &=\\{\\{\\varrho\\}\\}_{\\log}\\armean{u_1}, \\quad f^{\\mathrm num}_{\\varrho u_1,1} =\\armean{ u_1} f^{\\mathrm num}_{\\varrho,1} +\\armean{ p}, \\quad f^{\\mathrm num}_{\\varrho u_2,1}=\\armean{ u_2} f^{\\mathrm num}_{\\varrho} \\\\\n f^{\\mathrm num, x}_{E,1}&= \\left( \\{\\{\\varrho\\}\\}_{\\log} \\left( \\armean{ u_1}^2+\\armean{ u_2}^2 - \\frac{\\armean{ u_1+u_2}^2}{2} \\right) -\\frac{1}{\\gamma-1} \\frac{\\logmean{ \\varrho} }{\\logmean{ \\varrho/p} } +\n \\armean{ p} \\right) \\armean{u_1} \\\\\n &-\\frac{\\jump{p} \\jump{\\bu}}{4},\n \\end{aligned}\n \\end{equation}\n with $\\mathbf{f}^{\\operatorname{num}}_2$ defined analogously. Here, we have used the abbreviations $\\armean{\\varrho}=\\frac{\\varrho^{+}+\\varrho^-}{2} $ and\n $\\logmean{\\varrho}=\\frac{\\varrho^{+}-\\varrho^-}{\\log \\varrho^+-\\log \\varrho^-}$. \n\n\n\\begin{rmk}[Interpretation and solvability]\n\\begin{itemize}\n\\item Discretization \\eqref{eq:DG_standard_2} is classical entropy stable DGSEM method for the Euler equation equipped with two additional forcing terms due to the noise. The first term $\\bh$ has no stochastic component and can be interpreted as a simple forcing of the total energy whereas the second part $\\bg$ acts on the momentum equation and the total energy and takes the stochasticity into account.\n\\item The well-posedness assumption regarding \\eqref{eq:DG_standard_2} can be substituted with the condition that the coefficients of the nonlinear SDE system are Lipschitz continuous, as noted in \\cite{KuLu} for the deterministic parts. This substitution is directly deducible from the definition of the Euler fluxes and the assumptions concerning the positive values of pressure and density. The noise coefficients are linear and thus clearly Lipschitz as well. Hence standard results yield the existence of a solution defined on a given complete stochastic basis $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ with respect to the $(\\FF_t)$-Wiener process $W$ and given initial data (which we assume for simplicity to be deterministic).\n\\end{itemize}\n\\end{rmk}\n\n\\section{Concept of solutions and main results}\\label{se_concept}\n\n\\subsection{Measure-valued solutions}\\label{MVS}\n\n\nWe are now ready to introduce the concept of measure-valued martingale solutions to the \\textit{complete} stochastic Euler system written in entropy conservative variables (\\ref{eq:aEuler})--(\\ref{eq:cEuler}). From here onward, we denote by $\\mathcal{M}^+$ the space of non-negative Radon measures, and we denote by $A$ the space of ``dummy variables\":\n\n\\begin{equation}\\label{eq:space}\nA = \\bigg\\{[\\varrho', \\vec m',{S'}]\\bigg|\\varrho'\\geq 0, \\vec m' \\in \\R^3,S'\\in \\R\\bigg\\}\n\\end{equation}\nWe denote by $\\mathcal{P}(A)$ the space of probability measures on $A$. We follow \\cite{Mo} and define: \n\n\\begin{dfn}[Dissipative measure-valued martingale solution]\\label{E:dfn}Let $\\varrho_0\\in L^{\\gamma}(\\T)$, $\\vec m\\in L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)$, $S_0\\in L^{\\gamma}(\\T)$ and $\\phi \\in L_2(\\UU;L^2(\\T))$. Then \n\\[\n((\\Omega,\\FF, (\\FF_t)_{t\\geq 0},\\p),\\varrho,\\vec m,S, \\mathcal{R}_{\\text{conv}},\\mathcal{R}_{\\text{press}},\\mathcal{V}_{t,x},\\mathfrak t, W)\n\\]\nis called a dissipative measure-valued\nmartingale solution to (\\ref{eq:aEuler})--(\\ref{eq:cEuler}) with initial law $\\Lambda$, provided\\footnote{Some of our variables are not stochastic processes in the classical sense and we interpret their adaptedness in the sense of random distributions as introduced in \\cite{FrBrHo} (Chap. 2.2).}:\n\n\\begin{enumerate}\n \\item [(a)] $(\\Omega,\\FF, (\\FF_t)_{t\\geq 0},\\p)$ is a stochastic basis with complete right-continuous filtration;\n\\item[(b)] $\\mathfrak t$ is an $(\\FF_t)_{t\\geq 0}$-stopping time with $\\p(\\mathfrak t>0)=1$;\n \\item[(c)]$W$ is a $(\\FF_t)_{t\\geq 0}$-cylindrical Wiener process;\n \\item[(d)] The density $\\varrho$ is $(\\FF_t)_{t\\geq 0}$-adapted and satisfies $\\p$-a.s.\n \\[\n \\varrho(\\cdot\\wedge\\mathfrak t) \\in C_{\\text{loc}}([0,\\infty), W^{-4,2}(\\T))\\cap L_{\\text{loc}}^{\\infty}(0,\\infty;L^{\\gamma}(\\T));\n \\]\n \\item[(e)] The momentum $\\vec m$ is $(\\FF_t)_{t\\geq 0}$-adapted and satisfies $\\p$-a.s.\n \\[\n \\vec m(\\cdot\\wedge\\mathfrak t) \\in C_{\\text{loc}}([0,\\infty), W^{-4,2}(\\T))\\cap L_{\\text{loc}}^{\\infty}(0,\\infty;L^{\\frac{2\\gamma}{\\gamma +1}}(\\T));\n \\]\n \\item[(f)] The total entropy $S$ is $(\\FF_t)_{t\\geq 0}$-adapted and satisfies $\\p$-a.s.\n \\[\n S(\\cdot\\wedge\\mathfrak t) \\in L^\\infty([0,\\infty),L^{\\gamma}(\\T))\\cap BV_{w,\\text{loc}}(0,\\infty;W^{-l,2}(\\T)), l>\\frac{n+2}{2};\n \\]\n \\item[(g)] The parametrised measures $(\\mathcal{R}_{\\text{conv}},\\mathcal{R}_{\\text{press}},\\mathcal{V})$ are $(\\FF_t)_{t\\geq 0}$-progressively measurable and satisfy $\\p$-a.s.\n \\begin{eqnarray}\\label{eq:para}\n t\\mapsto \\mathcal{R}_{\\mathrm{conv}}(t\\wedge\\mathfrak t) &\\in &L_{\\text{weak-(*)}}^{\\infty}(0,\\infty;\\mathcal{M}^+(\\T, \\mathbb{R}^{3\\times 3}));\\\\\n t\\mapsto \\mathcal{R}_{\\mathrm{press}}(t\\wedge\\mathfrak t) &\\in &L_{\\text{weak-(*)}}^{\\infty}(0,\\infty;\\mathcal{M}^+(\\T, \\mathbb{R}));\\\\\n (t,x)\\mapsto \\mathcal{V}_{t\\wedge\\mathfrak t,x} &\\in &L_{\\text{weak-(*)}}^{\\infty}(Q;\\mathcal{P}(A));\n \\end{eqnarray}\n \\item[(h)] We have $\\varrho(0,\\cdot)=\\varrho_0$, $\\vec m(0,\\cdot)=\\vec m_0$ and $\\liminf_{\\tau\\rightarrow0+}S(\\tau,\\cdot)\\geq S(\\cdot,0)$ $\\mathbb P$-a.s.;\n \\item[(i)]For all $\\varphi \\in C^{\\infty}(\\T)$ and all $\\tau > 0$ there holds $\\p$-a.s.\n \\begin{equation}\\label{eq:cont}\n \\left[\\int_{\\T}\\varrho\\varphi\\, \\dd x\\right]_{t=0}^{t=\\tau\\wedge\\mathfrak t} = \\int_{0}^{\\tau\\wedge\\mathfrak t}\\int_{\\T} \\vec m \\cdot \\nabla \\varphi \\, \\dd x\\dd t;\n \\end{equation}\n \\item[(j)]For all $\\boldsymbol{\\varphi} \\in C^{\\infty}(\\T)$ with $\\boldsymbol{\\varphi}\\cdot\\mathbf n=0$ and all $\\tau > 0$ there holds $\\p$-a.s.\n \\begin{eqnarray}\\label{eq:mcxs}\n \\left[\\int_{\\T}\\vec m \\cdot \\boldsymbol{\\varphi}\\right]_{t=0}^{t=\\tau\\wedge\\mathfrak t}&=&\\int_{0}^{\\tau\\wedge\\mathfrak t}\\int_{\\T}\\left[\\frac{\\vec m \\otimes\\vec m}{\\varrho}:\\nabla\\boldsymbol{\\varphi}+\\varrho\\exp{\\left(\\frac{S}{c_v\\varrho}\\mathrm{div}\\boldsymbol{\\varphi}\\right)}\\right]\\, \\dd x\\, \\dd t\\\\\n &&+\\int_{0}^{\\tau\\wedge\\mathfrak t}\\nabla \\varphi:\\dd \\mathcal{R}_{\\text{conv}} \\dd t+\\int_{0}^{\\tau\\wedge\\mathfrak t}\\int_{\\T}\\mathrm{div} \\boldsymbol{\\varphi}\\,\\dd \\mathcal{R}_{\\text{press}} \\dd t\\nonumber\\\\\n &&+\\int_{0}^{\\tau\\wedge\\mathfrak t}\\boldsymbol{\\varphi}\\cdot \\varrho \\phi \\, \\dd x\\, \\dd W;\\nonumber\n \\end{eqnarray}\n \\item[(k)] The total entropy holds in the sense that \n \\begin{equation}\\label{eq:entr}\n \\int_{0}^{\\tau\\wedge\\mathfrak t}\\int_{\\T}\\left[\\langle\\mathcal{V}_{t,x};S'\\rangle\\partial_t \\varphi+\\langle\\mathcal{V}_{t,x},S'\\frac{\\vec m'}{\\varrho'}\\rangle\\cdot \\varphi\\right]\\, \\dd x\\dd t \\leq \\left[\\int_{\\T}\\langle \\mathcal{V}_{t,x};S'\\rangle \\varphi\\, \\dd x\\right]_{t=0}^{t=\\tau\\wedge\\mathfrak t},\n \\end{equation}\n for any $\\varphi \\in C^1([0,\\infty)\\times \\T), \\varphi \\geq 0, \\p$-a.s.\n \\item[(l)] The total energy satisfies\n \\begin{equation}\\label{eq:Ener}\n \\mathcal{E}_{t\\wedge\\mathfrak t}=\\mathcal{E}_{s\\wedge\\mathfrak t}+\\frac{1}{2}\\int_{s\\wedge\\mathfrak t}^{t\\wedge\\mathfrak t}\\|\\sqrt{\\varrho}\\phi\\|_{L_2(\\UU;L^2(\\T))}^2\\, \\dd \\sigma + \\int_{s\\wedge\\mathfrak t}^{t\\wedge\\mathfrak t}\\int_{\\T}\\vec m \\cdot \\phi \\, \\dd x \\dd W,\n \\end{equation}\n $\\p$-a.s. for a.a. $0\\leq s<t$, where\n \\[\n \\mathcal{E}= \\int_{\\T}\\left[\\frac{1}{2}\\frac{|\\vec m |^2}{\\varrho}+c_v\\varrho^{\\gamma}\\exp{\\left(\\frac{S}{c_v\\varrho}\\right)}\\right]\\, \\dd x + \\frac{1}{2}\\int_{\\T}\\dd \\,\\mathrm{tr}\\mathcal{R}_{\\text{conv}}(t) +c_v\\int_{\\T}\\dd \\,\\mathrm{tr}\\mathcal{R}_{\\text{press}}(t)\n \\]\n for $\\tau \\geq 0$ and \n \\[\n \\mathcal E_0= \\int_{\\T}\\left[\\frac{1}{2}\\frac{|\\vec m_0 |^2}{\\varrho_0}+c_v\\varrho_0^{\\gamma}\\exp{\\left(\\frac{S_0}{c_v\\varrho_0}\\right)}\\right]\\, \\dd x.\n \\]\n\\end{enumerate}\n\\end{dfn}\nThe following result is proved in \\cite{Mo} (with $\\mathfrak t=\\infty$) and ensures that dissipative measure-valued martingale solution exists globally.\n\\begin{thm}\\label{ExMainr}\nAssume (\\ref{eq:HS}) holds. Let $\\Lambda$ be a Borel probability measure on $L^{\\gamma}(\\T)\\times L^{\\gamma}(\\T)\\times L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)$ such that\n\\[\n\\Lambda\\bigg\\{(\\varrho,S,\\vec m)\\in L^{\\gamma}(\\T)\\times L^{\\gamma}(\\T)\\times L^{\\frac{2\\gamma}{\\gamma +1}}(\\T): 0<\\underline{\\varrho}<\\varrho<\\overline{\\varrho},0<\\underline{\\vartheta}<\\vartheta<\\overline{\\vartheta} \\bigg\\}=1,\n\\]\nwhere $\\underline{\\vartheta}, \\overline{\\vartheta},\\underline{\\varrho}, \\overline{\\varrho} $ are deterministic constants. Moreover, the moment estimate\n\n\\[\n\\int_{L^{\\gamma}(\\T)\\times L^{\\gamma}(\\T)\\times L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)}\\left\\|\\frac{1}{2}\\frac{|\\vec m|^2}{\\varrho}+c_v\\varrho^{\\gamma}\\exp{\\left(\\frac{S}{c_v\\varrho}\\right)}\\right\\|_{L^1(\\T)}^p\\,\\dd \\Lambda < \\infty,\n\\]\nholds for all $p\\geq 1$. Then there exists a dissipative measure-valued martingale solution to the \\textit{complete} stochastic Euler system (\\ref{eq:aEuler})--(\\ref{eq:cEuler}) in the sense of Definition \\ref{E:dfn} (with $\\mathfrak t=\\infty$) subject to the initial law $\\Lambda$. \n\nIt satisfies the entropy balance also in the renormalised sense, that is we have\n \\begin{equation}\\label{eq:entrren}\n \\int_{0}^{\\tau\\wedge\\mathfrak t}\\int_{\\T}\\left[\\langle\\mathcal{V}_{t,x};Z(S')\\rangle\\partial_t \\varphi+\\langle\\mathcal{V}_{t,x},Z(S')\\frac{\\vec m'}{\\varrho'}\\rangle\\cdot \\varphi\\right]\\, \\dd x\\dd t \\leq \\left[\\int_{\\T}\\langle \\mathcal{V}_{t,x};Z(S')\\rangle \\varphi\\, \\dd x\\right]_{t=0}^{t=\\tau\\wedge\\mathfrak t},\n \\end{equation}\n for any $\\varphi \\in C^1([0,\\infty)\\times \\T), \\varphi \\geq 0, \\p$-a.s.,{and any $Z \\in BC(\\R)$ non-decreasing.} \n\\end{thm}\n\n\n\\subsection{Strong solutions}\nIn the following, we will give the definition of a local strong pathwise solution. These solutions are strong in the probabilistic and PDE sense, at least locally in time. In particular, the Euler system (\\ref{Euler}) will\nbe satisfied pointwise in space-time on the given stochastic basis associated\nto the cylindrical Wiener process $W$.\n\n\\begin{dfn}[Strong Solution]\\label{def:compstrong}\nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0}, \\p)$ be a complete stochastic basis with a right-continuous filtration, let $W$ be an $(\\FF_t)_{t\\geq 0}$-cylindrical Wiener process, $\\phi\\in L_2(\\mathfrak U;L^2(\\T))$ and $\\ell\\geq4$. The triplet $[r, \\Theta, \\vec v]$ and a stopping time $\\mathfrak{s}$ is called a (local) strong pathwise solution to the system (\\ref{Euler}) provided:\n\n\\begin{itemize}\n \\item[(a)] the density $r> 0$ $\\p$-a.s., $t \\mapsto r (t\\wedge \\mathfrak{s},\\cdot)\\in W^{\\ell,2}(\\T)$ is $(\\FF_t)_{t\\geq 0}$-adapted,\n \\[\n \\E\\left[\\sup_{t\\in [0,T]}\\|r (t\\wedge \\mathfrak{s},\\cdot)\\|_{W^{\\ell,2}(\\T)}^q\\right]<\\infty \\quad\\text{for all $1\\leq q <\\infty$, $T>0$},;\n \\]\n \\item[(b)] the temperature $\\Theta > 0$ $\\p$-a.s., $t \\mapsto \\Theta (t\\wedge \\mathfrak{s},\\cdot)\\in W^{\\ell,2}(\\T)$ is $(\\FF_t)_{t\\geq 0}$-adapted,\n \\[\n \\E\\left[\\sup_{t\\in [0,T]}\\|\\Theta (t\\wedge \\mathfrak{s},\\cdot)\\|_{W^{\\ell,2}(\\T)}^q\\right]<\\infty \\quad\\text{for all $1\\leq q <\\infty$, $T>0$};\n \\]\n \\item[(c)] the velocity $t\\mapsto \\vec v(t\\wedge \\mathfrak{s},\\cdot) \\in W^{\\ell,2}(\\T)$is $(\\FF_t)_{t\\geq 0}$-adapted,\n \\[\n \\E\\left[\\sup_{t\\in [0,T]}\\|\\vec v (t\\wedge \\mathfrak{s},\\cdot)\\|_{W^{\\ell,2}(\\T)}^q\\right]<\\infty \\quad\\text{for all $1\\leq q <\\infty$, $T>0$};\n \\]\n \\item[(d)] for all $t\\geq0$ there holds $\\p$-a.s.\n \\[\n r(t\\wedge\\mathfrak{s}) = \\varrho(0)- \\int_{0}^{t\\wedge \\mathfrak{s}}\\mathrm{div}_x(r\\vec v)\\,\\dd t,\n \\]\n \\[\n (r\\vec v)(t\\wedge \\mathfrak{s})=(r\\vec v)(0)-\\int_{0}^{t\\wedge}\\mathrm{div}(r\\vec v\\otimes \\vec v)\\, \\dd t-\\int_{0}^{t\\wedge\\mathfrak{s}}\\nabla_x p(r,\\Theta)\\,\\dd t +\\int_{0}^{t\\wedge \\mathfrak{s}}r\\phi\\, \\dd W,\n \\]\n \\[\n (rs(r,\\Theta))(t\\wedge\\mathfrak{s}) = (rs(r,\\Theta))(0)- \\int_{0}^{t\\wedge \\mathfrak{s}}\\mathrm{div}_x(rs(r,\\Theta)\\vec v)\\,\\dd t,\n \\]\n where $s$ is the total entropy given by (\\ref{entropy}).\n\\end{itemize}\n\\end{dfn}\nSince density and temperature are strictly positive one can rewrite the system from Definition \\ref{def:compstrong} (d) equivalently in terms of $\\chi:=\\log(\\varrho)$, the velocity field $\\vec v$ and the temperature $\\Theta$ obtaining\n\\begin{align}\\label{EulerD}\n\\begin{aligned}\n\\dd \\Theta + \\vec v\\cdot\\nabla\\Theta\\dd t +\\Theta\\,\\mathrm{div}(\\vec v)\\dd t&=0 \\quad \\text{in}\\, Q\\\\\n\\dd \\chi + \\vec v\\cdot\\nabla\\chi+\\mathrm{div}_{x}(\\vec v)\\, \\dd t &=0 \\quad \\text{in}\\, Q,\\\\\n\\dd \\vec v+ (\\vec v\\cdot\\nabla)\\vec v\\, \\dd t + \\Theta\\nabla_x \\chi\\,\\dd t+\\,\\nabla\\Theta\\,\\dt&=\\phi \\,\\dd W\\quad \\text{in}\\, Q\n.\\end{aligned}\n\\end{align}\nThis is a symmetric hyperbolic system with additive noise. Given some initial data $(\\varrho_0,\\bfU_0,\\Theta_0)$ we expect the existence of a unique strong pathwise solution provided (setting $\\chi_0:=\\log(\\varrho_0)$)\n\\begin{align*}\n\\chi_0,\\bfU_0,\\Theta_0\\in W^{\\ell,2}(\\mathcal O),\\quad\\phi\\in L_2(\\mathfrak U;W^{\\ell,2}(\\mathcal O)),\n\\end{align*}\ntogether with no flux boundary conditions for the initial data.\nA corresponding result for the problem on the whole space follows from \\cite{Kim}.\n\nThe following weak (measure-valued)-strong uniqueness principle is proved in \\cite{Mo}.\n\\begin{thm}\\label{thm_a} The pathwise weak-strong uniqueness holds true for the system (\\ref{eq:aEuler})--(\\ref{eq:cEuler}) in the following sense.\n Let \n\\[((\\Omega , \\FF , (\\FF)_{t \\geq 0}, \\mathbb{P} ),\\varrho,\\vec m, S, \\mathcal{R}_{\\mathrm{conv}},\\mathcal{R}_{\\mathrm{press}},\\mathcal{V}_{t,x},W)\n\\]\n be a dissipative measure-valued martingale solution to (\\ref{eq:aEuler})-(\\ref{eq:cEuler}) in the sense of Definition (\\ref{E:dfn}) (with $\\mathfrak t=\\infty$) which satisfies additionally the relative entropy balance in the renormalised sense, cf.~\\eqref{eq:entrren}. Let the triplet $[r,\\Theta,\\vec v]$ and a stopping time $\\mathfrak{s}$ be a strong solution in the sense of Definition \\ref{def:compstrong} of the same problem; defined on the stochastic basis with the same Wiener process and with initial data\n \\begin{equation}\\label{Idata}\n \\varrho(0,\\cdot) =r(0,\\cdot),\\quad\\vec u (0,\\cdot)={\\vec v}(0,\\cdot),\\qquad \\vartheta (0,\\cdot)=\\vec \\Theta(0,\\cdot)\\quad\\p\\text{-a.s.}\n \\end{equation}\nThen\n \\[ [{\\varrho},{\\vartheta},{\\vec u}](\\cdot \\wedge \\mathfrak{s}) \\equiv [r,{\\Theta},{\\vec v}] (\\cdot \\wedge \\mathfrak{s}),\n\\]\nand\n\\[ \\mathcal{R}_{\\mathrm{conv}}=\\mathcal{R}_{\\mathrm{press}}=0, \\]\n $\\p$-a.s., and for any $(t,x)\\in (0,T)\\times\\T$\n \\[\n \\mathcal{V}_{t\\wedge \\mathfrak{s},x}= \\delta_{ s(r,\\Theta)},\n\\]\n$\\p$-a.s.\n\\end{thm}\n\\begin{rmk}\nIf $\\mathfrak p=0$ one can\nargue as in \\cite{feireisl2021numerical}\nto show that the solution obtained by our algorithm (cf. Theorem \\ref{thm:main1} below) satisfies the relative entropy balance in the renormalised sense, cf.~\\eqref{eq:entrren}, such that Theorem \\ref{thm_a} applies. For $\\mathfrak p\\geq1$ this is currently unclear. Note that this point does not seem to be affected by the noise, this issue appears already in the deterministic case.\n\\end{rmk}\n\n\\subsection{Consistency and convergence}\nIn this subsection we formulate our main results concerning the DG scheme from \\eqref{eq:DG_standard_2} with solution $(\\varrho^h,\\vec m^h,\\mathcal E^h)$ posed on a stochastic basis $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ regarding its approximation properties for the Euler equations \\eqref{Euler}. We work under the following hypothesis: There is an $(\\mathfrak F_t)$-stopping time $\\mathfrak t$ with $\\p(\\mathfrak t>0)=1$ and deterministic constants $K>0$ such that $\\p$-a.s.\n\\begin{align}\\label{ass:main}\n\\inf_{t\\in[0,\\mathfrak t],x\\in\\T}\\varrho^h(t,x)\\geq \\frac{1}{K},\\quad \\sup_{t\\in[0,\\mathfrak t],x\\in\\T}\\mathcal E^h(t,x)\\leq K,\n\\end{align}\nuniformly in $h$. This assumption is the natural counterpart of the corresponding hypothesis in the deterministic case made in various papers such as \\cite{AbLu,feireisl2021numerical, KuLu, LuYu}.\n\nSince the scheme \\eqref{eq:DG_standard_2} is formulated in terms of the total energy $\\mathcal E^h$, while Definition \\ref{E:dfn} is formulated in terms of the total entropy $S$ we need to introduce the approximate total entropy $S^h$ given by\n\\begin{align*}\nS^h=c_v\\varrho^h\\log\\bigg((\\gamma-1)\\bigg(\\mathcal E^h-\\frac{1}{2}\\frac{|\\vec m^h|^2}{\\varrho^h}\\bigg)\\bigg)-(c_v+1)\\varrho^h\\log(\\varrho^h).\n\\end{align*}\nAlso, the approximate temperature $\\vartheta^h$ is given as\n\\begin{align*}\n\\vartheta^h=\\frac{\\gamma-1}{\\varrho^h}\\bigg(\\mathcal E^h-\\frac{1}{2}\\frac{|\\vec m^h|^2}{\\varrho^h}\\bigg).\n\\end{align*}\nNow we formulate our main result concerning the consistency of the scheme \\eqref{eq:DG_standard_2}.\n\\begin{thm}\\label{thm:main1}\nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ be a complete stochastic basis with a probability measure $\\p$ on $(\\Omega,\\FF)$ and right-continuous filtration $(\\FF_t)_{t\\geq 0}$ and $W$ an $(\\FF_t)$-adapted Wiener process. Suppose that $\\phi\\in L_2(\\mathfrak U;L^2(\\T))$ satisfies \\eqref{eq:HS}. Let $(\\varrho^h,\\vec m^h,\\mathcal E^h)$ be the solution to \\eqref{eq:DG_standard_2}. Suppose there is an $(\\mathfrak F_t)$-stopping time $\\mathfrak t$ such that \\eqref{ass:main} holds. Suppose that the initial data $(\\varrho_0,\\vec m_0,S_0)$ satisfies\n\\begin{align*}\n\\varrho_0,S_0\\in\\ L^\\gamma(\\T),\\quad\\vec m_0\\in L^{\\frac{2\\gamma}{\\gamma+1}}(\\T),\\quad\\inf_{x\\in\\T}\\varrho_0>K,\\quad \\sup_{x\\in\\T}\\mathcal E_0<K.\n\\end{align*}\n Then it there is a null-sequence $(h_m)$ such that for all $q<\\infty$\n \\begin{equation}\n \\begin{cases}\n {\\varrho}^{h_m} \\to \\Tilde{\\varrho} \\,\\, \\text{in }\\,\\, C([0,\\tilde{\\mathfrak t}];W^{-4,2})\\cap C_w([0,\\Tilde{\\mathfrak t}]; L^{\\gamma}(\\T)),\\\\\n {S}_{h_m} \\to \\Tilde{S} \\,\\, \\text{in }\\,\\, L^q(0,T;W^{-\\mathfrak p-4,2}(\\T)),\\\\\n {S}_{h_m} \\rightharpoonup^\\ast \\Tilde{S} \\,\\, \\text{in }\\,\\, L^\\infty(0,T; L^{\\gamma}(\\T)),\\\\\n {\\vec{m}}^{h_m} \\to \\Tilde{\\vec m} \\, \\, \\text{in}\\, \\, C([0,\\tilde{\\mathfrak t}];W^{-4,2}(\\T))\\cap C_w ([0,\\Tilde{\\mathfrak t}];L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)),\\\\\n \\varrho^{h_m}\\vartheta^{h_m} \\to \\Tilde{\\varrho}\\Tilde{\\vartheta}+\\Tilde{\\mathcal{R}}_{\\text{press}} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(0,T;\\mathcal{M}^{+}(\\T,\\R)),\\\\\n \\frac{{\\vec m}^h\\otimes {\\vec m}^h}{\\varrho^h} \\to \\frac{\\Tilde{\\vec m}^h\\otimes \\Tilde{\\vec m}^h}{\\Tilde\\varrho^h}+\\Tilde{\\mathcal{R}}_{\\text{conv}} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(0,T;\\mathcal{M}^{+}(\\T,\\R^{3\\times 3})), \\\\\n\\delta_{(\\varrho^h,\\vec m^h,S^h)}\\to \\Tilde{\\mathcal{V}}_{t,x} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(Q;\\mathcal{P}(A)), \n \\end{cases}\n \\end{equation}\nin law as $h\\rightarrow0$, where, for a filtered probability space $(\\Tilde{\\Omega},\\Tilde\\FF, (\\Tilde\\FF_t)_{t\\geq 0},\\Tilde\\p)$, an $(\\Tilde\\FF_t)$-adapted Wiener process $\\Tilde W$ and an $(\\Tilde\\FF_t)$-stopping time $\\Tilde{\\mathfrak t}$,\n\\[\n((\\Tilde{\\Omega},\\Tilde\\FF, (\\Tilde\\FF_t)_{t\\geq 0},\\Tilde\\p),\\Tilde\\varrho,\\Tilde{\\vec m},\\Tilde S, \\Tilde{\\mathcal{R}}_{\\text{conv}},\\Tilde{\\mathcal{R}}_{\\text{press}},\\Tilde{\\mathcal{V}}_{t,x},\\Tilde{\\mathfrak t}, \\Tilde{W})\n\\]\nis a dissipative measure-valued solution to (\\ref{Euler}) in the sense of Definition \\ref{E:dfn} with initial law $\\delta_{\\varrho_0}\\otimes\\delta_{\\vec m_0}\\otimes\\delta_{S_0}$. It holds $\\tilde{\\mathfrak t}\\sim \\mathfrak t$ in law.\n\\end{thm}\n\\begin{rmk}\nA general framework for the numerical approximation of martingale solutions to nonlinear stochastic PDEs is given in \\cite{OPW}. However, it is rather designed for parabolic problems and does not include the possibility for the solution to develop defect measures which is crucial for hyperbolic problems as considered here.\n\\end{rmk}\nIf we have a strong pathwise solution (as described in Definition \\ref{def:compstrong}) we can obtain a stronger result. The strong solution exists on a given stochastic basis up to the stopping time $\\mathfrak s$. For the strong solution, there is no vaccum and the energy is bounded, i.e., a counterpart of \\eqref{ass:main} is satisfied. Hence one may hope that $\\mathfrak s$ is significantly smaller than $\\mathfrak t$. For $M\\in\\N$\nwe introduce the stopping time\n\\[\n\\mathfrak s_{M}=\\inf\\bigg \\{t\\in (0,\\mathfrak{s})|\\quad\\|(r,\\Theta,\\vec v)(s,\\cdot)\\|_{W^{\\mathfrak p+1,\\infty}(\\T)}>M\\bigg\\},\n\\]\nwhere $\\mathfrak p$ is the degree of the polynomials in the DG solution space, cf. Section \\ref{sec:DG}.\nSince $[r,\\Theta,\\vec v]$ is a strong solution,\n\\[\n\\p\\left[\\lim_{M\\to \\infty}\\mathfrak s_M=\\mathfrak{s}\\right]=1.\n\\]\nOur main result regarding convergence is the following theorem.\n\\begin{thm}\\label{thm:main2}\nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ be a complete stochastic basis with a probability measure $\\p$ on $(\\Omega,\\FF)$ and right-continuous filtration $(\\FF_t)_{t\\geq 0}$, $W$ an $(\\FF_t)$-adapted Wiener process and $\\phi\\in L_2(\\mathfrak U;W^{\\ell,2}(\\mathcal O))$ for some $\\ell\\geq\\max\\{4,\\mathfrak p+3\\}$. Let $\\mathbf{U}^h=(\\varrho^h,\\vec m^h,\\mathcal E^h)$ be the solution to \\eqref{eq:DG_standard_2}. Suppose there is an $(\\mathfrak F_t)$-stopping time $\\mathfrak t$ such that \\eqref{ass:main} holds.\nLet the triplet $[r,\\Theta,\\vec v]$ and a stopping time $\\mathfrak{s}$ be a strong solution in the sense of Definition \\ref{def:compstrong} with initial data\n\\begin{align*}\n\\varrho_0,\\vec m_0,\\Theta_0\\in W^{\\mathfrak p+3,2}(\\T),\\quad\\inf_{x\\in\\T}\\varrho_0>K,\\quad \\sup_{x\\in\\T}\\mathcal E_0<K.\n\\end{align*}\n\\begin{enumerate}\n\\item[(a)] Let $\\mathfrak p=0$. Then it holds for all $t\\in[0,T]$ and $M\\in\\N$\n\\begin{align}\\label{eq:errorA}\n\\E \\Big[\\Big(\\|\\varrho^h-r\\|^2_{L^2(\\T)}+\\|\\vec u^h-\\vec v\\|^2_{L^2(\\T)}+\\|\\vartheta^h-\\Theta\\|^2_{L^2(\\T)}\\Big)(t\\wedge\\mathfrak t\\wedge\\mathfrak s_M)\\Big]\\leq\\,c h^{\\frac{1}{2}}.\n\\end{align}\n\\item[(b)] For $\\mathfrak p\\geq1$, we further assume that there is some deterministic $\\mathfrak c>0$ independent of $h$ such that\n\\begin{equation}\\label{as_1}\n|\\mathbf{U}^h (\\mathbf{x}_j ,t\\wedge\\mathfrak t) - \\mathbf{U}^h (\\mathbf{x}_i ,t\\wedge \\mathfrak t) | \\leq \\mathfrak c h \\qquad \\forall x_j, x_i \\in \\mathcal{K}.\n\\end{equation}\n Then, it holds for all $t\\in[0,T]$ and $M\\in\\N$\n\\begin{align}\\label{eq:errorB}\n\\E \\Big[\\Big(\\|\\varrho^h-r\\|^2_{L^2(\\T)}+\\|\\vec u^h-\\vec v\\|^2_{L^2(\\T)}+\\|\\vartheta^h-\\Theta\\|^2_{L^2(\\T)}\\Big)(t\\wedge\\mathfrak t\\wedge\\mathfrak s_M)\\Big]\\leq\\,c h.\n\\end{align}\n\\end{enumerate}\nThe constant $c$ above depends on $M$ and $K$ from \\eqref{ass:main}. \n\\end{thm}\n\\begin{rmk}\nThere is hardly any literature on the numerical approximation of local strong solutions to SPDEs. In fact, the only result which is comparable to Theorem \\ref{thm:main2} we are aware of is \\cite{BrDo}. In \\cite{BrDo} local strong solutions to the 3D incompressible stochastic Navier--Stokes equations are considered.\n\\end{rmk}\n\\begin{rmk}\\label{rmk:noise}\nIn the case of a truly infinite dimensional noise, a practical implementation requires an approximation by a finite sum, i.e., replacing $W$ by $W^N=\\sum_{k=1}^Ne_k\\beta_k$ for some large $N\\in\\N$. This leads to the additional error term\n\\begin{align*}\n\\sum_{k=N+1}^\\infty\\int_0^{\\tau\\wedge\\mathfrak s_m\\wedge \\mathfrak t}\\int_{\\T}\\varrho^h|\\phi e_k|^2\\dx\\dt\\leq c(K)\\sum_{k=N+1}^\\infty\\int_{\\T}|\\phi e_k|^2\\dx\n\\end{align*}\nin the proof of Theorem \\ref{thm:main2}. Its size can be controlled by the choice of $N$.\n\\end{rmk}\n\n\\section{Consistency}\\label{mvsproof}\n\nThe aim of this section is to prove Theorem \\ref{thm:main1}. First of all, we can use the boundedness of $\\mathcal E^h$ from \\eqref{ass:main}. Arguing as in \n\\cite{HoFeBr}\nwe can deduce the following bounds\n\\begin{align}\\label{eq:2512}\\begin{aligned}\n \\varrho^h&\\in L^{\\infty}([0,\\mathfrak t];L^{\\gamma}(\\T)), \\\\\n \\vec m^h &\\in L^{\\infty}([0,\\mathfrak t];L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)),\\\\\n \\frac{\\vec m^h}{\\sqrt{\\varrho^h}}&\\in L^{\\infty}([0,\\mathfrak t];L^{2}(\\T)),\\\\\n \\frac{\\vec m^h \\otimes \\vec m^h}{\\varrho^h}&\\in L^{\\infty}([0,\\mathfrak t];L^{1}(\\T)),\\\\\n S^h&\\in L^{\\infty}([0,\\mathfrak t];L^{\\gamma}(\\T)),\\\\\n \\frac{ S^h}{\\sqrt{\\varrho^h}}&\\in L^{\\infty}([0,\\mathfrak t];L^{2\\gamma}(\\T)),\n\\end{aligned}\n\\end{align}\nwhich hold $\\p$-a.s. uniformly in $h$.\n\n\\subsection{The consistency formulation}\nDue to our shape regular mesh, cf. \\cite{KuLu}, and by following line by line the proof of Theorem 4.2 in \\cite{LuOe} we obtain for\n all $\\tau \\in (0,T]$: \n \\begin{itemize}\n \\item for all $\\varphi \\in C^{p+1}(\\overline{\\T})$:\n \\begin{equation}\\label{eq:consistency_rho}\n \\left[ \\int_{\\T} \\varrho^h \\varphi \\diff \\bx \\right]_{t=0}^{t=\\tau} =\\int_0^\\tau \\int_{\\T} \\bm^h \\cdot \\nabla_{\\bx} \\varphi \\diff \\bx \\diff t\n +\\int_0^\\tau e_{\\varrho^h} (t,\\varphi) \\diff t;\n \\end{equation}\n\n \n \\item for all $\\boldsymbol{\\varphi}\\in C^{p+1}(\\overline{\\T};\\R^2)$:\n \\begin{equation}\\label{eq:consistency_m}\n\\begin{aligned}\n \\left[ \\int_{\\T} \\bm^h \\boldsymbol{\\varphi} \\diff \\bx \\right]_{t=0}^{t=\\tau} &=\\int_0^\\tau \\int_{\\T} \\Big(\\frac{\\bm^h\\otimes\\bm^h}{\\varrho^h} : \\nabla_{\\bx}\\boldsymbol{\\varphi} +\\varrho^h\\vartheta^h \\mathrm{div}_{\\bx} \\boldsymbol{\\varphi} \\Big)\\diff \\bx \\diff t\\\\\n&+\\int_0^\\tau\\int_{\\T} \\varrho^h\\phi\\cdot\\Pi_h\\boldsymbol\\varphi\\diff \\bx \\,\\dd W + \\int_0^\\tau e_{\\bm^h} (t,\\boldsymbol{\\varphi}) \\diff t,\n\\end{aligned}\n \\end{equation}\n where $\\Pi_h$ is the projection onto the solution space $\\mathcal V^h$;\n \\item for all $\\psi \\in C^{p+1}( \\overline{\\T}), \\; \\psi\\geq 0$:\n \\begin{equation}\\label{eq:consistency_S}\n \\left[ \\int_{\\T} S^h \\psi \\diff \\bx \\right]_{t=0}^{t=\\tau} \\leq \\int_0^\\tau \\int_{\\T} S^h \\frac{\\bm^h}{\\varrho^h} \\cdot \\nabla_\\bx {\\psi} \\diff \\bx \\diff t + \\int_0^\\tau e_{S^h} (t,\\psi) \\diff t;\n \\end{equation}\n \\item We have the energy balance\n\\begin{align} \\label{eq:consistency_E}\n\\int_{\\T} \\mathcal E^h(\\tau)\\diff \\bx =\\int_{\\T} \\mathcal E^h_0 \\diff \\bx+\\frac{1}{2}\\int_0^\\tau\\|\\sqrt{\\varrho^h}\\phi\\|_{L_2}^2 \\,\\dd t+ \\int_0^\\tau\\int_{\\T}\\phi\\cdot \\vec m^h\\diff\\bx\\,\\dd W\n\\end{align}\n \\item The error $ \\mathbf e^h:=(e_{j\\varrho^h},e_{\\vec m^h}, e_{S^h})$ tends to zero under mesh refinement: We have $\\p$-a.s.\n \\begin{equation}\\label{eq:consistency_error}\n \\norm{\\mathbf e^h(\\varphi,\\boldsymbol{\\varphi},\\psi)}_{L^1(0,\\mathfrak t)} \\to 0 \\text{ if } h\\to 0. \n \\end{equation}\n \\end{itemize}\nNote that the proof of \\eqref{eq:consistency_error} in \\cite{LuOe} is only based on the bounds from \\eqref{ass:main}, which is the reason why it only holds up to the stopping time in our case. We also remark that, in fact, a stronger statement is proved in \\cite{LuOe} for the case $\\mathfrak p\\geq 1$ and under the additional assumption \\eqref{as_1}. In our case it can be written as\n \\begin{equation}\\label{eq:consistency_errorB}\n \\sup_{t\\in[0,\\mathfrak t]}|\\mathbf e^h(\\varphi,\\boldsymbol{\\varphi},\\psi)|\\leq\\,ch\\|(\\varphi,\\boldsymbol{\\varphi},\\psi)\\|_{W^{\\mathfrak p+1,\\infty}(\\T)},\n \\end{equation}\nwhere $c=c(\\mathcal{K})$ with $\\mathcal{K}$ from \\eqref{ass:main}.\n\nIf $\\mathfrak p=0$ instead, it is proved using the Godunov/(local) Lax-Friedrichs fluxes that\n \\begin{equation}\\label{eq:consistency_errorA}\n \\sup_{t\\in[0,\\mathfrak t]}|\\mathbf e^h(\\varphi,\\boldsymbol{\\varphi},\\psi)|\\leq\\,ch^{\\frac{1}{2}}\\|(\\varphi,\\boldsymbol{\\varphi},\\psi)\\|_{W^{1,\\infty}(\\T)}.\n \\end{equation}\nThe latter is based on a weak BV-estimate.\n\\begin{rmk}\nIt should be stressed out that in the case $\\mathfrak p=0$ we are in the classical FV context. The $L^1$ error in the relative energy with $1/2$ corresponds to $1/4$ in $L^2$ of the conserved (primitive) quantities. Under the additional assumption that the total variation of the numerical solution is uniformly bounded $1/2$ in the $L^2$-norm is achieved, cf. \\cite{LuSh}.\nThe study in \\cite{LuSh} employs the Godunov method. \nNonetheless, the findings can be applied to the Lax-Friedrichs FV method by following the same analysis.\nMoreover, it is important to highlight that, in the examination of consistency within the DG framework, the additional assumption \\eqref{as_1} was introduced regarding the variation among the numerical solution at different nodal points within a single element and their proportional adjustments concerning the grid parameter $h$. The application of limiters was also disregarded. Including the application of limiters, a result similar to the FV setting can only be expected.\\footnote{However, it is still unclear how limiters can be constructed and incorporated for the stochastic setting which will be part of future research.} This also is in line with the analysis of the flux corrected FE method approach using monolithic convex limiting and residual distribution schemes. In these contexts, one can directly follow the analysis presented in \\cite{LuSh}, expecting convergence rates of $1/4$ or $1/2$ only if the total variation of the numerical solution remains uniformly bounded.\nNote that it is currently work in progress to develop new deterministic a priori error bounds for the DG method (as well as RD and MCL) by means of the consistency analysis.\n\n\\end{rmk}\n\n In order to prepare the proof of compactness we need some time-regularity of the variables $\\varrho^h,\\vec m^h$ and $S^h$.\n For the stochastic term in the momentum equation we have\n using continuity of $\\Pi_h$\n\\begin{align*}\n\\E \\left[ \\left \\|\\int_{0}^{\\cdot} \\Pi_h(\\varrho^h \\phi )\\,\\dd W\\right\\|_{C^{\\alpha}([0,T];L^2(\\T))}^{q}\\right] \\leq\\, c \\,\\E \\left[ \\int_{0}^{T}\\Big\\|\\Pi_h(\\sqrt{\\varrho^h} \\phi)\\Big\\|^{q}_{L_2(\\UU,L^2(\\T))} \\, \\dd t \\right]\n\\end{align*}\n for all $\\alpha \\in (0,1/2-1/q)$ and $q>2$, see [\\cite{Brt_1}, Lemma 9.1.3. b)] or [\\cite{Hofm}, Lemma 4.6]). Using continuity of $\\Pi_h$ we end up with\n\\begin{align*}\n\\E \\left[ \\left \\|\\int_{0}^{\\cdot} \\Pi_h(\\varrho^h \\phi) \\,\\dd W\\right\\|_{C^{\\alpha}([0,T];L^2(\\T))}^{q}\\right] \\leq\\, c \\,\\E \\left[ \\int_{0}^{T}\\Big\\|\\sqrt{\\varrho^h} \\phi\\Big\\|^{q}_{L_2(\\UU,L^2(\\T))} \\, \\dd t \\right] \\leq\\, c(\\overline{\\varrho},q,\\phi,T).\n\\end{align*}\nThe final estimate follows from $\\int_{\\T}\\varrho^h\\dx=\\int_{\\T}\\varrho_0\\dx$ and \\eqref{eq:HS}.\n\nNow combining the deterministic estimates from \\eqref{eq:2512} and \\eqref{eq:consistency_errorB} with the stochastic estimate above, and using the embeddings $W^{1,2}_t \\hookrightarrow C^{\\alpha}_{t}$ and $L^2_x \\hookrightarrow W^{-3,2}_x$ shows \n \\begin{equation}\\label{estimate}\n\\E \\left[\\big\\|\\vec{m}^h\\big\\|_{C^{\\alpha}([0,\\mathfrak t];W^{-\\mathfrak p-3,2}(\\T))} \\right] \\leq c(T)\n \\end{equation}\nfor all $\\alpha < 1/2$ as a consequence of \\eqref{eq:consistency_m}. Similarly, the continuity equation \\eqref{eq:consistency_rho} together with \\eqref{eq:2512} and \\eqref{eq:consistency_errorB} yields $\\partial_t \\varrho_{\\varepsilon} \\in L^{\\infty}(0,\\mathfrak t;W^{-\\mathfrak p-3,2}(\\T))$ $\\p$-a.s. In particular, we obtain\n\n\\[\n\\E\\left[\\big\\| \\varrho^h\\big\\|_{C^{\\alpha}([0,\\mathfrak t];W^{-\\mathfrak p-3,2}(\\T))}\\right] \\leq C.\n\\]\nFinally, we deduce\n\\begin{align} \\label{eq:12.01}\n\\E\\left[\\| S^h\\|_{\\mathrm{BV}([0,\\mathfrak t];W^{-\\mathfrak p-3,2}(\\T))}\\right] \\leq C\n\\end{align}\nfrom \\eqref{eq:consistency_S}.\n\n\\subsection{Compactness Argument}\n\n\nWe set the spaces (where $q<\\infty$ is arbitrary):\n\n\\begin{eqnarray*}\n\\mathscr{X}_{\\Vec{m}}: = C([0,T];W^{-\\mathfrak p-4,2}(\\T))\\cap C_w ([0,T];L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)),&\\quad& \\mathscr{X}_{W} :=C([0,T];\\UU_0),\\\\\n\\mathscr{X}_{\\varrho}:=C([0,T];W^{-\\mathfrak p-4,2}(\\T))\\cap C_w([0,T]; L^{\\gamma}(\\T)), &\\quad& \\mathscr{X}_{\\text{C}}:=L^{\\infty}_{w*}(0,T;\\mathcal{M}(\\T,\\R^{3\\times3})),\\\\\n\\mathscr{X}_{ S}:=L^q(0,T;W^{-\\mathfrak p-4,2}(\\T))\\cap L^\\infty_{w^\\ast}(0,T; L^{\\gamma}(\\T)), &\\qquad& \\mathscr{X}_{\\text{Q}}:=L^{\\infty}_{w*}(0,T;\\mathcal{M}(\\T,\\R^3)),\\\\\n\\mathscr X_{\\mathfrak t}=\\R,\\qquad\\mathscr{X}_{\\text{P}}:=L^{\\infty}_{w*}(0,T;\\mathcal{M}^{+}(\\T,\\R)), &\\qquad&\n\\mathscr{X}_{\\text{e}}:=L^2(0,T;W^{-\\mathfrak p-4}(\\T)),\n\\end{eqnarray*}\n with respect to weak-* topology for all spaces with $L^{\\infty}(\\cdot,\\mathcal{M}^{\\cdot}(\\cdot))$. Furthermore, for $T>0$ fixed, we choose the product path space\n\\begin{equation}\n \\mathfrak{X}:=\\mathscr{X}_{\\varrho }\\times \\mathscr{X}_{ \\Vec{m}} \\times\\mathscr{X}_{ S} \\times\\mathscr{X}_{\\text{prss}} \\times\\mathscr{X}_{\\text{conv}}\\times\\mathscr X_{\\text{Q}}\\times \\mathscr{X}_{\\text{e}}\\times \\mathscr{X}_{W}\\times \\mathscr{X}_{\\mathfrak t},\n\\end{equation}\nwhere $\\mathscr{X}_{\\text{prss}}$, $\\mathscr{X}_{\\text{conv}}$ and $\\mathscr X_{\\text{Q}}$ are the path spaces for the nonlinear terms\n\\[\n \\mathrm{P}^h:=(\\varrho^h)^{\\gamma} \\exp{\\left( \\frac{ S^h}{c_v \\varrho^h}\\right)},\\qquad\\mathrm{C}^h:=\\frac{\\vec m^h\\otimes \\vec m^h}{\\varrho^h},\\qquad \\mathrm Q^h:= S^h \\frac{\\vec m^h}{\\varrho^h},\n\\]\nrespectively. \nWe denote by $\\mathscr B_{\\mathfrak X}$ the Borelian $\\sigma$-algebra on $\\mathfrak X$ and by $$\\mathcal{L}[\\varrho^h(\\cdot\\wedge\\mathfrak t),\\vec{m}^h(\\cdot\\wedge\\mathfrak t), \\mathrm S^h(\\cdot\\wedge\\mathfrak t), P^h(\\cdot\\wedge\\mathfrak t), \\mathrm C^h(\\cdot\\wedge\\mathfrak t), \\mathrm Q^h,\\mathbf e^h(\\cdot\\wedge\\mathfrak t), W,\\mathfrak t]$$ the probability law on $\\mathfrak{X}$. \nAs in \\cite{Mo} we can infer that it is a sequence of tight measures on $\\mathfrak{X}$. Note that the variables $\\mathfrak t$ and $\\mathbf e^h$ are not included in \\cite{Mo}. The law of $\\mathfrak t$ is clearly tight as being a Radon measure on the Polish space $\\R$, while tightness of $\\mu_{ \\mathbf{e}^h}$ follows from \\eqref{eq:consistency_errorB}. Also, tightness of the law of $S^h$ follows from \\eqref{eq:12.01} and Helly’s selection theorem.\nIn view of Jakubowksi's version of the Skorokhod representation theorem \\cite{Jaku} (see also Brzezniak et al. \\cite{Brzez}, and \\cite[Section 2.8]{FrBrHo} for property c), we have the following proposition.\n\n\\begin{prop} \\label{skorokhod}\nThere exists a nullsequence $(h_m)_{m \\in \\N}$, a complete probability space $(\\Tilde{\\Omega}, \\Tilde{\\FF},\\Tilde{\\p})$ with $(\\mathfrak{X},\\mathscr{B}_{\\mathfrak{X}})$-valued random variables $(\\Tilde{\\varrho}^{h_m},\\Tilde{\\vec{m}}^{h_m},\\Tilde{ S}^{h_m}, \\Tilde{P}^{h_m}, \\Tilde{C}^{h_m},\\Tilde{Q}^{h_m},\\Tilde{\\mathbf e^{h_m}} ,\\Tilde{W}^{h_m},\\Tilde{\\mathfrak t}^{h_m})$, $m \\in \\N$,\\\\ and $(\\Tilde{\\varrho},\\Tilde{\\vec{m}}, \\Tilde{ S},\\Tilde{P},\\Tilde{C},\\Tilde{Q},\\tilde{\\mathbf e}, \\Tilde{W},\\tilde{\\mathfrak t})$ such that\n\\begin{itemize}\n \\item [(a)] For all $m \\in \\N$ the law of $$(\\Tilde{\\varrho}^{h_m},\\Tilde{\\vec{m}}^{h_m},\\Tilde{ S}^{h_m}, \\Tilde{P}^{h_m}, \\Tilde{C}^{h_m},\\Tilde{Q}^{h_m},\\Tilde{\\mathbf e^{h_m}} ,\\Tilde{W}^{h_m},\\Tilde{\\mathfrak t}^{h_m})$$ on $\\mathfrak{X}$ coincides with $$\\mathcal{L}[\\varrho^{h_m}(\\cdot\\wedge\\mathfrak t),\\vec{m}^{h_m}(\\cdot\\wedge\\mathfrak t), \\mathrm S^{h_m}(\\cdot\\wedge\\mathfrak t), P^{h_m}(\\cdot\\wedge\\mathfrak t), \\mathrm C^{h_m}(\\cdot\\wedge\\mathfrak t), \\mathrm Q^{h_m},\\mathbf e^{h_m}(\\cdot\\wedge\\mathfrak t), W,\\mathfrak t];$$ \n \\item[(b)] The sequence \n$$(\\Tilde{\\varrho}^{h_m},\\Tilde{\\vec{m}}^{h_m},\\Tilde{ S}^{h_m}, \\Tilde{P}^{h_m}, \\Tilde{C}^{h_m},\\Tilde{Q}^{h_m},\\Tilde{\\mathbf e^{h_m}} ,\\Tilde{W}^{h_m},\\Tilde{\\mathfrak t}^{h_m}), \\,\\,m \\in \\N,$$ converges $\\Tilde{\\p}$-almost surely to $$(\\Tilde{\\varrho},\\Tilde{\\vec{m}}, \\Tilde{ S},\\Tilde{P},\\Tilde{C},\\Tilde{Q},\\tilde{\\mathbf e}, \\Tilde{W},\\tilde{\\mathfrak t})$$ in the topology of $\\mathfrak{X}$, i.e.\n \\begin{equation}\n \\begin{cases}\n \\Tilde{\\varrho}^{h_m} \\to \\Tilde{\\varrho} \\,\\, \\text{in }\\,\\, C([0,T];W^{-\\mathfrak p-4,2})\\cap C_w([0,T]; L^{\\gamma}(\\T)),\\\\\n \\Tilde{S}^{h_m} \\to \\Tilde{S} \\,\\, \\text{in }\\,\\, L^q(0,T;W^{-\\mathfrak p-4,2}(\\T)),\\\\\n \\Tilde{S}^{h_m} \\rightharpoonup^\\ast \\Tilde{S} \\,\\, \\text{in }\\,\\, L^\\infty(0,T; L^{\\gamma}(\\T)),\\\\\n \\Tilde{\\vec{m}}_{h_m} \\to \\Tilde{\\vec m} \\, \\, \\text{in}\\, \\, C([0,T];W^{-\\mathfrak p-4,2}(\\T))\\cap C_w ([0,T];L^{\\frac{2\\gamma}{\\gamma +1}}(\\T)),\\\\\n \\Tilde{P}^{h_m} \\to \\overline{\\Tilde{P}} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(0,T;\\mathcal{M}^{+}(\\T,\\R)),\\\\\n \\Tilde{C}^{h_m} \\to \\overline{\\Tilde{C}} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(0,T;\\mathcal{M}^{+}(\\T,\\R^{3\\times 3})), \\\\\n \\Tilde{Q}^{h_m} \\to \\overline{\\Tilde{Q}} \\,\\, \\text{in} \\,\\, L_{w^*}^{\\infty}(0,T;\\mathcal{M}(\\T,\\R^3)), \\\\\n \\Tilde{\\vec{e}}^{h_m} \\to 0 \\, \\, \\text{in}\\, \\, L^2(0,T;W^{-\\mathfrak p-4,2}(\\T)),\\\\\n \\Tilde{W}^{h_m} \\to \\Tilde{W} \\,\\, \\text{in} \\,\\, C([0,T];\\UU_0),\\\\\n \\Tilde{\\mathfrak t}^{h_m} \\to \\Tilde{\\mathfrak t} \\,\\, \\text{in} \\,\\, \\R,\n \\end{cases}\n \\end{equation}\n $\\Tilde{\\p}$-a.s.\n \\item[(c)] For any Carath\\'eodory function $H =H(t,x,\\varrho,\\vec m, S)$ where $(t,x)\\in (0,T)\\times\\T,\\, \\, (\\varrho,\\vec m,S)\\in \\R^5$, satisfying for some $q_1,q_2,q_3>0$ the growth condition\n \\[\n H(t,x,\\varrho,\\vec m, S)\\lesssim 1 +|\\varrho|^{q_1}+|\\vec m|^{q_2}+|S|^{q_2}\n \\]\n uniformly in $(t,x)$, we denote by $\\overline{H(\\varrho,\\vec m, S)}(t,x) =\\langle \\mathcal{V}_{t,x},H \\rangle$. Then it holds\n \\[\n H(\\tilde{\\varrho}^{h_m},\\tilde{\\vec m}^{h_m}, \\tilde{S}^{h_m}) \\rightharpoonup \\overline{H(\\tilde{\\varrho},\\tilde{\\vec m}, \\tilde{S})}\\quad\\text{in}\\,\\,L^{k}((0,T)\\times\\T)\n \\]\n $\\tilde{\\p}$-a.s. as $m \\to \\infty$ for all $1 < k\\leq\\frac{\\gamma+1}{q_1}\\wedge \\frac{2}{q_2}$.\n \n\\end{itemize}\n\\end{prop}\n\n\nTo guarantee adaptedness of random variables and to ensure that the stochastic integral is well-defined on the new probability space we introduce filtrations for correct measurability. We simplify notation and set\n$ \\Tilde{\\mathcal{X}}:=\\left[\\tilde{\\varrho},\\tilde{\\vec{m}},\\tilde{ S}\\right]$.\nLet $\\tilde{\\FF}_t$ and $\\tilde{\\FF}_{t}^{h_m}$ be the $\\tilde{\\p}$-augmented filtration of the correspnding random variables from Proposition \\ref{skorokhod}, i.e.\n\\begin{align*}\n\\tilde{\\FF}_t &= \\sigma (\\sigma(t\\vee\\tilde{\\mathfrak t},\\rr \\Tilde{\\mathcal{X}},\\rr\\tilde{W}) \\cup \\sigma_t( \\tilde{P},\\tilde{C},\\tilde{Q},\\tilde e)\\cup\\{ \\mathcal{N} \\in \\tilde{\\FF};\\tilde{\\p}(\\mathcal{N})=0\\}), t\\geq 0,\\\\\n\\tilde{\\FF}_{t}^{\\varepsilon_m}&=\\sigma(\\sigma(t\\vee\\Tilde{\\mathfrak t},\\rr\\Tilde{\\mathcal{X}}^{\\varepsilon_m},\\rr\\tilde{W}^{h_m})\\cup \\sigma_t (\\tilde{P}^{h_m},\\tilde{C}^{h_m},\\tilde{Q}^{h_m},\\tilde{\\mathbf e}^{h_m})\\cup\\{ \\mathcal{N} \\in \\tilde{\\FF};\\tilde{\\p}(\\mathcal{N})=0\\}), t\\geq 0.\n\\end{align*}\n\nHere $\\rr$ denotes the restriction operator to the interval $[0,t]$ on the path space and $\\sigma_t$\\footnote{The family of $\\sigma$-fields $(\\sigma_{t}[\\vec V])_{t\\geq 0}$ given as random distribution history of \n \\begin{equation*}\n \\sigma_t[\\vec V]:= \\bigcap_{s>t}\\sigma\\left(\\bigcup_{\\varphi\\in C_c^{\\infty}(Q;\\R^3)}\\{\\langle \\vec V, \\varphi \\rangle <1 \\}\\cup \\{N\\in \\FF, \\p(N)=0\\} \\right)\n \\end{equation*}\n is called the history of $\\vec V$. In fact, any random distribution is adapted to its history, see\\cite{FrBrHo} (Chap. 2.2).} denotes the history of a random distribution.\n\n\\subsection{Passage to the limit}\nNoticing that \\eqref{eq:DG_standard_2} is a finite dimensional problem we can write in the form\n\\begin{align}\\label{eq:1406a}\nX_t^m&=X^m(0)+\\int_0^t\\mu(X^m_s)\\,\\dd s+\\int_0^t\\Sigma(X_s)\\,\\dd W,\n\\end{align}\nwhere $(X_t^m)$ is an $\\R^N$-valued stochastic process (in, fact $X_t^m$ corresponds to $\\bvu$ from \\eqref{eq:DG_standard_2} with $h=h_m$), $\\mu:\\R^N\\rightarrow\\R^N$ and $\\Sigma:\\R^N\\rightarrow\\R^{N\\times N}$ are Lipschitz continuous functions in the range of $(X^m_{t\\wedge \\mathfrak t})$ due to \\eqref{ass:main}.\nWe have to show that $\\tilde\\p$-a.s.\n\\begin{align}\\label{eq:1406b}\n\\tilde X_{t\\wedge \\tilde{\\mathfrak t}}^m&=\\tilde X^m(0)+\\int_0^{t\\wedge \\mathfrak t}\\mu(\\tilde X^m_s)\\,\\dd s+\\int_0^{t\\wedge \\mathfrak t}\\Sigma(\\tilde X^m_s)\\,\\dd \\tilde W^m,\n\\end{align}\ni.e., the equation continues to hold on the new probability space. Here $\\tilde X_t^m$ relates to $(\\tilde\\varrho^{h_m},\\Tilde{\\vec{m}}^{h_m},\\Tilde{\\mathcal E}^{h_m})$ exactly as $X_t^m$ relates to $(\\varrho^{h_m},{\\vec{m}}^{h_m},{\\mathcal E}^{h_m})$ via \\eqref{eq:DG_standard_2}.\nBy Proposition \\ref{skorokhod} the processes $(X^m_{t\\wedge\\mathfrak t})$ and $(\\tilde X^m_t)=(\\tilde X^m_{t\\wedge\\tilde{\\mathfrak t}})$ coincide in law on $C^0([0,T];\\R^N)$ and the same holds true for $W$ and $\\tilde W$.\nThe mapping $X\\mapsto \\int_0^{\\cdot}\\mu(t,X_t)\\dt$ is continuous on the $C^0([0,T];\\R^N)$. However,\nthe mapping $(X,W)\\mapsto \\int_s^t\\Sigma(r,X_r)\\,\\diff W$ is not.\nSo, we can not identify it immediately. We will make use of the fact that a martingale is uniquely determined by its quadratic variations.\nThis can be done with the help of an\nelementary method by Brzezniak and Ondrej\\'at \\cite{BZ}.\nWe consider the functionals\n\\begin{align*}\n\\mathfrak M(Y,\\mathfrak r)_t&=Y_{t\\wedge \\mathfrak r}-Y(0)-\\int_0^{t\\wedge \\mathfrak r}\\mu(r,Y_r)\\diff r,\\\\\n\\mathfrak N(Y,\\mathfrak r)_t&=\\int_0^{t\\wedge \\mathfrak r}|\\Sigma(r,Y_r)|^2\\diff r,\\quad \\mathfrak L(Y,\\mathfrak r)_t=\\int_0^{t\\wedge \\mathfrak r}\\Sigma(r,Y_r)\\diff r\n\\end{align*}\nObviously $\\mathfrak M$, $\\mathfrak N$ and $\\mathfrak L$ are continuous on the pathspace. Consequently, by equality of laws, we have\n\\begin{align*}\n\\mathfrak M(X^m,\\mathfrak t)_{t}&\\sim^d \\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t})_{t},\\quad\n\\mathfrak N(X^m,\\mathfrak t)_{t} \\sim^d \\mathfrak N(\\tilde X^m,\\tilde{\\mathfrak t})_{t},\\quad\n\\mathfrak L(X^m,\\mathfrak t)_{t}\\sim^d \\mathfrak L(\\tilde X^m,\\tilde{\\mathfrak t})_{t}.\n\\end{align*}\nLet $\\mathfrak M(X^m,\\mathfrak t)_{s,t}$ denote the increment $\\mathfrak M(X^m,\\mathfrak t)_{t}-\\mathfrak M(X^m,\\mathfrak t)_{s}$ and similarly for $\\mathfrak N(X^m,\\mathfrak t)_{s,t}$ and $\\mathfrak N_k(X^m,\\mathfrak t)_{s,t}$. \nNote that the proof will be complete once we show that the process $\\mathfrak M(\\tilde X^m, \\tilde{\\mathfrak t})_t$ is an $(\\tilde{\\FF}_t^m)_{t\\geq0}$-martingale and its quadratic and cross variations satisfy, respectively,\n\\begin{equation}\\label{marta}\n\\begin{split}\n\\langle\\langle \\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t})\\rangle\\rangle_{t}&=\\mathfrak N(\\tilde X^m,\\tilde{\\mathfrak t})_{t},\\quad\\langle\\langle \\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t}),\\tilde W\\rangle\\rangle_{t}=\\mathfrak L(\\tilde X^m,\\tilde{\\mathfrak t})_{t}.\n\\end{split}\n\\end{equation}\nIndeed, in that case we have\n\\begin{align}\\label{neu3108a}\n\\Big\\langle\\Big\\langle \\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t})-\\int_0^{\\cdot\\wedge \\tilde{\\mathfrak t}}\\Sigma(s,\\tilde X^m_s)\\,\\dd\\tilde W\\Big\\rangle\\Big\\rangle_{t}=0\n\\end{align}\nwhich implies the desired equation \\eqref{eq:1406b}. Let us verify \\eqref{marta}. To this end, we fix times $s,t\\in[0,T]$ such that $s<t$ and let\n$$h:\\R\\times C^0([0,s];\\R^N)\\rightarrow [0,1]$$\nbe a continuous function.\nSince $\\mathfrak M(X^m,\\mathfrak t)$\nis a square integrable $(\\F_t)_{t\\geq0}$-martingale, we infer that\n$$\\big[\\mathfrak M(X^m,\\mathfrak t)\\big]^2-\\mathfrak N(X^m,\\mathfrak t),\\quad \\mathfrak M(X^m,\\mathfrak t)W-\\mathfrak L(X^m,\\mathfrak t),$$\nare $(\\FF_t)_{t\\geq0}$-martingales.\nLet $\\bfr_s$ be the restriction of a function to the interval $[0,s]$. Then it follows from the equality of laws that\n\\begin{equation}\\label{exp11}\n\\tilde{\\E}\\big[\\,h\\big(s\\vee\\tilde{\\mathfrak t},\\bfr_s\\tilde X^m,\\bfr_s\\tilde{W}^m\\big)\\mathfrak M(\\tilde{\\mathfrak t},\\tilde X^m,\\tilde{\\mathfrak t})_{s,t}\\big]=\\E \\big[\\,h\\big(s\\vee \\mathfrak t,\\bfr_sX^m,\\bfr_sW\\big)\\mathfrak M(X^m,\\mathfrak t)_{s,t}\\big]=0,\n\\end{equation}\n\\begin{equation}\\label{exp21}\n\\begin{split}\n&\\tilde{\\E}\\bigg[\\,h\\big(s\\vee\\tilde{\\mathfrak t},\\bfr_s\\tilde X^m,\\bfr_s\\tilde{W}^m\\big)\\Big([\\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t})^2]_{s,t}-\\mathfrak N(\\tilde X^m,\\tilde{\\mathfrak t})_{s,t}\\Big)\\bigg]\\\\\n&=\\E\\bigg[\\,h\\big(s\\vee{\\mathfrak t},\\bfr_sX^m,\\bfr_sW^m\\big)\\Big([\\mathfrak M(X^m,\\mathfrak t)^2]_{s,t}-\\mathfrak N(X^m,\\mathfrak t)_{s,t}\\Big)\\bigg]=0,\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{exp31}\n\\begin{split}\n&\\tilde{\\E}\\bigg[\\,h\\big(s\\vee\\tilde{\\mathfrak t},\\bfr_s\\tilde X^m,\\bfr_s\\tilde{W}^m\\big)\\Big([\\mathfrak M(\\tilde X^m,\\tilde{\\mathfrak t})\\tilde{W}^m]_{s,t}-\\mathfrak L(\\tilde X^m,\\tilde{\\mathfrak t})_{s,t}\\Big)\\bigg]\\\\\n&=\\E\\bigg[\\,h\\big(s\\vee{\\mathfrak t},\\bfr_sX^m,\\bfr_sW\\big)\\Big([\\mathfrak M(X^m,\\mathfrak t)W]_{s,t}-\\mathfrak L(X^m,\\mathfrak t)_{s,t}\\Big)\\bigg]=0.\n\\end{split}\n\\end{equation}\nSo we have shown \\eqref{marta} and hence (\\ref{neu3108a}). This finishes the proof of \\eqref{eq:1406b} and thus have shown that the equations\n \\begin{equation}\\label{eq:consistency_rhotilde}\n \\left[ \\int_{\\T} \\tilde\\varrho^h \\varphi \\diff \\bx \\right]_{t=0}^{t=\\tau\\wedge\\tilde{\\mathfrak t}} =\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\int_{\\T} \\tilde{\\bm}^h \\cdot \\nabla_{\\bx} \\varphi \\diff \\bx \\diff t\n +\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\tilde e_{\\varrho^h} (t,\\varphi) \\diff t;\n \\end{equation}\n \\begin{equation}\\label{eq:consistency_mtilde}\n\\begin{aligned}\n \\left[ \\int_{\\T} \\tilde{\\bm}^h \\boldsymbol{\\varphi} \\diff \\bx \\right]_{t=0}^{t=\\tau\\wedge\\tilde{\\mathfrak t}} &=\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\int_{\\T} \\Big(\\frac{\\tilde{\\bm}^h\\otimes\\tilde{\\bm}^h}{\\tilde\\varrho^h} : \\nabla_{\\bx}\\boldsymbol{\\varphi} +\\tilde\\varrho^h\\tilde\\vartheta^h \\mathrm{div}_{\\bx} \\boldsymbol{\\varphi} \\Big)\\diff \\bx \\diff t\\\\\n&+\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}}\\int_{\\T} \\tilde\\varrho^h\\phi\\cdot\\Pi_{h_m}\\boldsymbol\\varphi\\diff \\bx \\,\\dd \\tilde W + \\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\tilde e_{\\bm^h} (t,\\boldsymbol{\\varphi}) \\diff t,\n\\end{aligned}\n \\end{equation}\n \\begin{equation}\\label{eq:consistency_Stilde}\n \\left[ \\int_{\\T} \\tilde S^h \\psi \\diff \\bx \\right]_{t=0}^{t=\\tau\\wedge\\tilde{\\mathfrak t}} \\leq \\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\int_{\\T} \\tilde S^h \\frac{\\tilde\\bm^h}{\\tilde\\varrho^h} \\cdot \\nabla_\\bx {\\psi} \\diff \\bx \\diff t + \\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}} \\tilde e_{S^h} (t,\\psi) \\diff t;\n\\end{equation}\nhold $\\tilde\\p$-a.s. for all sufficiently smooth spatial test-functions. For the stochastic integral in we have by It\\^{o}-isometry, $\\phi\\in L_2(\\mathfrak U;L^2(\\T))$ and \\eqref{ass:main}\n\\begin{align*}\n\\tilde\\E\\bigg[\\bigg|\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}}\\int_{\\T} \\tilde\\varrho^h\\phi\\cdot(\\Pi_{h_m}-\\mathrm{id})\\boldsymbol\\varphi\\diff \\bx \\,\\dd \\tilde W\\bigg|^2\\bigg]&=\\tilde\\E\\bigg[\\int_0^{\\tau\\wedge\\tilde{\\mathfrak t}}\\bigg(\\int_{\\T} \\tilde\\varrho^h\\phi\\cdot(\\Pi_{h_m}-\\mathrm{id})\\boldsymbol\\varphi\\diff \\bx\\bigg)^2\\dt \\bigg]\\\\\n&\\leq\\,c(\\tau,K,\\phi)\\|(\\Pi_{h_m}-\\mathrm{id})\\boldsymbol\\varphi\\|^2_{L^2(\\T)},\n\\end{align*}\nwhich vanishes as $m\\rightarrow\\infty$.\nBy Proposition \\ref{skorokhod} we can pass to the limit in the remaining terms noticing that the error terms on the right-hand side vanish as $m\\rightarrow\\infty$. Hence the limit object is a dissipative solution in the sense of Definition \\ref{E:dfn} provided we can establish the energy balance. \n\n \n\nOn the original probability space, the approximate the energy equality holds in the sense that\n\\begin{eqnarray*}\n\\mathfrak E_{t\\wedge\\mathfrak t}^{h_m}= \\mathfrak E_{s\\wedge\\mathfrak t}^{h_m}+\\frac{1}{2}\\int_{s\\wedge\\mathfrak t}^{t\\wedge\\mathfrak t}\\|\\sqrt{\\varrho^{h_m}}\\phi\\|_{L_2(\\UU,L^2(\\T))}^{2}\\,\\dd \\sigma+ \\int_{s\\wedge\\mathfrak t}^{t\\wedge\\mathfrak t}\\int_{\\T}\\vec m^{h_m}\\phi \\,\\dd x\\dd W,\n\\end{eqnarray*}\n$\\p$-a.s for a.a $0\\leq s<t$ (including $s=0$), where\n\n\\[\n\\mathfrak E_{t}^{h_m} =\\int_{\\T}\\left[\\frac{1}{2}\\frac{|\\vec m^{h_m}|^2}{\\varrho^{h_m}}+ c_v(\\varrho^{h_m})^{\\gamma}\\exp{\\left(\\frac{S^{h_m}}{c_v\\varrho^{h_m}}\\right)} \\right]\\, \\dd x ,\n\\]\nfor a.a $t \\geq 0$. For any fixed $s$ this is equivalent to\n\n\\[\n-\\int_{s\\wedge\\mathfrak t}^{T\\wedge\\mathfrak t}\\partial_t \\varphi \\mathfrak E_{t\\wedge\\mathfrak t}^{h_m} \\, \\dd t- \\varphi(s)\\mathfrak E_{s\\wedge\\mathfrak t}^{h_m} = \\frac{1}{2}\\int_{s\\wedge\\mathfrak t}^{T\\wedge\\mathfrak t}\\varphi \\|\\sqrt{\\varrho^{h_m}}\\phi\\|_{L_2(\\UU,L^2(\\T))}^{2}\\,\\dd t+ \\int_{s\\wedge\\mathfrak t}^{T\\wedge\\mathfrak t}\\varphi\\int_{\\T}\\vec m^{h_m} \\cdot \\phi \\,\\dd x\\dd W,\n\\]\n$\\p$-a.s for all $\\varphi\\in C_{0}^{\\infty}([s,T))$. By virtue of Theorem 2.9.1 in \\cite{FrBrHo}, and in view of Proposition \\ref{skorokhod} the energy equality continues to hold on the new probability space and reads\n\n\\[\n\\tilde{\\mathfrak E}_{t\\wedge\\tilde{\\mathfrak t}}^{h_m}=\\tilde{\\mathfrak E}_{s\\wedge\\tilde{\\mathfrak t}}^{h_m}+\\frac{1}{2}\\int_{s\\tilde{\\wedge\\mathfrak t}}^{t\\wedge\\tilde{\\mathfrak t}}\\|\\sqrt{\\tilde{\\varrho}^{h_m}}\\phi\\|_{L_2(\\UU,L^2(\\T))}^{2}\\,\\dd \\sigma+ \\int_{s\\wedge\\tilde{\\mathfrak t}}^{t\\wedge\\tilde{\\mathfrak t}}\\int_{\\T}\\tilde{\\vec m}^{h_m}\\phi \\,\\dd x\\dd \\tilde{W}^{h_m},\n\\]\n$\\tilde{\\p}$-a.s. for a.a $s$(including $s=0$) and all $t \\geq s$. Averaging in $t$ and s, the above expression becomes continuous on the path space. Furthermore, fixing $s=0$ and Lemma 2.1 in \\cite {Debussche}, the bounds established in Proposition \\ref{skorokhod}, and higher moments to perform the limit $m\\to \\infty$ we obtain\n\n \n\\begin{equation}\\label{eq:newEner}\n \\tilde{\\mathfrak{E}}_{t\\wedge\\tilde{\\mathfrak t}}\\leq\\tilde{\\mathfrak{E}}_0+\\frac{1}{2}\\int_{s\\wedge\\tilde{\\mathfrak t}}^{t\\wedge\\tilde{\\mathfrak t}}\\|\\sqrt{\\tilde{\\varrho}}\\phi\\|_{L_2(\\UU;L^2(\\T))}^2\\, \\dd \\sigma + \\int_{s\\wedge\\tilde{\\mathfrak t}}^{t\\wedge\\tilde{\\mathfrak t}}\\int_{\\T}\\tilde{\\vec m} \\cdot \\phi \\, \\dd x \\dd \\tilde{W},\n\\end{equation}\n $\\p$-a.s. for a.a. $t\\in [0,T]$, where\n \\[\n \\tilde{\\mathfrak {E}}_t= \\int_{\\T}\\left[\\frac{1}{2}\\frac{|\\tilde{\\vec m} |^2}{\\tilde{\\varrho}}+c_v\\tilde{\\varrho}^{\\gamma}\\exp{\\left(\\frac{\\tilde{S}}{c_v\\tilde{\\varrho}}\\right)}\\right]\\, \\dd x + \\frac{1}{2}\\int_{\\T}\\dd\\, \\mathrm{tr}\\tilde{\\mathcal{R}}_{\\text{conv}}(t) +c_v\\int_{\\T}\\dd\\, \\tilde{\\mathcal{R}}_{\\text{press}}(t),\n \\]\n and \n \\[\n \\tilde{\\mathfrak E}_0= \\int_{\\T}\\left[\\frac{1}{2}\\frac{|{\\vec m}_0 |^2}{{\\varrho}_0}+c_v{\\varrho}_0^{\\gamma}\\exp{\\left(\\frac{S_0}{c_v\\varrho_0}\\right)}\\right]\\, \\dd x.\n \\]\nPerforming the limit $\\varepsilon_m\\to 0$ yields an energy inequality. \nHence, to convert (\\ref{eq:newEner}) to equality, it is sufficient to augment the term contributing to the internal energy ($\\tilde{\\mathcal{R}}_{\\text{press}}(t)$) by $h(t)\\dd x$ with A suitable spatially homogeneous $h\\geq 0$. NOte that $\\tilde{\\mathcal{R}}_{\\text{press}}(t)$ acts on $\\mathrm{div}_x\\varphi$ such that\n$\\int_{\\T}h(t)\\mathrm{div}_x \\varphi\\,\\dd x=0$\nand hence\n \\[\n-\\int_{s\\wedge\\tilde{\\mathfrak t}}^{T\\wedge\\tilde{\\mathfrak t}}\\partial_t \\varphi \\tilde{\\mathfrak E}_t \\, \\dd t- \\varphi(s)\\tilde{\\mathfrak E}_{s\\wedge\\tilde{\\mathfrak t}} = \\frac{1}{2}\\int_{s\\wedge\\tilde{\\mathfrak t}}^{T\\wedge\\tilde{\\mathfrak t}}\\varphi \\|\\sqrt{\\tilde\\varrho}\\phi\\|_{L_2(\\UU,L^2(\\T))}^{2}\\,\\dd t+ \\int_{s\\wedge\\tilde{\\mathfrak t}}^{T\\wedge\\tilde{\\mathfrak t}}\\varphi\\int_{\\T}\\tilde{\\vec m}\\cdot \\phi \\,\\dd x\\dd \\tilde W,\n\\]\n$\\p$-a.s for all $\\varphi\\in C_{0}^{\\infty}([s,T))$. This completes the proof of Theorem \\ref{thm:main1}. \n\n\\section{Convergence}\\label{se_Convergence}\nIn this section we prove Theorem \\ref{thm:main2}. The main tool is the relative entropy, which we derive in the next subsection.\n\n\\subsection{The relative entropy}\nWe consider the \\textit{relative entropy} for the discrete solution $(\\varrho^h,\\vec m^h, \\mathcal E^h)$ measuring its distance to a smooth comparison function $(r,\\Theta,\\vec v)$ (which will be chosen in the next subsection as the strong pathwise solution to \\eqref{Euler}). It is given by\n\\begin{align}\\label{relative}\n\\begin{aligned}\n\\mathcal{K}\\big(\\varrho^h,E^h,\\vec m^h\\big|r,\\Theta,\\vec v\\big) &= \\frac{1}{2}\\int_{\\T}\\varrho^h\\left|\\frac{\\vec m^h}{\\varrho^h} -\\vec v\\right|^2\\, \\dd x-\\int_{\\T}\\Theta\\varrho^h {s}(\\varrho^h,\\mathcal E^h)\\, \\dd x-\\int_{\\T}\\varrho^h\\partial_{\\varrho}H_{\\Theta}(r,\\Theta)\\,\\dd x\\\\&+\\int_{\\T}\\big(\\partial_{\\varrho}H_{\\Theta}(r,\\Theta)(r)-H_{\\Theta}(r,\\Theta)\\big)\\, \\dd x\n\\end{aligned}\n\\end{align}\nfor $t\\in[0,\\mathfrak t]$. Here we used\nthe \\textit{ballistic free energy}\n\\begin{equation}\\label{ballistic}\n H_{{\\Theta}}(\\varrho,\\vartheta)=\\varrho e(\\varrho,\\vartheta)-\\Theta \\varrho s (\\varrho,\\vartheta),\n\\end{equation}\nrecalling the formulas for $e$ and $s$ from Section \\ref{sec:cr}.\nWe obtain the following result.\n\\begin{prop}[Relative Entropy Inequality]\\label{propA}\nLet $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ be a complete stochastic basis with a probability measure $\\p$ on $(\\Omega,\\FF)$ and right-continuous filtration $(\\FF_t)_{t\\geq 0}$ and $W$ an $(\\FF_t)$-adapted Wiener process. Let $(\\varrho^h,\\vec m^h,\\mathcal E^h)$ be the solution to \\eqref{eq:DG_standard_2}. Suppose there is an $(\\mathfrak F_t)$-stopping time $\\mathfrak t$ such that \\eqref{ass:main} holds. Let $(r,\\Theta, \\vec v)$ be a trio of $(\\FF_t)_{t\\geq 0}$ -adapted stochastic processes defined on $(\\Omega, \\FF, (\\FF_t)_{t\\geq 0},\\p)$ such that for some $T>0$\n\\begin{align*}\n r,\\Theta,\\vec v \\in C([0,T]; C^{1} (\\T)) \\quad\\p \\text{-a.s.,} \\,\\,\\,\n \\E\\left[\\sup_{t\\in [0,T]}\\|(r,\\Theta,\\vec v)\\|_{W^{1,\\infty}(\\T)}\\right]^q < \\infty \\text{ for all }\\, 2\\leq q < \\infty,\n\\end{align*}\n\\[\n0<\\underline{r} \\leq r(t,x) \\leq \\overline{r}, \\quad\n0<\\underline{\\Theta} \\leq \\Theta(t,x) \\leq \\overline{\\Theta} \\quad \\p\\text{-a.s.}\n\\]\nSuppose further that it holds\n\\begin{align}\\label{eq:sass}\n\\begin{aligned}\n \\dd r &= D_t^dr\\,\\dd t,\\\\\n \\dd \\vec v &=D_t^d \\vec v \\,\\dd t+ \\mathbb{D}_t^s \\vec v\\, \\dd W,\\\\\n \\dd \\Theta &= D_t^d \\Theta\\,\\dd t,\n\\end{aligned}\n\\end{align}\nwhere\n\\[\nD^dr,D^d \\Theta,D^d\\vec v \\in L^{q}(\\Omega; C([0,T];C^{1}(\\T)))\\qquad \\mathbb{D}^s\\vec v \\in L^2(\\Omega; L^2(0,T;L_2(\\UU;L^2(\\T))),\n\\]\n\\begin{equation}\\label{eq:prop}\n \\left(\\sum_{k\\geq 1}|\\mathbb{D}^s\\vec v(e_k)|^q\\right)^{\\frac{1}{q}} \\in L^{q}(\\Omega; L^q(0,T;L^q(\\T))).\n\\end{equation}\nThen the relative entropy inequality:\n\\begin{align}\\label{REI}\n\\begin{aligned}\n \\mathcal{K}\\bigg(\\varrho^h,\\mathcal E^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)&\\leq\\,c \\,\\mathcal{K}\\bigg(\\varrho^h,\\mathcal E^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)(0)+c\\int_{0}^{\\tau}\\mathcal{Q}\\bigg(\\varrho^h,\\vartheta^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)\\, \\dd t+\\mathbb{M}\\\\\n&+c\\,\\mathbf e^h(\\partial_{\\varrho}H_{\\Theta}(r,\\Theta),\\vec v,\\Theta)\n\\end{aligned}\n\\end{align}\nholds $\\p$-a.s for all $\\tau \\in (0,T\\wedge\\mathfrak t)$, where\n\\begin{align*}\n\\mathcal{Q}\\bigg(\\varrho^h,\\vartheta^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)=\n&\\int_{\\T}\\varrho\\left(\\frac{\\vec m^h}{\\varrho^h}-\\vec v\\right)\\cdot\\nabla_x\\vec v\\cdot \\left(\\vec v-\\frac{\\vec m^h}{\\varrho^h}\\right)\\,\\dd x\\nonumber\\\\\n&+\\int_{\\T}[(D_t^d \\vec v + \\vec v\\cdot\\nabla_x \\vec v )\\cdot(\\varrho\\vec v- \\vec m^h) -p(\\varrho^h,\\vartheta^h)\\mathrm{div}_x\\vec v] \\, \\dd x \\\\\n&-\\int_{0}^{\\tau}\\int_{\\T}[\\varrho^h s(\\varrho^h,\\vartheta^h) D_t^d \\Theta + s(\\varrho^h,\\vartheta^h)\\vec m^h\\cdot \\nabla_x \\Theta ] \\, \\dd x \\dd t\\nonumber\\\\\n&+\\int_{0}^{T}\\int_{\\T}[\\varrho^h s (r,\\Theta) \\partial_{t}\\Theta+ \\vec m^h s(r,\\Theta)\\cdot\\nabla_x\\Theta]\\, \\dd x\\dd t \\nonumber\\\\\n&+\\int_{\\T} \\left(\\left(1-\\frac{\\varrho^h}{r}\\right)\\partial_{t}p(r,\\Theta)-\\frac{\\vec m^h}{r}\\cdot\\nabla_x p(r,\\Theta)\\right)\\,\\dd x\\nonumber,\\\\\n&-\\sum_{k\\geq 1}\\int_{\\T}\\mathbb{D}_t^s\\vec v(e_k)\\cdot \\varrho^h\\phi(e_k)\\, \\dd x \\\\\n&+\\frac{1}{2}\\|\\sqrt{{\\varrho^h}}\\phi\\|_{L_2(\\UU,L^2(\\T))}^{2} +\\frac{1}{2}\\sum_{k\\geq 1}\\int_{\\T}\\varrho^h|\\mathbb{D}_t^s\\vec v(e_k)|^2\\, \\dd x, \n\\end{align*}\nand\n\\begin{align*}\n\\mathbb{M}&= \\int_{0}^{\\tau}\\int_{\\T}{\\vec m^h}\\phi \\,\\dd x\\dd {W} \\\\\n& -\\int_{0}^{t}\\int_{\\T}\\bigg[\\vec m^h\\mathbb{D}_t^s\\vec v+ \\Pi_h\\vec v\\varrho^h \\phi\\bigg]\\,\\dd x \\dd W +\\int_{0}^{t}\\int_{\\T}\\varrho^h\\vec v\\cdot \\mathbb{D}_t^s\\vec v\\, \\dd x\\dd W.\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nBy It\\^{o}'s formula we obtain from \\eqref{eq:consistency_m} and \\eqref{eq:sass}\n\\begin{align}\\label{eq:PA}\n\\begin{aligned}\n\\dd \\left( \\int_{\\T}\\vec m^h\\cdot\\vec v \\,\\dd x \\right)&=\\left(\\int_{\\T}\\bigg[\\vec m^h\\cdot D_t^d\\vec v+ \\left(\\frac{\\vec m^h\\otimes\\vec m^h}{\\varrho^h}\\right)\\cdot\\nabla \\vec v+ p(\\varrho^h, s^h)\\mathrm{div}\\vec v\\bigg]\\,\\dd x \\right)\\, \\dd t\\\\\n&+\\sum_{k\\geq 1}\\int_{\\T}\\mathbb{D}_t^s\\vec v(e_k)\\cdot \\varrho^h\\phi(e_k)\\, \\dd x \\dd t\\\\\n&+e_{\\vec m^h}(t,\\vec v)\\dt+\\dd M_1,\\end{aligned}\n\\end{align}\nwhere \n\\[\nM_1 =\\int_{0}^{t}\\int_{\\T}\\bigg[\\vec m\\mathbb{D}_t^s\\vec v+ \\Pi_h\\vec v\\varrho \\phi\\bigg]\\,\\dd x \\dd W.\n\\]\n\nSimilarly to (\\ref{eq:PA}), we compute\n\\begin{align}\\label{eq:PB}\n\\begin{aligned}\n\\dd\\left(\\int_{\\T}\\frac{1}{2}\\varrho^h |\\vec v|^2\\,\\dd x\\right)&=\\int_{\\T}\\varrho^h\\vec v \\cdot \\nabla \\vec v\\cdot \\vec v\\, \\dd x \\dd t+ \\int_{\\T}\\varrho^h\\vec v\\cdot D_t^d\\vec v\\, \\dd x \\dd t\\\\\n&+\\frac{1}{2}\\sum_{k\\geq 1}\\varrho^h|\\mathbb{D}_t^s\\vec v(e_k)|^2\\, \\dd x \\dd t +\\dd M_2, \\end{aligned}\n\\end{align}\nwhere \n\\[\nM_2 = \\int_{0}^{t}\\int_{\\T}\\varrho^h\\vec v\\cdot \\mathbb{D}_t^s\\vec v\\, \\dd x\\dd W.\n\\]\n\nTesting the entropy balance \\eqref{eq:consistency_S} with $\\Theta$ we deduce\n\\begin{equation}\\label{eq:PD}\n\\dd \\left(\\int_{\\T} \\varrho^h s(\\varrho^h,\\vartheta^h) \\Theta \\dd x\\right) \n\\geq\\int_{\\T} \\vec m^h s(\\varrho^h,\\vartheta^h) \\cdot \\nabla_x \\Theta\\, \\dd x\\dd t+\\int_{\\T}\\varrho^h s(\\varrho^h,\\vartheta^h) \\partial_t \\Theta\\, \\dd x\\dd t+e_{S^h}(t,\\Theta)\\dt.\n\\end{equation}\nSimilarly, testing the continuity equation \\eqref{eq:consistency_rho} with $\\partial_{\\varrho}H_{\\Theta}(r,\\Theta)$ yields\n\\begin{equation}\\label{eq:PF}\n \\dd\\left(\\int_{\\T}\\varrho \\partial_{\\varrho}H_{\\Theta}(r,\\Theta)\\, \\dd x \\right) =\\int_{\\T} \\vec m\\cdot \\nabla_x(\\partial_{\\varrho}H_{\\Theta}(r,\\Theta)) \\,\\dd x+ \\int_{\\T}\\varrho^h \\partial_t(\\partial_{\\varrho}H_{\\Theta}(r,\\Theta))\\, \\dd x+e_{\\varrho^h}(t,\\partial_{\\varrho}H_{\\Theta}(r,\\Theta))\\dt.\n\\end{equation}\nFinally, we collect and sum the resulting expressions (\\ref{eq:PA})--(\\ref{eq:PF}), and add the energy balance from \\eqref{eq:consistency_S} to the sum.\n\\end{proof}\n\n\n\n\\subsection{Error estimates}\n\\begin{proof}[Proof of Theorem \\ref{thm:main2}]\nWe choose now $(r,\\vec v,\\Theta)$ in Proposition \\ref{propA} to be the local strong solution from Definition and $t\\in[0,\\mathfrak t\\wedge \\mathfrak s_M]$. The terms contained in $\\mathcal Q$ corresponding exactly to those in \\cite{Mo}\\footnote{In fact, in \\cite{Mo} also the defect measures appear which are not present for the discrete solution.} and we obtain as there\n\\begin{align*}\n\\mathcal{Q}\\bigg(\\varrho^h,\\vartheta^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)(\\tau\\wedge \\mathfrak s_M\\wedge \\mathfrak t)\\leq\\,c \\int_0^{\\tau\\wedge \\mathfrak s_M\\wedge \\mathfrak t} \\mathcal{K}\\bigg(\\varrho^h,\\mathcal E^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)\\,\\diff t\n\\end{align*}\n$\\p$-a.s. with a constant depending on $M$.\nThe right-hand side can be handled by Gronwall's lemma. The stochastic terms in \\eqref{REI} vanish after taking expectations. Finally, due to the regularity of $(r,\\vec v,\\Theta)$ we can use estimate \\eqref{eq:consistency_errorB} and \\eqref{eq:consistency_errorA} (depending on whether $\\mathfrak p\\geq 1$ or $\\mathfrak p=0$) to control the error term by $h$. By \\cite[Lemma 2.7]{LuSh} we have for some $c>0$ independent of $h$\n\\begin{align*}\n&\\E\\Big[\\mathcal{K}\\bigg(\\varrho^h,\\mathcal E^h,\\vec m^h\\bigg|r,\\Theta,\\vec v\\bigg)(\\tau\\wedge \\mathfrak s_M\\wedge \\mathfrak t)\\Big]\\\\&\\qquad\\qquad\\leq\\,c\\,\\E \\Big[\\Big(\\|\\varrho^h-r\\|^2_{L^2(\\T)}+\\|\\vec u^h-\\vec v\\|^2_{L^2(\\T)}+\\|\\vartheta^h-\\Theta\\|^2_{L^2(\\T)}\\Big)(t\\wedge\\mathfrak t\\wedge\\mathfrak s_M)\\Big],\n\\end{align*}\nwhere the constant depends on $K$ from \\eqref{ass:main}. The proof is complete.\n\\end{proof}\n\n\n\n\\section{Numerical simulation}\\label{sec_numerical}\nTo support our theoretical analysis, we derive some simulations solving the stochastic Euler system \\eqref{EulerC}. The numerical experiments are conducted using the Julia programming language \\cite{bezanson2017julia}. \nFor stochastic time integration we employ if nothing else is said the Euler–Maruyama method from the DifferentialEquations.jl package \\cite{rackauckas2017differentialequations}. \nSpatial discretization is handled using the Trixi.jl framework \\cite{ranocha2022adaptive,schlottkelakemper2021purely}.\nVisualization of the results is achieved through Plots.jl \\cite{christ2023plots}, and Random.jl package of Julia is used for generating the noise environment. \nAs described before, up-to-the-authors knowledge this is the first time to present simulations for this type of problems where discontinuities may rise and we lose regularity inside the path even if we start with smooth initial data. In our numerical testing, we focus therefore either on short time periods and/or weak noise strength meaning that if we apply \\eqref{eq:14.02}, we multiply with constants which we choose appropriately to our testing. These numerical simulations should only give a proof of concept where further investigations in this direction together with a more efficient implementation strategy is part of future research. Meanwhile we also refer to \\cite{breit2023mean} for a numerical investigation in context of the stochastic Navier-Stokes equations. \\\\\nWe are investigating two errors in detail. We are investigating the expected value of the classical discrete $L^2$ norm in terms of density and the expected value \\eqref{eq:errorA} which represents the estimation of relative energy. The error behaviour in other quantities, such as momentum and energy, follows a similar pattern to that of the density in our observations and is therefore omitted from further analysis. We analyse therefore\n\\begin{align}\n\\mathtt{E}_1&=\\E \\left[ \\|\\varrho^h-\\varrho\\|_{L^2(\\T)} \\right]; \\label{L2_error}\\\\\n\\mathtt{E}_2&=\\E\\Big[\\Big(\\|\\varrho^h-r\\|^2_{L^2(\\T)}+\\|\\vec u^h-\\vec v\\|^2_{L^2(\\T)}+\\|\\vartheta^h-\\Theta\\|^2_{L^2(\\T)}\\Big) \\Big]. \\label{relative_energy}\n\\end{align}\n at the end time. \nSince an analytical solution is unavailable, we compute a reference solution on the finest grid using the same set-up and project the numerical solution $ \\varrho$ onto this finer grid to evaluate the errors. For the spatial discretisation, we consistently use the local Lax-Friedrichs (LLF) or Rusanov flux for the surface flux, while for the volume flux, we apply the entropy-conservative and kinetic-energy-preserving flux \\eqref{eq:Ranocha_flux} from \\cite{ranocha2018}. With polynomial degree zero, this approach aligns with the classical cell-centered finite volume (FV) framework, where the FV method employs the LLF flux.\nFor the DG case, we restrict ourself in the following mainly to the polynomial degree one for simplicity and efficiency. \nAs stated before, this part is only a proof of concept, a more and detailed investigation will be left for future work together with additional tasks like the construction of limiting strategies for the stochastic time-integration method and a more efficient implementation. In the simulations, we set the number of samples to 1000 and consider one- and two-dimensional cases. \n\n\n\\subsection{One dimension}\\label{sec_one_d}\n\nWe consider the one-dimensional stochastic compressible Euler equations $\\gamma = 1.4$\n\\begin{equation}\\label{eq:euler}\n \\dd \\begin{pmatrix} \\varrho \\\\ m \\\\ \\varrho e \\end{pmatrix}\n + \\dd_x \\begin{pmatrix} m \\\\ m^2/\\varrho + p \\\\ (\\varrho e + p) v \\end{pmatrix} \\dd t\n = \\frac{1}{2}\\mu^2 \\begin{pmatrix} 0 \\\\ 0 \\\\ \\varrho \\end{pmatrix} \\dd t +\n \\mu \\begin{pmatrix} 0 \\\\ \\varrho \\\\ m \\end{pmatrix} \\dd \\beta\n\\end{equation}\nwhere $\\mu$ denotes a stochastic forcing factor and $\\beta$ is a standard one-dimensional Wiener process. If nothing else is said, we select $\\mu=1$ in the one dimensional tests.\n\n\\subsubsection{Grid Convergence}\\label{sub_sub_section_1}\nWe consider a simple density wave \n\\begin{equation}\n \\varrho(0, x) = 1 + 0.5 \\sin\\bigl(2\\cdot x \\bigr), \\quad v(0, x) = 0.1, \\quad p(0,x)= 10,\n\\end{equation}\nfür $t \\in [0, 0.5]$ and $x \\in [-1, 1]$. In the deterministic case, we obtain a simple density wave (advection behaviour). The reference solution is calculated using $2^{12}=4096$ elements and the time-step is set to $\\Delta t=10^{-5}$ where $T=0.5$.\\\\\nIn Table \\ref{FV_DG_density}, the errors are plotted. We observe error behaviour for the FV methods in density around 0.8 and the estimation in relative entropy is around 1.23 which is a little bit better than from our theoretical result but beyond the accuracy estimation for classical FV methods which is one. \\\\\nIn the case of DGSEM with order $\\mathfrak{p=1}$ we obtain second order for the density error (and all other quantities) and fourth order in terms of the estimation of relative energy. \nThese results are better than the one obtained through our analysis which relies on the consistency investigation and does take the regularity of the solution into account. \nWe stress that the results presented lies inside the behaviour which one expects for DGSEM methods applied to the deterministic Euler equations as demonstrated already in several works, for instance \\cite{gassner2016,zbMATH07517718,chen2017}.\n\n\n\n\n\n\n{\\small{\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n& \\multicolumn{4}{c|}{FV} & \\multicolumn{4}{c|}{DGSEM ($\\mathfrak{p}=1$)} \\\\\n\\cline{2-9}\nElements & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) \\\\\n\\hline\n64 & 0.2492 & - & 13.6617 & - & 0.0029& - & 0.0041 & -\\\\ \n128 & 0.1616 & 0.63& 7.4707 & 0.87& 0.0007 & 2.00 & 0.0003&3.98 \\\\\n256 & 0.0913& 0.82 & 3.1981 & 1.23 & 0.0002 & 2.00 & $1.63\\cdot 10^{-5}$ & 4.00 \\\\\n512 & 0.0464 & 0.98 & 1.0426 & 1.60 & $4.48\\cdot 10^{-5}$ & 2.00 & $1.01\\cdot 10^{-6}$ & 4.02 \\\\ \\hline\naverage EOC& - & 0.81 & - & 1.23 & -& 2.0 & -& 4.00 \\\\ \\hline\n\\end{tabular}\n\\caption{LLF FV and entropy stable DGSEM ($\\mathfrak{p}=1$) methods - error plot using 1000 samples}\n\\label{FV_DG_density}\n\\end{table}\n}}\n\n\n\n\n\n\n\n\\subsubsection{One-dimensional Riemann problem}\nNext, we examine one-dimensional Riemann problems in the domain \\( [0,1] \\). It is important to note that our Theorem \\ref{thm:main2} does not apply to these test cases, as the initial conditions do not satisfy the requirement of belonging to \\( W^{\\mathfrak{p}+3,2} \\); in fact, they are no longer even Lipschitz continuous. These types of problems are included for comparison with the results from \\cite{LuSh}, where the Godunov method and a particular FV method were numerically studied in the deterministic case. The reference solutions are again calculated using \\(4096\\) elements. The initial data are as follows:\n\\begin{itemize}\n \\item Rarefaction wave: $ (\\varrho, v, p) (0, x) = \\begin{cases}\n (0.5197, - 0.7259,0.4), \\qquad x <0.5,\\\\\n (1, 0, 1),\\qquad x>0.5.\n \\end{cases}$\n\\item Contact discontinuity: $ (\\varrho, v, p) (0, x) = \\begin{cases}\n (0.5,0.5,5), \\qquad x <0.5,\\\\\n (1, 5,0, 5),\\qquad x>0.5.\n \\end{cases}$\n \\item Shock wave: $ (\\varrho, v, p) (0, x) = \\begin{cases}\n (1,0.7276,1), \\qquad x <0.5,\\\\\n (0.5313, 0, 0.4),\\qquad x>0.5.\n \\end{cases}$\n\\end{itemize}\nWe use $\\Delta t=10^{-5}$ and $T=0.2$ in all three test cases. The boundary conditions are outflow conditions. \nIn Tables \\ref{FV_DG_rarefaction} and \\ref{FV_Contact_Shock}, \nthe errors and experimental orders of convergence are given for different settings. \nTo summarize the results: \n\n\\begin{itemize}\n\\item Rarefaction wave: For $\\mathfrak p=0$ we recognise that convergence rates in terms of density are slightly larger than $0.5$ (in average 0.62) whereas the \nestimation of relative energy is around 1.28 (larger than 1). This is in agreement with the deterministic case from \\cite{LuSh} where for Godunov method similar results have been presented and verified the theoretical result\\footnote{In \\cite{LuSh}, the assumption on the initial data was less restrictive and only an estimation in terms of projection from the initial data and the initial data itself was used.}. In the DGSEM setting with $\\mathfrak{p}=1$, we recognise that due to the lack of regularity and on account of the Gibbs phenomena (projection at the beginning of the simulation), the convergence error rates are lower in both quantities. Here, additional techniques have to be used like the application of modal filters, for instance similar to the works \\cite{offner2015zweidimensionale,zbMATH00036793}, to obtain better error rates. In the following, we will concentrate on the FV setting. \n\\item For the single contact wave the convergence rate for the density is in average around $0.43$ which is slightly better than $1/4$. In terms of the estimation of the relative energy we have $0.84$ where the most accurate results are obtained on the finest grid.\n\\item For the single shock wave the convergence rate of $\\mathtt{E}_1$ is around $0.52$. The error in $\\mathtt{E}_2$ is about $1.02$. This outcome is consistent with the deterministic case discussed in \\cite{LuSh}.\n\\end{itemize}\n\n\n{\\small{\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n& \\multicolumn{4}{c|}{FV} & \\multicolumn{4}{c|}{DGSEM ($\\mathfrak{p}=1$)} \\\\\n\\cline{2-9}\nElements & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) \\\\\n\\hline\n64 & 0.0321 & - & 0.0046 & - & 0.0141 & - & 0.00111 & -\\\\ \n128 & 0.0224 & 0.51 & 0.0021 & 1.09 & 0.0098 & 0.52 & 0.00077 & 0.52\\\\\n256 & 0.0148 & 0.61 & 0.0009 & 1.26 & 0.0074& 0.41 & 0.00066 & 0.23 \\\\\n512 & 0.0089 & 0.72 & 0.0003 & 1.47 & 0.0061 & 0.27 & 0.00061 & 0.08 \\\\ \\hline\naverage EOC& - & 0.61 & - & 1.27 & - & 0.4 & - & 0.28 \\\\ \\hline\n\\end{tabular}\n\\caption{LLF-FV and entr. dissipative DGSEM methods for the rarefaction wave - 1000 samples}\n\\label{FV_DG_rarefaction}\n\\end{table}\n}}\n\n\n\n\n\n\n\n{\\small{\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n & \\multicolumn{4}{c|}{Contact} & \\multicolumn{4}{c|}{Shock} \\\\\n\\cline{2-9}\nElements & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) \\\\\\hline\n64 & 0.0673 & - & 0.5913 & - & 0.049 & - & 0.010 & -\\\\ \n128 & 0.0525 & 0.36& 0.3620 & 0.71 & 0.037 & 0.45 & 0.005 & 0.86 \\\\\n256 & 0.0395 & 0.41 & 0.2058 &0.81 & 0.025 & 0.51 & 0.003 & 1.01 \\\\\n512 & 0.0279 & 0.50 & 0.1031 & 1.00 & 0.017 & 0.60 & 0.001 & 1.19\\\\ \\hline\naverage EOC& - & 0.42 & & 0.84& -& 0.52 & - & 1.02\\\\ \\hline\n\\end{tabular}\n\\caption{LLF-FV method - 1000 samples}\n\\label{FV_Contact_Shock}\n\\end{table}\n}}\n\n\n\n\n\n\n\n\n\n\\subsubsection{1D SOD}\nThis experiment is the famous SOD problem. The exact deterministic solutions contains a rarefaction, contact and shock waves. In this example, the final time is set to \n$T=0.15$ and the initial data is given by \n\\begin{equation*}\n (\\varrho, v, p) (0, x) = \\begin{cases}\n (1,0,1), \\qquad x <0.5,\\\\\n (0.125,0, 0.1),\\quad x>0.5.\n \\end{cases}\n\\end{equation*}\nWe use again $4096$ elements and $\\Delta t=10^{-5}$. The convergence rates can be seen in Table \\ref{FV_SOD} and we recognize that even without the regulartiy of the intitial data, we obtain convergence rates which align with the result itself inside the FV setting ($\\approx 0.5$ in $\\mathtt{E}_1$ and $\\approx 1$ in $\\mathtt{E}_2$). \n{\\small{\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nElements & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1)$ &$\\mathtt{E}_2$ &EOC($\\mathtt{E}_2)$ \\\\ \\hline \\hline\n64 & 0.0378 & - & 0.0203 & - \\\\ \n128 & 0.0287 & 0.40 & 0.0106 & 0.93 \\\\\n256 & 0.0200 & 0.51 & 0.0052& 1.05 \\\\\n512 & 0.0130 & 0.63 & 0.0022 & 1.20 \\\\ \\hline\nav. order & - & 0.51 & -&1.06 \\\\ \\hline\n\\end{tabular}\n\\caption{FV method for SOD - 1000 samples}\n\\label{FV_SOD}\n\\end{table}\n\n}}\nWe also illustrate the impact of the $\\mu$ parameter and provide examples demonstrating various noise behaviours. Figure \\ref{fig:SOD} presents the density profiles for different configurations of the (stochastic) Euler equations, with a zoomed-in view around the shock wave. The simulations utilise the FV framework with $2^{12}$ elements. The solid black line represents the deterministic solution obtained via the explicit Euler method with $\\Delta t = 10^{-4}$. For the stochastic equations, the Euler–Maruyama method is applied using the same $\\Delta t$.\n\nIn the left-hand panels, we display three simulations with distinct noise behaviors, while the right-hand panels compare two simulations with identical noise behavior but different $\\mu$ values: one (blue, dotted) and two (red, dash-dotted), alongside the deterministic solution. The parameter $\\mu$ significantly influences the deviation of the stochastic solution from the deterministic one, with further analysis on this effect provided in the subsequent subsection.\n\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{Figures/SOD_both.pdf}\n \\includegraphics[width=0.49\\textwidth]{Figures/SOD_real.pdf}\n \\caption{Different noise (left) and different noise strength(right)}\n \\label{fig:SOD}\n\\end{figure}\n\n\n\\subsection{Two dimension}\\label{sec_two_d}\n\nWe consider the two-dimensional stochastic compressible Euler equations \\eqref{EulerC_short} with the right hand side \n\n\\begin{equation*}\n\\mathbf{h}(\\bU)=\\mu^2\\varrho\\begin{pmatrix}0\\\\0\\\\ 0 \\\\1\\end{pmatrix},\\quad\\mathbf{g}(\\bU)\\dd W= \\mu\\begin{pmatrix}0\\\\\\varrho\\, \\dd \\beta_1\\\\\\varrho\\,\\dd \\beta_2 \\\\m_1\\,\\dd\\beta_1+m_2\\,\\dd\\beta_2\\end{pmatrix}.\n\\end{equation*}\nwhere $\\mu$ denotes again a stochastic forcing factor. The domain is divided into $n\\times n$ uniform quads for the elements ($n \\in \\N$). \n\n\\subsubsection{Convergence }\nSimilar to Section \\ref{sub_sub_section_1}, we consider now the two dimensional density wave \ngiven by \n\\begin{equation}\n \\varrho(0, x) = 1 + 0.5 \\sin\\bigl(2\\cdot x \\bigr), \\quad v_1(0, x) = 0.1, \\quad v_2(0, x) = 0.1, \\quad p(0,x)= 10,\n\\end{equation}\nfür $t \\in [0, 0.1]$ and $x \\in [-1, 1]^2$. Here, we use $\\mu=0.1$ and calculate everything with $\\Delta t=10^{-4}$. The reference solution is calculated using $256^2$ elements. \nIn Table \\ref{FV_DG_density_2d}, we see the error behaviour for the FV and DGSEM method with $\\mathfrak{p}=1$. We reach first and second order of accuracy which is in line with the\ndeterministic numerical simulations for a smooth test case, e.g. \\cite{oeff2023,gassner2016,zbMATH07517718,chen2017} and references therein. \n\n\n\n \n\n{\\small{\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n& \\multicolumn{4}{c|}{FV} & \\multicolumn{4}{c|}{DGSEM ($\\mathfrak{p}=1$)} \\\\\n\\cline{2-9}\nElements & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) & $\\mathtt{E}_1$ & EOC($\\mathtt{E}_1$) & $\\mathtt{E}_2$ & EOC($\\mathtt{E}_2$) \\\\\n\\hline\n$16^2$ & 0.2622 & - & 11.5589 & - & 0.0365 & - & 0.6069 & -\\\\ \n$32^2$ & 0.1816 & 0.53 & 6.6302 & 0.80 & 0.0093 & 1.97 & 0.0453 & 3.74 \\\\\n$64^2$ & 0.0974 & 0.90 & 2.4735 & 1.42 & 0.0022 & 2.05 & 0.0027 & 4.06 \\\\\n$128^2$ & 0.0368 & 1.41 & 0.4531 & 2.45 & 0.0001 & 2.30 & 0.0001 & 4.59 \\\\ \\hline\naverage EOC& - & 0.95 & - & 1.52 & - & 2.11 & - & 4.13 \\\\ \\hline\n\\end{tabular}\n\\caption{LLF FV and entropy stable DGSEM ($\\mathfrak{p}=1$) methods - error plot using 1000 samples}\n\\label{FV_DG_density_2d}\n\\end{table}\n}}\n\n\\subsubsection{Kelvin-Helmholtz and influence of the noise }\nIn the second test case in two-space dimensions, we consider the famous Kelvin-Helmholtz instability test. \nKelvin-Helmholtz describes a shear flow of three fluid layers with different densities. \nThe initial data are given by \n\\begin{equation}\\label{init_KH}\n (\\varrho, v_1, v_2, p)(\\bx, 0)=\\begin{cases}\n (2,-0.5,0,2.5), \\quad I_1 \\leq y \\leq I_2,\\\\\n (1,0.5,0,2.5), \\text{ otherwise 0},\n \\end{cases}\n\\end{equation}\nwhere the interface profiles $I_j=I_j(\\bx):= J_j+\\epsilon Y_j(\\bx),\\; j=1,2$ and $0<\\epsilon\\ll1$, are chosen to be small perturbations around the lower $J_1=0.25$ and the upper $J_2=0.75$ interfaces, respectively. Moreover, \n$Y_j= \\sum_{m=1}^M a_j^m \\cos \\left( b_j^m +2\\pi m x \\right)$, $j=1,2$, with $a_j^m\\in [0,1]$ and $b_j^m\\in[-\\pi, \\pi],\\; j=1,2,\\; m=1,\\cdots, M$ are arbitrary but fixed numbers. The coefficients $a_j^m$ have been normalized such that $\\sum_{m=1}^M a_j^m=1$ to guarantee that $|I_j-J_j|\\leq \\epsilon$ for $j=1,2.$ In the simulations, we have $M=10$, $\\epsilon=0.01$ and $T=1.5$. We solve the Kelvin-Helmholtz problem on $[0,1]\\times[0,1]$ with periodic boundary conditions. \nIt is known that no grid conference can be obtained as can be seen for instance in \\cite{Fjordholm2016, feireisl2020convergence,LuOe} and references therein. Vertices rise on different locations and fine scales which can also derive problems in terms of stability for entropy dissipative DG methods \\cite{glaubitz2024generalized,ranocha2025robustness}.\\\\\nHowever, to demonstrate now the differences driven by the noise and also by the forcing factor $\\mu$, we print in Figure \\ref{fig:main} different realisations using \nthe flux differencing DGSEM method with $\\mathfrak{p}=2$ on a grid with $64 \\times 64$ elements. The time step is set to $\\Delta t=10^{-4}$. We are using again explicit Euler in the deterministic case (for comparison) and the Euler–Maruyama in the stochastic setting. \nSubfigure \\ref{fig:sub1} depicts the deterministic solution, whereas subfigures \\ref{fig:sub2}–\\ref{fig:sub3} show two different realizations of the same test case with distinct noise configurations. Subfigures \\ref{fig:sub3}–\\ref{fig:sub4} illustrate the same problem but with varying noise strengths. These results clearly highlight the significant impact of noise strength on the solution. \nWe do not want to hide that if the strength is to high above $1$, we have obtained also stability issues in this test case for the tested realisations. This can possible also happen for other realisations with the smaller strength where the noise can have an stabilisation effect as well. A more detailed investigation of such context will left for future research. \n\n\n\n\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\linewidth]{Figures/Fig_DG_2_grid_6_det.pdf}\n \\caption{Deterministic}\n \\label{fig:sub1}\n \\end{subfigure}\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\linewidth]{Figures/Fig_DG_2_grid_6.pdf}\n \\caption{First realization with noise $\\mu=0.1$}\n \\label{fig:sub2}\n \\end{subfigure}\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\linewidth]{Figures/Fig_DG_2_grid_6_2.pdf}\n \\caption{Second realization with noise $\\mu=0.1$}\n \\label{fig:sub3}\n \\end{subfigure}\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\linewidth]{Figures/Fig_DG_2_grid_6_2_strong.pdf}\n \\caption{Second realization with noise $\\mu=0.25$}\n \\label{fig:sub4}\n \\end{subfigure}\n \\caption{Kelvin-Helmholtz and influence of the noise }\n \\label{fig:main}\n\\end{figure}\n \n\n\n\n\\section{Conclusion and outlook}\\label{se_outlock}\n\n\n\nIn this work, we have extended the entropy-dissipative DGSEM methods including the FV case to solve the stochastic Euler system, demonstrating convergence and providing error estimates across various settings. Our analysis builds on the consistency framework from \\cite{LuOe} and the theoretical results in \\cite{Mo}, leading to a first convergence result for the stochastic Euler equations. Additionally, we implemented these methods and examined error behavior across multiple experiments as a proof of concept.\\\\\nWe have seen that the method is stable and for sufficient smooth solutions (for the deterministic case), we obtain even better convergence rates than we would have expected. \n This is also in line with the purely deterministic framework where several optimal error estimations for different DG settings are known as references for instance in \\cite{zbMATH06698849,Jiao2022,Liu2020Liu,zbMATH07086311}. However, our analysis is relying on the consistency estimation where no regularity have been assumed. In the future, we plan to derive new error estimates using the consistency analysis and transfer it also from the purely deterministic setting to the stochastic one.\nMeanwhile, we have numerically analyzed the stochastic Euler equations using LLF-FV and DGSEM methods. We observe that LLF-FV methods exhibit error behavior in the stochastic setting similar to that derived in \\cite{LuSh} for the deterministic case. Furthermore, our testing of the DGSEM method shows that, with smooth initial data, numerical results resemble those of deterministic case. However, for cases involving discontinuities, we observe oscillations, indicating a need for the development of limiting strategies including stochasticity to address more complex test cases where a first step in this direction was recently done in \\cite{Wo}. Future research will focus on these limiting strategies, along with turbulence modeling aspects.\n\n\\section*{Acknowledgement}\n\nThe work of D.B.\nwas supported by the German Research Foundation (DFG) within the framework of the priority research program SPP 2410 under the grant BR 4302/3-1 (525608987) and under the personal grant BR 4302/5-1 (543675748).\\\\\nP.Ö. was supported by the DFG within SPP 2410, project OE 661/5-1 \n(525866748) and under the personal grant OE 661/4-1(520756621).\\\\\nP.Ö. thanks Hendrik Ranocha (JGU Mainz) for his assistance in implementing the stochastic forcing term in Trixi.jl.\n\n\\section*{Conflict of Interest}\nThe authors have no conflict of interest to declare.\n\n\\bibliographystyle{abbrv}\n\\bibliography{literature}\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":88,"cells":{"id":{"kind":"string","value":"2412.07667v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07667v1"},"title":{"kind":"string","value":"A symbolic computational approach to the generalized gambler's ruin problem in one and two dimensions"},"full_text":{"kind":"string","value":"\\documentclass[reqno]{amsart}\n\\usepackage[utf8]{inputenc}\n\\usepackage{xcolor,fullpage,hyperref,cleveref,enumitem}\n\\usepackage{tikz,amsmath,amssymb,color,mathtools}\n\\usetikzlibrary{calc}\n\\usepackage{float}\n\\usepackage{pgf}\n\\usepackage[backrefs]{amsrefs}\n\\usepackage{amsthm}\n\\usepackage{listings}\n\n\n\n\\lstset{frame=tb,\n language=Python,\n aboveskip=3mm,\n belowskip=3mm,\n showstringspaces=false,\n columns=flexible,\n basicstyle={\\small\\ttfamily},\n numbers=none,\n numberstyle=\\tiny\\color{gray},\n keywordstyle=\\color{blue},\n commentstyle=\\color{dkgreen},\n stringstyle=\\color{mauve},\n breaklines=true,\n breakatwhitespace=true,\n tabsize=3\n}\n\n\n\n\\newcommand{\\norm}[1]{\\left\\lVert#1\\right\\rVert}\n\n\n\\theoremstyle{definition} \n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}{Corollary}[theorem]\n\\newtheorem{proposition}{Proposition}[theorem]\n\\newtheorem{problem}{Problem}\n\\newtheorem{example}{Example}\n\\newtheorem{exercise}{Exercise}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{guess}[theorem]{Guess}\n\\theoremstyle{definition}\n\\newtheorem{definition}{Definition}[section]\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\\newtheorem{observation}[theorem]{Observation}\n\\newtheorem*{question}{Question}\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\PP}{\\mathbb{N}^+}\n\\newcommand{\\ZZ}{\\mathbb{Z}}\n\n\n\\DeclarePairedDelimiter\\ceil{\\lceil}{\\rceil}\n\\DeclarePairedDelimiter\\floor{\\lfloor}{\\rfloor}\n\n\n\\newcommand{\\seqnum}[1]{\\href{https://oeis.org/#1}{\\rm \\underline{#1}}}\n\n\\usepackage{thmtools} \n\\usepackage{thm-restate}\n\\declaretheorem[name=Theorem,numberwithin=section]{thm}\n\n\n\\newcommand{\\tcr}[1]{\\textcolor{red}{#1}}\n\\newcommand{\\tcb}[1]{\\textcolor{blue}{#1}}\n\n\\newcommand{\\lucy}[1]{{\\bf\\color{magenta}Lucy: #1}}\n\n\n\\title{A symbolic computational approach to the generalized gambler's ruin problem in one and two dimensions}\n\n\\author[Martinez]{Lucy Martinez}\n\\address[L.~Martinez]{Department of Mathematics, Rutgers University, Piscataway, NJ 08854}\n\\email{\\textcolor{blue}{\\href{mailto:lucy.martinez@rutgers.edu}{lucy.martinez@rutgers.edu}}}\n\n\n\\begin{document}\n\\begin{abstract}\nThe power of symbolic computation, as opposed\nto mere numerical computation, is illustrated with\nefficient algorithms for studying the generalized gambler's ruin problem in one and two dimensions. We also consider a new generalization of the classical gambler's ruin where we add a third step which we call the mirror step. In this scenario, we provide closed formulas for the probability and expected duration.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\\section{Introduction} \nThroughout we let $x$ be some positive integer such that $0< x < N$ where $N\\in \\NN=\\{1,2,3,\\ldots \\}$. Consider a gambler who starts with $x$ dollars. At each gamble, the gambler either wins a dollar with probability $\\frac{1}{2}$ or loses a dollar with probability $\\frac{1}{2}$. The gambler's goal is to reach $N$ dollars without first running out of money (i.e., hitting $0$ dollars). If the gambler reaches $N$ dollars, we say that they are a \\textit{winner}. The gambler continues to play until they either run out of money or win. This scenario is known as the \\textit{gambler's ruin problem}, first posed by Pascal in 1656 in a letter to Fermat, as noted by Edwards \\cite{Edwards1983}. In 1657, Christiaan Huygens restated the problem and published a solution for the probability of winning~\\cite{Huygens}. For additional historical context, we refer the reader to a paper of Seongjoo Song and Jongwoo Song~\\cite{Song}.\n\nIn this paper, we begin by providing an overview of the classical gambler's ruin problem, recalling results for both the probability of winning and the expected duration of the game. We also summarize analogous results on the generalized $1$-dimensional and $2$-dimensional versions of the gambler's ruin problem. Building on the $1$-dimensional version, we introduce a new generalization of the classical gambler's ruin game that includes an additional step called the \\textit{mirror step}. For the generalized model, we derive formulas for the probability of winning and expected duration of the game. The objective of this paper is to propose an approach to reduce the computational running time required to determine the probability of winning and the expected duration specifically for the generalized 1-dimensional and 2-dimensional versions.\n\n\n\n\\subsection{Classical gambler's ruin problem}\\label{subsec:classical}\nLet $f(x)$ be the probability that the gambler exits the game as a winner starting with $x$ dollars. For $0<x<N$, this probability satisfies the recurrence relation\n\\begin{equation} \\label{eq:probclass}\n f(x)=\\frac{1}{2}f(x-1)+\\frac{1}{2}f(x+1), \\quad f(0)=0, f(N)=1.\n\\end{equation}\nThat is, if the gambler starts with $x$ dollars, then in the next round, the gambler has $x-1$ dollars or $x+1$ dollars, each with probability $\\frac{1}{2}$. Using this recurrence relation and the boundary conditions, we can find the solution to be $f(x)=\\frac{x}{N}$. \n \n \nIf the gambler starts with $x$ dollars, let $g(x)$ be the expected number of steps (expected duration of the game) the gambler takes to exit the game (either with $N$ dollars or $0$ dollars). Similar to the probability, for $1<x<N$, \\[g(x)=\\frac{1}{2}g(x-1)+\\frac{1}{2}g(x+1) +1, \\quad g(0)=0, g(N)=0.\\] At each round, if the gambler has $x$ dollars, then in the next round, the gambler will have either $x-1$ dollars or $x+1$ dollars, each with probability $\\frac{1}{2}$. However, we add 1 to the count since the gambler has taken one extra step. Using the recurrence relation and the boundary conditions, we can find the solution to be $g(x)=x(N-x)$.\n\n\nBuilding upon the expected duration, we can obtain the probability generating function of the duration of the game. For a formal variable $t$ and $0<x<N$, \\[F(x,t)=t\\left(\\frac{1}{2}F(x-1,t)+\\frac{1}{2}F(x+1,t)\\right), \\quad F(0,t)=1, F(N,t)=1.\\]\nTaking the derivative of $F(x,t)$ with respect to $t$, and evaluating at $t=1$ recovers the expected duration of the game at $x$.\n\n\nConsider extending the game so that the probability of losing one dollar or winning one dollar are not the same. In other words, let $p$ be the probability of winning one dollar, and $q=1-p$ be the probability of losing one dollar. Let $f(x)$ be the probability of exiting the game as a winner starting with $x$ dollars. Similarly to \\Cref{eq:probclass}, we get\n\\[ f(x)=qf(x-1)+pf(x+1), \\quad f(0)=0, f(N)=1, \\text{ and } p+q=1. \\]\n\n\nEdwards gives a conjecture on how Pascal solved the above using a method of recursive formula \\cite{Edwards1983}. We provide Edwards' solution to $f(x)$. Rewrite $f(x+1)-f(x)$ and observe the following\n\n\\begin{equation*}\n f(x+1)-f(x)=\\frac{q}{p}\\left(f(x)-f(x-1)\\right)=\\frac{q^2}{p^2}\\left(f(x-1)-f(x-2)\\right)=\\cdots= \\frac{q^i}{p^i}\\left(f(1)-f(0)\\right).\n\\end{equation*}\n\nHence, \n\n\\begin{align*}\n f(x)&= \\left ( \\sum_{j=0}^{x-1}\\left(\\frac{q}{p}\\right)^j \\right ) f(1).\n\\end{align*}\n\nSince $p=1-q$ then $p\\neq q$, it follows that $\\frac{q}{p}\\neq 1$. Therefore, by the geometric series, we obtain\n\n\\begin{align*}\n 1=f(N)&=\\frac{1-\\left(\\frac{q}{p}\\right)^N}{1-\\frac{q}{p}} \\cdot f(1).\n\\end{align*}\n\nThus, we can recover the following\n\n\\begin{align*}\n f(1)=\\frac{1-\\frac{q}{p}}{1-\\left(\\frac{q}{p}\\right)^N}\n\\quad \\text{and} \\quad\n f(x)=\\frac{1-\\left(\\frac{q}{p}\\right)^x}{1-\\left(\\frac{q}{p}\\right)^N}.\n\\end{align*}\n\n\\subsection{Generalized $1$-dimensional gambler's ruin}\nThe gambler's ruin problem can be formulated as follows: A particle starts at a point $x$ on a line of length $N$ where $0<x<N$. The particle moves to the left from $x$ to $x-1$ with probability $\\frac{1}{2}$, or to the right from $x$ to $x+1$ with probability $\\frac{1}{2}$. \n\n\nConsider extending the 1-dimensional gambler's ruin game to include more than two steps on a line of length $N$. Let $r$ be a positive integer. Let $a_1,a_2,\\ldots , a_r$ be distinct integers such that $a_1<a_2<\\cdots<a_r$ where $a_1<0$ and $a_r>0$. Let $p_1, p_2,\\ldots,p_r$ be probabilities such that $p_1+p_2+\\ldots + p_r=1$, and let $P$ be the probability table $P=[[a_1,p_1], [a_2,p_2],\\ldots , [a_r,p_r]]$ where each pair $[a_i,p_i]$ represents the outcome $a_i$ occurring with probability $p_i$. The generalized 1-dimensional gambler's ruin problem states that if a particle is currently at some $x$ then the particle moves from $x$ to $x+a_1$ with probability $p_1$, or moves from $x$ to $x+a_2$ with probability $p_2$, or moves from $x$ to $x+a_3$ with probability $p_3$, and so on.\n\n\nSimilarly to the method of solving a system of linear equations as in \\Cref{subsec:classical}, we can obtain the probability of winning and the expected duration for the generalized case for any starting position. However, as $N$ grows, the computation time to solve a system of $N$ linear equations will be slower. In \\Cref{sec:general 1d}, we present a faster method to reduce the computational running time by going from a system of $N-1$ linear equations with $N-1$ unknowns to a system of $a_r$ linear equations with $a_r$ unknowns where $a_r$ is the maximum of the steps in the probability table $P=[[a_1,p_1], [a_2,p_2],\\ldots , [a_r,p_r]]$. Our method significantly drops the computational running time and we make comparisons between the direct approach and our strategy in \\Cref{subsec:time comparison in 1d}.\n\n\n\\subsection{$2$-dimensional gambler's ruin}\\label{subsec:2 dim}\nLet $M$ and $N$ be positive integers. Consider a particle starting at a point $(x,y)$ in the interior of a rectangular grid of size $M\\times N$, where $0<x<M$ and $0<y<N$. At each step, the particle moves in one of four directions, each with probability $\\frac{1}{4}$: $(x-1,y), (x,y+1), (x+1,y),(x,y-1)$. The particle stops moving once it hits one of the four boundaries, defined by $x=0, x=M, y=0$, or $y=N$. \nBy setting up a recurrence relation for the expected duration, we can obtain a system of $(M-1)\\times(N-1)$ linear equations with $(M-1)\\times(N-1)$ unknowns. Andrej Kmet and Marko Petkov\\v{s}ek gave an explicit formula involving a double sum, enabling direct computation of the expected duration for the $2$-dimensional gambler's ruin game without the need to solve systems of equations or use recursion \\cite{Kmet}. While Kmet and Petkov\\v{s}ek's formula expresses the expected duration as a double sum, our method reduces the computational running time by going from a system of $(M-1)\\times(N-1)$ linear equations with $(M-1)\\times(N-1)$ unknowns to a system of $N-1$ linear equations and $N-1$ unknowns. Our method is significantly faster than the direct approach and Kmet and Petkov\\v{s}ek' formula. We make comparisons in \\Cref{subsec:time comparison in 2d}.\n\n\nOne way to generalize the $2$-dimensional game is to change the probabilities of each of the four directions with probabilities $p_L, p_U, p_R,$ and $p_B$, corresponding to left, up, right and down movements, respectively where $p_L+p_U+p_R+p_B=1$. Although one can generalize the number of steps for either of the four directions, we focus on the case when the set of steps the particle can\nmove is $\\{[0,1],[0,-1],[1,0],[-1,0]\\}$,\n and remark that one can adapt our strategy for an {\\it arbitrary} (finite) set of allowed steps, and arbitrary probability distribution.\n\n\n\\subsection{A mirror step variant of gambler's ruin}\\label{subsec:mirror}\nWe consider a new generalization of the gambler's ruin problem. A particle starts at some point $x$ on a line of length $N$ where $0< x <N$. At each step, the particle either moves from $x$ to $x-1$ with probability $q_1$, or moves from $x$ to $x+1$ with probability $q_2$, or moves from $x$ to $N-x$ with probability $p$ where $0<p<1$ and $q_1+q_2+p=1$. We call this last step the \\textit{mirror step}. The particle continues to walk on the line until it reaches $0$ or $N$. In this paper, we focus on the case when $q_1=q_2=\\frac{1-p}{2}$ and we call this the \\textit{symmetric case}. \n\n\nWe begin with an example where the particle starts at $x=1$ and generate data for different $p$ values with fixed $N$. \n\n\n\\begin{example}\\label{ex: x=1 prob in intro}\nLet $N=100$ and $x=1$. We generate data for the probability that if the particle is currently at $x=1$, the particle eventually ends at $100$. Let $p\\in \\{\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\ldots, \\frac{1}{10}\\}$. \n\n\nWe use the procedure \\texttt{Lk(p,x,N)}, as described in \\Cref{appendix:mirror} in Maple which generates the following data in about $4.765$ seconds:\n\\begin{align*}\n T \\coloneqq &[0.4142135624, 0.3660254038, 0.3333333333, 0.3090169944, 0.2898979486, 0.2742918852, 0.2612038750, \\\\\n &0.2500000000,0.2402530734].\n\\end{align*}\n\n\\noindent The sequence $T$ reads as follows. If the particle is currently at $x=1$ and $p=\\frac{1}{2}$, the particle moves from $x$ to $x-1$ with probability $\\frac{1-p}{2}=\\frac{1}{4}$, or moves from $x$ to $x+1$ with probability $\\frac{1-p}{2}=\\frac{1}{4}$, or moves from $x$ to $100-x$ with probability $p=\\frac{1}{2}$. Then, the probability of the particle starting at $x=1$ and ending at $100$ is $T_1=0.4142135624$. Similarly, if the particle is currently at $x=1$ and $p=\\frac{1}{3}$, the particle moves from $x$ to $x-1$ with probability $\\frac{1-p}{2}=\\frac{1}{3}$, or moves from $x$ to $x+1$ with probability $\\frac{1-p}{2}=\\frac{1}{3}$, or moves from $x$ to $100-x$ with probability $p=\\frac{1}{3}$. Then, the probability of the particle starting at $x=1$ and ending at $100$ is $T_2=0.3660254038$. Thus, $T_i$ is the probability of the particle starting at $x=1$ and ending at $100$ for $p=\\frac{1}{i+1}$ where $1\\leq i\\leq 9$.\n\n\nWe then use the function \n\\texttt{identify} in Maple on the sequence $T$. The function \\texttt{identify} is based, in part, on the continued fraction expansion of any given numerical value. Using this function on the values of $T$, we conjecture that each of the probabilities in $T$ converges to\n\\begin{align*}\n M\\coloneqq&\\left[\\sqrt{2}-1, \\frac{\\sqrt{3}-1}{2}, \\frac{1}{3}, \\frac{\\sqrt{5}-1}{4}, \\frac{\\sqrt{6}-1}{5}, \\frac{\\sqrt{7}-1}{6},\\frac{2 \\sqrt{2}-1}{7}, \\frac{1}{4}, \\frac{\\sqrt{10}-1}{9} \\right].\n\\end{align*}\nThat is, the probability of the particle starting at $x=1$ and ending at $100$ converges to $M_i$ for $p=\\frac{1}{i+1}$ where $1\\leq i\\leq 9$.\n\\end{example}\n\n\nThe previous example illustrates that when the particle starts at $x=1$, the probability of ending at $N$ converges fast. Denote this probability by $f_N^{(p)}(x)$. We state the following guess for $x=1$ and in \\Cref{cor:probability at infinity} we provide a proof.\n\n\\begin{guess}\\label{guess:x=1}\nIf the particle starts at $x=1$, then\n\\[\\lim_{N\\to\\infty}f_N^{(p)}(1)=\\frac{\\sqrt{p}-p}{1-p}.\\]\n\\end{guess}\n\nIn \\Cref{sec:mirror step}, we provide other expressions for $\\lim_{N\\to \\infty} f_N^{(p)}(x)$ when $x=2$ and $x=N-2$ with fixed $N$, and in \\Cref{cor:probability at infinity} we provide the general formula for the limit.\n\n\n\nThis paper is structured as follows. In Section \\ref{sec:general 1d}, we present a new approach to compute the probability of winning and the expected duration of the game that reduces the computational running time for the generalized $1$-dimensional gambler's ruin \\footnote{All computations were performed using Maple on a laptop with an Intel Core i7-10510U processor (4 cores, 8 logical processors) and 8 GB of RAM.}. In Section \\ref{sec:2d case}, we provide the analogous approach for the generalized $2$-dimensional case and compare the computational running times to a formula provided by Andrej Kmet and Marko Petkov\\v{s}ek. In Section \\ref{sec:mirror step}, we consider a mirror step variant of gambler's ruin and provide closed formulas for both the probability of winning and the expected duration of the game. We conclude with future directions in Section \\ref{sec:future}.\n\n\n\n\\section{Generalized $1$-dimensional Gambler's Ruin}\\label{sec:general 1d}\n\nIn this section, we introduce the recurrence relation for the probability of winning in the generalized $1$-dimensional gambler's ruin game, and introduce symbolic variables to the recurrence equation of the probability and expected duration. Recall that $P=[[a_1,p_1], [a_2,p_2],\\ldots , [a_r,p_r]]$ denotes a probability table, where each pair $[a_i,p_i]$ represents the outcome $a_i$ occurring with probability $p_i$. To set up notation, we start with an example.\n\n\n\\begin{example}\\label{ex:probability notation}\nLet $N=5$ and $P$ be the probability table given by $P=\\left[[-2,\\frac{1}{2}],[1,\\frac{1}{4}],[2,\\frac{1}{4}]\\right]$. If the particle starts at some $x$ where $0<x<5$ on the line of length $5$, then it can move along the line as follows: from $x$ to $x-2$ with probability $\\frac{1}{2}$, or from $x$ to $x+1$ with probability $\\frac{1}{4}$, or from $x$ to $x+2$ with probability $\\frac{1}{4}$.\n\\end{example}\n\n\\subsection{Probability}\\label{subsec:prob}\nWe now establish the recurrence relation for the probability that the particle reaches some position $\\geq N$ starting from an initial position $x$. We then rewrite this recurrence and introduce new variables for the probabilities at each $x$.\n\nDefine $f(x)$ as the probability that a particle starting at $x$ will eventually reach a position $\\geq N$. For $0<x<N$, this probability satisfies the recurrence relation\n\\begin{equation}\\label{eq:generalprob}\n f(x)=\\sum_{i=1}^r p_if(x+a_i),\n\\end{equation}\nwhere $a_i\\in \\ZZ$, $a_1<a_2<\\cdots <a_r$, $a_1<0$ and $a_r>0$. Unlike the classical gambler’s ruin problem, the generalized 1-dimensional scenario has more than two boundary conditions. Certainly $f(0)=0$ and $f(N)=1$. Since $a_1< a_2< \\cdots< a_r$, the values $a_1$ and $a_r$ represent the minimum and maximum of all the integers $a_i$, respectively. For any integer $k$ such that $a_{1}+1 \\leq k \\leq 0$, it follows that $f(k)=0$. Indeed, if the particle is at $x=1$, it may move to $x+a_1=a_1+1$ with probability $p_1$. Given that $a_1<0$, this movement brings the particle to some position $k\\leq 0$, implying that $f(k)=0$ for $k=a_1+1, a_1+2, \\ldots, -1,0$. Similarly, for any integer $\\ell$ such that $N \\leq \\ell \\leq N+a_r-1$, we have $f(\\ell)=1$. If the particle is at $x=N-1$, it may move to $x+a_r=N+a_r-1$ with probability $p_r$. Since $a_r>0$, this movement brings the particle to some position $\\ell\\geq N$, so $f(\\ell)=0$ for $\\ell=N,N+1,N+a_r-2,N+a_r-1$. \nThus, there are $a_r-a_1$ boundary conditions.\n \n\n\\begin{example}\\label{ex:slow version N=5}(Continuing \\Cref{ex:probability notation})\nRecall the probability table $P=\\left[[-2,\\frac{1}{2}],[1,\\frac{1}{4}],[2,\\frac{1}{4}]\\right]$ and $N=5$. If $f(x)$ denotes the probability that the particle reaches some position $\\geq 5$, then\n\\begin{equation*}\n f(x)=\\frac{1}{2}f(x-2)+\\frac{1}{4}f(x+1)+\\frac{1}{4}f(x+2)\n\\end{equation*}\nwith initial and final conditions $f(-1)=f(0)=0$ and $f(5)=f(6)=1$. \n\n\\noindent This setup results in a system of 4 linear equations for $0<x<5$,\n\\begin{align*}\n f(1)&=\\frac{1}{2}f(-1)+\\frac{1}{4}f(2)+\\frac{1}{4}f(3)\\\\\n f(2)&=\\frac{1}{2}f(0)+\\frac{1}{4}f(3)+\\frac{1}{4}f(4)\\\\\n f(3)&=\\frac{1}{2}f(1)+\\frac{1}{4}f(4)+\\frac{1}{4}f(5)\\\\\n f(4)&=\\frac{1}{2}f(2)+\\frac{1}{4}f(5)+\\frac{1}{4}f(6).\n\\end{align*}\nSolving for the unknowns using the boundary conditions yields $f(1)= \\frac{1}{5}, f(2)=\\frac{13}{45}, f(3)=\\frac{23}{45}, f(4)=\\frac{29}{45}$.\n\\end{example}\n\nAlthough this direct method works for small values of $N$, solving the system of $N-1$ equations becomes computationally expensive as $N$ grows. To address this, we rewrite the recurrence relation \\Cref{eq:generalprob} as\n\\begin{equation}\\label{eq:bettergeneralprob}\nf(x)=\\frac{1}{p_r}f(x-a_r)-\\frac{1}{p_r}\\sum_{i=1}^{r-1}p_if(x+a_i-a_r)\n\\end{equation}\nobtained by the change of variables $x\\mapsto x-a_r$. The boundary conditions remain, $f(a_1+1)= f(a_1+2)= \\ldots= f(-1)=f(0)=0$ and $f(N)=f(N+1)=\\ldots = f(N+a_r-2)= f(N+a_r-1)=1$. For each $1\\leq j \\leq a_r$ define $d_j=f(j)$ and construct the set $S=\\{d_1,d_2,\\ldots, d_j\\}$ where $a_r$ is the maximum of the $a_i$'s. Using these variables, we express \n\\begin{align}\\label{eq:linear equations for better recurrence}\n f(a_r+1),f(a_r+2),\\ldots, f(N+a_r-1)\n\\end{align}\nas linear combinations of the elements in $S$.\n\n \nObserve that $a_1<a_2<\\cdots<a_r$ implies $a_i-a_r<0$ for any $1\\leq i \\leq r-1$. Hence, $x+a_i-a_r<x\\leq N+a_r-1$ for all $x\\in \\{a_r+1, a_r+2,\\ldots, N+a_r-1\\}$ since $0<a_r+1<a_r+2< \\cdots< N+a_r-1$. Thus, $x+a_i-a_r< N+a_r-1$. Also, for any $1\\leq i\\leq r-1$, $x+a_i-a_r>x+a_1-a_r$ since $a_i> a_1$. Now, $x+a_i-a_r> x+a_1-a_r\\geq a_r+1+a_1-a_r=a_1+1$ since $0<a_r+1<a_r+2<\\cdots < N+a_r-1$. It follows that $x+a_i-a_r> a_1+1$.\n\nCombining the above, we get $a_1+1<x+a_i-a_r<N+a_r-1$. This implies that $f(x+a_i-a_r)$ depends only on the following terms \n\\begin{align*}\n &f(a_1+1),f(a_1+2),\\ldots, f(0),\\\\\n &f(1), f(2), \\ldots, f(a_r),\\\\\n &f(a_r+1),f(a_r+2),\\ldots, f(N-1)\\\\\n &f(N), f(N+1), f(N+2), \\ldots , f(N+a_r-1).\n\\end{align*}\n\nWe know $0=f(a_1+1)=f(a_1+2)=\\cdots =f(0)$, $1=f(N)= f(N+1)=f(N+2)= \\cdots =f(N+a_r-1)$ and $f(1)=d_1,f(2)=d_2,\\ldots,f(j)=d_j$ where $j=a_r$. For any $a_r+1\\leq x \\leq N-1$,\n$f(a_r+1)$ is a linear combination of $f(1)$ and $f(a_1+1), f(a_2)+1,\\ldots, f(a_{r-1}+1)$. Recall that $a_1+1\\leq a_2, a_2+1\\leq a_3, \\ldots, a_{r-1}+1\\leq a_r$. Hence, $f(a_r+1)$ depends on at most the expression $f(a_r)$, which is known. Thus, $f(a_r+1)$ is a linear combination of the elements in $S$. Since $f(x)$ is a recursive formula, for any $x>a_r+1$, $f(x)$ will be a linear combination of the variables in $S$. Simultaneously, we have $f(N)=f(N+1)=\\ldots =f(N+a_r-2)=f(N+a_r-1)=1$. Therefore, $f(N), f(N+1), f(N+2), \\ldots , f(N+a_r-1)$ are linear combinations of $\\{d_1,d_2,\\ldots,d_j\\}$ where $j=a_r$ and are all equal to $1$. Hence, solving this system of $a_r$ equations with $a_r$ unknowns yields solutions for the variables in $S$ which provide solutions for the rest of the expressions, namely $f(a_r+1),f(a_r+2),\\ldots, f(N-1)$. \n\n\n\\begin{example}\\label{ex:faster version N=5}(Continuing Example \\ref{ex:slow version N=5})\nLet $N=5$ and $P=[[-2,1/2],[1,1/4],[2,1/4]]$ as in \\Cref{ex:slow version N=5}. Recall the boundary conditions: $f(-1)=f(0)=0$ and $f(5)=f(6)=1$. Note that $a_3=2$, the maximum of $\\{-2,1,2\\}$, introduces the new variables $\\{d_1,d_2\\}$ such that $f(1)=d_1$. Using the recurrence relation $f(x)=4f(x-2)-2f(x-4)-f(x-1)$, we construct the equations for $f(3), f(4) , f(5)$, and $f(6)$, with $r=3$:\n\\begin{align*}\n f(-1)&=f(0)=0 \\\\\n f(1)&=d_1\\\\\n f(2)&=d_2 \\\\\n f(3)&=4f(1)-2f(-1)-f(2)= 4d_1-d_2\\\\\n f(4)&=4f(2)-2f(0)-f(3)=5d_2-4d_1 \\\\\n f(5)&=4f(3)-2f(1)-f(4)= 18d_1-9d_1\\\\\n f(6)&=4f(4)-2f(2)-f(5)=27d_2-34d_1.\n\\end{align*}\nSince $f(5)=f(6)=1$, the system of equations $1= 18d_1-9d_1$ and $1=27d_2-34d_1$ has the solution $d_1=\\frac{1}{5}$ and $d_2=\\frac{13}{45}$. Substituting these values gives \n\\[ f(1)=\\frac{1}{5}, f(2)=\\frac{13}{45}, f(3)=\\frac{23}{45}, f(4)=\\frac{29}{45}.\\]\nThese results agree with \\Cref{ex:slow version N=5}.\n\\end{example}\n\n\n\n\n\n\\subsection{Expected duration}\\label{subsec:exp}\nIn the previous subsection, we considered the probability for the particle to reach some position $\\geq N$. In this section, we consider the expected duration for the particle to end at a position $\\leq 0$ or a position $\\geq N$. Since this is analogous to the probability case, we omit the details.\n\n\nDefine $g(x)$ as the expected number of steps that a particle starting at $x$ will eventually reach a position $\\leq 0$ or a position $\\geq N$. For $0<x<N$, this expected duration satisfies the recurrence relation\n\\begin{equation}\\label{eq:generalexp}\n g(x)=\\sum_{i=1}^r p_ig(x+a_i)+1,\n\\end{equation}\nwhere $a_i\\in \\ZZ$, $a_1<a_2<\\cdots <a_r$, $a_1<0$, $a_r>0$ and boundary conditions $0=g(a_1+1)= g(a_1+2)=\\cdots=g(-1)=g(0)$ and $0=g(N)=g(N+1)=\\cdots =g(N+a_r-2)=g(N+a_r-1)$.\n\nSimilar to the probability case, we rewrite \\Cref{eq:generalexp} to \n\\begin{equation}\\label{eq:bettergeneralexp}\ng(x)=\\frac{1}{p_r}g(x-a_r)-\\frac{1}{p_r}\\sum_{i=1}^{r-1}p_ig(x+a_i-a_r)-\\frac{1}{p_r}\n\\end{equation}\nwhere $0=g(a_1+1)= g(a_1+2)=\\cdots=g(-1)=g(0)$ and $0=g(N)=g(N+1)=\\cdots =g(N+a_r-2)=g(N+a_r-1)$.\n\nUsing \\Cref{eq:bettergeneralexp} reduces the computational running time for the expected duration of the game as we will see in the next subsection.\n\n\n\\subsection{Comparison between the slower method and the faster method}\\label{subsec:time comparison in 1d}\nIn this subsection, we compare the computational running times in Maple between the classical approach to solving $N-1$ linear equations and the faster method introduced in the previous section. Specically, we evaluate the performance of \\lstinline{GR1dLG} and \\lstinline{NewGR1dLG}, as described in \\Cref{appendix:ggr1d}, Using the commands \\lstinline{time(GR1dLG)} and \\lstinline{time(NewGR1dLG)}, we measure the execution time for varying values of $N$ with $P=[[-1,\\frac{1}{3}],[1,\\frac{1}{3}],[2,\\frac{1}{3}]]$. The measure timed in Maple are summarized below:\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$N$ & Slow Method & Faster Method\\\\\n&(seconds) & (seconds)\\\\\n\\hline\n$100$ & $0.796$ & $0.015$ \\\\ \\hline\n$105$ & $11.171$ & $0.015$ \\\\ \\hline\n$110$ & $21.093$ & $0.015$ \\\\ \\hline\n$115$ & $500.828$ & $0.015$ \\\\ \\hline\n$120$ & $2466.609$ & $0.046$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe data in the table suggests that introducing symbolic variables in \\Cref{eq:bettergeneralexp} dramatically reduces the running time for computing the expected durations of the game starting at all starting locations.\n\n\\subsection{Variance} In this subsection, we consider the variance of the duration of the generalized $1$-dimensional gambler's ruin game. The computation builds up the expected duration discussed in \\Cref{subsec:exp}. \n\nLet $P=[[a_1,p_1],[a_2,p_2],\\ldots,[a_r,p_r]]$ such that $p_1+\\ldots+p_r=1$ and define $F(x,t)$ as the probability generating function of the duration of the generalized $1$-dimensional gambler's ruin game. For $0<x<N$, this functions satisfies the recurrence relation\n\n\\begin{equation*}\nF(x,t)=t(p_1F(x+a_1,t)+p_2F(x+a_2,t)+\\cdots+p_rF(x+a_r,t)),\n\\end{equation*}\n\nwhere $F(0,t)=1$ and $F(N,t)=1$. Making the substitution $t\\mapsto z+1$ yields\n\n\\begin{equation}\\label{eq:1d gen fun in z}\nF(x,z)=(z+1)(p_1F(x+a_1,z)+p_2F(x+a_2,z)+\\cdots+p_rF(x+a_r,z),\n\\end{equation}\n\nwith $F(0,z)=1$ and $F(N,z)=1$. We derive an expression to estimate the second factorial moment. Expanding $F(x,z)$ as a Taylor series gives\n\n\\begin{equation}\\label{eq:taylor in 1d}\n F(x,z)=1+g(x)z+\\frac{h(x)}{2!}z^2+\\cdots\n\\end{equation}\n\nwhere $g(x)$ is the expected duration as defined as in \\Cref{subsec:exp}, and $h(x)$ represents the second factorial moment at $x$. Substituting \\Cref{eq:taylor in 1d} into \\Cref{eq:1d gen fun in z} results in\n\n\\begin{align*}\n 1+g(x)z+\\frac{h(x)}{2}z^2 + \\dots &=(1+z)\\left( p_1\\left(1+g(x+a_1)z+\\frac{h(x+a_1)}{2}z^2 +\\cdots \\right)\\right)\\\\\n &+(1+z)\\left(p_2\\left(1+g(x+a_2)z+\\frac{h(x+a_2)}{2}z^2+\\cdots\\right)\\right)\\\\\n &\\quad \\vdots\\\\\n &+(1+z)\\left(p_r\\left(1+g(x+a_r)z+\\frac{h(x+a_r)}{2}z^2+\\cdots\\right)\\right).\n\\end{align*}\n\n\\noindent From the previous expression, we extract the coefficient of $z^2$ to get an expression for $h(x)$. Hence,\n\n\\begin{equation*}\n h(x)-(p_1h(x+a_1)+p_2h(x+a_2)+\\cdots + p_rh(x+a_r))=2(p_1g(x+a_1)+p_2g(x+a_2)+\\cdots+p_rg(x+a_r))\n\\end{equation*}\nwhere $0=g(a_1+1)= g(a_1+2)=\\cdots=g(-1)=g(0)$ and $0=g(N)=g(N+1)=\\cdots =g(N+a_r-2)=g(N+a_r-1)$. The sum $g(x)+h(x)$ gives the second moment for $0<x<N$. \n\n\\noindent The variance at $x$, denoted $V(x)$, is computed as\n\\[ V(x)=g(x)+h(x)-(g(x))^2\\]\nwhere $g(x)$ and $h(x)$ are defined as follows\n\n\\begin{align*}\n &g(x)=\\frac{1}{p_r}g(x-a_r)-\\frac{1}{p_r}\\sum_{i=1}^{r-1}p_ig(x+a_i-a_r)-\\frac{1}{p_r}\n \\intertext{and}\n &h(x)-\\sum_{i=1}^{r}p_ih(x+a_i)=2\\sum_{i=1}^rp_ig(x+a_i).\n\\end{align*} \n\nWe conclude this section with a table comparing the expected duration and the standard deviation when the particle starts at $x=\\frac{N}{2}$ for various $N$ and probability table $P=[[-2,\\frac{1}{3}],[1,\\frac{1}{3}],[2,\\frac{1}{3}]]$ .\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$N$ & $x$ & Expected Duration& Standard Deviation \\\\\n\\hline\n$10$ & $5$ & $ 8.613479400$ & $ 6.321808669$\\\\\\hline\n$20$ & $10$ & $ 25.23344696$ & $18.44137538$\\\\ \\hline\n$30$ & $15$ & $42.94261730$ & $ 29.22243692$\\\\ \\hline\n$40$ & $20$ & $59.58246747$ & $ 37.26482832$ \\\\ \\hline\n$50$ & $25$ & $75.36543964$ & $ 43.30155080$\\\\ \\hline\n$60$ & $30$ & $90.70157954$ & $48.13687964$\\\\ \\hline\n$70$ & $35$ & $105.8379590$ & $52.27336865 $\\\\ \\hline\n$80$ & $40$ & $120.8913756$ & $55.98253147$\\\\ \\hline\n$90$ & $45$ & $135.9117966$ & $59.40593597$\\\\ \\hline\n$100$ & $50$ & $150.9194653$ & $62.61931702$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\section{$2$-Dimensional Gambler's Ruin}\\label{sec:2d case}\n\nIn this section, we consider the $2$-dimensional gambler's ruin and provide analogous results to the probability, expected duration and variance discussed in \\Cref{sec:general 1d}. We introduce symbolic variables to the recurrence equation of the probability and expected duration. We then compare the computational running times in Maple between our method and the formula provided by Kmet and Petkov\\v{s}ek. We conclude this section with an analysis of the variance.\n\nWe begin by recalling the setup for the $2$-dimensional case as described in \\Cref{subsec:2 dim}. Consider a particle starting at a point $(x,y)$ in the interior of a rectangular grid of size $M\\times N$, where $0<x<M$ and $0<y<N$. At each step, the particle moves in one of four directions, with probabilities $p_L, p_U, p_R,$ and $p_B$, corresponding to left, up, right and down movements, respectively. The particle stops moving once it hits one of the four boundaries, defined by $x=0, x=M, y=0$, or $y=N$. \n\n\n\n\n\\subsection{Probability}\nWe begin by considering the recurrence relation for the probability that the particle reaches some position at one of the four boundaries. We then rewrite this recurrence and introduce new variables for the probabilities at each $x$ and $y$.\n\nDefine $f(x,y)=f_L(x,y)L+f_R(x,y)R+f_U(x,y)U+f_D(x,y)D$, where $f_L(x,y)$ is the probability that the particle, starting at $(x,y)$ will exit the rectangle on the left side,\nand analogously for $f_R(x,y),f_U(x,y),f_D(x,y)$, and $L,R,U,D$ are formal variables.\nFor $0<x<M$ and $0<y<N$, this probability satisfies the recurrence relation\n\\begin{equation}\\label{eq:twodim prob}\n f(x,y)=p_Lf(x-1,y)+p_Uf(x,y+1)+p_Rf(x+1,y)+p_Bf(x,y-1)\n\\end{equation}\n\nwith boundary conditions $f(0,y)=L, f(x,N)=U, f(M,y)=R$ and $f(x,0)=B$ for all $0<x<M$ and $0<y<N$. \nThus, the coefficients of $L, U, R,$ and $B$ in $f(x,y)$ represent the respective probabilities that a particle starting at $(x,y)$ will end on each boundary. For any $0<x<M$ and $0<y<N$, $f(x,y)$ can be determined by solving a system of $(M-1)\\times(N-1)$ linear equations with $(M-1)\\times(N-1)$ unknowns.\n\nIn the next subsection, we will introduce symbolic variables into the recurrence relation for the expected duration, as outlined in \\Cref{sec:general 1d}. This method, applied to Equation \\ref{eq:twodim prob} , \nbut the same approach works for efficient computations of probabilities.\nThis is further detailed in \\Cref{appendix:ggr2d}.\n\n\\subsection{Expected duration}\\label{subsec:exp in 2d} We now consider the expected duration until the particle reaches some position at one of the four boundaries. We focus on the case when $p_W=p_N=p_E=p_S=\\frac{1}{4}$, and remark that one can adapt our strategy for $p_W\\neq p_N\\neq p_E\\neq p_S$ as described in \\Cref{appendix:ggr2d}. \n\n\nDefine $g(x,y)$ as the expected number of steps that a particle starting at $(x,y)$ will eventually take to reach a position at one of the boundaries. For $0<x<M$ and $0<y<N$, this expected duration satisfies the recurrence relation\n\\begin{equation}\\label{eq:2d expected duration}\n g(x,y)=\\frac{1}{4}g(x-1,y)+\\frac{1}{4}g(x,y+1)+\\frac{1}{4}g(x+1,y)+\\frac{1}{4}g(x,y-1)+1\n\\end{equation}\n\nwith boundary conditions $g(0,y)=g(M,y)=g(x,0)=g(x,N)=0$ for all $0<x<M$ and $0<y<N$. Similarly to the probability case, $g(x,y)$ can be determined by solving a system of $(M-1)\\times(N-1)$ linear equations with $(M-1)\\times(N-1)$ unknowns. \n\n\nOrr and Zeilberger provided a solution that reduces the number of linear equations from $(M-1)\\times(N-1)$ to $\\mathcal{O}(N+M)$. Their approach exploits symmetry by solving $g(0,y)=g(M-1,y)$ for $0<y<N$, and $g(x,1)=g(x,N-1)$ for $0<x<M$ \\cite{OrrDoron}. Kmet and Petkov\\v{s}ek gave an explicit solution involving a double sum, enabling direct computation of the expected duration without the need to solve systems of equations or use recursion \\cite{Kmet}. For the special case where $M=N$, they established the following result.\n\n\n\\begin{theorem}[\\cite{Kmet}, Equation (11)]\nConsider the 2-dimensional gambler's ruin problem as stated in this section. Then, the expected duration of the game when the particle starts at $(x,y)$ is given by\n\\begin{equation*}\n g(x,y)=\\frac{4}{M^2}\\sum_{\\substack{k=1 \\\\ k \\text{ odd}}}^{M-1} \\left(\\sin{\\left(\\frac{jk\\pi}{M}\\right)}\\cot{\\left(\\frac{k\\pi}{2M}\\right)} \\sum_{\\substack{\\ell=1 \\\\ \\ell \\text{ odd}}}^{M-1} \\frac{\\sin{(\\frac{i\\ell\\pi}{M})}\\cot{(\\frac{\\ell\\pi}{2M})}}{\\sin^2{(\\frac{k\\pi}{2M})}+\\sin^2{(\\frac{\\ell \\pi}{2M})}}\\right)\n\\end{equation*}\nfor any $0\\leq x,y\\leq M$.\n\\end{theorem}\n\nWe propose an alternative approach to those of Orr and Zeilberger, and Kmet and Petkov\\v{s}ek. By rewriting \\Cref{eq:2d expected duration}, we obtain the recurrence\n\\begin{equation}\\label{eq:2d new expected recurrence}\n g(x,y)=4g(x-1,y)-g(x-1,y-1)-g(x-1,y+1)-g(x-2,y)-4\n\\end{equation}\n\nwhere $x\\to x-1$. The boundary conditions remain $g(0,y)=g(M,y)=g(x,0)=g(x,N)=0$ for all $x,y$. For each $1\\leq j \\leq N-1$ define $d_j=g(1,j)$ and construct the set $S=\\{d_1,d_2,\\ldots, d_{N-1}\\}$. Using these variables, we reduce the system of equations. Initially, $g(x,y)$ is solved by constructing a system of $(M-1)\\times(N-1)$ linear equations with $(M-1)\\times(N-1)$ unknowns. By using $S$, we reformulate the system into:\n\n\\begin{itemize}[leftmargin=0.2 in]\n \\item $N-1$ equations for $x=1$,\n \\item $(M-2)\\times(N-1)$ equations for $2\\leq x\\leq M-1$, and\n \\item $N-1$ boundary equations for $x=M$.\n\\end{itemize}\n\nThe boundary equations, derived using \\Cref{eq:2d new expected recurrence}, are expressed in terms of the variables in $S$ and reduce the system to $N-1$ boundary equations with $N-1$ unknowns. Once these $N-1$ variables are solved, the remaining $(M-2)\\times(N-1)$ equations can be determined. We illustrate this process in \\Cref{ex:new approach}.\n\n\n\\begin{example}\\label{ex:new approach}\nLet $M=N=3$. This system consists of $4$ linear equations with $2$ unknowns: $d_1=g(1,1)$ and $d_2=g(1,2)$. The boundary conditions are $g(0,y)=g(M,y)=g(x,0)=g(x,N)=0$ for all $0\\leq x,y\\leq 3$. Using the recurrence in \\Cref{eq:2d new expected recurrence}, we derive\n\\begin{align*}\n d_1&=g(1,1)\\\\\n d_2&=g(1,2)\\\\\n g(2,1)&=4d_1-d_2-4\\\\\n g(2,2)&=4d_2-d_1-4\\\\\n g(3,1)&=0=16d_1-8d_2-16\\\\\n g(3,2)&=0=16d_2-8d_1-16.\n\\end{align*}\nSolving the last two equations yields $d_1=2$ and $d_2=2$. Substituting these values into the earlier equations gives $g(1,1)=g(1,2)=g(2,1)=g(2,2)=2$.\n\\end{example}\n\n\n\\subsection{Comparison between Kmet and Petkov\\v{s}ek formula and faster method}\\label{subsec:time comparison in 2d}\nWe compare the computational running times in Maple between Kmet and Petkov\\v{s}ek's formula for the expected duration and the faster method introduced in the previous subsection. Specifically, we evaluate the performance of \\lstinline{NewGR2dL} and \\lstinline{KmetPetkovsek}, as described in \\Cref{appendix:ggr2d}. Using the commands \\lstinline{time(evalf(NewGR2dL(M,M)))} and \\lstinline{time(KmetPetkovsek(M)))}, we measure the execution time for varying values of $M$. The results provide a direct comparison of the efficiency of the two methods. The measured times in Maple are summarized below:\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n$M$ & Faster Method & Kmet and Petkov\\v{s}ek & $M$ & Faster Method & Kmet and Petkov\\v{s}ek \\\\\n&(seconds) & (seconds)& &(seconds) & (seconds)\\\\\n\\hline\n10 & 0.015 & 0.015 & 90 & 9.843 & 181.953 \\\\ \\hline\n20 & 0.015 & 0.500 & 100 & 14.765 & 301.906\\\\ \\hline\n30 & 0.093 & 1.859 & 110 & 21.171 & 451.625\\\\ \\hline\n40 & 0.343 & 5.703 & 120 & 36.187 & 641.609\\\\ \\hline\n50 & 0.468 & 15.171 & 130 & 57.281 & 872.015 \\\\ \\hline\n60 & 1.171 & 35.125 & 140 & 80.140 & 1155.281\\\\ \\hline\n70 & 2.640 & 63.062 & 150 & 133.109 & 1564.515\\\\ \\hline\n80 & 5.062 & 114.234 & 160 & 226.312& 2023.125\\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nWhile Kmet and Petkov\\v{s}ek's formula expresses the expected duration as a double sum, it is computationally less efficient compared to our method.\n\n\n\\subsection{Variance} In this subsection, we analyze the variance of the duration of the $2$-dimensional gambler's ruin game under the condition $p_W=p_N=p_E=p_S=\\frac{1}{4}$. The computation builds upon the expected duration discussed \\Cref{subsec:exp in 2d}.\n\nDefine $F(x,y,t)$ as the probability generating function of the duration of the $2$-dimensional gambler's ruin game. For $0<x<M$ and $0<y<N$, this function satisfies the recurrence relation\n\n\\begin{equation}\\label{eq:2d variance eq in t}\n F(x,y,t)=\\frac{t}{4}(F(x-1,y,t)+F(x+1,y,t)+F(x,y-1,t)+F(x,y+1,t))\n\\end{equation}\n\nwhere $F(0,0,t)=1$ and $F(M,N,t)=1$. Making the substitution $t\\mapsto z+1$ yields\n\n\\begin{equation}\n F(x,y,z)=\\frac{1+z}{4}(F(x-1,y,z)+F(x+1,y,z)+F(x,y-1,z)+F(x,y+1,z)),\n\\end{equation}\n\nwith $F(0,0,z)=1$ and $F(M,N,z)=1$. We derive an expression to estimate the second factorial moment. Expanding $F(x,y,z)$ as a Taylor series gives\n\n\\begin{equation}\\label{eq:taylor in 2d}\n F(x,y,z)=1+g(x,y)z+\\frac{h(x,y)}{2!}z^2 +\\cdots\n\\end{equation}\n\nwhere $g(x,y)$ is the expected duration as defined in \\Cref{subsec:exp in 2d}, and $h(x,y)$ represents the second factorial moment at $(x,y)$. Substituting \\Cref{eq:taylor in 2d} into \\Cref{eq:2d variance eq in t} and extracting the coefficient of $z^2$ yields an expression for $h(x,y)$:\n\n\n\\[4h(x,y)-h(x-1,y)-h(x+1,y)-h(x,y-1)-h(x,y+1)=2(g(x-1,y)+g(x+1,y)+g(x,y-1)+g(x,y+1))\\]\nwhere $g(0,y)=g(M,y)=g(x,0)=g(x,N)=0$ for all $x,y$. The sum $g(x,y)+h(x,y)$ gives the second moment for $0<x<M$ and $0<y<N$. \n\nThe variance at $(x,y)$, denoted by $V(x,y)$, is computed as\n\\[ V(x,y)=g(x,y)+h(x,y)-(g(x,y))^2\\]\nwhere $g(x,y)$ and $h(x,y)$ are defined by\n\\[g(x,y)=4g(x-1,y)-g(x-1,y-1)-g(x-1,y+1)-g(x-2,y)-4\\]\nand\n\\[\n 4h(x,y)-h(x-1,y)-h(x+1,y)-h(x,y-1)-h(x,y+1)=2(g(x-1,y)+g(x+1,y)+g(x,y-1)+g(x,y+1)).\\]\n\n\nWe conclude this section with a table comparing the expected duration and the standard deviation when the particle starts at $(x,y)=(\\frac{M}{2},\\frac{M}{2})$ for various $M$ under probabilities $P=[\\frac{1}{6},\\frac{1}{3},\\frac{1}{6},\\frac{1}{3}]$.\n\n\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$M$ & $(x,y)$ & Expected Duration& Standard Deviation \\\\\n\\hline\n$2$ & $(1,1)$ & $1$ & $0$\\\\\\hline\n$4$ & $(2,2)$ & $4.470588235$ & $ 2.891342524$\\\\ \\hline\n$6$ & $(3,3)$ & $10.3030$ & $7.102295958$\\\\ \\hline\n$8$ & $(4,4)$ & $18.47746573 $ & $12.97689858$ \\\\ \\hline\n$10$ & $(5,5)$ & $28.99020033 $ & $20.52455308$\\\\ \\hline\n$12$ & $(6,6)$ & $ 41.84019550$ & $29.74741677$\\\\ \\hline\n$14$ & $(7,7)$ & $ 57.02707373$ & $40.64621816$\\\\ \\hline\n$16$ & $(8,8)$ & $ 74.55067222$ & $53.22126050$\\\\ \\hline\n$18$ & $(9,9)$ & $94.41091165 $ & $67.47268859$\\\\ \\hline\n$20$ & $(10,10)$ & $116.6077497 $ & $83.40057864$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{A Mirror Step Variant of Gambler's Ruin}\\label{sec:mirror step}\n\nIn this section, we begin by formulating a new generalization of the gambler's ruin problem in $1$-dimension. A particle starts at some point $x$ on a line of length $N$ where $0< x <N$. At each step, the particle moves from $x$ to $x-1$ with probability $q_1$, or moves from $x$ to $x+1$ with probability $q_2$, or moves from $x$ to $N-x$ with probability $p$ where $0<p<1$ and $q_1+q_2+p=1$. We call this last step the \\textit{mirror step}. The particle continues to walk on the line until it reaches $0$ or $N$. We focus on the case when $q_1=q_2=\\frac{1-p}{2}$ and we call this the \\textit{symmetric case}.\n\n\n\\subsection{Probability}\nDefine $f(x)=f_N^{(p)}(x)$ as the probability that a particle starting at $x$ will eventually reach $N$. For $0<x<N$, this probability satisfies the recurrence relation\n\\begin{equation} \\label{eq:prob recurrence mirror}\n f(x)=\\frac{1-p}{2}f(x-1)+\\frac{1-p}{2}f(x+1)+pf(N-x)\n\\end{equation}\nwhere $f(0)=0$ and $f(N)=1$. Before providing the solution to \\Cref{eq:prob recurrence mirror}, we will try to guess the limit\nas $N$ goes to infinity of the probability when the particle starts at some $x$ and ends at $N$, by using a fixed, large, $N$ as discussed in \\Cref{subsec:mirror}. We are interested in\n\n\\begin{equation}\\label{eq:limit of fN(x)}\n \\lim_{N\\to \\infty} f_N^{(p)}(x)\n\\end{equation}\n\nand we hope to get expressions for when $x=2$ and $x=N-2$. First, we describe the approach that will generate data for fixed values of $N$ and $x$. Using the data, we can make some guesses for the limit in \\Cref{eq:limit of fN(x)}.\n\n\\begin{itemize}[leftmargin=0.2 in]\n \\item Fix $N$ as large as possible and use Maple to generate $N-1$ linear equations with \\Cref{eq:prob recurrence mirror}.\n \\item Solve for the $N-1$ linear equations.\n \\item Generate data for $f_N^{(p)}(x)$ for different $p$ values with fixed $N$ and a fixed location $x$. \n \\item We use the function \\texttt{identify} in Maple to try to identify the numerical values given by $f_N^{(p)}(x)$. The function \\texttt{identify} is based, in part, on the continued fraction expansion of the numerical values. As $N$ grows the numerical value will converge to some number.\n \\item Guess a formula for the number with fixed $x$ and varying $p$.\n\\end{itemize}\n\n\\noindent We begin with an example on how to generate data in Maple using the code accompanying this article.\n\n\\begin{example}\\label{ex: x=1 prob}\nLet $N=100$ and $x=2$. We generate data for the probability that if the particle starts at $x=2$, the particle eventually ends at $100$. Let $p\\in \\{\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\ldots, \\frac{1}{10}\\}$. \n\nWe use the procedure \\texttt{Lk(p,x,N)}, as described in \\Cref{appendix:mirror}, in Maple which generates the following data in about $8.390$ seconds:\n\\begin{align*}\n T \\coloneqq &[0.48528137423857029281, 0.46410161513775458705, 0.444, 0.42705098312484227231,\\\\ &0.41171425595857973499, 0.39811169380648470689, 0.38595282133533513790, 0.375, \\\\\n &0.36506306819388080622].\n\\end{align*}\n\n\\noindent The sequence $T$ reads as follows. If the particle starts at $x=2$ and $p=\\frac{1}{2}$, the particle moves from $x$ to $x-1$ with probability $\\frac{1-p}{2}=\\frac{1}{4}$, or moves from $x$ to $x+1$ with probability $\\frac{1-p}{2}=\\frac{1}{4}$, or moves from $x$ to $100-x$ with probability $p=\\frac{1}{2}$. Then, the probability of the particle starting at $x=2$ and ending at $100$ is $T_1=0.48528137423857029281$. Similarly, if the particle starts at $x=2$ and $p=\\frac{1}{3}$, the particle moves from $x$ to $x-1$ with probability $\\frac{1-p}{2}=\\frac{1}{3}$, or moves from $x$ to $x+1$ with probability $\\frac{1-p}{2}=\\frac{1}{3}$, or moves from $x$ to $100-x$ with probability $p=\\frac{1}{3}$. Then, the probability of the particle starting at $x=2$ and ending at $100$ is $T_2=0.46410161513775458705$. Thus, $T_i$ is the probability of the particle starting at $x=2$ and ending at $100$ for $p=\\frac{1}{i+1}$ where $1\\leq i\\leq 9$.\n\nUsing \\texttt{identify} in Maple for the sequence $T$, we conjecture that each of the probabilities in $T$ converge to \n\\begin{align*}\n M\\coloneqq&\\left[-8+6 \\sqrt{2}, -3+2 \\sqrt{3}, \\frac{4}{9}, \n\\frac{-5+3 \\sqrt{5}}{4}, \n\\frac{-24 + 14\\sqrt{6}}{25}, \n\\frac{-7+4 \\sqrt{7}}{9}, \n\\frac{-32+36 \\sqrt{2}}{49}, {\\frac{3}{8}}, \n\\frac{-40+22 \\sqrt{10}}{81}\\right].\n\\end{align*}\nThat is the probability of the particle starting at $x=2$ and ending at $N$ as $N$ grows converges to $M_i$ for $p=\\frac{1}{i+1}$ where $1\\leq i\\leq 9$.\n\\end{example}\n\nThe previous example illustrates that when the particle starts at $x=2$, $\\{f_N^{(p)}(2)\\}$ converges fast. We state the following guess for the expression of the limit when $x=2$ and in \\Cref{cor:probability at infinity} we provide a proof.\n\n\\begin{guess}\\label{guess:x=2}\n If the particle starts at $x=2$, then\n\\[\\lim_{N\\to\\infty}f_N^{(p)}(2)=\\frac{2\\sqrt{p}(1+p-2\\sqrt{p})}{(1-p)^2}.\\]\n\\end{guess}\n\nUsing the same approach from above, we can obtain data for $x=N-2$. We guess the following expression of the limit when $x=N-2$ and provide a proof in \\Cref{cor:probability at infinity}.\n\n\\begin{guess}\\label{guess:x=N-2}\n If the particle starts at $x=N-2$, then\n\\[\\lim_{N\\to\\infty}f_N^{(p)}(N-2)=\\frac{(1+p)(1+p-2\\sqrt{p})}{(1-p)^2}.\\]\n\\end{guess}\n\n\nWe were able to guess more expressions for $\\lim_{N\\to\\infty} f_N(x)$ with other $x$ values. After we made these guesses, we established the following key lemma which provides a relation between $f(x)$ and $f(N-x)$ for any $0\\leq x\\leq N$. We stress that this is only true for the symmetric case when the probability of the particle moving from $x$ to $x-1$ is the same as the probability of the particle moving from $x$ to $x+1$.\n\n\\begin{lemma}\\label{lem:mirror probability equal to 1}\nConsider the symmetric case when $f(x)=\\frac{1-p}{2}f(x-1)+\\frac{1-p}{2}f(x+1)+pf(N-x)$ with boundary conditions $f(0)=0, f(N)=1$ for some $0<p<1$. For any $0\\leq x\\leq N$, the following identity holds \\[f(x) + f(N-x)=1.\\]\n\\end{lemma}\n\n\\begin{proof}\n Call $g(x)=1-f(N-x)$. We will show that $f(x)+f(N-x)=1$ by proving that $g(x)=f(x)$. Note that $g(0)=0$ and $g(N)=1$, and $f(x)=1-g(N-x)$. Using the recurrence in \\Cref{eq:prob recurrence mirror}, we substitute $1-g(N-x)$ for $f(x)$ and obtain:\n \n \\begin{equation*}\n g(N-x)=\\frac{1-p}{2}g(N-(x-1))+\\frac{1-p}{2}g(N-(x+1))+pg(x).\n \\end{equation*}\n That is,\n \\begin{equation*}\n g(x)=\\frac{1-p}{2}g(x+1))+\\frac{1-p}{2}g(x-1)+pg(N-x)\n \\end{equation*}\n which has boundary conditions $g(0)=0$ and $g(N)=1$. Therefore, $g(x)=f(x)$ as desired.\n\\end{proof}\n\nLemma \\ref{lem:mirror probability equal to 1} establishes that the sums of the probabilities when the particle starts at $x$ and when the particle starts at $N-x$ equals to 1. Using this identity, we rewrite \\Cref{eq:prob recurrence mirror} to\n\n\\begin{equation}\\label{eq:inhomogeneous eq}\n f(x)=\\frac{p}{1+p}+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f(x-1)+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f(x+1), \\quad f(0)=0, f(N)=1.\n\\end{equation}\n\nWe can now derive the probability of the particle ending at $N$ if it starts at some $x$, for general $N, x$ and $p$.\n\n\\begin{theorem}\\label{thm:probability mirror}\nConsider the generalization of the gambler's ruin problem when we add a mirror step. Then, the probability of ending at $N$ starting at $x$ is given by\n\\begin{equation*}\n f(x)=\\frac{1}{2}\\frac{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\n +\\frac{1}{2}\\frac{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N-\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x+\\frac{1}{2}\n\\end{equation*}\nwhenever we restrict the particle moves by either moving from $x$ to $x-1$ with probability $q_1$, or from $x$ to $x+1$ with probability $q_2$, or from $x$ to $N-x$ with probability $p$ where $q_1=q_2=\\frac{1-p}{2}$.\n\\end{theorem}\n\n\n\\begin{proof}\n We can solve for \\Cref{eq:inhomogeneous eq} because it is an inhomogeneous recurrence relation. Hence, we find a homogeneous and an inhomogeneous solution. \n The general solution to the homogeneous equation $f(x)=\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f(x-1)+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f(x+1)$ is\n \n \\begin{equation*}\n f(x)=A\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^x+B\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^x\n \\end{equation*}\n \n for some numbers $A$ and $B$. \n Next, we find the particular solution to the inhomogeneous relation by setting $f^*(x)=C$ for some constant $C$. Then,\n \n \\begin{equation*}\n f^*(x)=\\frac{p}{1+p}+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f^*(x-1)+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)f^*(x+1)\n \\end{equation*}\n \n becomes\n \n \\begin{equation*}\n C=\\frac{p}{1+p}+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)C+\\frac{1}{2}\\left(\\frac{1-p}{1+p}\\right)C\n \\end{equation*}\n \n which has solution $C=\\frac{1}{2}$. \n \n Therefore, $f^*(x)=\\frac{1}{2}$ is the particular solution, and the general inhomogeneous solution is\n \n \\begin{equation}\\label{eq:general inhomogeneous sln}\n f(x)=A\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^x+B\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^x+\\frac{1}{2}.\n \\end{equation}\n \n Using Maple, we find $A$ and $B$ by using the boundary conditions to get a system of two linear equations. Namely,\n \n \\begin{align*}\n 0&=A+B+\\frac{1}{2} \\intertext{and}\n \\frac{1}{2}&=A\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^N+B\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^N.\n \\end{align*}\n \n Solving for the above linear equations, we get\n \\begin{align}\n A&=\\frac{1}{2}\\frac{\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^N+1}{\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^N-\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^N} \\label{eq: A eq}\\intertext{and}\n B&=\\frac{1}{2}\\frac{\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^N+1}{\\left(\\frac{1+p-2\\sqrt{p}}{1-p}\\right)^N-\\left(\\frac{1+p+2\\sqrt{p}}{1-p}\\right)^N}.\\label{eq: B eq}\n \\end{align}\n \n Rewriting Equations \\ref{eq:general inhomogeneous sln}, \\ref{eq: A eq} and \\ref{eq: B eq} gives\n \\begin{align*}\n f(x)&=A\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x+B\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x+\\frac{1}{2} \n \\end{align*}\nas desired.\n\\end{proof}\n\n\\Cref{thm:probability mirror} provides a formula for $\\lim_{N\\to\\infty}f_N(x)$ for any $x$ where $0<x<N$.\n\n\\begin{corollary}\\label{cor:probability at infinity}\nIf the particle starts at some $x$ where $0<x<N$, then\n\\[\\lim_{N\\to \\infty} f_N^{(p)}(x)=\\frac{1}{2}-\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\]\nwhenever we restrict the particle moves by either moving from $x$ to $x-1$ with probability $q_1$, or from $x$ to $x+1$ with probability $q_2$, or from $x$ to $N-x$ with probability $p$ where $q_1=q_2=\\frac{1-p}{2}$.\n\\end{corollary}\n\n\\begin{proof}\nLet \n\n\\begin{equation}\\label{eq: fNx}\n f_N^{(p)}(x)=\\frac{1}{2}\\frac{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\n +\\frac{1}{2}\\frac{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N-\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x+\\frac{1}{2},\n\\end{equation}\n\n and \\[g_N^{(p)}(x)=\\frac{1}{2}\\frac{1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x+\\frac{1}{2}-\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x.\\]\n We will show that $\\lim_{N\\to\\infty}f_N^{(p)}(x)-g_N^{(p)}(x)=0$.\n Observe that\n \n \\begin{align*}\n f_N^{(p)}(x)-g_N^{(p)}(x)&=\\frac{1}{2}\\frac{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\n +\\frac{1}{2}\\frac{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N-\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\\\\n & \\quad -\n \\frac{1}{2}\\frac{1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x+\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\\\\n &=\\frac{1}{2}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\\left(\\frac{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N\\right)\\\\\n &\\quad+\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\left(\\frac{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N+1}{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N-\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N}+\\frac{1}{2}\\right)\\\\\n &=\\frac{1}{2}\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\\left(\\frac{\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N+\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^{2N}}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\right)-\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\left(\\frac{1+\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\right)\\\\\n &=\\frac{1}{2}\\frac{\\left(1+\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N\\right)\\left( \\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N - \\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x\\right)}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^N-\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^N}\\\\\n &=\\frac{1}{2}\\frac{\\left(1+ \\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^{N}\\right)\\left(\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x-1\\right)}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^{2N}-1}.\n \\end{align*}\n\n Since\n \n \\begin{align*}\n \\lim_{N\\to \\infty} \\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^{N}=0 \\intertext{and}\n \\lim_{N\\to \\infty}\\frac{1}{2}\\frac{\\left(\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^x-1\\right)}{\\left(\\frac{1+\\sqrt{p}}{1-\\sqrt{p}}\\right)^{2N}-1}=0,\n \\end{align*}\nIt follows that \\[\\lim_{N\\to\\infty}f_N^{(p)}(x)-g_N^{(p)}(x)=0.\\]\n\nThus,\n\\[\\lim_{N\\to \\infty} f_N^{(p)}(x) =\\frac{1}{2}-\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)^x.\\]\n\\end{proof}\n\nUsing \\Cref{cor:probability at infinity} provides a proof for \\Cref{guess:x=1}. See the following example.\n\\begin{example}\nSetting $x=1$,\n\\[\\lim_{N\\to \\infty} f_N^{(p)}(1)=\\frac{1}{2}-\\frac{1}{2}\\left(\\frac{1-\\sqrt{p}}{1+\\sqrt{p}}\\right)=\\frac{\\sqrt{p}-p}{1-p}\\] as expected.\n\\end{example}\n\n\n\\subsection{Expected duration}\n\nWe now consider the expected duration of the gambler's ruin problem with a mirror step. Define $g(x)$ as the expected number of steps that a particle starting at $x$ will eventually reach a position $0$ or $N$. For $0<x<N$, this expected duration satisfies the recurrence relation\n \\begin{equation} \\label{eq:mirror exp}\n g(x)=\\frac{1-p}{2}g(x-1)+\\frac{1-p}{2}g(x+1) + pg(N-x) +1, \\quad g(0)=0, g(N)=0.\n \\end{equation}\nWe use this recurrence relation to find a closed formula for the expected duration of the game.\n\\begin{theorem}\\label{thm:expected duration mirror}\nConsider the generalization of the gambler's ruin problem when we add a mirror step. Then, the expected duration of ending at $0$ or $N$ starting at $x$ is given by\n\\begin{equation*}\n g(x)=\\frac{1}{1-p}x(N-x)\n\\end{equation*}\nwhenever we restrict the particle moves by either moving from $x$ to $x-1$ with probability $q_1$, or from $x$ to $x+1$ with probability $q_2$, or jumps to $N-x$ with probability $p$ where $q_1=q_2=\\frac{1-p}{2}$.\n\\end{theorem}\nRemark: When $p=0$, \\Cref{thm:expected duration mirror} recovers the formula for the expected duration of the classical gambler's ruin game.\n\n\n\\begin{proof}\nLet $h(x)=\\frac{1}{1-p}x(N-x)$ and observe that $h(0)=0, h(N)=0$. We prove that $h(x)$ satisfies the same recurrence relation as $g(x)$. Applying the recurrence from \\Cref{eq:mirror exp} to $h(x)$ and simplifying yields \n\\begin{align*}\n &\\frac{1-p}{2}h(x-1) +\\frac{1-p}{2}h(x+1)+ph(N-x)+1\\\\\n &= \\frac{1-p}{2}\\left(\\frac{1}{1-p}(x-1)(N-x+1)\\right)+\\frac{1-p}{2}\\left(\\frac{1}{1-p}(x+1)(N-x-1)\\right)+p\\left(\\frac{1}{1-p}x(N-x)\\right)+1\\\\\n &=\\frac{1}{1-p}x(N-x).\n\\end{align*}\n\nHence, $h(x)$ satisfies the following recurrence relation \n\\begin{equation*}\n h(x)=\\frac{1-p}{2}h(x-1) +\\frac{1-p}{2}h(x+1)+ph(N-x)+1, \\quad h(0)=0, h(N)=0.\n\\end{equation*}\nThus, we get that $h(x)=g(x)$ which completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\\section{Future Work }\\label{sec:future} \n\nIn \\Cref{sec:mirror step}, we consider a generalization of the gambler's ruin problem where the particle starts at some point $x$ on a line of length $N$ where $0< x <N$. At each step, the particle either moves to the left by one step with probability $q_1$, moves to the right by one step with probability $q_2$, or moves to $N-x$ with probability $p$ where $0<p<1$ and $q_1+q_2+p=1$. We focus on the case when $q_1=q_2=\\frac{1-p}{2}$, and we provide formulas for the probability that the particle ends at $N$ and the expected number of steps to finish the game. Thus, it is an open problem to give formulas for general $q_1, q_2$ and $p$. \n\nComputational evidence suggests the following conjecture when the probability of moving from $x$ to $x-1$ is the same as the probability of moving from $x$ to $N-x$. \n\\begin{conjecture}\nConsider the generalization of the gambler's ruin problem when we add a mirror step. If the particle starts at $x=1$, then\n\\[\\lim_{N\\to \\infty} f_N^{(p)}(1)= \\frac{\\sqrt{(p+1)(1-3p+4p^2)}-(1-2p)(p+1)}{2p(p+1)}\\]\nwhenever we restrict the particle moves by either moving from $x$ to $x-1$ with probability $q_1$, or from $x$ to $x+1$ with probability $p$, or from $x$ to $N-x$ with probability $q_2$ where $q_1=q_2=\\frac{1-p}{2}$.\n\\end{conjecture}\n\n\n\n\\section*{Acknowledgements}\nThe author thanks her advisor Dr. Doron Zeilberger for the introduction\nto the problem and feedback on an earlier draft. The author was supported by the NSF Graduate Research Fellowship Program under Grant No. 2233066.\n\n\n\n\n\\bibliographystyle{plain}\n\\bibliography{bibliography.bib}\n\n\\appendix\n\\section{Computational Tools for Analyzing the Gambler's Ruin Problem}\\label{appendix:maple}\n\\subsection{Method Descriptions}\nThere are four text files accompanying this article: \\lstinline{GGR.txt, GGR1d.txt, GGR2d.txt,} and \\lstinline{GGR1dMirror.txt} which can be found in the GitHub repository \\\\\n\\href{https://github.com/marti310/Gamblers-Ruin}{https://github.com/marti310/Gamblers-Ruin}.\n\nIn this section, we will describe the functionality of\nsome of the main procedures. These text files should be saved in the same directory. All procedures were written and tested for Maple 20.\n\n\\subsubsection{\\lstinline{GGR.txt}}\nThe \\lstinline{GGR.txt} file contains the following main procedures.\n\n\\begin{itemize}\n\\item \\lstinline{ProbN(N)}\n\n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and computes the probability of ending at $N$ for every $1\\leq x \\leq N-1$.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR.txt`:\n N:=10;\n ProbN(N);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10]}!\n \n \\item \\lstinline{ExpN(N)}\n \n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and computes the expected number of steps of ending at either $0$ or $N$ for every $1\\leq x \\leq N-1$.\n\\end{itemize}\n\n\\subsubsection{\\lstinline{GGR1d.txt}}\\label{appendix:ggr1d}\nThe \\lstinline{GGR1d.txt} file contains procedures for the classical gambler's ruin game. In addition, it contains procedures for the $1$-dimensional case. We provide the main procedures.\n\n\\begin{itemize}\n\\item \\lstinline{GR1dPG(N,P)}\n\n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[[a_1,p_1],[a_2,p_2],...,[a_r,p_r]]$ and computes the probability of the particle ending at some position $\\geq N$ for every $1\\leq x \\leq N-1$.\n \n Remark: This procedures uses the classical approach of solving for $N-1$ linear equations.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR1d.txt`:\n P:=GR1dPG(10,[[-1,1/2],[1,1/2]]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10]}!\n\n \\item \\lstinline{GR1dLG(N,P)}\n \n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[[a_1,p_1],[a_2,p_2],...,[a_r,p_r]]$ and computes the expected number of steps for the particle to end at $\\leq 0$ or $\\geq N$ for every $1\\leq x \\leq N-1$.\n\n Remark: This procedures uses the classical approach of solving for $N-1$ linear equations.\n\n \\item \\lstinline{NewGR1dPG(N,P)}\n \n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[[a_1,p_1],[a_2,p_2],...,[a_r,p_r]]$ and computes the probability for the particle to end at $\\geq N$ for every $1\\leq x \\leq N-1$.\n\n Remark: This procedures uses the faster method.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR1d.txt`:\n P:=NewGR1dPG(10,[[-1,1/2],[1,1/2]]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10]}!\n \n \\item \\lstinline{NewGR1dLG(N,P)}\n \n Returns a list $L$ of length $N-1$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[[a_1,p_1],[a_2,p_2],...,[a_r,p_r]]$ and computes the expected number of steps for the particle to end at $\\leq 0$ or $\\geq N$ for every $1\\leq x \\leq N-1$.\n\n Remark: This procedures uses the faster method.\n\\end{itemize}\n\n\\subsubsection{\\lstinline{GGR2d.txt}}\\label{appendix:ggr2d}\nThe \\lstinline{GGR2d.txt} file contains procedures for the $2$-dimensional gambler's ruin game. We provide the main procedures.\n\n\\begin{itemize}\n\\item \\lstinline{GR2dP(M,N,L,U,R,B)}\n\n Returns an $(M-1) \\times (N-1)$ matrix whose entries are linear combinations of L,U,R,B.\n \n This function inputs positive integers $M, N$ and symbols L,U,R,B where L is the left edge, U is the top edges, R is the right edge and B is the bottom edge of the $M\\times N$ rectangle, and computes the probability of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the classical approach of solving for $(M-1)\\times(N-1)$ linear equations.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n GR2dP(3,3,L,U,R,B);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[(3L)/8 + (3B)/8 + U/8 + R/8, (3L)/8 + B/8 + (3U)/8 + R/8], [L/8 + (3B)/8 + U/8 + (3R)/8, L/8 + B/8 + (3U)/8 + (3R)/8]]}!\n\n \\item \\lstinline{GR2dL(M,N)}\n \n Returns an $(M-1)\\times (N-1)$ matrix $M$.\n \n This function inputs positive integers $M, N$ and computes the expected number of steps of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the classical approach of solving for $(M-1)\\times(N-1)$ linear equations.\n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n GR2dL(3,3);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[2, 2], [2, 2]]}!\n\n \\item \\lstinline{NewGR2dP(M,N,L,U,R,B)}\n\n Returns an $(M-1) \\times (N-1)$ matrix whose entries are linear combinations of L,U,R,B.\n \n This function inputs positive integers $M, N$ and symbols L,U,R,B where L is the left edge, U is the top edges, R is the right edge and B is the bottom edge of the $M\\times N$ rectangle, and computes the probability of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the faster method. \n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n NewGR2dP(3,3,L,U,R,B);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[(3L)/8 + (3B)/8 + U/8 + R/8, (3L)/8 + B/8 + (3U)/8 + R/8], [L/8 + (3B)/8 + U/8 + (3R)/8, L/8 + B/8 + (3U)/8 + (3R)/8]]}!\n\n \n \\item \\lstinline{NewGR2dL(M,N)}\n \n Returns an $(M-1)\\times (N-1)$ matrix $M$.\n \n This function inputs positive integers $M, N$ and computes the expected number of steps of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the faster method.\n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n NewGR2dL(3,3);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[2, 2], [2, 2]]}!\n\n \\item \\lstinline{NewGR2dPG(M,N,L,U,R,B,P)}\n\n Returns an $(M-1) \\times (N-1)$ matrix whose entries are linear combinations of L,U,R,B.\n \n This function inputs positive integers $M, N$, symbols L,U,R,B where L is the left edge, U is the top edges, R is the right edge and B is the bottom edge of the $M\\times N$ rectangle, and a probability table $P=[p_L,p_U,p_R,p_B]$ such that the particle moves left by one step with probability $p_L$, or moves up by one step with probability $p_U$, or moves right by one step with probability $p_R$, or moves down by one step with probability $p_B$, and computes the probability of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the faster method. \n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n NewGR2dPG(3,3,L,U,R,B,[1/4,1/4,1/4,1/4]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[R/8 + (3L)/8 + U/8 + (3B)/8, R/8 + (3L)/8 + (3U)/8 + B/8], [(3R)/8 + L/8 + U/8 + (3B)/8, (3R)/8 + L/8 + (3U)/8 + B/8]]}!\n \\item \\lstinline{NewGR2dLG(M,N,P)}\n\n Returns an $(M-1) \\times (N-1)$ matrix.\n \n This function inputs positive integers $M, N$, and a probability table $P=[p_L,p_U,p_R,p_B]$ such that the particle moves left by one step with probability $p_L$, or moves up by one step with probability $p_U$, or moves right by one step with probability $p_R$, or moves down by one step with probability $p_B$, and computes the expected number of steps of the particle starting at some point $(a,b)$ and ending on L, U, R or B for every $1\\leq a \\leq M-1$ and $1\\leq b \\leq M-1$.\n\n Remark: This procedures uses the faster method. \n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n NewGR2dLG(3,3,[1/4,1/4,1/4,1/4]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[2, 2], [2, 2]]}!\n\n\n \n \\item \\lstinline{KmetPetkovsek(N)}\n\n Returns an $(N-1) \\times (N-1)$ matrix.\n \n This function inputs a positive integer $N$ and implements Kmet and Petkovsek's formula for the expected duration of the 2-dimensional gambler's ruin game for the $M=N$ case. \n \n Example:\n \n \\begin{lstlisting}\n read `GGR2d.txt`:\n KmetPetkovsek(3);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[[2, 2], [2, 2]]}!\n\\end{itemize}\n\n\\subsubsection{\\lstinline{GGR1dMirror.txt}}\\label{appendix:mirror}\nThe \\lstinline{GGR1dMirror.txt} file contains the following main procedures for the new generalization of gambler's ruin where we add a third step.\n\n\\begin{itemize}\n\\item \\lstinline{ProbN2(N,P)}\n\n Returns a list $L$ of length $N$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[p_1,p_2,p_3]$ where $p_1+p_2+p_3=1$ and outputs a list $L$ of length $N$ such that $L[x]$ is the probability of the particle ending at $N$ when it starts at $x$ where the particle can move from $x$ to $x-1$ with probability $p_1$, or $x$ to $x+1$ with probability $p_2$, or $x$ to $N-x$ with probability $p_3$.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR1dMirror.txt`:\n ProbN2(5,[1/3,1/3,1/3]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[7/19, 9/19, 10/19, 12/19, 1]}!\n\n \\item \\lstinline{ExpN2(N,P)}\n\n Returns a list $L$ of length $N$.\n \n This function inputs a positive integer $N$ and a probability table $P$ where $P=[p_1,p_2,p_3]$ where $p_1+p_2+p_3=1$ and outputs a list $L$ of length $N$ such that $L[x]$ is the expected number of steps that the particle takes to end at $N$ or $0$ when it starts at $x$ where the particle can move from $x$ to $x-1$ with probability $p_1$, or $x$ to $x+1$ with probability $p_2$, or $x$ to $N-x$ with probability $p_3$.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR1dMirror.txt`:\n ExpN2(5,[1/3,1/3,1/3]);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{[6, 9, 9, 6, 0]}!\n\n\n \\item \\lstinline{Lk(p,x,N)}\n\n Returns a number.\n \n This function inputs a probability value $p, 0<p<1$, and positive integers $x,N$ where $0<x<N$ and outputs the exact probability of the particle ending at $N$ when it starts at location $x$ where with probability $(1-p)/2$ the particle moves to $x-1$, with probability $(1-p)/2$ the particle moves to $x+1$ and with probability $p$ the particle moves to $N-x$.\n\n Example:\n \n \\begin{lstlisting}\n read `GGR1dMirror.txt`:\n Lk(1/3,4,100);\n \\end{lstlisting}\n Output:\n \n \\lstinline!{0.4974226119}!\n \n\\end{itemize}\n\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":89,"cells":{"id":{"kind":"string","value":"2412.07644v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07644v1"},"title":{"kind":"string","value":"Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations"},"full_text":{"kind":"string","value":"\\documentclass[11pt]{article}\n\\usepackage[centertags]{amsmath}\n\\usepackage{mathtools}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{graphicx}\n\\usepackage{upquote}\n\\usepackage{cases}\n\\usepackage{hyperref}\n\\usepackage{xcolor}\n\\usepackage{multicol}\n\n\\usepackage{scalerel}\n\n\\usepackage{scalerel}\n\n\\newcommand\\reallywidehat[1]{\\arraycolsep=0pt\\relax\\begin{array}{c}\n\\stretchto{\n \\scaleto{\n \\scalerel*[\\widthof{\\ensuremath{#1}}]{\\kern-.5pt\\bigwedge\\kern-.5pt}\n {\\rule[-\\textheight/2]{1ex}{\\textheight}} }{\\textheight} }{0.5ex}\\\\ #1\\\\ \\rule{-1ex}{0ex}\n\\end{array}\n}\n\n\\textwidth=15cm \\topmargin -1.5cm\n\\textheight=22.5cm\n\\evensidemargin 0.4cm\n\\oddsidemargin 0.6cm\n\\frenchspacing\n\n\\newtheorem{tw}{Theorem}[section]\n\\newtheorem{lem}[tw]{Lemma}\\newtheorem{cor}[tw]{Corollary}\\newtheorem{prop}[tw]{Proposition}\\newtheorem{definition}{Definition}[section]\n\\newtheorem{rem}{Remark}\n\\newtheorem{example}{Example}[section]\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\nn{{\\hfill$\\square$}}\n\\def\\X{{\\mathcal X}}\n\\def\\G{{\\mathcal G}}\n\\def\\R{{\\mathcal R}}\n\\def\\Rn{{\\mathbb R}^n}\n\\def\\N{{\\mathbb N}}\n\\def\\t{{\\mathbf t}}\n\\def\\s{{\\mathbf s}}\n\\def\\L{{\\pmb{\\mbox{L}}}}\n\\def\\={\\hspace{-3mm}&=&\\hspace{-3mm}}\n\\def\\ll{\\left|\\hspace{-0.5mm}\\left|}\n\\def\\rr{\\right|\\hspace{-0.5mm}\\right|}\n\\renewcommand{\\baselinestretch}{1,2}\n\n\n\n\n\\title{\\texttt{ Characterizations of multidimensional compact almost automorphic functions and applications to Poisson's and heat equations}}\n\\author{ Alan Chávez$^1$\\footnote{E-mail: ajchavez@unitru.edu.pe (corresponding author)}~, Jolbyn Casta\\~neda $^2$\\footnote{E-mail: jocastanedar@unitru.edu.pe}, Alexis R. Carranza $^3$\\footnote{E-mail: alexxx@unitru.edu.pe}, Kamal Khalil $^4$\\footnote{E-mail: kamal.khalil.00@gmail.com.}\\\\\n\\small{$^{1,3}$ OASIS research group, Instituto de investigaci\\'on en Matem\\'aticas, }\\\\\n\\small{ Departamento de Matem\\'aticas, FCFYM, Universidad Nacional de Trujillo,}\\\\\n\\small{Av. Juan Pablo II S/N, Trujillo-Per\\'u}\\\\\n\\small{$^{2}$ OASIS research group \\& Escuela de Matem\\'aticas - FCFYM, Universidad Nacional de Trujillo,}\\\\\n\\small{Av. Juan Pablo II S/N, Trujillo-Per\\'u}\\\\\n\\small{$^{4}$ LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France,}\\\\\\\\\n\\small{TO THE MEMORY OF MANUEL PINTO}\\\\\n}\n\n\\bibliographystyle{acm}\n\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\n\n\nLet \\(\\mathcal{G}\\) be a non-empty subset of the Euclidean space \\(\\mathbb{R}^m\\) (\\(m \\geq 1\\)). This work is dedicated to further exploring the properties of \\(\\mathcal{G}\\)-multi-almost automorphic functions defined on \\(\\mathbb{R}^m\\) with values in a Banach space \\(\\mathbb{X}\\). Using the theory of \\(\\mathcal{G}\\)-multi-almost automorphic functions, we provide two new characterizations of compact almost automorphic functions. In the first characterization, \\(\\mathcal{G}\\) corresponds to the lattice subgroup \\(\\mathbb{Z}^m \\subset \\mathbb{R}^m\\); in the second, \\(\\mathcal{G}\\) is taken to be a dense subset of \\(\\mathbb{R}^m\\). Furthermore, we establish the invariance of the space of bounded and compactly \\(\\mathcal{G}\\)-multi-almost automorphic functions under integral operators with Bi-almost automorphic kernels. Finally, we present applications to the analysis of the almost automorphic dynamics of Poisson's equation and the heat equation.\n\n\\end{abstract}\n\n\\tableofcontents\n\n\\section{Introduction}\\label{sec:Introduction}\n\\setcounter{equation}{0}\n\\color{red}\n\n\n\\color{black}\nSince their introduction by S. Bochner in 1966 \\cite{03,04,05}, almost automorphic functions have garnered significant attention from mathematicians. This interest arises, on one hand, because these functions are a natural generalization of periodic and almost periodic functions, making them an ideal framework for studying the dynamics of differential, integral, or integro-differential equations, as demonstrated in works such as \\cite{campos2014almost, campos2020barycentric, rezoug2024asymptotically} and other related references. On the other hand, the concept of almost automorphy provides a fertile ground for investigations in abstract topological dynamical systems, as evidenced by studies like \\cite{fuhrmann2020tameness, garcia2021mean, lenz2024abstract, lenz2024pure} and the references cited therein.\n\nIn this work, we contribute to the ongoing study of the almost automorphic dynamics of partial differential equations, specifically to the Poisson's and heat equations.\n\nThe concept of complex-valued almost automorphic functions on a topological group \\( G \\) was first developed by Veech \\cite{18, veech1965almost}. Let \\( \\mathcal{X} \\) be a Banach space, \\( G = \\mathbb{R}^m \\), and let \\( \\mathrm{R} \\) denote the collection of all sequences in \\( \\mathbb{R}^m \\). The notion of almost automorphy for a function \\( f: \\mathbb{R}^m \\to \\mathcal{X} \\) is described in the following definition:\n\n\\begin{definition}(Bochner)\\label{dfAA}\nLet \\( f: \\mathbb{R}^m \\rightarrow \\mathcal{X} \\) be a continuous function. Then, \\( f \\) is almost automorphic if and only if, for any sequence \\( \\mathbf{b}_k = (b_k^1, b_k^2, \\dots, b_k^m) \\in \\mathrm{R} \\), there exist a subsequence \\( \\mathbf{b}_{k_l} = (b_{k_l}^1, b_{k_l}^2, \\dots, b_{k_l}^m) \\) of \\( \\mathbf{b}_k \\) and a function \\( f^* : \\mathbb{R}^m \\to \\mathcal{X} \\) such that the following pointwise limits hold:\n\\[\nf^*(\\mathbf{t}) = \\lim_{l \\to \\infty} f(\\mathbf{t} + \\mathbf{b}_{k_l}), \\quad\nf(\\mathbf{t}) = \\lim_{l \\to \\infty} f^*(\\mathbf{t} - \\mathbf{b}_{k_l}).\n\\]\n\\end{definition}\n\nAdditionally, if the previous limits are replaced by the following:\n\\[\n\\lim_{l \\to \\infty} \\sup_{\\mathbf{t} \\in \\mathcal{K}} \\| f(\\mathbf{t} + \\mathbf{b}_{k_l}) - f^*(\\mathbf{t}) \\|_{\\mathcal{X}} = 0, \\quad\n\\lim_{l \\to \\infty} \\sup_{\\mathbf{t} \\in \\mathcal{K}} \\| f^*(\\mathbf{t} - \\mathbf{b}_{k_l}) - f(\\mathbf{t}) \\|_{\\mathcal{X}} = 0,\n\\]\nwhere \\( \\mathcal{K} \\) is any compact subset of \\( \\mathbb{R}^m \\), then \\( f \\) is called compact almost automorphic.\n\nNote that in the provided definition, the sequence \\( (\\mathbf{b}_k)_{k \\in \\mathbb{N}} \\) can be any sequence in the Euclidean space \\( \\mathbb{R}^m \\). However, a new approach explored in works such as \\cite{chavez2022multi, chavez2023almost} involves considering these sequences within a set \\( \\mathrm{R} \\), which is a non-empty collection of sequences in \\( \\mathbb{R}^m \\) (rather than the collection of all sequences in \\( \\mathbb{R}^m \\)). This perspective led the first author and collaborators to introduce the concept of \\( \\mathrm{R} \\)-multi-almost periodic/automorphic functions, a more general notion than the classical concept of almost periodicity/automorphy. In \\cite{chavez2022multi, chavez2023almost}, the authors extensively studied the fundamental structural properties of \\( (\\mathrm{R}, \\mathcal{B}) \\)-multi-almost periodic/automorphic functions and provided applications to both partial differential equations and integral equations.\n\nLet \\( \\mathcal{G}\\) be a non empty subset of \\(\\mathbb{R}^m\\). In the present work, we will denote by \\( \\mathrm{R}_\\mathcal{G} \\) a non empty collection of sequences in \\( \\mathcal{G} \\), while \\( \\mathrm{R}_\\mathcal{G}^a \\) will denote the collection of all sequences in \\( \\mathcal{G} \\). What is involved in this work is the notion of \\( \\mathrm{R}_\\mathcal{G} \\)-multi-almost automorphic function, which is as follows (see also Definition \\ref{eovako} below)\n\n\\begin{definition}\\label{eovako00}\\index{function!(compactly) $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic}\nLet $F : {\\mathbb R}^{m} \\rightarrow \\mathcal{X}$ be a continuous function. $F(\\cdot)$ is ${\\mathrm{R}}_{{\\mathcal{G}}}$-multi-almost automorphic if, for every sequence $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m})) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ there exist a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k})$ and a function\n$F^{\\ast} : {\\mathbb R}^{m} \\rightarrow \\mathcal{X}$ such that\n\\begin{align}\\label{love1234567800}\n\\lim_{l\\rightarrow +\\infty}F\\bigl({\\bf t} +(b_{k_{l}}^{1},\\cdot \\cdot\\cdot, b_{k_{l}}^{m}) \\bigr)=F^{\\ast}({\\bf t} ) \n\\end{align}\nand\n\\begin{align}\\label{love12345678900}\n\\lim_{l\\rightarrow +\\infty}F^{\\ast}\\bigl({\\bf t} -(b_{k_{l}}^{1},\\cdot \\cdot\\cdot, b_{k_{l}}^{m}) \\bigr)=F({\\bf t} ),\n\\end{align}\npointwise for ${\\bf t}\\in {\\mathbb R}^{m}$.\n\\end{definition}\n\nWhen \\(\\mathrm{R}_\\mathcal{G}=\\mathrm{R}_\\mathcal{G}^a \\), by way of simplifying things, \\( \\mathrm{R}_\\mathcal{G}^a \\)-multi-almost automorphic functions will be referred to as \\( \\mathcal{G} \\)-multi-almost automorphic functions.\n\nThe concept of \\( \\mathcal{G} \\)-multi-almost automorphic functions is not entirely new. In the one-dimensional case where \\( \\mathcal{G} = \\mathbb{Z} \\) (the discrete subgroup of \\( \\mathbb{R} \\)), the first author and collaborators introduced the class of piecewise continuous almost automorphic functions (also known as \\( \\mathbb{Z} \\)-almost automorphic functions) between 2013 and 2014. Initially, this class of discontinuous functions was developed to study differential equations with piecewise constant arguments \\cite{chavez2014discontinuous}. Later, the approach was extended to investigate differential equations with deviating arguments, as demonstrated in works such as \\cite{ding2017asymptotically, qi2022piecewise}.\n\nMore recently, the theory of \\( \\mathbb{Z} \\)-almost automorphic functions with values in a general Banach space has been explored in \\cite{Alanpiecewise}. With the help of this theory, it is proved that every compact almost automorphic function is \\( \\mathbb{Z} \\)-almost automorphic and uniformly continuous, and vice versa, providing a new characterization of the compact almost automorphic function space. Additionally, in the same work, \\( \\mathbb{Z} \\)-almost automorphic functions were utilized to offer a straightforward proof of the characterization of discrete almost automorphic functions as restrictions to \\( \\mathbb{Z} \\) of compact almost automorphic functions. Consequently, new applications beyond the theory of differential equations for \\( \\mathbb{Z} \\)-almost automorphic functions have emerged. In the work \\cite{es2022compact} the study of the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces is provided.\n\nBuilding on the aforementioned preliminaries, in the applications of the class of \\( \\mathbb{Z} \\)-almost automorphic functions and recognizing the significance of almost automorphic functions in the study of differential equations, this work makes two primary contributions:\n\\begin{itemize}\n\\item First, we present two new characterizations of compact almost automorphic functions (Section \\ref{section char}), expanding the theoretical framework established in earlier studies.\n\\item Second, using the new characterizations, we analyze the almost automorphic dynamics of Poisson's and heat equations.\n\\end{itemize}\n\n\\noindent Let us develop into more detail regarding the second point mentioned previously. We begin by considering the Poisson's equation:\n\n\\begin{equation}\\label{Poisson 0}\n\\Delta u=f,\n\\end{equation}\nwhere \\( \\Delta \\) denotes the Laplace operator in the spatial variable \\( x = (x_1, \\ldots, x_m) \\), defined as \\( \\Delta = \\sum_{i=1}^m \\partial^2_{x_i} \\). After characterizing compact almost automorphic functions, we establish, for equation (\\ref{Poisson 0}), the following result: {\\it if \\( f \\) is bounded and \\( \\mathcal{G} \\)-multi-almost automorphic, where \\( \\mathcal{G} \\) is either the lattice subgroup \\( \\mathbb{Z}^m \\) or a dense subset of \\( \\mathbb{R}^m \\), then every bounded and continuous function that is a distributional solution of the Poisson's equation (\\ref{Poisson 0}) is compact almost automorphic}. \n\n\nNow, le us consider the following initial value problem for the heat equation: \n\\begin{equation}\\label{Heat equation 1}\n \\left\\{\n \\begin{array}{lll}\n \\partial_t u(t,x)&=&\\Delta u(t,x), \\quad t>0,\\, x\\in \\mathbb{R}^m,\\\\\n u(0,x) &= &f(x), \\quad x\\in \\mathbb{R}^m,\n \\end{array}\\right.\n \\end{equation}\nwhere \\( \\partial_t \\) denotes the partial derivative with respect to time \\( t \\) and \\(\\Delta \\) is the Laplace operator in space variable. Analogously to the Poisson's equation, we prove that: {\\it if the initial data \\( f \\) in (\\ref{Heat equation 1}) is \\( \\mathcal{G} \\)-multi-almost automorphic, where \\( \\mathcal{G} \\) is either the lattice subgroup \\( \\mathbb{Z}^m \\) or a dense subset of \\( \\mathbb{R}^m \\), then the solution \\( u(t,x) \\) of the heat equation (\\ref{Heat equation 1}) is also compact almost automorphic in space variable}. \n\nIn summary, our work extends and generalizes results from previous studies such as \\cite{CHAKPPINTO2023, sibuya1971almost}, emphasizing the persistence of compact almost automorphic properties in solutions to the Poisson's and heat equations under weaker conditions on the source or initial data (respectively).\n\nThe organization of the present work is as follows: In Section \\ref{section maa}, we provide the definition and some basic facts about \\( \\mathcal{G} \\)-multi-almost automorphic functions. In Section \\ref{section dense}, we study the case in which \\( \\mathcal{G} \\) is a dense subgroup of \\( \\mathbb{R}^m \\). In Section \\ref{section char}, we present new characterizations of compact almost automorphic functions. In Section \\ref{section conv}, we prove the invariance under convolution products with compact Bi-almost automorphic kernels of the bounded and compactly \\( \\mathrm{R}_{\\mathcal{G}} \\)-multi-almost automorphic function space. In Section \\ref{section appl}, we provide applications of the developed theory to Poisson's and heat equations. Finally, the appendices contain additional results and proofs related to almost automorphic functions on topological groups, as well as conditions under which a subgroup of the Euclidean space becomes dense.\n\n\\section{On $\\mathcal{G}$-multi-almost automorphic functions}\\label{section maa}\n\nThroughout the present work, $\\mathcal{X}, \\mathcal{Y}$ denote Banach spaces, we assume that $m\\in {\\mathbb N}$, ${\\mathcal B}$ is a non-empty collection of subsets of $\\mathcal{X}$, \\( {\\mathcal{G}} \\) is a non-empty subset of ${\\mathbb R}^{m}$, ${\\mathrm{R}}_{{\\mathcal{G}}}$ is a non-empty set of sequences in $\\mathcal{G}$ and ${\\mathrm{R}}_{{\\mathcal{G}}}^a$ is the collection of all the sequences in $\\mathcal{G}$. Usually, ${\\mathcal B}$ denotes the collection of all bounded subsets of $\\mathcal{X}$ or all compact subsets of $\\mathcal{X}$. We will always assume that for every $x\\in \\mathcal{X}$, there exist $B\\in {\\mathcal B}$ such that $x\\in B.$ $BUC(\\mathbb{R}^m; \\X)$ is the space of bounded and uniformly continuous functions from $\\mathbb{R}^m$ to $ \\X$, while $ UC(\\mathbb{R}^m; \\X)$ is the space of uniformly continuous functions from $\\mathbb{R}^m$ to $ \\X$.\n\n\\begin{definition}\\label{eovako}\\index{function!(compactly) $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic}\nSuppose that $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ is a continuous function. Then \nwe say that the function $F(\\cdot;\\cdot)$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic if, for every $B\\in {\\mathcal B}$ and for every sequence $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m})) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ there exist a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k})$ and a function\n$F^{\\ast} : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ such that\n\\begin{align}\\label{love12345678}\n\\lim_{l\\rightarrow +\\infty}F\\bigl({\\bf t} +(b_{k_{l}}^{1},\\cdot \\cdot\\cdot, b_{k_{l}}^{m});x\\bigr)=F^{\\ast}({\\bf t};x) \n\\end{align}\nand\n\\begin{align}\\label{love123456789}\n\\lim_{l\\rightarrow +\\infty}F^{\\ast}\\bigl({\\bf t} -(b_{k_{l}}^{1},\\cdot \\cdot\\cdot, b_{k_{l}}^{m});x\\bigr)=F({\\bf t};x),\n\\end{align}\npointwise for ${\\bf t}\\in {\\mathbb R}^{m}$ and uniformly for $x\\in B$. If the above limits converge uniformly on compact subsets of ${\\mathbb R}^{m}$, then we say that \n$F(\\cdot ; \\cdot)$ is compactly $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic. \n\\end{definition}\nIn the particular case in which ${\\mathrm{R}}_{{\\mathcal{G}}}={\\mathrm{R}}_{{\\mathcal{G}}}^a$, we say that $F(\\cdot ; \\cdot)$ is $({\\mathcal{G}},{\\mathcal B})$-multi-almost automorphic (res. compactly $({\\mathcal{G}},{\\mathcal B})$-multi-almost automorphic). Also note that, if $\\mathcal{G}=\\mathbb{R}^m$, then we are in the class of $({\\mathrm{R}},{\\mathcal B})$-multi-almost automorphic functions, see \\cite{chavez2022multi}.\n\n\nAlso, if the function $F({\\bf t};x) =F( {\\bf t} )$ (i.e., $F : {\\mathbb R}^{m} \\rightarrow \\mathcal{Y}$), then we say that $F$ is ${\\mathrm{R}}_{{\\mathcal{G}}}$-multi-almost automorphic (resp. compactly ${\\mathrm{R}}_{{\\mathcal{G}}}$-multi-almost automorphic) and that space is denoted by $ {{\\mathrm{R}}_{{\\mathcal{G}}}}AA({\\mathbb R}^{m} ; \\mathcal{Y})$ (resp. $ \\mathcal{K} {\\mathrm{R}}_{{\\mathcal{G}}}AA({\\mathbb R}^{m} ; \\mathcal{Y})$). In this situation, if ${\\mathrm{R}}_{{\\mathcal{G}}}={\\mathrm{R}}_{{\\mathcal{G}}}^a$, $F(\\cdot)$ will be called ${\\mathcal{G}}$-multi-almost automorphic (resp. compactly ${\\mathcal{G}}$-multi-almost automorphic), see Definition \\ref{eovako00}; and the space is denoted by $ {{\\mathcal{G}}}AA({\\mathbb R}^{m} ; \\mathcal{Y})$ (resp. $ \\mathcal{K} {\\mathcal{G}}AA({\\mathbb R}^{m} ; \\mathcal{Y})$). Also, $ AA({\\mathbb R}^{m} ; \\mathcal{Y})$ (resp. $ \\mathcal{K}AA({\\mathbb R}^{m} ; \\mathcal{Y})$) will denote the space of (classical) almost automorphic functions (resp. compactly-almost automorphic functions).\n\n\\begin{prop}\\label{eovako54321}\\index{function!(compactly) $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic}\nSuppose that $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ is a continuous function. Then \nthe function $F(\\cdot;\\cdot)$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic if and only if for every $B\\in {\\mathcal B}$ and for every sequence $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m})) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ there exists a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k})$ such that\n\\begin{align}\\label{snajkamlada}\n\\lim_{l\\rightarrow +\\infty}\\lim_{n\\rightarrow +\\infty} F\\Bigl({\\bf t} - {\\bf b}_{k_{l}}+{\\bf b}_{k_{n}} ;x\\Bigr)=F({\\bf t};x) ,\n\\end{align}\npointwise for ${\\bf t}\\in {\\mathbb R}^{m}$ and all $x\\in B$.\n\\end{prop}\n\\begin{proof}\nIt is not difficult to see that if $F$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic, then given any set $B\\in {\\mathcal B}$ and any sequence $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m})) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$, there exists a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k})$ such that (\\ref{snajkamlada}) holds. On the other hand, let $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m})) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ and $B\\in {\\mathcal B}$, by hypothesis, there exists a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k})$ such that (\\ref{snajkamlada}) holds. Now, let us define, for each fixed $l$, each ${\\bf t}\\in \\mathbb{R}^m$ and any $x\\in B$, the function\n$$F^*({\\bf t} -{\\bf b}_{k_{l}};x):=\\lim_{n\\rightarrow +\\infty} F\\Bigl({\\bf t} -{\\bf b}_{k_{l}}+{\\bf b}_{k_{n}} ;x\\Bigr)\\, . $$\nThen, for each ${\\bf t} \\in \\mathbb{R}^m$, since ${\\bf t}={\\bf t} +{\\bf b}_{k_{l}} - {\\bf b}_{k_{l}}$ we have\n$$F^*({\\bf t};x)=\\lim_{n\\rightarrow +\\infty} F\\Bigl({\\bf t} + {\\bf b}_{k_{n}} ;x\\Bigr)\\, ,$$\nuniformly for $x\\in B$. On the other hand, we have\n$$ \\lim_{n\\rightarrow +\\infty}F^*({\\bf t}- {\\bf b}_{k_{n}};x)=\\lim_{l\\rightarrow +\\infty}F^*({\\bf t} - {\\bf b}_{k_{l}};x)=$$\n$$=\\lim_{l\\rightarrow +\\infty}\\lim_{n\\rightarrow +\\infty} F\\Bigl({\\bf t} - {\\bf b}_{k_{l}}+{\\bf b}_{k_{n}} ;x\\Bigr)=F\\Bigl({\\bf t } ;x\\Bigr)\\, ,$$\npoint-wisely in ${\\bf t} \\in \\mathbb{R}^m$ and uniformy for $x\\in B$. \nThus, $F$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic.\n\\end{proof}\n\n\\begin{definition}\\label{UnifContF}\nLet $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$. Then, the function $F(\\cdot;\\cdot)$ is called uniformly continuous in $\\mathbb{R}^m$, uniformly for any $B\\in {\\mathcal B}$ if, for every $B\\in {\\mathcal B}$ and every\n$\\epsilon>0$, there exists $\\delta =\\delta (\\epsilon,B)>0$, such that, if $\\t,\\s \\in \\mathbb{R}^m$ with $||\\t-\\s||<\\delta$, then \n$$\n\\sup_{x\\in B}\\| F \\bigl({\\bf t};x\\bigr) - F\\bigl({\\bf s};x\\bigr) \\|_{\\mathcal{Y}}<\\epsilon\\, .\n$$\n\\end{definition}\nIn this situation, we will say that $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ is uniformly continuous in the first variable.\n\nIn the following definition, we introduce the notion of $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-uniformly continuous function.\n\n\\begin{definition}\\label{(R,B) uniformly continuous}\nLet $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$. Then, the function $F(\\cdot;\\cdot)$ is called $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-uniformly continuous if, for every $B\\in {\\mathcal B}$ and for every two sequences $({\\bf a}_{k}=(a_{k}^{1},a_{k}^{2},\\cdots ,a_{k}^{m}))$, $ ({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m}))$ $ \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ such that $ ||{\\bf a}_{k}-{\\bf b}_{k} || \\rightarrow 0$, as $k\\to +\\infty$, we have \n\\begin{align}\n\\lim_{k\\rightarrow +\\infty } \\sup_{x\\in B}\\| F \\bigl({\\bf t}+{\\bf a}_{k};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k};x\\bigr) \\|_{\\mathcal{Y}} \\rightarrow 0,\n\\end{align}\npointwise for ${\\bf t} \\in \\mathbb{R}^m$.\\end{definition}\nNote that, if ${\\mathcal{G}}=\\mathbb{R}^m$ and ${\\mathrm{R}}_{{\\mathcal{G}}}={\\mathrm{R}}_{{\\mathcal{G}}}^a$, then Definition \\ref{UnifContF} and Definition \\ref{(R,B) uniformly continuous} are equivalent. \n\nThe next theorem gives a useful consequence for functions that are compactly $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic. Its proof is inspired by \\cite[Lemma 3.3]{chavez2024compact}. The theorem is used to prove, for instance, the uniform continuity of functions defined by integrals with kernels depending on two variables (see for example Theorems \\ref{NewThmappl} and \\ref{InvConv1}).\n\n\\begin{tw}\\label{ThRKAA}\nLet $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ be bounded and continuous. If $F$ is compactly $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic, then $F$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic and $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-uniformly continuous.\n\\end{tw}\n\\begin{proof}\n\nSince \\( F \\) is compactly \\( ({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal{B}}) \\)-multi-almost automorphic, it follows that \\( F \\) is \\( ({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal{B}}) \\)-multi-almost automorphic. Now, let us prove that \\( F \\) is \\( ({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal{B}}) \\)-uniformly continuous.\n\nLet \\( B \\) be any set in \\( \\mathcal{B} \\), and let us pick the arbitrary sequences \\( {\\bf a}_{k} = (a_{k}^{1}, a_{k}^{2}, \\cdots , a_{k}^{m}) \\) and \\( {\\bf b}_{k} = (b_{k}^{1}, b_{k}^{2}, \\cdots , b_{k}^{m}) \\in {\\mathrm{R}}_{{\\mathcal{G}}} \\) which satisfy \\( ||{\\bf a}_{k} - {\\bf b}_{k} || \\rightarrow 0 \\) as \\( k \\to \\infty \\). We must show that, for each \\( {\\bf t} \\in \\mathbb{R}^m \\):\n\n\\begin{align*}\n\\lim_{k\\to \\infty}\\sup_{x\\in B}\\| F \\bigl({\\bf t}+{\\bf a}_{k};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k};x\\bigr) \\|_{\\mathcal{Y}} = 0\\, .\n\\end{align*}\nLet us define for each $({\\bf t};x)$ the uniformly bounded sequence \n$$ \\alpha_{k}({\\bf t};x):= \\| F \\bigl({\\bf t}+{\\bf a}_{k};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k};x\\bigr) \\|_{\\mathcal{Y}}\\, .$$\n We claim that $\\alpha_{k}({\\bf t},x) \\to 0$ as $k\\to \\infty$. In fact, let $( \\alpha_k'({\\bf t},x)) \\subset (\\alpha_k({\\bf t},x))$ be a convergent subsequence which converges to $\\psi_0({\\bf t},x)$, where\n$$ \\alpha_k'({\\bf t},x)= \\| F \\bigl({\\bf t}+{\\bf a}_{k}^{'};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k}^{'};x\\bigr) \\|_{\\mathcal{Y}}$$\nand $ ({\\bf a}_{k}^{'}) \\subset ({\\bf a}_{k})$, $({\\bf b}_{k}^{'}) \\subset ({\\bf b}_{k})$ are subsequences coming from the definition of $ \\alpha_k'({\\bf t},x)$. \nSince $F$ is compactly $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic, there exist a subsequence $({\\bf b}_{k_{l}}=(b_{k_{l}}^{1},b_{k_{l}}^{2},\\cdot \\cdot\\cdot , b_{k_{l}}^{m}))$ of $({\\bf b}_{k}^{'})$ (also a subsequence $({\\bf a}_{k_{l}})$ of $({\\bf a}_{k}^{'})$) and a continuous (in the first variable) function\n$F^{\\ast} : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ such that the following limits hold\n\n\\begin{align}\\label{love12345678}\n\\lim_{l\\rightarrow +\\infty} \\sup_{t\\in \\mathcal{E}} \\| F\\bigl({\\bf t} +{\\bf b}_{k_{l}};x\\bigr)-F^{\\ast}({\\bf t};x) \\|\n\\end{align}\nand\n\\begin{align}\\label{love123456789}\n\\lim_{l\\rightarrow +\\infty} \\sup_{t\\in \\mathcal{E}} \\| F^{\\ast}\\bigl({\\bf t} -{\\bf b}_{k_{l}};x\\bigr)-F({\\bf t};x)\\|,\n\\end{align}\nwhich are also uniformly in $x\\in B$ and, $\\mathcal{E}$ is any compact subset of $\\mathbb{R}^m$ \n\nOn the other hand, because $(||{\\bf a}_{k}-{\\bf b}_{k}||)$ is bounded, there exists $R>0$ such that for all $l\\in \\mathbb{\nN}$, ${\\bf c}_{k_{l}}:={\\bf a}_{k_{l}}-{\\bf b}_{k_{l}} \\in \\overline{ B(0,R)}$ (the closed ball with center $0$ and radius $R$). Therefore, we have:\n\\begin{align*}\n\\| F \\bigl({\\bf t}+{\\bf a}_{k_{l}};x\\bigr) &- F\\bigl({\\bf t}+{\\bf b}_{k_{l}};x\\bigr) \\|_{\\mathcal{Y}} \\leq \\\\\n&\\leq \\| F \\bigl({\\bf t}+{\\bf c}_{k_{l}}+{\\bf b}_{k_{l}};x\\bigr) - F^{\\ast}\\bigl({\\bf t}+{\\bf c}_{k_{l}};x\\bigr) \\|_{\\mathcal{Y}} +\\\\ \n&+ \\|F \\bigl({\\bf t}+{\\bf b}_{k_{l}};x\\bigr)- F^{\\ast}\\bigl({\\bf t};x\\bigr) \\|_{\\mathcal{Y}} +\\|F^{\\ast} \\bigl({\\bf t};x\\bigr)- F^{\\ast}\\bigl({\\bf t}+{\\bf c}_{k_{l}} ;x\\bigr) \\|_{\\mathcal{Y}} \\\\\n&\\leq \\sup_{{\\bf z}\\in \\overline{ B(0,R)}+{\\bf t}}\\| F \\bigl({\\bf z}+{\\bf b}_{k_{l}};x\\bigr) - F^{\\ast}\\bigl({\\bf z};x\\bigr) \\|_{\\mathcal{Y}} +\\\\ \n&+ \\|F \\bigl({\\bf t}+{\\bf b}_{k_{l}};x\\bigr)- F^{\\ast}\\bigl({\\bf t};x\\bigr) \\|_{\\mathcal{Y}} +\\|F^{\\ast} \\bigl({\\bf t};x\\bigr)- F^{\\ast}\\bigl({\\bf t}+{\\bf c}_{k_{l}} ;x\\bigr) \\|_{\\mathcal{Y}} .\n\\end{align*}\n\nNow, since $F(\\cdot,\\cdot)$ is compactly $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphy and $F^{\\ast} \\bigl(\\cdot;x\\bigr)$ is continuous, we have (using the previous inequality)\n$$\\lim_{l \\to \\infty}\\sup_{x\\in B} \\| F \\bigl({\\bf t}+{\\bf a}_{k_{l}};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k_{l}};x\\bigr) \\|_{\\mathcal{Y}} = 0\\, .$$\nTherefore, $\\psi_0({\\bf t},x)=0$. Now, since every convergent subsequence of the bounded sequence $\\alpha_{k}({\\bf t};x)$ converge towards zero, we conclude \n$$\\lim_{k \\to \\infty} \\sup_{x\\in B} \\| F \\bigl({\\bf t}+{\\bf a}_{k};x\\bigr) - F\\bigl({\\bf t}+{\\bf b}_{k};x\\bigr) \\|_{\\mathcal{Y}} =0\\, .$$\n\n\\end{proof}\n\n\nOf course, as in the classical situation, we have the next two results\n\n\\begin{prop}\\label{prou}\n Let $F:\\mathbb{R}^m \\times\\X \\to \\mathcal{Y}$ be $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic and uniformly continuous in the first variable; then, its limit function $F^*$(see definition \\ref{eovako}) is also uniformly continuous in the first variable.\n\\end{prop}\n\\begin{proof}\nLet $(b_k) \\in {\\mathrm{R}}_{{\\mathcal{G}}}$ and $B\\in \\mathcal{B}$, since $F$ is $({\\mathrm{R}}_{{\\mathcal{G}}},{\\mathcal B})$-multi-almost automorphic there exist a subsequence $(b_{k_l})\\subset(b_k)$ and a function $F^*:\\mathbb{R}^m \\times \\X \\to \\mathcal{Y}$ such that the following limits, point-wise in ${\\bf t}$ and uniformly in $x\\in B$, hold\n$$\\lim_{l\\rightarrow\\infty}{F(\\t+b_{k_l};x)=F^*(\\t;x)}\\, ,$$\n$$\\lim_{l\\rightarrow\\infty}{F^*(\\t-b_{k_l};x)=F(\\t;x)}.$$\nNow, let $\\epsilon > 0$, since $F$ is uniformly continuous in the first variable, there exists $\\delta > 0$ such that if $\\t,\\s \\in \\mathbb{R}^m$ with $||\\t-\\s||<\\delta$, then $\\sup_{x\\in B}||F(\\t;x) - F(\\s ;x)||_{\\mathcal{Y}}<\\epsilon/2$. Therefore, taking the limit when $l\\rightarrow\\infty$ in the following inequality\n\\begin{eqnarray*}\n ||F^*(\\t;x) - F^*(\\s;x)||_{\\mathcal{Y}} &\\leq & || F^*(\\t;x) - F(\\t+b_{k_l};x)||_{\\mathcal{Y}} + ||F(\\t+b_{k_l};x) - F(\\s + b_{k_l};x)||_{\\mathcal{Y}} +\\\\\n &+& ||F(\\s + b_{k_l};x)-F^*(\\s ;x)||_{\\mathcal{Y}}\\, ,\n\\end{eqnarray*}\nwe have:\n$$\\sup_{x\\in B}||F^*(\\t,x)-F^*(\\s,x)||_{\\mathcal{Y}} < \\epsilon\\, ,$$\nprovided that $\\t,\\s \\in \\mathbb{R}^m$ with $||\\t-\\s||<\\delta$. Thus, $F^*$ is uniformly continuous in the first variable.\n\\end{proof}\n\nThe proof of the following proposition is analogous to that of the previous one, so we omit its proof.\n\\begin{prop}\nSuppose that \\( F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y} \\) is an \\(({\\mathrm{R}}_{{\\mathcal{G}}}, {\\mathcal{B}})\\)-multi-almost automorphic function with limit function \\( F^* \\) (see Definition \\ref{eovako}). If for every \\( B \\in {\\mathcal{B}} \\) there exists a real number \\( L_{B} > 0 \\) such that, for every \\( x \\in B \\) and \\( {\\bf s}, {\\bf t} \\in {\\mathbb R}^{m} \\), we have\n\\[\n\\bigl\\| F({\\bf s}; x) - F({\\bf t}; x) \\bigr\\|_{\\mathcal{Y}} \\leq L_{B} \\|{\\bf s} - {\\bf t}\\|_{\\mathcal{X}},\n\\]\nthen, we also have \n\\[\n\\bigl\\| F^*({\\bf s}; x) - F^*({\\bf t}; x) \\bigr\\|_{\\mathcal{Y}} \\leq L_{B} \\|{\\bf s} - {\\bf t}\\| , \n\\]\nfor every $x\\in B$ and ${\\bf s}, {\\bf t} \\in \\mathbb{R}^m$.\n\\end{prop}\n\n\\noindent Following the work \\cite{chavez2022multi}, we have:\n\\begin{tw}\\label{Follprev}\nThe following properties hold\n\\begin{enumerate}\n\\item The space of bounded and ${\\mathrm{R}}_{{\\mathcal{G}}}$-multi-almost automorphic functions is a Banach space under the supremum norm.\n\n\\item Suppose that $F : {\\mathbb R}^{m} \\times \\mathcal{X} \\rightarrow \\mathcal{Y}$ is $(\\mathcal{G},{\\mathcal B})$-multi-almost automorphic, where \n ${\\mathcal B}$ denotes any collection of compact subsets of $\\mathcal{X}.$ If for every $B\\in {\\mathcal B}$ there exists a real number $L_{B}>0$ such that, for every $x,\\ y\\in B$ and ${\\bf t}\\in {\\mathbb R}^{m},$\nwe have\n\\begin{align}\\label{elbe}\n\\bigl\\| F({\\bf t};x) -F({\\bf t} ;y)\\bigr\\|_{\\mathcal{Y}}\\leq L_{B}\\|x-y\\|_{\\mathcal{X}} ;\n\\end{align} \nthen, for every set $B\\in {\\mathcal B},$ the set $\\{F({\\bf t},x) : {\\bf t}\\in \\mathcal{G},\\ x\\in B\\}$ is relatively compact in $\\mathcal{Y}.$\n\\end{enumerate}\n\\end{tw}\nIn Theorem \\ref{Follprev}, the proof of part {\\it 1} is analogous to the classical version of almost automorphy, see for instance \\cite{CHAKPPINTO2023}, while the proof of item {\\it 2} is analogous to \\cite[Proposition 2.5-(i)]{chavez2022multi}.\n\nWith respect to the previous proposition, we clarify the following: in item {\\it 1}, the completeness is of the space \\( BC(\\mathbb{R}^m; \\mathcal{X}) \\cap \\mathrm{R}_{\\mathcal{G}} A A(\\mathbb{R}^m; \\mathcal{X}) \\) and not just of the space \\( \\mathrm{R}_{\\mathcal{G}} A A(\\mathbb{R}^m; \\mathcal{X}) \\), this is due to the fact that, in general, the elements of \\( \\mathrm{R}_{\\mathcal{G}} A A(\\mathbb{R}^m; \\mathcal{X}) \\) are not bounded (see the examples presented in \\cite{chavez2022multi}). With respect to item {\\it 2}, we clarify that we are working with all the sequences in \\( \\mathcal{G}\\), so that when \\( \\mathcal{G} = \\mathbb{R}^m \\), we are dealing with the classical situation of almost automorphy, as in \\cite[Proposition 2.5-(i)]{chavez2022multi}. In the next section we prove that if \\( \\mathcal{G} \\) is a dense subgroup of \\( \\mathbb{R}^m \\), then every function in \\( \\mathcal{G} A A(\\mathbb{R}^m; \\mathcal{X}) \\) is bounded and has relatively compact range. \n\n\\section{$\\mathcal{G}$-multi-almost automorphic functions, where $\\mathcal{G}$ is a dense subgroup of $\\mathbb{R}^m$}\\label{section dense} \n\nAs quoted in the previous section, we know that an $({\\mathrm R}_{\\mathcal{G}},{\\mathcal B})$-multi-almost automorphic function is not necessarily bounded and, therefore, does not have a relatively compact range. In the next theorem, we overcome this situation by requiring $\\mathcal{G}$ to be a dense subgroup of $\\mathbb{R}^m$.\n\\begin{tw}\\label{teo1}\nLet $\\mathcal{G} \\subset \\mathbb{R}^m$ be a dense subgroup and $f \\in \\mathcal{G}AA(\\mathbb{R}^m ; \\mathcal{X})$, then:\n\\begin{enumerate}\n\\item $f$ is bounded.\n\\item $f$ has relatively compact range. Thus, $\\mathcal{R}(f):=\\{f({\\bf t}) : {\\bf t}\\in \\mathbb{R}^m\\}$ is relatively compact in $\\mathcal{X}$.\n\\end{enumerate}\n\\end{tw}\n\\begin{proof}\nLet us see:\n\\begin{enumerate}\n\\item [{\\it 1.}] Let $f_r := f|_{\\mathcal{G}} : \\mathcal{G} \\rightarrow \\X$ be the restriction of $f$ to $\\mathcal{G}$, which is a continuous function. We note that $f_r$ is almost automorphic on $\\mathcal{G}$ and, according to Theorem \\ref{pro1}, $f_r$ is bounded on $\\mathcal{G}$. Consequently, due to the continuity of $f$, the boundedness of $f_r$ and the density of $\\mathcal{G}$ in $\\mathbb{R}^m$, it follows that $f$ is bounded on $\\mathbb{R}^m$.\n\n\\item [{\\it 2.}] Since $\\mathcal{G}$ is dense in $\\mathbb{R}^m$, according to Proposition \\ref{grp}, we have: \n\\begin{equation}\\label{Eqnclos}\n\\overline{f(\\mathbb{R}^m)} = \\overline{f(\\overline{\\mathcal{G}})} = \\overline{f_r(\\mathcal{G})},\n\\end{equation}\nwhere $f_r$ is the restriction of $f$ to the group $\\mathcal{G}$. Moreover, since $f_r: \\mathcal{G} \\rightarrow \\X$ is almost automorphic, by Theorem \\ref{pro2}, $f_r$ has a relatively compact range, that is $\\overline{f_r(\\mathcal{G})}$ is compact in $\\mathcal{X}$. Therefore, from (\\ref{Eqnclos}), we conclude that $\\overline{f(\\mathbb{R}^m)}$ is compact in $\\mathcal{X}$.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{cor} Let $\\mathcal{G}$ be a dense subgroup of $\\mathbb{R}^m$, then\n\\begin{enumerate}\n\\item $\\mathcal{G}AA(\\mathbb{R}^m ; \\mathcal{X}) \\subset BC(\\mathbb{R}^m ; \\mathcal{X});$ \nwhere $BC(\\mathbb{R}^m ; \\mathcal{X})$ is the space of bounded continuous functions.\n\\item $\\mathcal{G}AA(\\mathbb{R}^m ; \\X)$ is a Banach space under the supremum norm.\n\\end{enumerate}\n\\end{cor}\n\nIn the following result, we establish that if $\\mathcal{G}$ is a dense subgroup of $\\mathbb{R}^m$, then any $\\mathcal{G}$-multi-almost automorphic function which decays asymptotically across $\\mathcal{G}$ must be the null function.\n\\begin{tw}\\label{teo6}\nLet $\\mathcal{G}$ be a dense subgroup of $\\mathbb{R}^m$ and $f\\in \\mathcal{G}AA(\\mathbb{R}^m ; \\X)$ such that \n$$\\displaystyle\\lim_{\\t \\in \\mathcal{G},\\, ||\\t||\\rightarrow\\infty}{||f(\\t)||_{\\X}} = 0\\, ;$$ \nthen, $f \\equiv 0$. \\end{tw}\n\\begin{proof}\nSince $f\\in \\mathcal{G} AA(\\mathbb{R}^m ;\\mathcal{X})$, then its rectriction map $f_r:\\mathcal{G}\\to \\X$ is almost automorphic on the group $\\mathcal{G}$. On the other hand, according to Lemma \\ref{lem8}, $\\mathcal{G}$ is unbounded. Therefore, there exist a sequence $(b_k)_{k\\in\\mathbb{N}}$ in $\\mathcal{G}$ which is unbounded. Now, since $f_r$ is almost automorphic, there exit a subsequence $(b_{k_l})_{l\\in\\mathbb{N}}\\subset (b_k)_{k\\in\\mathbb{N}}$ and a function $f_r^*:\\mathcal{G} \\to \\mathcal{X}$ such that for each $\\t \\in \\mathcal{G}$, the following limits hold:\n\\begin{equation}\\label{eq1}\n f_r^*(\\t) = \\lim_{l \\rightarrow \\infty}{f_r(\\t+b_{k_l})}\\, ,\n\\end{equation}\n\\begin{equation}\\label{eq2}\n \\lim_{l \\rightarrow \\infty}{f_r^*(\\t-b_{k_l})} = f_r(\\t)\\, .\n\\end{equation}\nBut, $\\displaystyle\\lim_{\\t \\in \\mathcal{G},\\, ||\\t||\\rightarrow\\infty}{||f(\\t)||_{\\X}} = 0;$ then, in light of (\\ref{eq1}), we have:\n$$f_r^*(\\t) = 0\\, , \\t\\in \\mathcal{G}\\, .$$\nNow using (\\ref{eq2}), we have:\n$$f_r(\\t)=\\lim_{l \\rightarrow \\infty}{f_r^*(\\t-b_{k_l})}=0\\, , \\t\\in \\mathcal{G}\\, . $$\nThus, $f_r \\equiv 0$ in $\\mathcal{G}$. Now the conclusion follows from the continuity of $f$ and the denseness of the group $\\mathcal{G}$. \\end{proof}\n\n\\section{New characterizations of compact almost automorphic functions}\\label{section char}\n\nIn this section, we present the main results of the present work. We begin with the following well-known characterization of multidimensional compact almost automorphic functions presented in \\cite{CHAKPPINTO2023}.\n\\begin{tw}\\label{crt} ({\\bf First characterization}) A function $f:\\mathbb{R}^{m}\\to \\mathcal{X}$ is compact almost automorphic if and only if it is almost automorphic and uniformly continuous. Thus,\n$$\\mathcal{K}AA(\\mathbb{R}^m ; \\X) = AA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X) .$$\n\\end{tw}\n\nThe next characterization, where we consider $\\mathcal{G}=\\mathbb{Z}^m$, is a generalization to the multidimensional setting of \\cite[Theorem 3.16]{Alanpiecewise}.\n\n\n\\begin{tw} ({\\bf Second characterization}) \n A function $f: \\mathbb{R}^m \\rightarrow \\X$ is compact almost automorphic if and only if it is $\\mathbb{Z}^m$-almost automorphic and uniformly continuous, that is:\n $$\\mathcal{K}AA(\\mathbb{R}^m ; \\X) = \\mathbb{Z}^mAA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X) .$$\n\\end{tw}\n\\begin{proof}\nLet us see\\\\\n\\noindent i) Let $f$ be a compact almost automorphic function, then $f$ is $\\mathbb{Z}^m$-almost automorphic and, according to Theorem \\ref{crt}, it is uniformly continuous. Therefore we have the inclusion $\\mathcal{K}AA(\\mathbb{R}^m ; \\X)\\subset \\mathbb{Z}^mAA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X).$ \\\\\n\\noindent ii) Let us prove the reverse inclusion. Consider $f$ a $\\mathbb{Z}^m$-almost automorphic and uniformly continuous function; then, according to Theorem \\ref{crt}, we need to prove that $f$ is almost automorphic.\\\\\nLet $({\\bf b}_{k}=(b_{k}^{1},b_{k}^{2},\\cdot \\cdot\\cdot ,b_{k}^{m}))_{k\\in \\mathbb{N}}$ be an arbitrary sequence in $\\mathbb{R}^m$. Denoting by $[|\\cdot |]$ the integer part of a real umber and by $\\{\\cdot \\}$ the decimal part, then for each $j\\in \\{1,2,\\cdots , m\\}$ we have the decomposition \n$b_k^j=[|b_k^j|]+\\{b_k^j\\}$, where $[|b_k^j|]\\in \\mathbb{Z}$ and $\\{b_k^j\\} \\in [0,1)$. In this way we have \n\\begin{eqnarray*}\n {\\bf b}_k &=& \\underbrace{([|b_k^1|],[|b_k^2|],\\cdots,[|b_k^m|])}_{\\in \\mathbb{Z}^m} + \\underbrace{(\\{b_k^1\\},\\{b_k^2\\},\\cdots,\\{b_k^m\\})}_{\\in [0,1)^m}.\n\\end{eqnarray*}\nSince $f \\in \\mathbb{Z}^mAA(\\mathbb{R}^m,\\X)$, there exist a subsequence $(\\alpha_k)_{k \\in\\mathbb{N}}$ of $([|b_k^1|],[|b_k^2|],\\cdots,[|b_k^m|])_{k \\in\\mathbb{N}}$ and a function $f^*:\\mathbb{R}^m \\rightarrow \\X$ such that the following pointwise limits hold\n$$\\lim_{k\\rightarrow\\infty}{f(\\t+\\alpha_k)=f^*(\\t)}\\, , \\t \\in \\mathbb{R}^m\\, ,$$\n$$\\lim_{k\\rightarrow\\infty}{f^*(\\t-\\alpha_k)=f(\\t)}\\, , \\t \\in \\mathbb{R}^m\\, .$$ \nOn the other hand, since for each $k\\in \\mathbb{N}$, $(\\{b_k^1\\},\\{b_k^2\\},\\cdots,\\{b_k^m\\}) \\in [0,1)^m$, then the sequence $((\\{b_k^1\\},\\{b_k^2\\},\\cdots,\\{b_k^m\\}))_{k\\in \\mathbb{N}}$ is bounded. Therefore, there exists a convergent subsequence $(\\beta_k=(\\beta_k^1,\\beta_k^2,\\cdots,\\beta_k^m))_{k\\in \\mathbb{N}}$ of $(\\{b_k^1\\},\\{b_k^2\\},\\cdots,\\{b_k^m\\}))_{k\\in \\mathbb{N}}$ which converges to the point $\\beta_0=(\\beta_0^1,\\beta_0^2,\\cdots,\\beta_0^m) \\in [0,1]^m$. \n\nIn this way, it is possible to construct a subsequence $(\\eta_k)_{k \\in \\mathbb{N}} \\subset ({\\bf b}_k)_{k\\in \\mathbb{N}}$, defined by\n$$\\eta_k := (\\alpha_k^1,\\alpha_k^2,\\cdots,\\alpha_k^m)+(\\beta_k^1,\\beta_k^2,\\cdots,\\beta_k^m)=\\alpha_k + \\beta_k \\, .$$\nNow, define the function $h:\\mathbb{R}^m\\to \\X$ by $h(\\t):= f^*(\\t+\\beta_0)$.\n\n\\noindent Let $\\t \\in \\mathbb{R}^m$ be fixed, we have\n\\begin{eqnarray*}\n ||f(\\t+\\eta_k)-h(\\t)||_{\\X}\\leq ||f(\\t+\\alpha_k+\\beta_k)-f(\\t+\\alpha_k+\\beta_0)||_{\\X}+||f(\\t+\\alpha_k+\\beta_0)-h(\\t)||_{\\X}\\, .\n\\end{eqnarray*}\nSince \\(\\beta_k \\to \\beta_0\\) and \\(||f(\\t + \\beta_0 + \\alpha_k) - h(\\t)||_{\\mathcal{X}} \\rightarrow 0\\) when \\(k \\rightarrow \\infty\\) and due to the uniform continuity of \\(f\\), we conclude (according to the previous inequality) that:\n$$\\lim_{k\\rightarrow \\infty}{f(\\t+\\eta_k)=h(\\t)}.$$\nOn the other hand, according to Proposition \\ref{prou}, we know that \\( f^* \\) is uniformly continuous. Then, arguing as before, it can be proved that\n$$\\lim_{k\\rightarrow \\infty}{h(\\t-\\eta_k)=f(\\t)}\\, .$$\n\\end{proof}\n\n\\begin{tw} ({\\bf Third characterization})\n Let $f: \\mathbb{R}^m\\to \\mathcal{X}$ be a continuous function and $\\mathcal{G}$ a dense subset of $\\mathbb{R}^m$. Then, $f$ is compact almost automorphic if and only if it is $\\mathcal{G}$-almost automorphic and uniformly continuous. Thus,\n $$\\mathcal{K}AA(\\mathbb{R}^m ; \\X) = \\mathcal{G}AA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X) .$$\n\\end{tw}\n\\begin{proof}\nThe inclusion $\\mathcal{K}AA(\\mathbb{R}^m ; \\X) \\subset \\mathcal{G}AA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X)$ follows from Theorem \\ref{crt}. Let us prove the reverse inclusion.\n\nLet $f \\in \\mathcal{G}AA(\\mathbb{R}^m ; \\X) \\cap UC(\\mathbb{R}^m ; \\X)$, according to Theorem \\ref{crt} we need to prove that $f$ is almost automorphic. Let $({\\bf b}_k) \\subset \\mathbb{R}^m$ be an arbitrary sequence and $\\epsilon>0$. For each $k\\in \\mathbb{N}$, let $B^*({\\bf b}_k,\\epsilon/2^k):=B({\\bf b}_k,\\epsilon/2^k)\\setminus \\{{\\bf b_k} \\}$, where $B({\\bf b}_k,\\epsilon/2^k)$ is the open ball with center $ {\\bf b}_k$ and radius $\\epsilon /2^k$. Since $\\mathcal{G}$ is dense in $\\mathbb{R}^m$, for each $k\\in \\mathbb{N}$, there exists a point $\\gamma_k \\in B^*({\\bf b}_k,\\epsilon/2^k)\\cap \\mathcal{G}$. Equivalently, there exists a sequence $(\\gamma_{k})$ in $\\mathcal{G}$ such that $\\gamma_{k}\\not = {\\bf b}_k $ and $||\\gamma_{k}-{\\bf b}_k||<\\epsilon/2^k$, which implies that $\\lim_{k\\to \\infty}||\\gamma_{k}-{\\bf b}_k||=0$.\n\n Since $f$ is $\\mathcal{G}$-almost automorphic, there exist a subsequence $(\\gamma_{k_l}) \\subset (\\gamma_{k})$ and a function $f^*:\\mathbb{R}^m \\rightarrow \\X$ such that the following pointwise limits hold\n\\begin{equation}\\label{EeQqNnWw}\n \\lim_{l\\rightarrow\\infty}{f(\\t+\\gamma_{k_l})=f^*(\\t)}\\, , \\, \\, \\, \\lim_{l\\rightarrow\\infty}{f^*(\\t-\\gamma_{k_l})=f(\\t)}\\, , \\, \\, {\\bf t} \\in \\mathbb{R}^m\\, .\n\\end{equation}\nFrom Proposition \\ref{prou}, it follows that $f^*$ is also uniformly continuous. On the other hand, there exists a subsequence $({\\bf b}_{k_l}) \\subset ({\\bf b}_k)$ such that $\\lim_{l\\to \\infty}||\\gamma_{k_l}-{\\bf b}_{k_l}||=0$. Now, from the following inequalities\n\\begin{eqnarray*}\n ||f(\\t+{\\bf b}_{k_l})-f^*(\\t)||_{\\mathcal{X}}&\\leq & ||f(\\t+{\\bf b}_{k_l})-f(\\t+\\gamma_{k_l})||_{\\mathcal{X}}+\\\\\n &+& ||f(\\t+\\gamma_{k_l})-f^*(\\t) ||_{\\mathcal{X}}\\, ,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n ||f^*(\\t-{\\bf b}_{k_l})-f(\\t)||_{\\mathcal{X}}&\\leq & ||f^*(\\t-{\\bf b}_{k_l})-f^*(\\t-\\gamma_{k_l})||_{\\mathcal{X}}+\\\\\n &+& ||f^*(\\t-\\gamma_{k_l})-f(\\t) ||_{\\mathcal{X}}\\, ,\n\\end{eqnarray*}\nthe uniform continuity of $f$, of $f^*$ and the limits in (\\ref{EeQqNnWw}) we conclude that\n$$\\lim_{l\\rightarrow\\infty}{f(\\t+ {\\bf b}_{k_l})=f^*(\\t)}\\, ; $$\nand,\n$$\\lim_{l\\rightarrow\\infty}{f^*(\\t-{\\bf b}_{k_l})=f(\\t)}.$$\nTherefore $f$ is almost automorphic.\n\n\\end{proof}\n\n\\section{Convolution products}\\label{section conv}\n\nUsing the notation $\\mathcal{I}_{{\\bf t}}=(-\\infty,t_1]\\times (-\\infty,t_2]\\times \\cdots \\times (-\\infty,t_m]$ for a point ${\\bf t} = (t_1,t_2,\\cdots , t_m) \\in \\mathbb{R}^m$, we have:\n\n\\begin{tw}\\label{TI0101} \\cite{chavez2022multi}\n\n\\begin{enumerate}\n\\item Let $f:\\mathbb{R}^{m}\\rightarrow \\X$ be a bounded ${\\mathrm R}_{\\mathcal{G}}$-multi-almost automorphic function. Let \n$(K({\\bf t}))_{{\\bf t}\\in (0,\\infty)^{m}}\\subseteq L(\\X, \\mathcal{Y})$ be a strongly continuous operator family and \n$$\\int_{(0,\\infty)^{m}}\\| K({\\bf t})\\|_{L(\\X,\\mathcal{Y})}\\, d {\\bf t}\\, < +\\infty .$$\nDefine\n\\begin{equation}\\label{rep01}\nF({\\bf t}):=\\int_{\\mathcal{I}_{\\bf t}}K({\\bf t}-\\eta)f(\\eta)\\, d\\eta,\\quad {\\bf t}\\in {\\mathbb R}^{m}.\n\\end{equation}\n\nThen $F(\\cdot)$ is a bounded ${\\mathrm R}_{\\mathcal{G}}$-multi-almost automorphic function. \n\n\\item Let $f:\\mathbb{R}^{m}\\rightarrow \\X$ be a bounded (compactly) ${\\mathrm R}_{\\mathcal{G}}$-multi-almost automorphic function. Let \n$(K({\\bf t}))_{{\\bf t}\\in \\mathbb{R}^{m}}\\subseteq L(\\X, \\mathcal{Y})$ be a strongly continuous operator family and \n$$\\int_{\\mathbb{R}^{m}}\\| K({\\bf t})\\|_{L(\\X,\\mathcal{Y})}\\, d {\\bf t}\\, < +\\infty .$$\nDefine\n\\begin{equation}\\label{rep02}\nF({\\bf t}):=\\int_{\\mathbb{R}^m}K({\\bf t}-\\eta)f(\\eta)\\, d\\eta,\\quad {\\bf t}\\in {\\mathbb R}^{m}.\n\\end{equation}\nThen $F(\\cdot)$ is a bounded (compactly) ${\\mathrm R}_{\\mathcal{G}}$-multi-almost automorphic function. \n\\end{enumerate}\n\\end{tw}\n\\begin{proof}\nLet us prove the second item under the hypothesis that $f$ is compactly ${\\mathrm R}_{\\mathcal{G}}$-multi-almost automorphic. Let $({\\bf b}_{k})$ be an arbitrary sequence in ${\\mathrm R}_{\\mathcal{G}}$, then there exist a subsequence $({\\bf b}_{k_{l}})$ of $({\\bf b}_{k})$ and a function\n$f^{\\ast} : {\\mathbb R}^{m}\\rightarrow \\X$ such that the following limits hold\\begin{align}\\label{love1234567801}\n\\lim_{l\\rightarrow +\\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||f\\bigl({\\bf t} +{\\bf b}_{k_{l}})-f^{\\ast}({\\bf t}) ||_{\\mathcal{X}}=0\n\\end{align}\nand\n\\begin{align}\\label{love12345678902}\n\\lim_{l\\rightarrow +\\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||f^{\\ast}\\bigl({\\bf t}-{\\bf b}_{k_{l}})-f({\\bf t}) ||_{\\mathcal{X}}=0\\, ;\n\\end{align}\nwhere $\\mathcal{E}$ is a compact subset of $\\mathbb{R}^m$. Now, let us define the function\n$$\nF^{\\ast}({\\bf t}):=\\int_{\\mathbb{R}^m}K({\\bf t}-\\eta)f^{\\ast}(\\eta)\\, d\\eta,\\quad {\\bf t}\\in {\\mathbb R}^{m}\\, .\n$$\nTake $\\mathcal{E}\\subset \\mathbb{R}^m$ a compact set; then, for any ${\\bf t} \\in \\mathcal{E} $ we have\n\\begin{eqnarray*}\n||F({\\bf t}+{\\bf b}_{k_{l}})-F^{\\ast}({\\bf t})||_{\\mathcal{Y}} &=&|| \\int_{\\mathbb{R}^m}K(\\eta)( f({\\bf t}-\\eta + {\\bf b}_{k_{l}})-f^{\\ast}({\\bf t}-\\eta))\\, d\\eta||_{\\mathcal{Y}}\\\\\n&\\leq & \\int_{\\mathbb{R}^m}||K(\\eta)||_{L(\\mathcal{X},\\mathcal{Y})} \\sup_{z\\in \\mathcal{E}_{\\eta}}|| f(z+{\\bf b}_{k_{l}})-f^{\\ast}(z)||_{\\mathcal{X}}d\\eta\\, ,\n\\end{eqnarray*}\nwhere, $\\mathcal{E}_{\\eta}:=\\mathcal{E} -{\\eta}$ is a compact subset of $\\mathbb{R}^m$. Using \n(\\ref{love1234567801}) and the Lebesgue's Dominated Convergence Theorem, we conclude\n$$\\lim_{l \\to \\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||F({\\bf t}+{\\bf b}_{k_{l}})-F^{\\ast}({\\bf t})||_{\\mathcal{Y}}=0\\, .$$\nAnalogously, we have\n$$\\lim_{l \\to \\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||F^{\\ast}({\\bf t}-{\\bf b}_{k_{l}})-F({\\bf t})||_{\\mathcal{Y}}=0\\, .$$\n\\end{proof}\n\nBased on the characterization theorems of compactly multi-almost automorphic functions presented in Section \\ref{section char}, we derive the following corollary:\n\\begin{cor}\nLet $\\mathcal{G}$ be either $\\mathbb{Z}^m$ or a dense subset of $\\mathbb{R}^m$ and $f:\\mathbb{R}^{m}\\rightarrow \\X$ be a bounded and ${\\mathcal{G}}$-multi-almost automorphic function. Let \n$(K({\\bf t}))_{{\\bf t}\\in \\mathbb{R}^{n}}\\subseteq L(\\X, \\mathcal{Y})$ be a strongly continuous operator family and \n$$\\int_{\\mathbb{R}^{m}}\\| K({\\bf t})\\|_{L(\\X,\\mathcal{Y})}\\, d {\\bf t}\\, < +\\infty .$$\nDefine\n\\begin{equation}\\label{rep021}\nF({\\bf t}):=\\int_{\\mathbb{R}^m}K({\\bf t}-\\eta)f(\\eta)\\, d\\eta,\\quad {\\bf t}\\in {\\mathbb R}^{m}.\n\\end{equation}\nIf $F$ is uniformly continuous, then $F$ is compactly multi-almost automomorphic; that is $F \\in \\mathcal{K} AA(\\mathbb{R}^m; \\mathcal{Y})$.\n\\end{cor}\n\nIn what follows, we study the invariance of the bounded and compact $\\mathrm{R}_{\\mathcal{G}}$-multi-almost automorphic function space in the context where the kernel \\( K \\) in (\\ref{rep01}) or (\\ref{rep02}) depends on two variables, i.e., is of the form \\( K(\\cdot, \\cdot) \\). For this purpose, we introduce the notion of a (compactly) $\\mathrm{R}_{\\mathcal{G}}$-multi-Bi-almost automorphic function in the subsequent definition.\n \n\\begin{definition}\\label{defBaa} A jointly continuous function $K:\\mathbb{R}^m\\times \\mathbb{R}^m\\times \\X \\to \\mathcal{Y}$ is $(\\mathrm{R}_{\\mathcal{G}},\\mathcal{B})$-multi-Bi-almost automorphic if for any $B \\in \\mathcal{B}$ and any sequence $(b_k)_{k \\in \\mathbb{N}} \\in \\mathrm{R}_{\\mathcal{G}}$, there exist a subsequence $(b_{k_l})\\subset (b_k)$ and a function $K^*:\\mathbb{R}^m\\times \\mathbb{R}^m\\times \\X \\to \\mathcal{Y}$ such that\n\\begin{equation}\\label{EqNew01}\n\\lim_{l \\to +\\infty} K({\\bf t}+b_{k_l}, {\\bf s}+b_{k_l},x)=K^*({\\bf t}, {\\bf s},x)\\, ,\n\\end{equation}\nand \n\\begin{equation}\\label{EqNew02}\n\\lim_{l \\to +\\infty}K^*({\\bf t}-b_{k_l}, {\\bf s}-b_{k_l},x)=K({\\bf t}, {\\bf s},x)\\, ,\n\\end{equation}\nhold pointwisely for $({\\bf t}, {\\bf s}) \\in \\mathbb{R}^m \\times \\mathbb{R}^m$ and any $x \\in B$. \nIf the limits in (\\ref{EqNew01}) and (\\ref{EqNew02}) are uniform on compact subsets of $\\mathbb{R}^m \\times \\mathbb{R}^m$, then we say that $K$ is compactly $(\\mathrm{R}_{\\mathcal{G}},\\mathcal{B})$-multi-Bi-almost automorphic.\n\\end{definition}\n\nThe next definition, was given in \\cite{chavez2021almostaut} in order to study almost automorphic type solutions to integral equations of advanced and delayed type\n\\begin{definition}\nWe say that a jointly continuous function $K:\\mathbb{R}^m\\times\\mathbb{R}^m\\times \\X \\to \\mathcal{Y}$ is $\\lambda$-bounded if there exists a non negative function $\\lambda:\\mathbb{R}^m\\times\\mathbb{R}^m \\to \\mathbb{R}$ such that for every $\\tau \\in \\mathbb{R}^m$ we have\n\\begin{equation}\\label{Eeqqlambda}\n||K(\\t+\\tau,\\s+\\tau,x)||_{\\mathcal{Y}} \\leq \\lambda({\\bf t},{\\bf s})\\, ,\n\\end{equation}\nwhere the inequality is for each $(\\t,\\s)\\in \\mathbb{R}^m\\times\\mathbb{R}^m$ and any $x\\in \\X$.\n\\end{definition}\n\nAs it was proved for the one-dimensional case, see \\cite[Lemma 2.3]{chavez2024compact}, the inequality (\\ref{Eeqqlambda}) also holds for the limit function $K^*$ of a $\\lambda$-bounded and multi-Bi-almost auromorphic function $K$. That is, we have the following lemma\n\n\\begin{lem}\\label{lemBou}\nLet us suppose that $K:\\mathbb{R}^m \\times \\mathbb{R}^m \\times \\X\\to \\mathcal{Y}$ is $(\\mathrm{R}_{\\mathcal{G}}, \\mathcal{B})$-multi-Bi-almost automorphic and $\\lambda$-bounded. Then, its limit function $K^*:\\mathbb{R}^m\\times \\mathbb{R}^m\\times \\X\\to \\mathcal{Y}$ (see definition \\ref{defBaa}) is also $\\lambda$-bounded; that is, for each $(\\t,\\s)\\in \\mathbb{R}^m\\times\\mathbb{R}^m$ and any $x\\in \\X$ we have\n$$||K^*(\\t+\\tau,\\s+\\tau,x)||_{\\mathcal{Y}} \\leq \\lambda({\\bf t},{\\bf s}),\\, \\, \\forall \\tau \\in \\mathbb{R}^m\\, .$$\n\\end{lem}\n\n\n\\noindent {\\sl Property ({\\bf SC})}. We say that the operator valued function \\( K: \\mathbb{R}^m \\times \\mathbb{R}^m \\to \\mathcal{L}(\\X; \\mathcal{Y}) \\) satisfies the property of being \\textbf{(SC)} if, for each \\( x \\in \\X \\), the function \\( (\\mathbf{t}, \\mathbf{s}) \\mapsto K(\\mathbf{t}, \\mathbf{s}) x \\) is continuous.\n\n\n\\begin{tw}\\label{TeoComb}\nLet $\\mathrm{R}_{\\mathcal{G}}$ be the set of sequences such that all of its subsequences are also in $\\mathrm{R}_{\\mathcal{G}}$ and let the operator valued function $K:\\mathbb{R}^m \\times \\mathbb{R}^m\\to \\mathcal{L}(\\X; \\mathcal{Y})$ satisfies ({\\bf SC}), is compactly $\\mathrm{R}_{\\mathcal{G}}$-multi-Bi-almost automorphic and is $\\lambda$-bounded with $\\lambda({\\bf t},{\\bf s})=\\phi({\\bf t}-{\\bf s})$ and $\\phi \\in L^1(\\mathbb{R}^m)$. \nThen, the operator $\\Gamma$, defined by\n$$\\Gamma u({\\bf t}):= \\int_{\\mathbb{R}^m} K({\\bf t},\\eta)u(\\eta)d\\eta\\, ,$$\nmaps the Banach space $\\mathcal{K} \\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m ; \\mathcal{X})\\cap BC(\\mathbb{R}^m ; \\mathcal{X})$ to the Banach space $\\mathcal{K} \\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m ; \\mathcal{X})\\cap BC(\\mathbb{R}^m ; \\mathcal{Y})$. \\end{tw}\n\n\\begin{proof} Let $u \\in \\mathcal{K} \\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\X)\\cap BC(\\mathbb{R}^m ; \\mathcal{X})$ and define $v:=\\Gamma u$. \nSince $K(\\cdot,\\cdot)$ is compactly $\\mathrm{R}_{\\mathcal{G}}$-multi-Bi-almost automorphic and $u$ is compactly $\\mathrm{R}_{\\mathcal{G}}$-multi-almost automorphic we can ensure that given any sequence $(b_k)\\subset \\mathrm{R}_{\\mathcal{G}}$ there exist a subsequence $(b_{k_l})\\subseteq (b_k)$ and functions $K^*(\\cdot,\\cdot)$ and $u^*$ such that the following limits, uniformly on compact subsets of $\\mathbb{R}^m\\times \\mathbb{R}^m$, hold:\n$$\\lim\\limits_{l\\to +\\infty} K(\\t+b_{k_l},\\s+b_{k_l})=K^*(\\t,\\s),\\quad \\lim\\limits_{l\\to +\\infty} K^*(\\t-b_{k_l},\\s-b_{k_l})=K(\\t,\\s);$$\nand the following limits, uniformly on compact subsets of $\\mathbb{R}^m$, also hold\n$$\\lim\\limits_{l\\to +\\infty} u(\\t+b_{k_l})=u^*(\\t),\\quad \\lim\\limits_{l\\to +\\infty} u^*(\\t-b_{k_l})=u(\\t).$$\nNow, using the Lebesgue's Dominated Convergence Theorem we see that\n\\begin{equation}\\label{Eq001}\n\\lim\\limits_{l\\to +\\infty} v(\\t+b_{k_l})=v^*(\\t),\\, \\, \\lim_{l\\to \\infty}v^*(\\t-b_{k_l})=v(\\t)\\, ,\n\\end{equation}\nwhere\n$$v^*(\\t):=\\int_{\\mathbb{R}^m} K^*(\\t,\\eta)u^*(\\eta)d\\eta\\, .$$\nThat is, $v$ is $\\mathrm{R}_{\\mathcal{G}}$-multi-almost automorphic. In what follows we prove that the limits in (\\ref{Eq001}) are uniform for ${\\bf t}$ in compact subsets of $\\mathbb{R}^m$.\n\n\nLet $E$ be a compact subset of $\\mathbb{R}^m$. Since $\\phi \\in L^1(\\mathbb{R}^m)$, there exist a strictly increasing sequence of positive real numbers $(R_n)_{n\\in \\mathbb{N}}$ such that:\n$$\\lim_{n \\to \\infty}\\int_{\\mathbb{R}^m \\setminus B[R_n,0]}\\phi(z)dz =0\\, , $$\nwhere $B[R_n,0]$ is the closed ball of center $0$ and radii $R_n$.\n\nNote that, if $n$ is sufficiently large, then there exists a subsequence $(\\tilde{R}_n)$ of $(R_n)$ such that $ E+\\mathbb{R}^m \\setminus B[R_n,0] \\subset \\mathbb{R}^m \\setminus B[\\tilde{R}_n,0] $. Now, let us take ${\\bf t} \\in E$, then we have\n\\begin{eqnarray*}\n||v(\\t+{\\bf b}_{k_l})-v^*(\\t)||_{\\mathcal{X}}&\\leq & \\int_{\\mathbb{R}^m}||K(\\t+{\\bf b}_{k_l}, \\eta+{\\bf b}_{k_l})-K^*(\\t,\\eta)||\\, ||u(\\eta+\\t_k)||_{\\mathcal{X}}d\\eta \\\\\n&+& \\int_{\\mathbb{R}^m}||K^*({\\bf t},\\eta)||\\, ||u(\\eta+{\\bf b}_{k_l})-u^*(\\eta)||d\\eta\\\\\n&\\leq & \\int_{B[R_n,0]}||K(\\t+{\\bf b}_{k_l}, \\eta+{\\bf b}_{k_l})-K^*(\\t,\\eta)||\\, ||u(\\eta+\\t_k)||_{\\mathcal{X}}d\\eta \\\\\n&+& \\int_{\\mathbb{R}^m \\setminus B[R_n,0] }||K(\\t+{\\bf b}_{k_l}, \\eta+{\\bf b}_{k_l})-K^*(\\t,\\eta)||\\, ||u(\\eta+\\t_k)||_{\\mathcal{X}}d\\eta \\\\\n&+& \\int_{\\mathbb{R}^m}||K^*({\\bf t},\\eta)||\\, ||u(\\eta+{\\bf b}_{k_l})-u^*(\\eta)||d\\eta\\\\\n&\\leq & ||u||_{\\infty} \\sup_{({\\bf t}, {\\bf s})\\in \\mathcal{E}\\times B[R_n,0]} ||K(\\t+{\\bf b}_{k_l}, \\eta+{\\bf b}_{k_l})-K^*(\\t,\\eta)||\\, \\mathcal{L}(B[R_n,0])\\\\\n&+& 2||u||_{\\infty} \\int_{\\mathbb{R}^m \\setminus B[R_n,0]}\\phi({\\t}-\\eta)d\\eta\\\\\n&+& \\int_{\\mathbb{R}^m} \\phi(\\eta) \\, \n\\sup_{z\\in E_{\\eta}}|| u(z+{\\bf b}_{k_{l}})-u^{\\ast}(z)||_{\\mathcal{X}}d\\eta\\, ,\\\\\n\\end{eqnarray*}\nwhere, $E_{\\eta}:=E -{\\eta}$ is a compact subset of $\\mathbb{R}^m$. From the previous inequalities, we have\n\\begin{eqnarray*}\n\\sup_{{\\bf t}\\in \\mathcal{E}}||v(\\t+{\\bf b}_{k_l})-v^*(\\t)||_{\\mathcal{X}}&\\leq & ||u||_{\\infty} \\sup_{({\\bf t}, {\\bf s})\\in \\mathcal{E}\\times B[R_n,0]} ||K(\\t+{\\bf b}_{k_l}, \\eta+{\\bf b}_{k_l})-K^*(\\t,\\eta)||\\, \\mathcal{L}(B[R_n,0])\\\\\n&+& 2||u||_{\\infty} \\int_{\\mathbb{R}^m \\setminus B[R_n,0]}\\phi({\\t}-\\eta)d\\eta\\\\\n&+& \\int_{\\mathbb{R}^m} \\phi(\\eta) \\, \n\\sup_{z\\in E_{\\eta}}|| u(z+{\\bf b}_{k_{l}})-u^{\\ast}(z)||_{\\mathcal{X}}d\\eta\\, .\n\\end{eqnarray*}\nNow, taking the limit in the last inequality when $l\\to \\infty$, we obtain\n$$\\lim_{l\\to \\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||v(\\t+{\\bf b}_{k_l})-v^*(\\t)||_{\\mathcal{X}} \\leq 2||u||_{\\infty} \\int_{\\mathbb{R}^m \\setminus B[R_n,0]}\\phi({\\t}-\\eta)d\\eta .$$\n\nOn the other hand, the integral of the right hand side satisfies:\n$$\\int_{\\mathbb{R}^m \\setminus B[R_n,0]}\\phi({\\t}-\\eta)d\\eta\\leq \\int_{E+\\mathbb{R}^m \\setminus B[R_n,0]}\\phi(z)dz \\leq \\int_{\\mathbb{R}^m \\setminus B[\\tilde{R}_n,0]}\\phi(z)dz\\, \\to 0, \\, \\, n\\to \\infty\\, .$$\nTherefore,\n$$\\lim_{l\\to \\infty}\\sup_{{\\bf t}\\in \\mathcal{E}}||v(\\t+{\\bf b}_{k_l})-v^*(\\t)||_{\\mathcal{X}}=0\\, .$$\n\nAnalogously, we have\n$$\\lim_{l\\to \\infty}\\sup_{{\\bf t}\\in E}||v^*(\\t-{\\bf b}_{k_l})-v(\\t)||_{\\mathcal{X}} =0\\, . $$\n\n\\end{proof}\n\n\n\\begin{tw}\\label{NewThmappl}\nLet $\\mathrm{R}_{\\mathcal{G}}$ be the set of sequences such that all of its subsequences are also in $\\mathrm{R}_{\\mathcal{G}}$ and let the operator valued function $K:\\mathbb{R}^m \\times \\mathbb{R}^m\\to \\mathcal{L}(\\X; \\mathcal{Y})$ satisfies ({\\bf SC}), is compactly multi-Bi-almost automorphic and is $\\lambda$-bounded with $\\lambda({\\bf t},{\\bf s})=\\phi({\\bf t}-{\\bf s})$ and $\\phi \\in L^1(\\mathbb{R}^m)$. \nThen, the operator $\\Gamma$, defined by\n$$\\Gamma u({\\bf t}):= \\int_{\\mathbb{R}^m} K({\\bf t},\\eta)u(\\eta)d\\eta\\, ,$$\nmaps the space $\\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\mathcal{X})\\cap B UC(\\mathbb{R}^m ; \\mathcal{X})$ to the space $\\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\mathcal{X})\\cap B UC(\\mathbb{R}^m ; \\mathcal{Y})$.\n\\end{tw}\n\\begin{proof}\nUsing the hypothesis we can see that, if $u$ is bounded then $\\Gamma u$ is also bounded. Also, if $u \\in \\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\mathcal{X})$, then (as in the previous theorem) we conclude that $\\Gamma u \\in \\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\mathcal{X})$. It only rest to prove that $\\Gamma u$ is uniformly continuous.\n\nLet $(\\t_k),(\\s_k)$ be two sequences in $\\mathbb{R}^m$ such that $||\\t_k-\\s_k|| \\to 0$ when $k\\to +\\infty$, then\n\\begin{eqnarray*}\n||v(\\t_k)-v(\\s_k)||&\\leq & \\int_{\\mathbb{R}^m}||K(\\t_k, \\eta+\\t_k)-K(\\s_k,\\eta+\\s_k)||\\, ||u(\\eta+\\t_k)||d\\eta \\\\\n&+& \\int_{\\mathbb{R}^m}||K(\\s_k,\\eta+\\s_k)||\\, ||u(\\eta+\\t_k)-u(\\eta+\\s_k)||d\\eta\\\\\n&:=&I_1(k)+I_2(k)\\, .\n\\end{eqnarray*}\nNow, by Lemma \\ref{lemBou}, Theorem \\ref{ThRKAA}, the uniform continuity of $u$ and the Lebesgue's Dominated Convergence Theorem, we conclude\n$$\\lim_{k\\to +\\infty}I_1(k)=0 \\, =\\, \\lim_{k\\to +\\infty}I_2(k) \\, .$$\n\\end{proof}\n\n\n\\begin{cor}\nLet the operator valued function $K:\\mathbb{R}^m \\times \\mathbb{R}^m\\to \\mathcal{L}(\\X; \\mathcal{Y})$ satisfies ({\\bf SC}), is compactly multi-Bi-almost automorphic and is $\\lambda$-bounded with $\\lambda({\\bf t},{\\bf s})=\\phi({\\bf t}-{\\bf s})$ and $\\phi \\in L^1(\\mathbb{R}^m)$. \nThen, the operator $\\Gamma$, defined by\n$$\\Gamma u({\\bf t}):= \\int_{\\mathbb{R}^m} K({\\bf t},\\eta)u(\\eta)d\\eta\\, ,$$\nmaps the Banach space $\\mathcal{K} AA(\\mathbb{R}^m, \\mathcal{X})$ to the Banach space $\\mathcal{K} AA(\\mathbb{R}^m, \\mathcal{Y})$. \n\\end{cor}\n\nUsing the previous results and the characterization theorems of the space of compactly multi-almost automorphic functions presented in section \\ref{section char}, it is not difficult to prove the next corollary\n\\begin{cor}\nLet $\\mathcal{G}$ be either $\\mathbb{Z}^m$ or a dense subset of $\\mathbb{R}^m$ and let $\\mathrm{R}_{\\mathcal{G}}$ be the set of sequences such that all of its subsequences are also in $\\mathrm{R}_{\\mathcal{G}}$ and let the operator valued function $K:\\mathbb{R}^m \\times \\mathbb{R}^m\\to \\mathcal{L}(\\X; \\mathcal{Y})$ satisfies ({\\bf SC}), is $\\mathrm{R}_{\\mathcal{G}}$-multi-Bi-almost automorphic and is $\\lambda$-bounded with $\\lambda({\\bf t},{\\bf s})=\\phi({\\bf t}-{\\bf s})$ and $\\phi \\in L^1(\\mathbb{R}^m)$. \nThen, if $u$ is bounded and belongs to $\\mathrm{R}_{\\mathcal{G}} AA(\\mathbb{R}^m, \\mathcal{X})$, and $\\Gamma u$ is uniformly continuous, where\n$$\\Gamma u({\\bf t}):= \\int_{\\mathbb{R}^m} K({\\bf t},\\eta)u(\\eta)d\\eta\\, ;$$\nthen, $\\Gamma u \\in \\mathcal{K} AA(\\mathbb{R}^m, \\mathcal{Y})$.\n\n\\end{cor}\n\n\nIn the proof of the next result, we use the fact that every multi-Bi-almost automorphic function is, according to Theorem \\ref{ThRKAA}, ${\\mathrm{R}}_{{\\mathcal{G}}}$-uniformly continuous, where $\\mathcal{G}$ is the set $\\{({\\bf t},{\\bf s})\\in \\mathbb{R}^m \\times \\mathbb{R}^m\\, :\\, {\\bf t}={\\bf s} \\}$. We omit the details.\n\\begin{tw}\\label{InvConv1}\nLet $K:\\mathbb{R}^m \\times \\mathbb{R}^m\\to \\mathcal{L}(\\X; \\mathcal{Y})$ satisfies ({\\bf SC}), is compactly multi-Bi-almost automorphic and is $\\lambda$-bounded with $\\lambda({\\bf t},{\\bf s})=\\phi({\\bf t}-{\\bf s})$ and $\\phi \\in L^1([0,\\infty)^m)$. Then, the operator $\\Pi$ defined by \n\\begin{equation}\\label{Operator2}\n\\Pi u ({\\bf t}):=\\int_{-\\infty}^{{\\bf t}} K({\\bf t},\\eta)u(\\eta)d\\eta\\, ,\n\\end{equation}\nmaps the Banach space $\\mathcal{K}AA(\\mathbb{R}^m; \\mathcal{X})$ to the Banach space $\\mathcal{K}AA(\\mathbb{R}^m; \\mathcal{Y})$.\n\\end{tw}\n\n\\color{black}\n\nApplications of Theorems \\ref{TeoComb} and \\ref{InvConv1} to ordinary differential equations with exponential dichotomy can be found in \\cite{chavez2024compact}.\n\n\\section{Applications to the Poisson's and heat equations}\\label{section appl}\n\n\n\n\\subsection{Compact almost automorphy of the solution to Poisson's equation}\n\\begin{definition}\nLet $ f\\in BC(\\mathbb{R}^m,\\mathbb{R}) $. A function $u:\\mathbb{R}^m \\to \\mathbb{R}$ is solution in the sense of distributions of the Poisson's equation \n$$\\Delta u=f,$$\nif\n\\begin{equation}\n\\notag\\int_{\\mathbb{R}^{m}}u(x)\\Delta\\phi(x)dx=\\int_{\\mathbb{R}^{m}} f(x)\\phi(x)dx,\\hspace{0.5cm}\\forall\\phi\\in C^{\\infty}_{0}(\\mathbb{R}^{m}).\n\\end{equation}\n\\end{definition}\nFrom the the theory of distributional solutions for the Laplace equation, we have\n\\begin{tw} \\label{Thmf01} \\cite{Jost02}\n\\begin{enumerate}\n\\item (Weyl) If $u\\in L^{1}_{loc}(\\mathbb{R}^{m},\\mathbb{R})$ and $u$ is a solution in the sense of distributions of $\\Delta u=0$, then $u$ is harmonic, i.e., $u\\in C^{\\infty}(\\mathbb{R}^{m})$ and $\\Delta u=0$ in $\\mathbb{R}^{m}$.\n\\item (Lioville) If $u:\\mathbb{R}^{m}\\to \\mathbb{R}$ is harmonic and bounded, then $u$ is constant.\n\\end{enumerate}\n\\end{tw}\n\nTheorem \\ref{Thmf01} and the Arzel'a-Ascolí Theorem were used in \\cite{CHAKPPINTO2023} to prove that, every bounded and continuous function with almost automorphic distributional laplacian is in fact compact almost automorphic.\nThe next Theorem is the main result of the present subsection.\n\n\\begin{tw}\\label{main theorem} Let $\\mathcal{G}$ be either $\\mathbb{Z}^m$ or a dense subset of $\\mathbb{R}^m$ and $f\\in \\mathcal{G}AA(\\mathbb{R}^{m},\\mathbb{R})$. If $u:\\mathbb{R}^{m}\\to \\mathbb{R}$ is a bounded continuous function which is a solution in the sense of distributions of Poisson's equation:\n\\begin{equation}\n\\notag\\Delta u=f\\, ,\n\\end{equation}\nthen $u\\in \\mathcal{K}AA(\\mathbb{R}^{m},\\mathbb{R})$.\n\\end{tw}\n\\begin{proof}\nIndeed, as demonstrated in the proof of \\cite[Theorem 20]{CHAKPPINTO2023}, we establish that \\( u \\) is uniformly continuous on \\( \\mathbb{R}^m \\) and compactly \\( \\mathcal{G} \\)-multi almost automorphic. Consequently, the conclusion that \\( u \\) is a compact almost automorphic solution follows from either the second characterization theorem (in the case where \\( \\mathcal{G} = \\mathbb{Z}^m \\)) or the third characterization theorem (when \\( \\mathcal{G} \\) is a dense subset of \\( \\mathbb{R}^m \\)) presented in Section \\ref{section char}.\n\\end{proof}\n\nThis theorem emphasizes that, even if the source function \\( f \\) is not fully almost automorphic, but belongs to \\( \\mathcal{G}\\text{AA}(\\mathbb{R}^m, \\mathbb{R}) \\) with \\( \\mathcal{G} \\) as specified, its solution remains compact almost automorphic. Consequently, compact almost automorphic solutions to Poisson's equation are preserved under a broader class of source functions, extending beyond the almost periodic/automorphic case presented in \\cite{sibuya1971almost,CHAKPPINTO2023}.\n\\subsection{Compact almost automorphy of the solution to heat equation}\n\nLe us consider the heat equation:\n\\begin{equation}\\label{Heat equation}\n \\left\\{\n \\begin{array}{lll}\n \\partial_t u(t,x)&=&\\Delta u(t,x), \\quad t>0,\\, x\\in \\mathbb{R}^m,\\\\\n u(0,x) &= &f(x), \\quad x\\in \\mathbb{R}^m,\n \\end{array}\\right.\n \\end{equation}\nwhere $ f \\in BC(\\mathbb{R}^m,\\mathbb{R}) $. The maximal domain of the Laplace operator $ \\Delta $ in the Banach space $Z= BC(\\mathbb{R}^m,\\mathbb{R}) $ or $ BUC(\\mathbb{R}^m,\\mathbb{R}) $, is given by $$ D(\\Delta):=\\lbrace v\\in Z : \\Delta v \\text{ exists in } Z\\rbrace . $$\n\\begin{definition}\nLet $ f\\in BC(\\mathbb{R}^m,\\mathbb{R}) \\cap \\overline{D(A)}$, by a solution of equation \\eqref{Heat equation} we mean a function $u: [0,+\\infty)\\times\\mathbb{R}^m \\longrightarrow \\mathbb{R}$, such that $u \\in C^{1}((0,+\\infty),BC(\\mathbb{R}^m,\\mathbb{R}))$, $ u(t,\\cdot) \\in D(\\Delta)$ for all $t>0$ and $u$ satisfies \\eqref{Heat equation} pointwisely.\\\\\nIn particular, if $f \\in BUC(\\mathbb{R}^m,\\mathbb{R}) \\cap \\overline{D(A)} $, the solution $u$ satisfies $u \\in C([0,+\\infty), BC(\\mathbb{R}^m,\\mathbb{R}))\\cap C^{1}((0,+\\infty),BC(\\mathbb{R}^m,\\mathbb{R}))$, $ u(t,\\cdot) \\in D(\\Delta) $ for all $t>0$ and $u$ satisfies \\eqref{Heat equation} poitnwisely.\n\\end{definition}\nIt is well known that, the heat equation \\eqref{Heat equation} admits the following solution \n$$ u(t,x)=T(t)f(x), \\quad t>0, \\, x\\in \\mathbb{R}^m, \\quad \\text{ for } f\\in \\overline{D(A)}.$$\nand $ u(0,x)=f(x) $, where $ (T(t))_{t\\geq 0} $ is the Gaussian semigroup in $ BC(\\mathbb{R}^m,\\mathbb{R}) $ given by: \n\\begin{eqnarray}\\label{Gaussian semigroup}\n T(t)f(x)&=& \\dfrac{1}{(4\\pi t)^{m/2}} \\int_{\\mathbb{R}^m} e^{-\\|x-z\\|^2 /(4t)} f(z) dz= \\dfrac{1}{(4\\pi t)^{m/2}} \\int_{\\mathbb{R}^m} e^{-\\|z\\|^2 /(4t)} f(x-z) dz \\nonumber \\\\\n &=&(K(t,\\cdot)*f)(x), \\quad x\\in \\mathbb{R}^m\\, ,\n\\end{eqnarray}\nfor $t>0$, $x\\in \\mathbb{R}^m$ (see for instance \\cite[Example 3.7.6]{Arendt}).\n\nThe following is the main result concerning the heat equation in this work\n\\begin{tw}\nLet $f\\in \\mathcal{G}AA(\\mathbb{R}^m,\\mathbb{R}) \\cap \\overline{D(A)}$, where $\\mathcal{G}$ be either $\\mathbb{Z}^m$ or a dense subset of $\\mathbb{R}^m$ and $f\\in \\mathcal{G}AA(\\mathbb{R}^{m},\\mathbb{R})$. Then, equation \\eqref{Heat equation} admits a unique solution $u$ such that, for each $t>0$, $u(t,\\cdot) \\in \\mathcal{K}AA(\\mathbb{R}^m,\\mathbb{R})$.\n\\end{tw}\n\\begin{proof}\nLet $f \\in \\mathcal{G} AA(\\mathbb{R}^m,\\mathbb{R})$ and let $t>0$, from the invariance Theorem \\ref{TI0101}, we have that $u(t,\\cdot)\\in \\mathcal{G}AA(\\mathbb{R}^m,\\mathbb{R})$. Now, using the charaterization theorems in section \\ref{section char}, it only rest to prove that $u(t,\\cdot)$ is uniformly continuous, and this follows as in ithe proof of \\cite[Theorem 25]{CHAKPPINTO2023}. In fact, let $x,y\\in \\mathbb{R}^m$; then \n\\begin{align}\\label{Lipch}\n|u(t,x)-u(t,y) | \\leq \\dfrac{1}{(4\\pi t)^{m/2}} \\int_{\\mathbb{R}^m} |e^{-\\|x-z\\|^2 /(4t)}-e^{-\\|y-z\\|^2 /(4t)} |dz \\, \\|f\\|_{\\infty}.\n\\end{align}\nAlso, for fixed $t>0$ and $z\\in \\mathbb{R}^m$, the function $x\\longmapsto e^{-\\|x-z\\|^2 /(4t)}$ is globally Lipschitz continuous, i.e., \n$$| e^{-\\|x-z\\|^2 /(4t)}-e^{-\\|y-z\\|^2 /(4t)}| \\leq \\sup_{w\\in \\mathbb{R}^m} \\dfrac{c_m}{2t} \\| w-z \\|e^{-\\|w-z\\|^2 /(4t)} \\| x-y\\|, $$\nwhere $c_m > 0$ is a constant. Note that\n$$\\sup_{w\\in \\mathbb{R}^m} \\| w-z \\|e^{-\\|w-z\\|^2 /(4t)}=\\| w_0-z \\|e^{-\\|w_0-z\\|^2 /(4t)}\\, \\, , w_0 \\in \\mathcal{D}\\, ,$$\nwhere \n$$\\mathcal{D}:=\\{ w\\in \\mathbb{R}^m\\, : \\, \\|w-z \\| =\\sqrt{2t}\\}\\, .$$\nNow, from (\\ref{Lipch}) and using invariance under translation of the Lebesgue integral, we have\n\\begin{align*}\n|u(t,x)-u(t,y) | &\\leq c'_m(t) \\|f\\|_{\\infty} \\|x-y\\|.\n\\end{align*}\n\\end{proof}\nThus, we have proved that the $\\mathcal{G}$-multi-almost automorphy of the initial data $f$ of equation \\eqref{Heat equation}, is sufficient to obtain the compact almost automorphy in space variable of the solution $u$. \n\n\n\n\n\n\n\n\\appendix\n\\section{Sequentially almost automorphic functions on topological groups}\nLet $\\mathcal{G}$ be a topological group (commutative, locally compact). We will write the group operation on $\\mathcal{G}$ additively, that is by $+$, and the inverse operation by $-$. \n\\begin{definition}\\label{Defi01}\nLet $f: \\mathcal{G} \\to \\mathcal{X}$ be a continuous function. $f$ is sequentially almost automorphic, if for any sequence $( s^{\\prime}_{m})$ of $\\mathcal{G}$ there exist a subsequence $( s_{m})$ of $(s^{\\prime}_{m})$ and a function $f^*:\\mathcal{G}\\to \\mathcal{X}$ such that the following pointwise limits holds:\n\\begin{equation}\n\\notag\\lim_{m\\to \\infty}f(t+s_{m})=f^*(t),\n\\end{equation}\nand:\n\\begin{equation}\n\\notag\\lim_{n\\to \\infty}f^*(t-s_{m})=f(t)\\, .\n\\end{equation}\n\\end{definition}\n\\noindent $AA(\\G,\\X)$ will denote the space of sequentially almost automorphic functions. The next two theorems are of importance in the present work.\n\n\\begin{tw}\\label{pro1}\nIf $f \\in AA(\\G,\\X)$, then $f$ is bounded.\n\\end{tw}\n\\begin{proof}\n \n\nSuppose that $f$ is not bounded, that is, there exists a sequence $(s'_m)_{m\\in \\mathbb{N}} \\subset \\G$ such that\n\\begin{equation*}\n \\lim_{n\\rightarrow\\infty}{||f(s'_m)||_{\\mathcal{X}}} = \\infty,\n\\end{equation*}\nas $f\\in AA(\\G,\\mathcal{X})$, there exists a subsequence $(s_m)_{m\\in \\mathbb{N}}\\subset(s'_m)_{m\\in \\mathbb{N}}$ and a function $h:\\G \\rightarrow \\mathcal{X}$ such that the following pointwise limits holds:\n\\begin{equation*}\n f^*(\\mathbf{t}) = \\lim_{m\\rightarrow \\infty}f(\\mathbf{t}+s_m)\\,\\,, \\,\\,\nf(\\mathbf{t}) = \\lim_{n\\rightarrow \\infty}f^*(\\mathbf{t}-s_m).\n\\end{equation*}\nIf $\\textbf{t} = 0$, in the first limit we have:\n\\begin{equation*}\n \\lim_{m\\rightarrow\\infty}{f(s_m)} = f^*(0)\\,\\,; \\,\\, f^*(0) \\in \\mathcal{X},\n\\end{equation*}\nwith what we have\n\\begin{equation*}\n \\lim_{m\\rightarrow \\infty}{||f(s_m)||_{\\mathcal{X}}} = ||f^*(0)||_{\\mathcal{X}} < \\infty,\n\\end{equation*}\nwhich is a contradiction, therefore, $f$ is bounded. \n\\end{proof}\n\\begin{tw}\\label{pro2}\nIf $f \\in AA(\\G,\\X)$, then the range of $f$ is relatively compact in $\\X$.\n\\end{tw}\n\\begin{proof}\n \nLet $(y'_m)_{m\\in \\mathbb{N}} \\subset \\mathrm{R}(f) := \\{y \\in X: \\exists \\t \\in \\G \\,\\,\\text{whit}\\,\\, f(\\t) = y\\}$ be an arbitrary sequence. By the definition of $\\R(f)$ there must be a sequence $(s'_m)_{m\\in \\N}\\subset \\G$ such that $y'_m = f(s'_m)$, as $f \\in AA(\\G,\\X)$ there exist a subsequence $(s_m)_{m\\in \\N} \\subset (s'_m)_{m\\in \\N}$ and a function $f^*:\\G \\rightarrow \\X$ such that \n\\begin{equation*}\n \\lim_{m\\rightarrow\\infty}{y_m} = \\lim_{m\\rightarrow\\infty}{f(s_m)} = f^*(0).\n\\end{equation*}\nTherefore, there exists a subsequence $(s_m)_{m\\in \\N} \\subset (s'_m)_{m\\in \\N}$ wich converges to $\\X$, which means that $\\R(f)$ is relatively compact.\n\\end{proof}\n\n\n\\section{Subgroups of $\\mathbb{R}^m$}\nHere we summarize important results on dense subgroups of the group $\\mathbb{R}^m$. We start with the following assertion:\n\\begin{lem}\\label{lem8}\nEvery nontrivial subgroup of $\\mathbb{R}^m$ is unbounded.\n\\end{lem}\n\\begin{proof}\n Let $ H $ be a nontrivial subgroup of $\\mathbb{R}^m $. \nSince $H $ is nontrivial, it contains at least one element $v$ other than the zero vector. Now, for eack $k\\in \\mathbb{N}$, define $ v_k := k \\cdot v \\in H$, where $k \\cdot v =v+v+\\cdots +v $ (k-times). Then, for each $ k $, we have $ \\|v_k\\| = \\|k \\cdot v\\| = k \\cdot \\|v\\| $. Therefore, the sequence $ \\{v_k\\} $ is unbounded. This proves that $ H $ is unbounded.\n\\end{proof}\n\\subsection{Dense subgroups of $\\mathbb{R}^m$}\nLet $\\mathbb{Q}$ be the rational numbers. Some dense subgroups of $\\mathbb{R}^m$ are, for example, $\\mathbb{Q}^m$, $\\mathbb{Q}^k\\times \\mathbb{R}^{m-k}$, $\\mathbb{Q}^{k_1}\\times G^{k_2} \\times \\mathbb{R}^{k_3}$ where $k_1+k_2+k_3=m$ and $G^{k_2}$ is a dense subgroup of $\\mathbb{R}^{k_2}$, etc. In the following Theorem and in the subsequent Proposition, it is described some conditions under which a subgroup of $\\mathbb{R}^m$ becomes dense. \n\n\\begin{tw}\\label{ApB1}\n Let $\\theta_1, \\theta_2, \\cdots,\\theta_m$ real numbers. For the subgroup \n $$\\mathbb{Z}^{m} + \\mathbb{Z}(\\theta_1, \\theta_2, \\cdots,\\theta_m)= \\{(s_1 + s_0\\theta_1, \\cdots,s_m+s_0\\theta_m); \\,\\,(s_0,s_1,\\cdots,s_m)\\in \\mathbb{Z}^{m+1}\\}\\subset \\mathbb{R}^m $$ \n to be dense in $\\mathbb{R}^m$, it is necessary and sufficient that the $m+1$ numbers $1,\\theta_1, \\theta_2, \\cdots,\\theta_m$ be linearly independent on $\\mathbb{Q}$.\n\\end{tw}\n\\begin{prop}\\label{ApB2}\n Let $G$ be a finitely generated subgroup of $\\mathbb{R}^m$. The following conditions are equivalent: \n \\begin{enumerate}\n \\item $G$ is dense in $\\mathbb{R}^m$.\n \\item For any nonzero linear form $\\phi: \\mathbb{R}^m\\rightarrow\\mathbb{R}$ we have $\\phi(G)\\not\\subset\\mathbb{Z}$.\n \\end{enumerate}\n\\end{prop}\nFor a proof of Theorem \\ref{ApB1} and Proposition \\ref{ApB2}, the reader is invited to consult the works \n\\cite{elghaoui2015rational,waldschmidt1995topologie}.\n\nFinally, we also present the next important fact on continuous functions:\n\\begin{prop}\\label{grp}\n Let $f:\\mathbb{R}^m \\rightarrow \\X$ be a continuous function and $A \\subset \\mathbb{R}^m$, then $$\\overline{f(\\overline{A})} = \\overline{f(A)}.$$\n\\end{prop}\n\n\\subsection*{Acknowledgments}\nThe authors would like to express their gratitude to the anonymous referees for their careful reading and helpful comments.\n\n\n\\subsection*{Funding}\nA. Ch\\'avez and J. Casta\\~neda were supported by CONCYTEC through the PROCIENCIA program under the E041-2023-01 competition, according to contract PE501082885-2023 \n\n\n\\section*{Declarations}\n\n\n\\subsection*{Conflict of interest}\nNo potential conflict of interest was reported by the authors.\n\n\n\n\\begin{thebibliography}{100}\n\n\\bibitem{Arendt}\n{\\sc Arendt, W., Batty, C.~J., Hieber, M., and Neubrander, F.}\n\\newblock {\\em Vector-valued Laplace Transforms and Cauchy Problems}, vol.~96.\n\\newblock Springer Science \\& Business Media, Basel, 2011.\n\n\\bibitem{03}\n{\\sc Bochner:, S.}\n\\newblock Curvature and {B}etti numbers in real and complex vector bundles.\n\\newblock {\\em Univ. e Politec. Torino Rend. Sem. Mat. 15\\/} (1955/56),\n 225--253.\n\n\\bibitem{04}\n{\\sc Bochner, S.}\n\\newblock A new approach to almost periodicity.\n\\newblock {\\em Proc. Nat. Acad. Sci. U.S.A. 48\\/} (1962), 2039--2043.\n\n\\bibitem{05}\n{\\sc Bochner, S.}\n\\newblock Continuous mappings of almost automorphic and almost periodic\n functions.\n\\newblock {\\em Proc. Nat. Acad. Sci. U.S.A. 52\\/} (1964), 907--910.\n\n\\bibitem{campos2014almost}\n{\\sc Campos, J., and Tarallo, M.}\n\\newblock Almost automorphic linear dynamics by {F}avard theory.\n\\newblock {\\em Journal of Differential Equations 256}, 4 (2014), 1350--1367.\n\n\\bibitem{campos2020barycentric}\n{\\sc Campos, J., and Tarallo, M.}\n\\newblock Barycentric solutions of linear almost periodic equations: {B}aire\n class and almost automorphy.\n\\newblock {\\em Journal of Dynamics and Differential Equations 32}, 3 (2020),\n 1475--1509.\n\n\\bibitem{chavez2024compact}\n{\\sc Ch{\\'a}vez, A., Aragon{\\'e}s, N., Pinto, M., and Zavaleta, U.}\n\\newblock Compact almost automorphic dynamics of non-autonomous differential\n equations with exponential dichotomy and applications to biological models\n with delay.\n\\newblock {\\em arXiv preprint arXiv:2404.16758 (\n https://doi.org/10.48550/arXiv.2404.16758)\\/} (2024).\n\n\\bibitem{chavez2014discontinuous}\n{\\sc Ch{\\'a}vez, A., Castillo, S., and Pinto, M.}\n\\newblock Discontinuous almost automorphic functions and almost automorphic\n solutions of differential equations with piecewise constant argument.\n\\newblock {\\em Electronic Journal of Differential Equations 2014}, 56 (2014),\n 1--13.\n\n\\bibitem{Alanpiecewise}\n{\\sc Ch{\\'a}vez, A., and Huatangari, L.~Q.}\n\\newblock Vector valued piecewise continuous almost automorphic functions and\n some consequences.\n\\newblock {\\em Journal of Mathematical Analysis and Applications 542}, 2\n (2025), 128768.\n\n\\bibitem{chavez2022multi}\n{\\sc Ch{\\'a}vez, A., Khalil, K., Kosti{\\'c}, M., and Pinto, M.}\n\\newblock Multi-dimensional almost automorphic type functions and applications.\n\\newblock {\\em Bulletin of the Brazilian Mathematical Society, New Series 53},\n 3 (2022), 801--851.\n\n\\bibitem{chavez2023almost}\n{\\sc Ch{\\'a}vez, A., Khalil, K., Kosti{\\'c}, M., and Pinto, M.}\n\\newblock Almost periodic type functions of several variables and applications.\n\\newblock {\\em Journal of Mathematical Analysis and Applications 525}, 1\n (2023), 127115.\n\n\\bibitem{CHAKPPINTO2023}\n{\\sc Ch{\\'a}vez, A., Khalil, K., Pereyra, A., and Pinto, M.}\n\\newblock Compact almost automorphic solutions to {P}oisson’s and heat\n equations.\n\\newblock {\\em Bulletin of the Malaysian Mathematical Sciences Society 47}, 2\n (2024), 1--22.\n\n\\bibitem{chavez2021almostaut}\n{\\sc Ch{\\'a}vez, A., Pinto, M., and Zavaleta, U.}\n\\newblock On almost automorphic type solutions of abstract integral equations,\n a {B}ohr-{N}eugebauer type property and some applications.\n\\newblock {\\em Journal of Mathematical Analysis and Applications 494}, 1\n (2021), 124395.\n\n\\bibitem{ding2017asymptotically}\n{\\sc Ding, H.-S., and Wan, S.-M.}\n\\newblock Asymptotically almost automorphic solutions of differential equations\n with piecewise constant argument.\n\\newblock {\\em Open Mathematics 15}, 1 (2017), 595--610.\n\n\\bibitem{elghaoui2015rational}\n{\\sc Elghaoui, M., and Ayadi, A.}\n\\newblock Rational criterion testing the density of additive subgroups of\n $\\mathbb{R}^n$ and $\\mathbb{C}^n$.\n\\newblock {\\em Applied general topology 16}, 2 (2015), 127--139.\n\n\\bibitem{es2022compact}\n{\\sc Es-sebbar, B., Ezzinbi, K., and Khalil, K.}\n\\newblock Compact almost automorphic solutions for semilinear parabolic\n evolution equations.\n\\newblock {\\em Applicable Analysis 101}, 7 (2022), 2553--2579.\n\n\\bibitem{fuhrmann2020tameness}\n{\\sc Fuhrmann, G., and Kwietniak, D.}\n\\newblock On tameness of almost automorphic dynamical systems for general\n groups.\n\\newblock {\\em Bulletin of the London Mathematical Society 52}, 1 (2020),\n 24--42.\n\n\\bibitem{garcia2021mean}\n{\\sc Garc{\\'\\i}a-Ramos, F., J{\\\"a}ger, T., and Ye, X.}\n\\newblock Mean equicontinuity, almost automorphy and regularity.\n\\newblock {\\em Israel Journal of Mathematics 243}, 1 (2021), 155--183.\n\n\\bibitem{Jost02}\n{\\sc Jost, J.}\n\\newblock {\\em Partial {D}ifferential {E}quations}, vol.~2.\n\\newblock Springer, New York, 2007.\n\n\\bibitem{lenz2024abstract}\n{\\sc Lenz, D., Spindeler, T., and Strungaru, N.}\n\\newblock Abstract almost periodicity for group actions on uniform topological\n spaces.\n\\newblock {\\em Canadian Journal of Mathematics 76}, 3 (2024), 798--829.\n\n\\bibitem{lenz2024pure}\n{\\sc Lenz, D., Spindeler, T., and Strungaru, N.}\n\\newblock Pure point spectrum for dynamical systems and mean, {B}esicovitch and\n {W}eyl almost periodicity.\n\\newblock {\\em Ergodic Theory and Dynamical Systems 44}, 2 (2024), 524--568.\n\n\\bibitem{qi2022piecewise}\n{\\sc Qi, L., and Yuan, R.}\n\\newblock Piecewise continuous almost automorphic functions and {F}avard’s\n theorems for impulsive differential equations in honor of {R}ussell\n {J}ohnson.\n\\newblock {\\em Journal of Dynamics and Differential Equations\\/} (2022), 1--43.\n\n\\bibitem{rezoug2024asymptotically}\n{\\sc Rezoug, N., Salim, A., and Benchohra, M.}\n\\newblock Asymptotically almost automorphy for impulsive integrodifferential\n evolution equations with infinite time delay via {M}{\\\"o}nch fixed point.\n\\newblock {\\em Evolution Equations and Control Theory\\/} (2024), 0--0.\n\n\\bibitem{sibuya1971almost}\n{\\sc Sibuya, Y.}\n\\newblock Almost periodic solutions of {P}oisson’s equation.\n\\newblock {\\em Proceedings of the American mathematical society 28}, 1 (1971),\n 195--198.\n\n\\bibitem{18}\n{\\sc Veech, W.~A.}\n\\newblock Almost automorphic functions.\n\\newblock {\\em Proceedings of the National Academy of Sciences 49}, 4 (1963),\n 462--464.\n\n\\bibitem{veech1965almost}\n{\\sc Veech, W.~A.}\n\\newblock Almost automorphic functions on groups.\n\\newblock {\\em American Journal of Mathematics 87}, 3 (1965), 719--751.\n\n\\bibitem{waldschmidt1995topologie}\n{\\sc Waldschmidt, M.}\n\\newblock {\\em Topologie des points rationnels}.\n\\newblock Univ. P. et M. Curie, 1995.\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n\n\n"}}},{"rowIdx":90,"cells":{"id":{"kind":"string","value":"2412.07886v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07886v1"},"title":{"kind":"string","value":"Convergence estimates for the Magnus expansion IE. 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\\addtolength{\\textwidth}{#1}\\addtolength{\\textwidth}{#1}}\n\\newcommand{\\updownextend}[1]{ \\addtolength{\\topmargin}{-#1} \\addtolength{\\textheight}{#1}\n\\addtolength{\\textheight}{#1}}\n\\marginextend{1cm}\n\\updownextend{0cm}\n\\allowdisplaybreaks[4]\n\\title[Convergence estimates for the Magnus expansion IE]{Convergence estimates for the Magnus expansion IE.\nFinite dimensional Banach algebras}\n\\author{Gyula Lakos}\n\\address{Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, H--1053, Hungary}\n\\email{lakos@renyi.hu}\n\\keywords{Magnus expansion, Baker--Campbell--Hausdorff expansion, convergence estimates, resolvent method}\n\\subjclass[2020]{Primary: 46H30, Secondary: 16W80.}\n\\begin{document}\n\\begin{abstract}\nWe review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates.\nObservations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made.\n\nIn this Part IE, we consider the case of finite dimensional Banach algebras.\nWe show that Magnus expansion is convergent (and works in logarithmic sense) if the cumulative norm\n$< 2+\\varepsilon_{n}$, where $\\varepsilon_{n}$ is a positive number depending on the\ndimension $n$ of the Banach algebra.\nWe also show concrete finite-dimensional counterexamples of multiple Baker--Campbell--Hausdorff type\nfor any cumulative norm $>2$ (necessarily of possibly great dimension).\n\\end{abstract}\n\\maketitle\n\\section*{Introduction}\nThis paper is a direct continuation of Part I \\cite{L1}.\nHere we will consider finite dimensional Banach algebras but which are\n not far from the extremal case of Part I.\n\n\\textbf{Introduction to the setting of finite dimensional Banach algebras.}\nSince its inception, Magnus \\cite{M},\n finite dimensional normed algebras, which are automatically Banach algebras,\n play a prominent role in the study and applications of the Magnus expansion,\n cf.~Blanes, Casas, Oteo, Ros \\cite{BCOR}.\nIn the case of operators on Hilbert spaces, finitely dimensionality is not a strong restriction\n with respect to the convergence radius in terms of the cumulative norm:\nThe counterexamples by Sch\\\"affer \\cite{Scha} (indirectly), Vinokurov \\cite{V}, Moan \\cite{Ma},\n and the convergence results of Moan, Niesen \\cite{MN}, Casas \\cite{Ca} show\n that the critical case for the convergence of the Magnus expansion for Hilbert space operators\n is the cumulative norm $\\pi$, and it is represented by concrete counterexamples over $2\\times2$ real matrices.\n(See Part II \\cite{L2} for a detailed discussion.)\nIn the case of general, possibly infinite dimensional Banach algebras, results of Moan, Oteo \\cite{MO} and Part I \\cite{L1}\n show that the critical convergence radius in terms of the cumulative norm is $2$,\n and counterexamples for convergence with cumulative norm $2$ must need to be somewhat wild.\nIt is not immediately clear that how finite dimensionality affects these results; but, as it turns out, positively.\nAn ease in the finite dimensional case that issues of convergence do not depend on the choice of the norm, as all norms are equivalent.\nSelecting among the convergent or divergent measures in terms of the possible norms is a different matter, however.\n\n\\snewpage\n\\textbf{Outline of content.}\nIn Section \\ref{sec:DivFin}, we start with considering the ``Minimal Examples'',\n which are counterexamples to the convergence of the BCH expansion in the setting of $2\\times2$\n upper triangular matrix algebras; these are of cumulative norm $\\pi+\\varepsilon$.\nNext, we exhibit examples of multiple Baker--Campbell--Hausdorff type for divergence\n with cumulative norm $2\\frac{n}{n-1}$ in some $n\\times n$ matrix algebras (which are, of course, of dimension $d=n^2$), where $n\\geq2$.\nAlthough examples are sometimes results of lucky choices, here will explain how these latter finite dimensional\n counterexamples relate to the infinite dimensional ones of Part I \\cite{L1}.\nIn Section \\ref{sec:ConFin},\n we show that Magnus expansion is convergent and works in logarithmic sense if the cumulative norm\n $< 2+\\varepsilon_{n}$, where $\\varepsilon_{n}$ is a positive number depending on the (real)\n dimension $n$ of the Banach algebra.\nThis section draws heavily on the resolvent method introduced in the formal case by\n Mielnik, Pleba\\'{n}ski \\cite{MP}, and developed further in Part I \\cite{L1}.\nInterestingly, methods of discrete geometry are used; notably,\n we use the estimates of Rogers \\cite{R} on packing densities of centrally convex bodies.\n\n\n\\textbf{Acknowledgements.}\nThe author is grateful to Fernando Casas and Kurusch Eb\\-ra\\-hi\\-mi-Fard for organizing venues for furthering research,\n and also to Pierre-Louis Giscard, Ilya Kuprov, and Stefano Pozza for inspiring conversations.\n\n\n\n\n\\begin{commentx}\n\\tableofcontents\n\\end{commentx}\n\n\\snewpage\n\n\\section{Divergence for finite dimensional Banach algebras}\n\\plabel{sec:DivFin}\nHere we will see how the higher dimensionality leads to divergent Magnus series with measures of cumulative norm\n approaching $2$ from above.\n\n\\subsection{Low dimensional counterexamples}\n\\plabel{ss:low}\n~\\\\\n\nReal and complex algebras of low dimension can be classified, cf. Study \\cite{Stu}, Scheffers \\cite{Sche},\n Kobayashi, Shirayanagi, Takahasi, Tsukada \\cite{KSTT}.\nIt is trivial that if the dimension $d$ of the unital algebra $\\mathfrak A$ is $1$ or $2$, then the algebra is commutative.\nRecall that in the commutative case the Magnus series is trivially convergent;\n the logarithmic version of Magnus formula holds if the cumulative norm is $<\\pi$,\n and beyond that it may or may not hold (compare, say, $\\mathbb R$ and $\\mathbb C$).\nIt is still easy to see that if the dimension $d$ of the unital algebra $\\mathfrak A$ is $3$, then the algebra is either\n commutative or isomorphic to the algebra of upper triangular matrices.\n\nNow, upper triangular matrices already lead to some divergent Magnus series:\n\n\\begin{lemma}\\plabel{lem:rys1}\nAssume that $-\\pi\\leq\\alpha\\leq\\pi$ and $\\varepsilon\\neq0$.\nLet\n\\[M_1^{[\\alpha,\\varepsilon]}=\\begin{bmatrix}\n\\frac{\\pi-\\alpha}2& -\\frac{\\pi+\\alpha}2 \\varepsilon\\\\&- \\frac{\\pi-\\alpha}2\n\\end{bmatrix}\n\\qquad\\text{and}\\qquad\nM_2^{[\\alpha,\\varepsilon]}=\\begin{bmatrix}\n\\frac{\\pi+\\alpha}2 &\n \\frac{\\pi-\\alpha}2 \\varepsilon\\\\\n&- \\frac{\\pi+\\alpha}2\n\\end{bmatrix}.\\]\nThen the Magnus (BCH) series of the ordered measure $M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)}$ is not absolutely convergent.\n\\begin{proof}\nIf the series \n $k\\mapsto\\mu_{k,\\mathbb R}\\left( M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)}\\right)$\n were absolute convergent, then so would be\n $k\\mapsto\\mu_{k,\\mathbb R}\\left(\\mathrm i\\cdot(M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)})\\right)$ in the complex(ified) algebra.\nAssume that $M^{[\\alpha,\\varepsilon]}$ is the absolute convergent sum of the Magnus series.\nThen\n\\begin{multline*}\\exp M^{[\\alpha,\\varepsilon]}=\n\\Rexp\\left(\\mathrm i\\cdot(M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)})\\right)=\\\\=\\exp (\\mathrm iM_1^{[\\alpha,\\varepsilon]})\\exp (\\mathrm iM_2^{[\\alpha,\\varepsilon]})=\n\\begin{cases}\\begin{bmatrix}-1&-\\frac{ \\varepsilon(\\pi^2+\\alpha^2)(1+\\mathrm e^{-\\mathrm i \\alpha }) }{\\pi^2- \\alpha^2}\\\\&-1\\end{bmatrix}\n&\\text{for } -\\pi < \\alpha < \\pi ,\\\\\\\\\n\\begin{bmatrix}-1&\\pm\\pi\\mathrm i\\varepsilon \\\\&-1\\end{bmatrix}\n&\\text{for } \\alpha =\\pm \\pi .\n\\end{cases}\n\\end{multline*}\nIn any case, the result is not diagonalizable, not even in the larger algebra of $2\\times 2$ complex matrices.\nIt implies that $M^{[\\alpha,\\varepsilon]}$ must have Jordan form $\\begin{bmatrix}a&1 \\\\&a\\end{bmatrix}$ in the larger algebra of $2\\times 2$ complex matrices\nwith $a\\in2\\pi\\mathrm i\\left(\\mathbb Z+\\frac12 \\right)$.\nThis, however, contradicts to the fact that the terms of the Magnus (BCH) series \n should be traceless in the larger algebra of $2\\times 2$ complex matrices.\n(The first term is known, and the other ones are sums of commutators.\nOr, alternatively, this also follows directly from considering the homomorphisms given by the diagonal positions.)\n\\end{proof}\n\\end{lemma}\n\nIn the previous proof, instead of ``proper analysis'', only linear algebra was used.\nThe following lemma, of more analytical nature, indicates the oscillatory nature of the coefficients:\n\n\\begin{lemma}\\plabel{lem:rys2}\nIn the previous lemma, for the terms of the Magnus expansion, $k\\geq1$,\n\\begin{multline*}\n\\mu_{k,\\mathrm R}\\left( M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)}\\right)=\\begin{bmatrix}\\pi& \\\\&-\\pi\\end{bmatrix}\\cdot\\delta_{k,1}+\\\\+\n\\begin{bmatrix}0&1 \\\\&0\\end{bmatrix}\\cdot\n\\begin{cases}(-1)^{\\frac{k }2}\\cdot-\n\\frac{2(\\pi^2+ \\alpha ^2)(1+\\cos\\alpha) \\varepsilon}{\\pi^2-\\alpha ^2}+O(2^{-k})&\\text{for $k$ even, } -\\pi < \\alpha < \\pi ,\\\\\\\\\n(-1)^{\\frac{k+1}2}\\cdot-\n\\frac{2(\\pi^2+ \\alpha^2)(\\sin\\alpha)\\varepsilon}{\n\\pi^2- \\alpha^2}+O(2^{-k})&\\text{for $k$ odd, } -\\pi < \\alpha < \\pi ,\\\\\\\\\nO(2^{-k})&\\text{for $k$ even, } \\alpha =\\pm \\pi ,\\\\\\\\\n\\pm(-1)^{\\frac{k-1}2}2\\pi\\varepsilon+ O(2^{-k})&\\text{for $k$ odd, } \\alpha =\\pm \\pi .\n\\end{cases}\n\\end{multline*}\nIn particular, the terms of the Magnus expansion limit to\n$\\begin{bmatrix}0& \\pm c^{[\\alpha]}_{\\mathrm{odd}}\\varepsilon\\\\&0\\end{bmatrix}$\nand\n$\\begin{bmatrix}0& \\pm c^{[\\alpha]}_{\\mathrm{even}}\\varepsilon\\\\&0\\end{bmatrix}$\nwhere $c^{[\\alpha]}_{\\mathrm{odd}},c^{[\\alpha]}_{\\mathrm{even}}\\geq0 $, but at least one of them is nonzero.\n\n\\begin{proof}\nLet\n\\[M^{[\\alpha,\\varepsilon]}(t)=\\begin{cases}\\begin{bmatrix}\\pi t&\n{\\dfrac {t\\varepsilon\\,{\\pi } \\left(\n (\\pi^2+ \\alpha^2) \\left(\n\\cosh \\left( t\\pi\\right) -\\mathrm e^{ - t \\alpha } \\right)-\n 2 \\alpha \\pi\\,\\sinh \\left( t\\pi \\right)\n \\right)\n }{ \\left( {\\pi }^{2}- \\alpha^{2} \\right)\\sinh \\left( t\\pi \\right) }}\n\\\\&-\\pi t\\end{bmatrix}\n\\\\\\hfill\\text{for }-\\pi < \\alpha < \\pi ,\\\\\n\\begin{bmatrix}\\pi t&\\mp\\pi t\\varepsilon\\left(\\boldsymbol\\beta(\\pm2\\pi t) \\right) \\\\&-\\pi t\\end{bmatrix}\n \\hfill\\text{for } \\alpha =\\pm \\pi ;\\\\\n\\end{cases}\\]\nwhere we used the abbreviation $\\boldsymbol\\beta(x)=\\frac{x}{\\mathrm e^x-1}$.\nThen $M^{[\\alpha,\\varepsilon]}(t)$ is meromorphic in $t$ with possible poles in $\\mathrm i\\mathbb Z$,\nand it can be checked that\n\\[\\exp M^{[\\alpha,\\varepsilon]}(t)=\\Rexp\\left(t\\cdot(M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)})\\right).\\]\nIn fact, $M^{[\\alpha,\\varepsilon]}(t)$ extends to $t=0$ holomorphically with $M^{[\\alpha,\\varepsilon]}(0)=\\Id_2$, and the exponential identity above also holds there.\nAs the continuation of $\\log$ is unique near the identity,\n\\[M^{[\\alpha,\\varepsilon]}(t)=\\log\\Rexp\\left(t\\cdot(M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)})\\right)\n=\\sum_{k=1}^\\infty t^k\\mu_{k,\\mathrm R}\\left( M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)} \\right)\\]\nholds for $t\\sim 0$.\nNow, $\\left(M^{[\\alpha,\\varepsilon]}(t)\\right)_{1,2}$ has residue\n\\begin{align*}\n&{\\frac { ({\\pi }^{2}+ \\alpha ^2) \\varepsilon\\, \\left(\n\\mathrm i\\right)\n(1+\\mathrm e^{-\\mathrm i\\alpha}) }{{\n\\pi }^{2}- \\alpha ^{2}}}\n&&\\text{at } t=\\mathrm i \\text{ for } -\\pi < \\alpha < \\pi ,\\\\\n&\n{\\frac { ({\\pi }^{2}+ \\alpha ^2 )\\varepsilon\\, \\left(-\\mathrm i\\right)\n(1+\\mathrm e^{\\mathrm i\\alpha}) }{{\n\\pi }^{2}- \\alpha ^{2}}}\n&&\\text{at } t=-\\mathrm i \\text{ for } -\\pi < \\alpha < \\pi ,\\\\\n& \\pm \\pi \\varepsilon\n&&\\text{at } t= \\mathrm i \\text{ for } \\alpha =\\pm \\pi ,\\\\\n& \\pm \\pi \\varepsilon\n&&\\text{at } t= -\\mathrm i \\text{ for } \\alpha =\\pm \\pi .\n\\end{align*}\nThen the residues at $t=\\pm\\mathrm i$ give the macroscopic contributions in the power series,\nthe residues at $t=\\pm 2\\mathrm i$ give $O(2^{-k})$, and the rest is altogether even smaller.\n\\end{proof}\n\\end{lemma}\n\n\\begin{example}\\plabel{ex:rys}\nIn the (proof of) the previous lemma, the cases\n$\\alpha=0$ (the ``balanced cases'')\n and\n$\\alpha=\\pm \\pi $ (the ``totally unbalanced cases'')\nare particularly instructive.\n\nHere we find for $t\\sim 0$, or even for $|t|<1$, or even just symbolically in $t$,\n\\[\n\\sum_{k=1}^\\infty t^k\\mu_{k,\\mathrm R}\\left( M_1^{[\\alpha,\\varepsilon]}\\mathbf 1_{[0,1)}\\boldsymbol. M_2^{[\\alpha,\\varepsilon]}\\mathbf 1_{[1,2)}\\right)\n=\\begin{cases}\\begin{bmatrix}\\pi t&\n\\varepsilon\\pi t \\tanh\\left(\\frac\\pi2 t\\right)\n\\\\&-\\pi t\\end{bmatrix}\n&\\text{for } \\alpha=0,\\\\\\\\\n\\begin{bmatrix}\\pi t&\\mp \\pi t\\varepsilon\\left(\\boldsymbol\\beta(\\pm2\\pi t) \\right) \\\\&-\\pi t\\end{bmatrix}\n&\\text{for } \\alpha =\\pm \\pi \\\\\n\\end{cases}\n\\]\\[\n=\\begin{cases}\\begin{bmatrix}\\pi t&\n\\sum_{j=1}^\\infty -4(-1)^j (1-2^{-2j})\\zeta(2j)\\varepsilon t^{2j}\n\\\\&-\\pi t\\end{bmatrix}\n&\\text{for } \\alpha=0,\\\\\\\\\n\\begin{bmatrix}\\pi t&\\mp \\pi\\varepsilon t + \\pi^2 \\varepsilon t^2\\pm\\sum_{j=1}^\\infty (-1)^j2\\pi\\zeta(2j)\\varepsilon t^{2j+1} \\\\&-\\pi t\\end{bmatrix}\n&\\text{for } \\alpha =\\pm \\pi .\n\\end{cases}\n\\]\n(One may recall that as $1<2j\\nearrow+\\infty$, one has $\\zeta(2j)\\searrow1$, and even $(1-2^{-2j})\\zeta(2j)\\searrow1$.)\n\\end{example}\nThe example(s) above are also instructive in illustrating that in the few cases when the\nMagnus expansion of $\\phi$ is explicitly known, it comes typically not from the direct evaluation of the\niterated integrals but from the knowledge of $\\log(\\Rexp(t\\cdot\\phi))$ for $t\\sim 0$.\n\\begin{lemma}\\plabel{lem:rys3}\nAssume that the algebra of $2\\times2$ (real or complex) upper triangular\nmatrices is endowed by an algebra norm $\\|\\cdot\\|$ such that $\\left\\| \\begin{bmatrix}1&\\\\&-1\\end{bmatrix} \\right\\|=1$.\nThen\n\\[\\|M_1^{[\\alpha,\\varepsilon]}\\|+\\|M_2^{[\\alpha,\\varepsilon]}\\|\\leq \\pi+\\pi|\\varepsilon|\n \\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\|. \\]\nIn the totally unbalanced cases,\n\\[\\|M_1^{[-\\pi,\\varepsilon]}\\|=\\pi, \\qquad \\|M_2^{[-\\pi,\\varepsilon]}\\|= \\pi|\\varepsilon|\n \\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\|,\\]\nand\n\\[\\|M_1^{[\\pi,\\varepsilon]}\\|=\\pi|\\varepsilon|\n \\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\|, \\qquad \\|M_2^{[\\pi,\\varepsilon]}\\|= \\pi.\\]\n\\begin{proof}\nThese are trivial norm estimates.\n\\end{proof}\n\\end{lemma}\n\\snewpage\n\nThe main point is,\n\\begin{theorem}\\plabel{thm:rys1}\nAssume that the algebra of $2\\times2$ (real or complex) upper triangular\n matrices is endowed by an $\\|\\cdot\\|_{\\ell^p}$ norm with $1\\leq p\\leq+\\infty$,\n or even just with any algebra norm $\\|\\cdot\\|$ such that $\\left\\| \\begin{bmatrix}1&\\\\&-1\\end{bmatrix} \\right\\|=1$.\nLet $r>0$ and $\\delta>0$. Then\n\n(a) There exist an example of type Lemma \\ref{lem:rys1} such that\n$\\|M_1^{[\\alpha,\\varepsilon]}\\|:\\|M_2^{[\\alpha,\\varepsilon]}\\|=r$ and\n$\\|M_1^{[\\alpha,\\varepsilon]}\\|+\\|M_2^{[\\alpha,\\varepsilon]}\\|<\\pi+\\delta$.\n\n(b) There exist a counterexample of BCH type $M_1\\mathbf 1_{[0,1)}\\boldsymbol. M_2\\mathbf 1_{[1,2)}$ to\nthe convergence of the Magnus expansion such that\n$\\|M_1 \\|:\\|M_2 \\|=r$ and\n$\\|M_1 \\|+\\|M_2 \\|=\\pi+\\delta$, and the coefficients of the Magnus expansion are unbounded.\n\\begin{proof}\n(a) We can choose a small $\\varepsilon>0$ such that\n$r\\in\\left[ \\varepsilon \\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\|,\n\\varepsilon^{-1} \\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\|^{-1}\n\\right]$\nand $\\pi \\varepsilon\\left\\| \\begin{bmatrix}0&1\\\\&0\\end{bmatrix}\\right\\| <\\delta$.\nAccording the previous lemma, the first relation implies that\nas we pass from one totally unbalanced case to the other, the norm ratio $r$ will be realized.\nThe second relation implies that the cumulative norm is still small.\n\n(b) We can linearly upscale (a).\n\\end{proof}\n\\end{theorem}\n\\begin{remark}\\plabel{rem:rys}\nThe $\\|\\cdot\\|_{\\ell^p}$ norms with $p=1,2,+\\infty$ can be computed relatively explicitly.\nFor example, regarding the cumulative norm\n\\[\\|M_1^{[\\alpha,\\varepsilon]}\\|_{\\ell^p}+\\|M_2^{[\\alpha,\\varepsilon]}\\|_{\\ell^p}=\\pi+\\pi|\\varepsilon|\\qquad\\text{if }p=1,\\infty;\\]\nand\n\\begin{multline*}\n\\|M_1^{[\\alpha,\\varepsilon]}\\|_{\\ell^2}+\\|M_2^{[\\alpha,\\varepsilon]}\\|_{\\ell^2}=\\\\\n=\\frac12\\pi|\\varepsilon|+ \\sqrt{\\left(\\frac{\\pi-\\alpha}2\\right)^2+\\left(\\frac{\\pi+\\alpha}2\\right)^2\\frac{|\\varepsilon|^2}4}\n+ \\sqrt{\\left(\\frac{\\pi+\\alpha}2\\right)^2+\\left(\\frac{\\pi-\\alpha}2\\right)^2\\frac{|\\varepsilon|^2}4}\n\\\\\n\\leq \\pi+\\pi|\\varepsilon|\n.\\eqedremark\n\\end{multline*}\n\\end{remark}\n\\begin{remark}\\plabel{rem:rys11}\nThe counterexamples in the previous theorem fit to the line \nof Wei \\cite{W} in using the argument of parabolicity (but we use it in a slightly more sophisticated way),\nof Michel \\cite{Mi} in finding a whole range of balanced and unbalanced counterexamples (but we consider a different setting),\nand of Vinokurov \\cite{V} in finding relatively simple counterexamples (but we use an even simpler algebra and\n exhibit a bigger range of counterexamples).\n\\qedremark\n\\end{remark}\n\\snewpage\nI prefer to think about the construction of Lemma \\ref{lem:rys1} as the ``Minimal Examples''.\nThis is not about any mathematical minimality property but that the construction realizes counterexamples in a quite minimal setting.\nThere are variants of it, but one cannot go below cumulative norm $\\pi$:\n\\begin{theorem}\\plabel{thm:rys2}\nAssume that $\\mathfrak A$ is the algebra of the $2\\times2$ upper triangular matrices\n (real or complex), endowed by a Banach algebra norm $|\\cdot|_{\\mathfrak A}$.\nAssume that $\\phi$ is a continuous $\\mathfrak A$-valued measure of cumulative norm $\\leq\\pi$.\nThen the Magnus series of $\\phi$ is (absolute) convergent.\n\\begin{proof}\nUsing standard Neumann series arguments, it is easy to see that the absolute value of the diagonal elements is dominated by the norm.\nMore precisely, if $X\\in \\mathfrak A$, then $|(X)_{1,1}|\\leq |X|_{\\mathfrak A}$ and $|(X)_{2,2}|\\leq |X|_{\\mathfrak A}$.\nIf $\\int|\\phi|_{\\mathfrak A}<\\pi$, then in the complex(ified) setting for $t\\in\\Dbar(0,1+\\varepsilon)$,\n $M(t)=\\log\\Rexp(t\\cdot\\phi)$ exists as one can apply the definition\n $\\log A=\\int_{\\lambda\\in{[0,1]}} \\frac{A-1}{\\lambda+(1-\\lambda)A}\\,\\mathrm d\\lambda$\n based on the critical behavior in the diagonals.\nThe power series expansion of $M(t)$ around $t=0$ gives the convergent Magnus expansion for $t=1$.\nIn fact, the same applies even if $\\int|\\phi|_{\\mathfrak A}=\\pi$ but $\\left|\\int(\\phi)_{1,1}\\right|<\\pi$\n and $\\left|\\int(\\phi)_{2,2}\\right|<\\pi$.\nLet us now consider the measure $\\tilde \\phi$ such that $\\phi=\\frac12(\\int(\\phi)_{1,1}+\\int(\\phi)_{2,2})\\cdot\\Id_2+\\tilde \\phi$.\nI. e. $\\tilde\\phi$ is the traceless modification of $\\phi$.\nThe Magnus expansion of $\\tilde \\phi$ is equiconvergent to the Magnus expansion of $ \\phi$.\nThis traceless modification is convergent as the integrated modified diagonals get off from the boundary of $\\Dbar(0,\\pi)$\nexcept if $C=\\int(\\phi)_{1,1}=-\\int(\\phi)_{2,2}$ with $|C|=\\int|\\phi|_{\\mathfrak A}=\\pi$.\nIt is sufficient to consider this latter case.\nReparametrizing the measure $\\phi$ in terms of variation, and multiplying it a complex unit vector, we can assume that\n$\\phi$ is supported on $[0,\\pi)$ and\n$\\phi=\\begin{bmatrix}\\mathbf 1_{[0,\\pi)}&\\mathbf c\\\\&-\\mathbf 1_{[0,\\pi)}\\end{bmatrix}$ and $|\\psi|_{\\mathfrak A}=\\mathbf 1_{[0,\\pi)}$.\nNow, $\\mathbf c$ may be essentially constant i. e. a scalar times $\\mathbf 1_{[0,\\pi)}$.\nIn this case the Magnus expansion is convergent.\nIf this is not so, then there are elements\n$\\begin{bmatrix}1&a\\\\&-1\\end{bmatrix}$ and $\\begin{bmatrix}1&b\\\\&-1\\end{bmatrix}$ of unit norm\nsuch that $a\\neq b$.\nBut then the norm of $n\\begin{bmatrix}0& (b-a)\\\\&0\\end{bmatrix}\n=\\left(\\begin{bmatrix}1&a\\\\&-1\\end{bmatrix}\\begin{bmatrix}1&b\\\\&-1\\end{bmatrix}\\right)^n-\\begin{bmatrix}1&\\\\&1\\end{bmatrix}$\nwould be bounded by $2$, which is an absurdum.\n\\end{proof}\n\\end{theorem}\nIf dimension $4$ is allowed for an algebra $\\mathfrak A$,\nthen there are already counterexamples to cumulative $\\pi$ with respect to the convergence of the Magnus expansion.\nIndeed, in the case of real $2\\times 2$ matrices with the $\\ell^2$ operator norm\nthe Moan--Schäffer example (see Schäffer \\cite{Scha}, Moan \\cite{Ma}, Moan, Niesen \\cite{MN})\n or the Magnus critical example (see Part II \\cite{L2} for a detailed discussion) are like that.\n\nTaking $\\oplus\\mathbb R$ and the joint maximum norm, we can always increase the dimension of the\nBanach algebras while leaving the cumulative norm of the counterexample the same.\nHowever, we have promised a series of counterexamples with cumulative norm $\\searrow2$.\n\\snewpage\n~\n\n\n\\subsection{Higher dimensional counterexamples}\n\\plabel{ss:high}\n~\\\\\n\nLet us start with a particular example, which we will generalize.\n\\begin{example}\n\\plabel{ex:div}\nHere we will construct an example for the divergence of Magnus expansion\n which is of mBCH type, using $5\\times5$ real matrices of cumulative norm $5\\cdot\\frac2{5-1}=\\frac52$.\nWe will use the matrices\n\\[M^{(5)}_1=\\frac24\\left[\\begin{matrix}0&1&1&1&1\\\\&0&&&\\\\&&0&&\\\\&&&0&\\\\&&&&0\\end{matrix}\\right],\\,\\,\\,\nM^{(5)}_2=\\frac24\\left[\\begin{matrix}0&&&&\\\\-1&0&1&1&1\\\\&&0&&\\\\&&&0&\\\\&&&&0\\end{matrix}\\right],\\,\\,\\,\nM^{(5)}_3=\\frac24\\left[\\begin{matrix}0&&&&\\\\&0&&&\\\\-1&-1&0&1&1\\\\&&&0&\\\\&&&&0\\end{matrix}\\right],\\]\n\\[M^{(5)}_4=\\frac24\\left[\\begin{matrix}0&&&&\\\\&0&&&\\\\&&0&&\\\\-1&-1&-1&0&1\\\\&&&&0\\end{matrix}\\right],\\,\\,\\,\nM^{(5)}_5=\\frac24\\left[\\begin{matrix}0&&&&\\\\&0&&&\\\\&&0&&\\\\&&&0&\\\\-1&-1&-1&-1&0\\end{matrix}\\right].\\]\n\nThe measure we will consider is\n$\\psi_5=M^{(5)}_1\\mathbf 1_{[0,1)}\\boldsymbol.\nM^{(5)}_2\\mathbf 1_{[1,2)}\\boldsymbol.\nM^{(5)}_3\\mathbf 1_{[2,3)}\\boldsymbol.\nM^{(5)}_4\\mathbf 1_{[3,4)}\\boldsymbol.\nM^{(5)}_5\\mathbf 1_{[4,5)}$.\nFor the matrices we will use the $\\ell^1$ operator norm $\\|\\cdot\\|_{\\ell^1}$.\nThen the cumulative norm of $\\psi_5$ is\n\\[\\int\\|\\psi_5\\|_{\\ell^1}=\\|M^{(5)}_1\\|_{\\ell^1}+\\ldots+\\|M^{(5)}_5\\|_{\\ell^1}=5\\cdot\\frac24=\\frac52.\\]\nOn the other hand,\n\\begin{align*}\n\\Rexp(\\psi_5)&=(\\exp M^{(5)}_1)\\ldots(\\exp M^{(5)}_5)\n=(\\Id_5+M^{(5)}_1)\\ldots(\\Id_5+M^{(5)}_5)\n\\\\\n&= \\left[ \\begin {matrix} -{\\dfrac {33}{32}}&-{\\dfrac {41}{32}}&-{\n\\dfrac {21}{32}}&{\\dfrac {9}{32}}&{\\dfrac {27}{16}}\\\\\\\\ -\n{\\dfrac {27}{16}}&-\\dfrac{3}{16}&-{\\dfrac {7}{16}}&\\dfrac3{16}&{\\dfrac {9}{8}}\n\\\\\\\\ -{\\dfrac {9}{8}}&-{\\dfrac {9}{8}}&\\dfrac38&\\dfrac18&\\dfrac34\n\\\\\\\\ -\\dfrac34&-\\dfrac34&-\\dfrac34&\\dfrac34&\\dfrac12\\\\\\\\\n -\\dfrac12&-\n\\dfrac12&-\\dfrac12&-\\dfrac12&1\\end {matrix} \\right]\n=\n \\left[ \\begin {matrix} 1&-7&-\\sqrt {15}&5&-2\n\\\\ -1&-2&2\\,\\sqrt {15}&5&-1\\\\ 1&8&0\n&5&0\\\\ -1&-2&-2\\,\\sqrt {15}&5&1\\\\\n1&-7&\\sqrt {15}&5&2\\end {matrix} \\right]\n\\cdot\\\\&\\qquad\\cdot\n\\underbrace{\\left[ \\begin {matrix} 1&0&0&0&0\\\\ 0&{\\frac {\n61}{64}}&-{\\frac {5\\,\\sqrt {15}}{64}}&0&0\\\\ 0&{\n\\frac {5\\,\\sqrt {15}}{64}}&{\\frac {61}{64}}&0&0\\\\ 0&0\n&0&-1&1\\\\ 0&0&0&0&-1\\end {matrix} \\right]}_{=F_5}\n \\cdot\n \\left[ \\begin {matrix} 1&-7&-\\sqrt {15}&5&-2\n\\\\ -1&-2&2\\,\\sqrt {15}&5&-1\\\\ 1&8&0\n&5&0\\\\ -1&-2&-2\\,\\sqrt {15}&5&1\\\\\n1&-7&\\sqrt {15}&5&2\\end {matrix} \\right]^{-1}\n\\end{align*}\ninforms us about the real Jordan form $F_5$ of $\\Rexp(\\psi_5)$.\nWe see that the geometric multiplicity of the eigenvalue $-1$ is $1$,\ntherefore it cannot be the exponential of a real matrix $M$.\n(The eigenvalue $-1$ is not forbidden for the exponential of a real matrix but\nits geometric multiplicity should be even.)\nOn the other hand, the sum $M$ of the Magnus series should be exactly a matrix like that.\n\\qedexer\n\\end{example}\nThat will be our general strategy:\n we will consider measures of real matrices with time-ordered exponential with\n eigenvalue $-1$ of geometric multiplicity $1$.\nIn order to establish this behaviour we, in fact, do not have to deal with\n the Jordan form, a simple rank computation for $\\Rexp(\\phi)+\\Id$ is sufficient.\n\n Now, we start the general construction.\nLet $Q^{(n)}_s$ be the $n\\times n$ real matrix such that its elements ($i$th row, $j$th column)\nare given by\n\\[\\left(Q^{(n)}_u\\right)_{i,j}=\\delta_{i,u}\\sgn(j-u)\\]\n(where the Kronecker's delta notation is used).\n\\begin{lemma}\\plabel{lem:parprod}\nFor any $1\\leq k\\leq n$, and any scalar $s$,\n\\begin{multline*}(\\exp sQ^{(n)}_1)\\ldots(\\exp sQ^{(n)}_k)=(\\Id_n+ sQ^{(n)}_1)\\ldots(\\Id_n+ sQ^{(n)}_k)\n=\\\\=\n\\left[\\begin{array}{c|c}\nA^{(k)} (s)+B^{(k)}(s)&C^{(k,n-k)}(s)\\\\\\hline 0_{(n-k)\\times k}&\\Id_{n-k}\n\\end{array}\\right];\n\\end{multline*}\nsuch that\n$A^{(k)}(s)$ is a $k\\times k$ matrix whose elements ($i$th row, $j$th column)\nare given by\n\\[\\left(A^{(k)}(s)\\right)_{i,j}=\n\\begin{cases}\n(s+1)^{j-i+1}-(s+1)^{j-i-1}&\\text{if }\\quad i<j,\\\\\ns+1&\\text{if }\\quad i=j,\\\\\n0&\\text{if }\\quad i>j;\n\\end{cases}\\]\n$B^{(k)}(s)$ is a $k\\times k$ matrix whose elements ($i$th row, $j$th column)\nare given by\n\\[\\left(B^{(k)}(s)\\right)_{i,j}=(s+1)^{k-i}-(s+1)^{k-i+1};\\]\n$C^{(k,n-k)}(s)$ is a $k\\times (n-k)$ matrix whose elements ($i$th row, $j$th column)\nare given by\n\\[\\left(C^{(k,n-k)}(s)\\right)_{i,j}=-(s+1)^{k-i}+(s+1)^{k-i+1}.\\]\n(The elements $B^{(k)}(s)$ and $C^{(k,n-k)}(s)$ depend only on the row number.)\n\\begin{proof}\nOne can prove this by induction in $k$.\n\\end{proof}\n\\end{lemma}\n\\begin{lemma}\\plabel{lem:parprod2}\n(a) For any $1\\leq n$, and any scalar $s$,\n\\[\nP^{(n)}(s):=(\\exp sQ^{(n)}_1)\\ldots(\\exp sQ^{(n)}_n)=(\\Id_n+ sQ^{(n)}_1)\\ldots(\\Id_n+ sQ^{(n)}_n)\n=A^{(n)} (s)+B^{(n)}(s)\n\\plabel{eq:proder}\n\\]\n\n(b) If $s+1\\neq \\pm 1$, then he eigenvalues of product matrix $P^{(n)}(s)$ have geometric multiplicity at most $1$.\nIf $s=0$, then $\\lambda=s+1=1$ has geometric multiplicity $n$;\nif $s=2$, then $\\lambda=s+1=-1$ has geometric multiplicity $n-1$.\n\n(c) If $s=2+\\frac2{n-1}$, then $-1$ is an eigenvalue of $P^{(n)}(s)$.\n\\begin{proof}\n(a) This is an immediate consequence of the previous lemma.\n\n(b) Let us consider the matrix $P^{(n)}(s)-\\lambda\\Id_n=(A^{(n)} (s)-\\lambda\\Id_n) +B^{(n)}(s)$.\n\nIf $\\lambda\\neq s+1$, then $(A^{(n)} (s)-\\lambda\\Id_n)$ is an invertible triangular matrix\nof full rank, while $B^{(n)}(s)$ is rank $0$ or $1$. Therefore the rank of the sum is $n-1$ of $n$.\n\nIf $\\lambda= s+1\\neq -1$, then (we also know that $s+1\\neq1$) we have\n$(s+1)^2-1\\neq 0$, thus\nthe column space $(A^{(n)} (s)-\\lambda\\Id_n)$ contains the\ncolumn vectors whose last entry is $0$,\nand $(A^{(n)} (s)-\\lambda\\Id_n)$ also has a $0$ column.\nMeanwhile the columns of $B^{(n)}(s)$\nare uniformly a column vector, whose last entry is $1-(s+1)=-s\\neq0$.\nThen it is easy to see that the columns of $(A^{(n)} (s)-\\lambda\\Id_n) +B^{(n)}(s)$\nmust be independent.\n\nThe special cases $\\lambda= s+1=\\pm1$ can be checked directly.\n\n(c) using (a) one can check that the column vector with uniform entries $1$ is an eigenvector with eigenvalue $-1$.\n\\end{proof}\n\\end{lemma}\nNow it is clear what to do, set $M^{(n)}_k=\\frac2{n-1}Q^{(n)}_k$, and\n\\[\\psi_n=M^{(n)}_1\\mathbf 1_{[0,1)}\\boldsymbol.\n\\ldots\\boldsymbol.\nM^{(n)}_k\\mathbf 1_{[k-1,k)}\\boldsymbol.\n\\ldots\\boldsymbol.\nM^{(n)}_n\\mathbf 1_{[n-1,n)}.\\]\nThen we obtain\n\\begin{theorem}\\plabel{th:parprod} Let $n\\geq2$.\nThe cumulative norm of $\\psi_n$ in the $\\ell^1$ operator norm $\\|\\cdot\\|_{\\ell^1}$ of $n\\times n$ matrices is\n\\[\\int \\|\\psi_n\\|_{\\ell^1}=\\frac{2n}{n-1},\\]\nwhile the Magnus expansion of $\\psi_n$ is not absolutely convergent.\n\\begin{proof}\nThe statements about the cumulative norms are straightforward.\nAs $-1$ is an eigenvalue of geometric multiplicity $1$ of $\\Rexp (\\psi_n)$,\nit cannot an exponential of a real matrix.\nMeanwhile, the sum $M$ of the Magnus series should be a matrix like that.\n\\end{proof}\n\\end{theorem}\n\n\\begin{remark}\\plabel{rem:parprod}\nOne can see that the eigenvalues of $\\Rexp (\\psi_n)$ are on the unit circle,\nwith the only nontrivial Jordan block\n$\\begin{bmatrix}-1&1\\\\&-1\\end{bmatrix}$ in complex form\n(corresponding to column vectors of arithmetic progressions).\n\nIndeed, one can argue as follows.\nLet $\\mathbf v$ be the column vector of length $n$ containing only entries $1$.\nThen the matrix $\\Id_n-\\frac1n\\mathbf v\\mathbf v^\\top$ defines a positive semidefinite\n quadratic form $S_n$ on $\\mathbb R^n$, but it descends to a positive definite\n quadratic form $\\tilde S_n$ on the factor space $\\mathbb R^n/\\mathbb R\\mathbf v=\\tilde{\\mathbb R}^n$.\nThen $P_n=\\Rexp(\\psi_n)$ leaves $S_n$ invariant, and it also descends to linear map $\\tilde P_n $ on $\\tilde{\\mathbb R}^n$,\n which is therefore orthogonal with respect to $\\tilde S_n$.\nThus the eigenvalues of $\\tilde P_n$ are unit complex numbers with trivial Jordan blocks.\nGetting back to $\\mathbb R^n$, can gives one extra eigenvalue from $\\mathbb R\\mathbf v$\nwhich we know to be $-1$, and it can give only one extra nontrivial Jordan block, which\nmust be as indicated because the effect of $P_n$ is easy to check on column vectors of arithmetic progressions.\n\\qedremark\n\\end{remark}\n\\snewpage\n\\begin{commenty}\n\\begin{example}\\plabel{ex:parprod}\n\\[\\Rexp(\\psi_2)=\\left[ \\begin {array}{cc} -3&2\\\\ \\noalign{\\medskip}-2&1\\end {array}\\right]=\n\\left[ \\begin {array}{cc} 2&-\\frac12\\\\ \\noalign{\\medskip}2&\\frac12\\end {array} \\right]\n\\left[ \\begin {array}{cc} -1&1\\\\ \\noalign{\\medskip}0&-1\\end {array}\\right]\n\\left[ \\begin {array}{cc} 2&-\\frac12\\\\ \\noalign{\\medskip}2&\\frac12\\end {array} \\right]^{-1}\n;\\]\n\\[\n\\Rexp(\\psi_3)=\n\\left[ \\begin {array}{ccc} -2&-1&2\\\\ \\noalign{\\medskip}-2&0&1\n\\\\ \\noalign{\\medskip}-1&-1&1\\end {array} \\right]\n=\n\\left[ \\begin {array}{ccc} 1&3&-1\\\\ \\noalign{\\medskip}-1&3&0\n\\\\ \\noalign{\\medskip}1&3&1\\end {array} \\right]\n\\left[ \\begin {array}{ccc} 1&0&0\\\\ \\noalign{\\medskip}0&-1&1\n\\\\ \\noalign{\\medskip}0&0&-1\\end {array} \\right]\n\\left[ \\begin {array}{ccc} 1&3&-1\\\\ \\noalign{\\medskip}-1&3&0\n\\\\ \\noalign{\\medskip}1&3&1\\end {array} \\right]^{-1}\n;\\]\n\\begin{multline*}\n\\Rexp(\\psi_4)=\n \\left[ \\begin {array}{cccc} -{\\frac {115}{81}}&-{\\frac {106}{81}}&-{\n\\frac {10}{81}}&{\\frac {50}{27}}\\\\ \\noalign{\\medskip}-{\\frac {50}{27}}\n&-{\\frac {5}{27}}&-{\\frac {2}{27}}&{\\frac {10}{9}}\n\\\\ \\noalign{\\medskip}-{\\frac {10}{9}}&-{\\frac {10}{9}}&\\frac59&\\frac23\n\\\\ \\noalign{\\medskip}-\\frac23&-\\frac23&-\\frac23&1\\end {array} \\right]\n= \\left[ \\begin {array}{cccc} 7&\\sqrt {5}&4&-\\frac32\\\\ \\noalign{\\medskip}-3\n&-3\\,\\sqrt {5}&4&-\\frac12\\\\\n\\noalign{\\medskip}-3&3\\,\\sqrt {5}&4&\\frac12\n\\\\ \\noalign{\\medskip}7&-\\sqrt {5}&4&\\frac32\\end {array} \\right]\n\\cdot\\\\\\cdot\n\\left[ \\begin {array}{cccc} {\\frac {79}{81}}&-{\\frac {8\\,\\sqrt {5}}{\n81}}&0&0\\\\ \\noalign{\\medskip}{\\frac {8\\,\\sqrt {5}}{81}}&{\\frac {79}{81\n}}&0&0\\\\ \\noalign{\\medskip}0&0&-1&1\\\\ \\noalign{\\medskip}0&0&0&-1\n\\end {array} \\right]\n\\left[ \\begin {array}{cccc} 7&\\sqrt {5}&4&-\\frac32\\\\ \\noalign{\\medskip}-3\n&-3\\,\\sqrt {5}&4&-\\frac12\\\\\n\\noalign{\\medskip}-3&3\\,\\sqrt {5}&4&\\frac12\n\\\\ \\noalign{\\medskip}7&-\\sqrt {5}&4&\\frac32\\end {array} \\right]^{-1}.\n\\eqedexer\n\\end{multline*}\n\\end{example}\n\\end{commenty}\n\nFor $d\\geq3$, let $\\mathrm C^{\\{\\{d\\}\\}}_{\\mathbb K}$ denote the infimum\n of the cumulative norms of $\\mathfrak A$ valued ordered measures whose Magnus expansion is not absolutely convergent\n and $\\mathfrak A$ is a $d$ dimensional Banach algebra over $\\mathbb K$.\nWe know that $\\mathrm C^{\\{\\{3\\}\\}}_{\\mathbb K}=\\pi$, and $\\mathrm C^{\\{\\{d\\}\\}}_{\\mathbb K}$\n is (possibly not strictly) decreasing in $d$, but $\\geq2$.\nBy Theorem \\ref{th:parprod}, for dimension $d=n^2$ ($n\\geq2$), the continuous measure $\\psi_n$ provides a counterexample\n to the convergence of the Magnus expansion.\nThus, for $d\\geq4$,\n\\begin{equation}\n\\mathrm C^{\\{\\{d\\}\\}}_{\\mathbb R}\\leq 2+\\frac2{ \\lfloor\\sqrt d \\rfloor-1}.\n\\plabel{eq:ram}\n\\end{equation}\n(Meanwhile, $\\mathrm C^{\\{\\{d\\}\\}}_{\\mathbb R}\\geq \\mathrm C^{\\{\\{d\\}\\}}_{\\mathbb C}\\geq \\mathrm C^{\\{\\{2d\\}\\}}_{\\mathbb R}$ is trivial.)\n\nNote, however, that the cumulative of $\\psi_2$ is $4$ and the cumulative of $\\psi_3$ is $3$.\nSo, these counterexamples get lower cumulative norms than $\\pi$ only for dimensions $d\\geq9$.\n\nFor $4\\leq\\dim_{\\mathbb K}\\mathfrak A\\leq8$,\n it would be interesting to see counterexamples with cumulative norm less than $\\pi$, if they exist.\nIn particular, most prominently, it is not clear how far one can go when $\\mathfrak A$ is isomorphic\n to the algebra of real $2\\times2$ matrices (allowing other norms than the $\\ell^2$ operator norm).\n\\snewpage\n~\n\n\n\\subsection{Relationship to the free $L^1$ mBCH counterexamples}\n\\plabel{ss:rel}\n~\\\\\n\nIn Part I \\cite{L1} we have considered counterexamples of mBCH type for the convergence of the Magnus expansion.\nThere, the tautological measure $2\\cdot\\mathrm Z^1_{[0,1)}$ was replaced by\n\\begin{multline*}\n\\hat\\psi_n=\\log\\Rexp(2\\cdot\\mathrm Z^1_{[0,\\frac1n)} )\\mathbf1_{[0,1)}\\boldsymbol.\\ldots\\\\\n\\ldots\\boldsymbol.\n\\log\\Rexp(2\\cdot\\mathrm Z^1_{[\\frac{k-1} n,\\frac{k}n )})\\mathbf1_{[k-1,k)}\\boldsymbol.\\ldots\\boldsymbol.\n\\log\\Rexp(2\\cdot\\mathrm Z^1_{[\\frac{n-1} n,\\frac{n}n)} )\\mathbf1_{[n-1,n)}\n\\end{multline*}\n(with $n\\geq2$).\nIn that the cumulative norm of the measure increases only slightly, but the time-ordered exponentals\n will be the same in $\\mathrm F^{1,\\mathrm{loc}}([0,1))$ but not lying in $\\mathrm F^{1 }([0,1))$.\nThese counterexample can be understood in the algebra finitely generated by the\n $\\log\\Rexp(2\\cdot\\mathrm Z^1_{[\\frac{k-1} n,\\frac{k}n )})$\n but which is, nevertheless infinite dimensional and not particularly manageable.\nAlso, the cumulative norm\n $\\int|\\hat\\psi_n|=n\\Theta\\left(\\frac2n\\right)=2+\\frac2n+O\\left(\\frac1{n^2}\\right) $\n and the cumulative norm $\\int\\|\\psi_n\\|_{\\ell^1}=2+\\frac2{n-1}$ are quite comparable.\nAccording to these, the counterexamples $\\psi_n$ compare preferably to the $\\hat\\psi_n$.\n\nHere we would like to argue that the $\\psi_n$ and $\\hat\\psi_n$ closely related to each other.\nThe point is that $\\psi_n$ can be obtained from $\\hat\\psi_n$ by ``reducing'' the algebra $\\mathrm F^{1 }([0,1))$.\nAlthough the following discussion can be made completely precise, for the sake of ease, we keep it informal.\nIn particular, we will sometimes pretend that $\\mathrm F^{1 }([0,1))$ is the superposition of\n elements $X_{t_1}\\ldots X_{t_k}$ with $t_i\\in[0,1)$, even if this is not so simple.\n The precise idea is that $\\mathrm F^{1 }([0,1))$ can be subjected to contractive homomorphisms but which\n can be set up so that the terms of the Magnus series are kept seen relatively large.\n\nThe first thing one can do is to replace $X_{t_1}\\ldots X_{t_k}$ by\n $\\mathsf{a}^{\\asc(t_1,\\ldots,t_k)}\\mathsf{d}^{\\des(t_1,\\ldots,t_k)}X_{t_1,t_2}$,\n i.~e. the internal structure of the expression gets ignored, only the number of ascents and descents get recorded.\nActually, it is sufficient to have $\\mathsf{a}\\equiv1$ and $\\mathsf{d}\\equiv-1$ here; this setup still keeps\n the size of the terms of the Magnus expansion.\nThis leads to an ``abstract composition kernel approach''.\nWhat can make this more down-to-earth is to consider it as a representation.\nWe have the representation space generated by superpositions of $Y_t$ ($t\\in [0,1)$),\nwhere the representation rule is $X_{t_1} Y_{t_2}=Y_{t_1}$ for $t_1<t_2$ and $X_{t_1} Y_{t_2}=-Y_{t_1}$ for $t_1>t_2$.\nIf the superpositions of $Y_t$ is coded as an $L^1$ function $f(t)$, then the effect of $Z_{[a,b)}$ on $f(t)$ leads to $\\tilde f(t)$\nwhere\n\\[\\tilde f(t)=\\int_{s\\in[0,1)} \\underbrace{\\chi_{[a,b)}(t) \\sgn(s-t)}_{=K_{[a,b)}(s,t)}\\,f(s)\\,\\mathrm ds.\\]\nThus $Z_{[a,b)}$ gets represented by kernel $K_{[a,b)}(s,t)$ ($L^1$ in $s$, $L^\\infty$ in $t$).\nThis representation is still quite strange, and indeed, there is no loss in norm with respect to terms of the Magnus expansion.\n\\snewpage\n\nWhat one can make the situation tamer is the introduction of additional eliminating relations $X_{t_1}X_{t_2}=0$ for $t_1,t_2\\in[\\frac{k-1}n,\\frac{k}n )$.\nThis \\textit{will} improve the convergence of the Magnus expansion but hopefully not so much.\nThis is compatible to the representation process of the previous paragraph.\nIt leads to the kernels $K^{[1/n]}_{[a,b)}(s,t)$ where the values on the squares $[\\frac{k-1}n,\\frac{k}n )\\times [\\frac{k-1}n,\\frac{k}n ) $\nare killed.\nThen, the image of the time-ordered exponential of $2\\cdot\\mathrm Z^1_{[0,1)}$, and of $\\hat \\psi_n$ and the computations \nwith the latter one can be\nviewed in terms of kernels which are linear combinations of characteristic functions of $[\\frac{k-1}n,\\frac{k}n )\\times [\\frac{l-1}n,\\frac{l}n )$.\nI. e. in terms of $n\\times n$ matrices.\nIndeed, this is exactly the process how $\\psi_n$ was obtained from $\\hat \\psi_n$.\n\\snewpage\n\n\\section{Convergence for finite dimensional Banach algebras}\n\\plabel{sec:ConFin}\n\nWe recall certain estimates due to Rogers, which are standard material.\nLet us consider any centrally symmetric compact convex body $H$ in the $n$-dimensional space $\\mathbb R^n$.\nLet $\\vartheta(H)$ be the infimum of the covering density of $\\mathbb R^n$ by translates\n of $H$; and let $\\vartheta_L(H)$ be the same with respect to lattice coverings.\nThen, by Rogers \\cite{R}, for $n\\geq3$,\n\\begin{align}\n\\vartheta(H)\\leq \\vartheta_L(H)\\leq&\\min_{0<\\eta<1/n}(1+\\eta)^n(1+n\\log(1/\\eta))\\plabel{eq:r1}\\\\\n&=-n{ W}_{-1} \\left(-{\\tfrac1n}\\right) \\left( 1-{\\frac {1}{n { W}_{-1} \\left(-{\\tfrac1n}\\right)} } \\right) ^{n+1}\\notag\\\\\n<& \\left(1+\\frac1{n\\log n}\\right)^n(1+n\\log(n\\log n))\\plabel{eq:r2}\\\\\n<& n\\log n +n\\log\\log n+2n+1 \\plabel{eq:r3}\\\\\n<& n\\log n +n\\log\\log n+5n.\\plabel{eq:r4}\n\\end{align}\nHere \\eqref{eq:r1} is the proper result of \\cite{R} (which is expressed explicitly using the Lambert $W_{-1}$ function);\n \\eqref{eq:r2} reflects the choice $\\eta=\\frac1{n\\log n}$; and\n while, for example, \\eqref{eq:r3} is still true,\n \\eqref{eq:r4} is the much quoted estimate;\n see G. Fejes Tóth \\cite{F} for more on this.\nThe main point is that there is a dimension-dependent but otherwise universal quantity $\\vartheta_n$\n (which can be chosen as any expression on the RHS),\n which is nearly linear in $n$, estimating the minimal translative covering densities from above.\nMoreover, as it is explained in Rogers, Zong \\cite{RZ}, if $0<r<1$, then $H$ can be covered by at\n most $(1+r^{-1})^n\\vartheta(H)\\leq (1+r^{-1})^n\\vartheta_n$\n translates of $rH$ (i. e. by homothetical copies of $H$ with ratio $r$).\nFor a recent review on these topics, see Naszódi \\cite{N}.\n\\begin{theorem}\\plabel{thm:gain2}\nAssume that $\\mathfrak A$ is Banach algebra of finite real dimension $n$.\nAssume that $\\phi$ is an ordered measure of cumulative norm $\\int|\\phi|=\\omega$.\nLet $\\lambda\\in[0,1]$.\nThen\n\\begin{multline}\n\\left|\\mu_{2,\\mathrm R}(\\phi)\\right|=\n\\left|\\int_{t_1,t_2\\in[0,1]} \\lambda^{\\asc(t_1,t_2)} (\\lambda-1)^{\\des(t_1,t_2)}\\phi(t_1)\\phi(t_2)\\right|\n\\\\\n\\leq\\frac{\\omega^2}2\\left(1- {\\frac {{2}^{1-n}}{n} \\left( {\\frac {1-\\frac2n}{\\mathrm e}} \\right) }\\frac1{\\vartheta_n}\n\\min(\\lambda,1-\\lambda)\\right).\n\\plabel{eq:gar3}\n\\end{multline}\n\n\\begin{proof}\nThe measure $\\phi$ can be approximated by measures of mBCH type.\nTherefore it is sufficient to prove the statement in the case\n\\[\\phi=u_1\\mathbf1_{ [\\tau_0,\\tau_1)}\\boldsymbol.\\ldots \\boldsymbol.u_k\\mathbf1_{\\mathbf [\\tau_{k-1},\\tau_k)}\\]\n such that $u_i\\in\\mathfrak A$, $|u_i|=1$, $0=\\tau_0<\\tau_1<\\ldots<\\tau_k=\\omega$.\nWe also write $\\phi(t)=u(t)\\mathbf 1_{[0,\\omega)}$, where $u(t)$ is the corresponding piecewise constant function.\n\\snewpage\n\nLet us apply the argument of Rogers, Zong \\cite{RZ} where $ H$ is the closed unit ball $K$ of $\\mathfrak A\\simeq\\mathbb R^n$.\nThen $K$ can be covered by at most $s= (1+r^{-1})^n\\vartheta_n $ many copies of $rK$, say $K_1,\\ldots, K_{\\lfloor s\\rfloor}$.\nLet $L_1=K\\cap K_1$, and $L_i=(K\\cap K_i )\\setminus (K_1\\cup \\ldots \\cup K_{i-1})$.\nLet $N_i$ be the union of those $[\\tau_{p-1},\\tau_p)$ such that $u_p\\in L_i$.\n(I. e., formally, $N_i=u^{-1}(L_i)$.)\nLet $v_i$ be the Lebesgue measure of $N_i$.\nClearly, $v_1+\\ldots+v_s=\\omega$.\nWe cut $N_i$ into a lower part $N_i^-$ and lower part $N_i^+$, each of them of Lebesgue measure $v_i/2$.\nLet $P_i^-:[0,v_i/2)\\rightarrow N_i^-$ and $P_i^+:[0,v_i/2)\\rightarrow N_i^+$\n be the corresponding monotone increasing measure preserving transformations.\nIt is easy to see that $P_i^+(q)-P_i^-(q)\\geq v_i/2$ for any $q$.\nOn the other hand, $|u(P_i^+(q))-u(P_i^-(q))|\\leq 2r$, by the construction of $L_i\\subset K_i$.\n\nOne can write\n\\begin{multline*}\n\\mu_{2,\\mathrm R}(\\phi)\n=\\int_{t_1,t_2\\in[0,\\omega]} \\lambda^{\\asc(t_1,t_2)} (\\lambda-1)^{\\des(t_1,t_2)} u(t_1)u(t_2)\\, \\mathrm dt_1\\,\\mathrm dt_2\\\\\n=\\sum_i\\int_{q\\in [0,v_i/2),t_2\\in[0,\\omega]} \\lambda^{\\asc(P_i^-(q),t_2)} (\\lambda-1)^{\\des( P_i^-(q),t_2)} u(P_i^-(q))u(t_2)\n+\\\\+ \\lambda^{\\asc(P_i^+(q),t_2)} (\\lambda-1)^{\\des( P_i^+(q),t_2)} u(P_i^-(q))u(t_2)\n\\, \\mathrm dq\\,\\mathrm dt_2.\n\\end{multline*}\n\n\nThe norm of $(\\lambda^{\\asc(t_{11},t_2)} (\\lambda-1)^{\\des(t_{11},t_2)} u(t_{11})+\n\\lambda^{\\asc(t_{12},t_2)} (\\lambda-1)^{\\des(t_{12},t_2)} u(t_{12})\n)u(t_2)$ can be estimated\n$(\\lambda^{\\asc(t_{11},t_2)} (1-\\lambda)^{\\des(t_{11},t_2)}\\cdot1 +\n\\lambda^{\\asc(t_{12},t_2)} (1-\\lambda)^{\\des(t_{12},t_2)}\\cdot1 )\\cdot1\n) $, but this leads only to the trivial estimate\n\\[|\\mu_{2,\\mathrm R}(\\phi)|\\leq\\frac12\\omega^2.\\]\nHowever, when $t_{11}<t_2<t_{12}$ or $t_{12}<t_2<t_{11}$ holds, then there is gain,\n as a component $\\min(\\lambda,1-\\lambda)\\cdot 2$ in the estimate can be replaced by\n $\\min(\\lambda,1-\\lambda)\\cdot | u(t_{11})-u(t_{12})|$.\nThis is a gain $\\min(\\lambda,1-\\lambda)\\cdot (2-| u(t_{11})-u(t_{12})|)$ in the estimate.\nTherefore, we have a better overall estimate\n\\begin{equation}\n|\\mu_{2,\\mathrm R}(\\phi)|\\leq\\frac{\\omega^2}2- \\sum_i\\min(\\lambda,1-\\lambda)\\cdot(2-2r)\\cdot\\left(\\frac{v_i}2\\right)^2.\n\\plabel{eq:gar1}\n\\end{equation}\nThe inequality between the arithmetic and square means implies that\n$\\sum (v_i)^2\\geq \\frac{\\omega^2}{\\lfloor s \\rfloor}\\geq \\frac{\\omega^2}s$.\nTherefore \\eqref{eq:gar1} implies\n\\begin{equation}\n|\\mu_{2,\\mathrm R}(\\phi)|\\leq\n\\frac{\\omega^2}2\\left(1- \\frac{1-r}s\\min(\\lambda,1-\\lambda)\\right).\n\\plabel{eq:gar2}\n\\end{equation}\n\nIn the previous argument the value of $0<r<1$ (and thus $s$) was unfixed.\nNow $r$ can be optimized to $r=\\frac12\\left(\\sqrt{n^2+6n+1}-n-1\\right)$, but, for the sake of simplicity, we take $r=1-\\frac2n$.\nWith this latter choice,\n\\[\\frac{1-r}s\\frac1{\\vartheta_n}= {\\frac {{2}^{1-n}}{n} \\left( {\\frac {n-2}{n-1}} \\right) ^{ n}}\\frac1{\\vartheta_n} >\n {\\frac {{2}^{1-n}}{n} \\left( {\\frac {1-\\frac2n}{\\mathrm e}} \\right) }\\frac1{\\vartheta_n} .\n\\]\n\nPutting this into \\eqref{eq:gar2}, we obtain \\eqref{eq:gar3}.\n\\end{proof}\n\n\\end{theorem}\n\\snewpage\nFrom Part I \\cite{L1} we may recall:\nFor $\\lambda\\in[0,1]$, the convergence radius of $\\Theta^{(\\lambda)}(x)$ around $x=0$ is\n\\[{\\mathrm C}_\\infty^{(\\lambda)}=\\begin{cases}\n2&\\text{if }\\lambda=\\frac12,\n\\\\\n\\dfrac{2\\artanh (1-2\\lambda)}{1-2\\lambda}=\\dfrac{\\log\\dfrac{1-\\lambda}{\\lambda}}{1-2\\lambda}&\\text{if }\\lambda\\in(0,1)\n\\setminus\\{\\frac12\\},\\\\\n+\\infty&\\text{if }\\lambda\\in\\{0,1\\}.\n\\end{cases}\\]\nThis is a strictly convex, nonnegative function in $\\lambda\\in(0,1)$,\nsymmetric for $\\lambda\\mapsto1-\\lambda$; its minimum is ${\\mathrm C}_\\infty^{(1/2)}=2$.\nIn particular, in $\\lambda\\in[0,1]$, it yields a $[2,+\\infty]$-valued strictly convex continuous function.\n\nFor $d\\geq3$, let $\\Lambda_d$ be the number such that $0<\\Lambda_d\\leq\\frac12$ and\n${\\mathrm C}_\\infty^{(\\Lambda_d )}={\\mathrm C}^{\\{\\{d\\}\\}}_{\\mathbb R}$.\nWe know that $\\Lambda_d$ is increasing in $d$ but $\\leq\\frac12$.\nThen\n\\[\\Lambda_d\\geq \\Lambda_3=0.0588740902\\ldots>\\frac1{17}.\\]\nFor a better appreciation, asymptotically,\n\\begin{lemma} \n\\plabel{lem:appr}\nFor $d\\geq25$,\n\\begin{equation}\n\\Lambda_d\\geq \\frac12\\left(1-\\sqrt{\\frac{3}{\\lfloor\\sqrt{d}\\rfloor-1}}\\right).\n\\plabel{eq:emo}\n\\end{equation}\n\\begin{proof}\nLet $\\tilde\\Lambda_d$ denote the RHS of \\eqref{eq:emo}.\nIt is sufficient to prove that ${\\mathrm C}_\\infty^{(\\tilde\\Lambda_d )}\\geq{\\mathrm C}^{\\{\\{d\\}\\}}_{\\mathbb R}$.\nIn turn, by \\eqref{eq:ram},\n for that, it is sufficient to prove that ${\\mathrm C}_\\infty^{(\\tilde\\Lambda_d )}\\geq2+\\frac2{\\lfloor\\sqrt d\\rfloor-1}$.\nOn the other hand, it is not hard to prove that\n${\\mathrm C}_\\infty^{(\\lambda)}\\geq2+\\frac83(\\lambda-1/2)^2$, leading to the desired result.\n\\end{proof}\n\\end{lemma}\nIn general, however, the estimate $\\Lambda_d>\\frac1{17}$ may be convenient.\nNow we can state\n\\begin{theorem}\n\\plabel{thm:fin}\nAssume that $\\mathfrak A$ is Banach algebra of finite real dimension $n$,\n and $\\phi$ is an $\\mathfrak A$ valued ordered measure such that\n\\begin{equation}\n\\int|\\phi|<\\frac{2}{1-\\left( {\\dfrac {{2}^{-2-n}}{n}\n\\left( {\\dfrac {1-\\frac2n}{\\mathrm e}} \\right) }\\dfrac{\\Lambda_n}{\\vartheta_n}\\right)}\n.\n\\plabel{eq:dana}\n\\end{equation}\nThen $\\psi$ is $M$-controlled, in particular, its Magnus expansion convergent, and the logarithmic Magnus formula holds.\n\\begin{proof}\nAssume that $\\phi$ is subject to \\eqref{eq:dana}.\nAssume $\\int|\\phi|\\geq2$.\nLet us decompose $\\phi=\\phi_1\\boldsymbol.\\phi_2\\boldsymbol. \\phi_3$, where\n$\\int|\\phi_1|=1$ and $\\int|\\phi_2|=2$.\nLet $\\lambda\\in[\\Lambda_n,1-\\Lambda_n]$.\nBy Theorem \\ref{thm:gain2},\n\\[\\Theta^{(\\lambda)}(\\phi_1)\\leq \\Theta^{(\\lambda)}( 1)\n-\\underbrace{\\frac{1}2\\left( {\\frac {{2}^{1-n}}{n}\n\\left( {\\frac {1-\\frac2n}{\\mathrm e}} \\right) }\\frac{\\Lambda_n}{\\vartheta_n}\\right)}_{r:=}\n.\\]\nThat means that $\\phi_1$ has delay\n\\[1- \\left(\\Theta^{(\\lambda)} \\right)^{-1}(\\Theta^{(\\lambda)}( 1)-r)\\geq\n1- \\left(\\Theta^{(1/2)} \\right)^{-1}(\\Theta^{(1/2)}( 1)-r)\\equiv\\frac{r}{4-r}.\n\\]\n(See the ``Delay estimate reduction principle'' of Part I.)\nThe same applies to $\\phi_2$, consequently $\\phi_1\\cdot\\phi_2$ collects delay at least $\\frac{2r}{4-r}$.\nThus the resolvent expansion will be convergent (thus the resolvent $\\mathcal R^{(\\lambda)}(\\Rexp \\phi)$ exists) as long as\n$ \\int|\\phi_3|<\\frac{2r}{4-r}$, i. e. if $\\int|\\phi |<2+\\frac{2r}{4-r}=\\frac{2}{1-\\frac14r}$.\nBut that was exactly our assumption.\n$\\mathcal R^{(\\lambda)}(\\Rexp \\phi)$ also exists if $\\int|\\phi|<2$.\nMore generally, the statement about the existence of the resolvent for\n$\\lambda\\in[\\Lambda_n,1-\\Lambda_n]$\nalso holds if $\\phi$ is replaced by $t\\cdot \\phi$ where $t$ is from the closed complex unit disk.\n\nNow, we claim, in general,\n\\begin{equation}\n\\frac{2}{1-\\frac14r}\\leq \\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}.\n\\plabel{eq:anto}\n\\end{equation}\nIndeed, otherwise, $\\frac{2}{1-\\frac14r}> \\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}$, and we could chose an $\\delta>0$ such that\n\\[\\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}<\\frac{2}{1-\\frac14r(1-\\delta)}.\\]\nWe also know that\n \\[\\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}=\\mathrm C_\\infty^{(\\Lambda_n)} <\\mathrm C_\\infty^{(\\Lambda_n(1-\\delta))}.\\]\nFinally, we could choose a counterexample $\\psi$ to the convergence of the Magnus expansion such that\n\\begin{equation}\\int|\\psi|<\\frac{2}{1-\\frac14r(1-\\delta)}\\plabel{eq:hora}\\end{equation}\nand\n\\begin{equation}\\int|\\psi|<\\mathrm C_\\infty^{(\\Lambda_n(1-\\delta))} .\\plabel{eq:xora}\\end{equation}\nRepeating the arguments of the previous paragraph,\n\\eqref{eq:hora} causes $\\mathcal R^{(\\lambda)}(\\Rexp (t\\cdot\\psi))$ to exist for\n$\\lambda\\in[\\Lambda_n(1-\\delta),1-\\Lambda_n(1-\\delta)]$ and $t\\in\\Dbar(0,1)$;\nwhile \\eqref{eq:xora} causes $\\mathcal R^{(\\lambda)}(\\Rexp (t\\cdot\\psi))$ to exist for $\\lambda\\in[0,\\Lambda_n(1-\\delta))$\nand $\\lambda\\in(1-\\Lambda_n(1-\\delta),1]$.\nUltimately, $\\psi$ is $M$-controlled, and its Magnus expansion is convergent; this is a contradiction.\n\nReturning to the main statement now, \\eqref{eq:anto} implies\n\\[\\int|\\phi|\\leq \\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}\\equiv \\mathrm C_\\infty^{(\\Lambda_n)}.\\]\nThis, in turn, implies that $\\mathcal R^{(\\lambda)}(\\Rexp (t\\cdot\\phi))$ to exist for $\\lambda\\in[0,\\Lambda_n )$\nand $\\lambda\\in(1-\\Lambda_n ,1]$.\nUltimately, $\\phi$ is $M$-controlled.\n\\end{proof}\n\\end{theorem}\nIf we ignore $M$-controlledness and the logarithmic issues, and we concentrate on the\nconvergence of the Magnus expansion, then the previous theorem says\n\\begin{equation}\n\\mathrm C_{\\mathbb R}^{\\{\\{n\\}\\}}\\geq\n\\frac{2}{1-\\left( {\\dfrac {{2}^{-2-n}}{n}\n\\left( {\\dfrac {1-\\frac2n}{\\mathrm e}} \\right) }\\dfrac{\\Lambda_n}{\\vartheta_n}\\right)}.\n\\plabel{eq:subdana}\n\\end{equation}\nNote, however, that there is a huge gap between \\eqref{eq:ram} and \\eqref{eq:subdana}, which would be interesting to close.\n\n\nOne can improve upon the cumulative convergence radius\n$\\mathrm C_2=2.89847930\\ldots$ of the BCH expansion by similar arguments.\n\n\n\\snewpage\n\\begin{thebibliography}{MM}\n\n\n\\gbibitem{BCOR}\nBlanes, S.; Casas, F.; Oteo, J. A.; Ros, J.:\nThe Magnus expansion and some of its applications.\n\\textit{Phys. Rep.} \\textbf{470} (2009), 151--238.\n\n\n\\gbibitem{Ca} Casas, Fernando: Sufficient conditions for the convergence of the Magnus expansion.\n \\textit{J. Phys. A } \\textbf{40} (2007), 15001--15017.\n\n\\gbibitem{F}\nFejes Tóth, Gábor:\nA note on covering by convex bodies. \\textit{Can. Math. Bull.} \\textbf{52} (2009), 361--365.\n\n\\gbibitem{KSTT}\nKobayashi, Yuji; Shirayanagi, Kiyoshi; Takahasi, Sin-Ei; Tsukada, Makoto:\nA complete classification of three-dimensional algebras\n over $\\mathbb R$ and $\\mathbb C$ --- \\begin{CJK}{UTF8}{gbsn} 温故知新\\end{CJK} [OnKoChiShin] (visiting old, learn new).\n \\textit{Asian-European Journal of Mathematics}\n\\textbf{14} (2021), 2150131 (25 pages).\n\n\\gbibitem{L1} Lakos, Gyula: \\textit{Convergence estimates for the Magnus expansion I. Banach algebras.} arXiv:1709.01791\n\n\\gbibitem{L2} Lakos, Gyula: \\textit{Convergence estimates for the Magnus expansion II. $C^*$-algebras.} arXiv:1910.03328\n\n\\gbibitem{M}\nMagnus, Wilhelm:\nOn the exponential solution of differential equations for a linear operator.\n\\textit{Comm. Pure Appl. Math.} \\textbf{7} (1954), 649--673.\n\n\n\\gbibitem{Mi} Michel, Jean: Bases des alg\\`{e}bres de Lie et s\\'erie de Hausdorff.\n\\textit{S\\'eminaire Dubreil. Alg\\`ebre,} \\textbf{27}\nn.1 (1973-1974), exp. n. 6, 1-9 (1974).\n\n\n\\gbibitem{MP}\nMielnik, Bogdan; Pleba\\'{n}ski, Jerzy:\nCombinatorial approach to Baker-Campbell-Hausdorff exponents.\n\\textit{Ann. Inst. H. Poincar\\'e Sect. A (N.S.)} \\textbf{12} (1970), 215--254.\n\n\n\n\\gbibitem{Ma} Moan, Per Christian: \\textit{On backward error analysis and Nekhoroshev stability in the\nnumerical analysis of conservative systems of ODEs.} PhD thesis, University\nof Cambridge, 2002.\n\n\\gbibitem{MN} Moan, Per Christian; Niesen, Jitse: Convergence of the Magnus series.\n\\textit{Found. Comput. Math.} \\textbf{8} (2008), 291--301.\n\n\\gbibitem{MO}\nMoan, P. C.; Oteo, J. A.:\nConvergence of the exponential Lie series.\n\\textit{J. Math. Phys.} \\textbf{42} (2001), 501--508.\n\n\\gbibitem{N}\nNaszódi, Márton:\nFlavors of translative coverings. In: \\textit{New Trends in Intuitive Geometry.} Springer Verlag, New York, Berlin, Heidelberg, 2018. Pages 335--358.\n\n\n\\gbibitem{R}\nRogers, C. A.: A note on coverings.\\textit{ Mathematika} \\textbf{4} (1957), 1--6.\n\n\\gbibitem{RZ}\nRogers C. A.; Zong C.: Covering convex bodies by translates of convex bodies. \\textit{Mathematika} \\textbf{44} (1997), 215--218.\n\n\\gbibitem{Scha}\nSch\\\"affer, Juan Jorge:\nOn Floquet's theorem in Hilbert spaces.\n\\textit{Bull. Amer. Math. Soc.} \\textbf{70} (1964), 243--245.\n\n\\gbibitem{Sche}\nScheffers, Georg: Zurückführung complexer Zahlensysteme auf typische Formen.\n\\textit{Math. Ann.} \\textbf{39} (1891), 293--390.\n\n\\gbibitem{Stu}\nStudy, E.: Über Systeme complexer Zahlen und ihre Anwendung in der Theorie der\n Transformationsgruppen. \\textit{Monatsh. Math. Physik} \\textbf{1} (1890), 283--354.\n\n\\gbibitem{V}\nVinokurov, V. A.:\nLogarithm of the solution of a linear differential equation, the Hausdorff formula and conservation laws. (In Russian)\n\\textit{Dokl. Akad. Nauk SSSR} \\textbf{319} (1991), 792--797; translation in\n\\textit{Soviet Math. Dokl.} \\textbf{44} (1992), 200--205.\n\n\n\\gbibitem{W} Wei, James:\nNote on the global validity of the Baker-Hausdorff and Magnus theorems.\n\\textit{J. Math. Phys.} \\textbf{4} (1963), 1337--1341.\n\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n"}}},{"rowIdx":91,"cells":{"id":{"kind":"string","value":"2412.07928v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.07928v1"},"title":{"kind":"string","value":"Renormalization for Bruin-Troubetzkoy ITMs"},"full_text":{"kind":"string","value":"\\documentclass[english]{amsart}\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{csquotes, xpatch}\n\\usepackage{microtype}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{mathrsfs}\n\\usepackage{mathtools}\n\\usepackage[all]{xy}\n\\usepackage{tikz}\n\\usepackage{caption}\n\\usepackage{subfig}\n\\usepackage{booktabs}\n\\usepackage{graphicx}\n\\usetikzlibrary{automata,positioning,calc,intersections,through,backgrounds,patterns,fit,external}\n\\usepackage[style=alphabetic, maxnames=5, backend=biber,sorting=nyt]{biblatex}\n\t\\addbibresource{biblio.bib}\n\t\\renewbibmacro{in:}{\\ifentrytype{article}{}{\\printtext{\\bibstring{in}\\intitlepunct}}}\n\\usepackage{url}\n\\usepackage[colorlinks]{hyperref}\n\t\\hypersetup{hidelinks}\t\t\\usepackage{bookmark}\n\\usepackage[nameinlink,capitalise,noabbrev]{cleveref}\n\n\\theoremstyle{plain}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\n\\newcommand{\\numberset}{\\mathbb}\n\\newcommand{\\CC}{\\numberset{C}} \n\\newcommand{\\DD}{\\numberset{D}}\n\\newcommand{\\HH}{\\numberset{H}}\n\\newcommand{\\NN}{\\numberset{N}}\n\\newcommand{\\PP}{\\numberset{P}}\n\\newcommand{\\QQ}{\\numberset{Q}}\n\\newcommand{\\RR}{\\numberset{R}}\n\\renewcommand{\\SS}{\\numberset{S}}\n\\newcommand{\\TT}{\\numberset{T}}\n\\newcommand{\\ZZ}{\\numberset{Z}}\n\n\\newcommand{\\cC}{\\mathcal{C}}\n\\newcommand{\\cD}{\\mathcal{D}}\n\\newcommand{\\cE}{\\mathcal{E}}\n\\newcommand{\\cH}{\\mathcal{H}}\n\\newcommand{\\cL}{\\mathcal{L}}\n\n\\renewcommand{\\epsilon}{\\varepsilon}\n\n\\newcommand{\\alphabet}{\\mathbf{A}} \n\\newcommand{\\rauzy}{\\mathcal{R}}\n\\newcommand{\\rauzygraph}{\\mathcal{G}_{\\rauzy}}\n\\newcommand{\\cone}{\\RR_+^{\\alphabet\\cup\\gap}}\n\\newcommand{\\gasket}{\\mathbf{R}}\n\\renewcommand{\\sl}{\\mathfrak{sl}}\n\\newcommand{\\bX}{\\mathbf{X}}\n\\newcommand{\\sW}{\\mathscr{W}}\n\n\\newcommand*{\\transpose}[2][-7mu]{\\ensuremath{\\mskip1mu\\prescript{\\smash{\\top\\mkern#1}}{}{\\mathstrut#2}}}\\providecommand{\\norm}[2][]{#1\\lVert#2#1\\rVert} \\providecommand{\\abs}[2][]{#1\\lvert#2#1\\rvert} \n\n\n\\DeclareMathOperator{\\GL}{GL}\n\\DeclareMathOperator{\\SL}{SL}\n\\DeclareMathOperator{\\SO}{SO}\n\\DeclareMathOperator{\\Mat}{Mat}\n\n\\DeclareMathOperator{\\diam}{diam}\n\\DeclareMathOperator{\\area}{area}\n\n\\begin{document}\n\n\\title[Renormalization for Bruin-Troubetzkoy ITMs]{Renormalization for Bruin-Troubetzkoy ITMs}\n\\date{\\today}\n\n\\author{Mauro Artigiani}\n\\address{Universidad Nacional de Colombia - Bogotá}\n\\email[M.~Artigiani]{martigiani@unal.edu.co}\n\n\\author{Pascal Hubert}\n\\address{Aix-Marseille Universit\\'e, CNRS\\\\ Institut de Math\\'ematiques de Marseille, I2L, 13453, Marseille, France}\n\\email[P.~Hubert]{pascal.hubert@univ-amu.fr}\n\n\\author{Alexandra Skripchenko}\n\\address{Faculty of Mathematics\\\\\n\tNational Research University Higher School of Economics\\\\\n\tUsacheva St. 6\\\\\n\t119048 Moscow, Russia}\n\\email[A.~Skripchenko]{sashaskrip@gmail.com}\n\n\\subjclass[2020]{Primary: 37E05; Secondary: 37A05, 37A44, 11J70}\n\n\\keywords{interval translation mappings, renormalization, multidimensional \n\tcontinued fraction algorithm, Pisot property, Hausdorff dimension}\n\n\\begin{abstract}\n\tWe study a class of interval translation mappings introduced by Bruin and\n\tTroubetzkoy, describing a new renormalization scheme, inspired by the\n\tclassical Rauzy induction for this class. We construct a measure, invariant\n\tunder the renormalization, supported on the parameters yielding infinite\n\ttype interval translation mappings in this class. With respect to this\n\tmeasure, a.e.\\ transformation is uniquely ergodic. We show that this set has\n\tHausdorff dimension between $1.5$ and $2$, and that the Hausdorff dimension\n\tcoincides with the affinity dimension. Finally, seeing our renormalization\n\tas a multidimensional continued fraction algorithm, we show that it has\n\talmost always the Pisot property.\n\t\n\tWe discover an interesting phenomenon: the dynamics of this class of\n\ttransformations is often (conjecturally: almost always) weak mixing, while\n\tthe renormalizing algorithm typically has the Pisot property.\n\\end{abstract}\n\n\\maketitle\n\n\\section{Introduction}\nThis paper is focused on the ergodic properties of two classes of related\ndynamical systems. The first class we are interested in is a particular family\nof interval translation mappings (or ITMs, for short), the second one is a\nMarkovian multidimensional continued fraction algorithm (MCF).\n\nITMs were introduced in~\\cite{BK} as a natural generalization of \\emph{interval\nexchange transformations} (IETs). IETs and their ergodic properties were widely\nstudied in the last decades, see, e.g.,~\\cites{Viana:IETs, Yoccoz:IEMs} and the\nreferences therein. Typical IETs are known to be uniquely\nergodic~\\cites{Masur:ue, Veech:ue} and weakly mixing~\\cite{AvilaForni}, while\nergodic properties of certain special classes of IETs can be remarkably\ndifferent (for example, Arnoux-Rauzy IETs are almost never minimal and those who\nare minimal are typically not weakly mixing~\\cite{ACFH}). All these results were\nachieved by the study of the properties of the renormalization algorithm called\n\\emph{Rauzy induction} (and variations of it). This algorithm can be seen as a\nrepresentative of the class of Markovian multidimensional continued fraction\nalgorithms. \n\nThe key difference between IETs and ITMs is that the latter are not necessarily\nsurjective: the images of the intervals do not need to form a partition, they\nsimply form a collection of subintervals of the original interval,\nsee~\\cref{fig:BT_ITM} for an example. More formally,\n\n\\begin{definition}\n\tAn \\emph{interval translation map} is a piecewise translation map $T$\n\tdefined on an open interval $I \\subset \\RR$ with values in $I$. We call $T$\n\ta $n$-interval translation map (or $n$-ITM) if $I$ has $n$ maximal open\n\tsub-intervals to which the restriction of the $T$ is a translation. The\n\tendpoints of theses intervals are called \\emph{singularities} of the map,\n\tand the endpoints of the image of the intervals are the images of the\n\tsingularities.\n\\end{definition}\n\nIt was noticed already in~\\cite{BK} that each ITM is either of \\emph{finite} or\n\\emph{infinite} type. This classification is based on the properties of the\n\\emph{attractor} of the ITM. Namely, for a given mapping $T$ we consider the\nsequence $\\Omega_n = I\\cap TI\\cap T^2I\\cdots\\cap T^{n}I$. If this sequence\nstabilizes for some $N\\in\\NN$, i.e., $\\Omega_k = \\Omega_{k+1}$ for all $k\\ge N$,\nthen the ITM $T$ is of \\emph{finite type}. If there is no such $N$ and the limit\nset $\\Omega =I\\cap TI\\cap T^2I\\cdots$ is a Cantor set, then the ITM is of\n\\emph{infinite type}, see also~\\cite{ST}.\n\nDynamics of ITMs of finite type basically coincides with the one of IETs.\nHowever, ITMs of infinite type are remarkably different. M.~Boshernitzan and\nI.~Kornfeld described the first example of ITM of infinite type. In the same\npaper, they formulated the following\n\n\\begin{conjecture}\\label{conjecture} \n\tThe set of parameters that give rise to ITMs of infinite type has zero\n\tLebesgue measure.\n\\end{conjecture}\n\nTo the best of our knowledge, this conjecture is currently completely open. The\nonly known cases are for ITMs on $2$ and $3$ intervals, see~\\cites{BK, Volk}\nrespectively, and for a (very special) family of $n$-ITMs, which generalizes the\none we study in this paper to an arbitrarily high number of intervals of\ncontinuity, see~\\cite{Br}. Very recently, a \\emph{topological} version of the\nconjecture has been proven in~\\cite{DSvS}.\n\nIn this paper, we focus on a special subclass of ITMs, which was defined by\nH.~Bruin and S.~Troubetzkoy in~\\cite{BT} and which was, historically, the first\nconcrete example of a family of ITMs. The class is described as follows: let\n$U:= \\{(\\alpha, \\beta): 0\\le\\beta\\le\\alpha\\le1 \\} $ and $L:= \\{(\\alpha, \\beta):\n0\\le\\alpha\\le\\beta+1\\le1 \\} $ and $R:=U\\cup L;$ for the internal point $(\\alpha,\n\\beta)\\in U$ we define \n\\begin{equation}\nT_{\\alpha,\\beta}(x) = \\begin{cases}\n x+\\alpha, & x\\in[0,1-\\alpha)\\\\\n x+\\beta, & x\\in[1-\\alpha, 1-\\beta)\\\\\n x+\\beta-1, & x\\in[ 1-\\beta, 1),\n \\end{cases}\n\\end{equation}\nsee~\\cref{fig:BT_ITM} for an example.\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=.6\\textwidth]{BT_ITM.pdf}\n\t\\caption{An example of a Bruin-Troubetzkoy ITM. The intervals below are images of the ones above, color coded.}\n\t\\label{fig:BT_ITM}\n\\end{figure}\n\nThe transformation $T(x) = T_{\\alpha,\\beta}(x)\\colon [0,1)\\to[0,1)$ is a\n$3$-ITM. By identifying the points $0$ and $1$ we get an interval translation\nmap on a circle with two intervals. We remark that the original example of ITM\nin the paper by Boshernitzan and Kornfeld also belongs to this family. In their\npaper, Bruin and Troubetzkoy proved \\cref{conjecture} for this special family of\nITMs (see~\\cite[Theorem~6]{BT}). They also showed that, considering the set of\n$3$-ITMs $T_{\\alpha, \\beta}$, the set $B$ of parameters that gives rise to\nuniquely ergodic ITMs of infinite type is a dense $G_\\delta$ subset of the set\n$A$ of parameters that give rise to ITMs of infinite type\n(see~\\cite[Corollary~13]{BT}). In this paper, we improve their result in the\nfollowing way: \n\n\\begin{theorem}\\label{thm:uniquelyergodic}\n\tThere exists a natural measure $\\mu$, whose support set coincides with $A$,\n\tsuch that for $\\mu$-almost all $(\\alpha, \\beta)\\in A$, the transformation \n\t$T_{\\alpha, \\beta}$ is uniquely ergodic. \n\\end{theorem} \n\nWe remark that the result by Bruin and Troubetzkoy was later generalized by\nBruin for a slightly more extended subclass of ITMs (see~\\cite[Theorem~1]{Br}).\nAlso, one can see Bruin-Troubetzkoy ITMs as a special type of double rotations,\nwhich were introduced in~\\cite{SIA} and studied in~\\cites{BC, AFHS}. \n\nBruin and Troubetzkoy obtained their results by describing a special type of\nrenormalization procedure (a Gauss-like map) for their class of ITMs. Using\nthis procedure, they found a symbolic (more precisely, substitutional)\npresentation of the interval translation mappings they were interested in, and\nused it to prove their Theorem~6 and to construct the non-uniquely ergodic\nexamples. \n\nOur strategy is quite different. In fact, we treat Bruin-Troubetzkoy family as a\nparticular class of \\emph{systems of isometries}. First, we introduce a new\nrenormalization procedure that is based on the induction that I.~Dynnikov\ndefined for systems of isometries (see \\cite{D}). Our renormalization algorithm\nis a projectivization of the linear map defined by the induction procedure and\ncan be seen as a Markovian multidimensional continued fraction (MCF) algorithm.\nOur \\cref{thm:uniquelyergodic} is hence an immediate corollary of the general\nstatement proved by C.~Fougeron in~\\cite{Fougeron:Simplicial} for a broad class\nof MCF.\n\nOur approach also allows us to get another improvement for the result by\nBruin and Troubetzkoy. Namely, we prove the following estimations on the\nHausdorff dimension of the set $A$ mentioned above: \n\n\\begin{theorem}\\label{thm:hdimboundaries}\n\tLet $A$ be the set of parameters $(\\alpha, \\beta)$ yielding infinite type\n\tBruin-Troubetzkoy ITMs. Then, its Hausdorff dimension can be bounded by\n\t\\[\n\t\t1.5 \\le \\dim_H (A) < 2.\n\t\\]\n\tMoreover, the Hausdorff dimension of the set $A$ is equal to its affinity\n\tdimension.\n\\end{theorem}\n\nWe refer to \\cref{sec:HausdorffEstimate} for the definition of affinity\ndimension. In the previous result, the upper bound follows from the application\nof Fougeron's criterion, in a fashion similar to how we obtain\n\\cref{thm:uniquelyergodic}. The lower bound is achieved by applying the strategy\ndeveloped in~\\cite{JiaoLiPanXu:Applications} for another fractal, of rather\nsimilar origin, called the Rauzy gasket. The Rauzy gasket was widely studied in\nthe literature for several reasons, including symbolic dynamics, Arnoux-Rauzy\ninterval exchange transformations, pseudogroups of rotations and\n$\\mathbb{R}$-trees as well as Novikov's problem of asymptotic behavior of plane\nsections of triply periodic surfaces, see \\cite{DHS} for more details and\nreferences. In fact, to prove that the Hausdorff dimension is equal to the\naffinity dimension, we follow a very recent paper by N.~Jurga who obtains a\nsimilar result for the Rauzy gasket~\\cite{Jurga:Gasket}.\n\n\\begin{remark}\n\tThe upper bound can be slightly improved using the strategy developed by\n\tPolicott and Sewell for the Rauzy gasket~\\cite{PollicottSewell},\n\tsee~\\cite{Zernikov:Diplom}. \n\\end{remark}\n\nIn view of~\\cref{thm:hdimboundaries}, we call the set $A$ the\n\\emph{Bruin-Troubetzkoy gasket}, see \\cref{fig:BTGasket,fig:BTGasket_simplex} on\n\\cpageref{fig:BTGasket_simplex}. The topological similarity between this fractal\nand the Rauzy gasket follows from the structure of the renormalization algorithm\nwe construct, which is quite similar to well-known and well studied MCF\nalgorithms, such as the Arnoux-Rauzy map, the Cassaigne algorithm and the\nArnoux-Rauzy-Poincar\\'e algorithms, see~\\cite{CLL} and the references therein\nfor the details. In order to reflect these features we give to our algorithm a\nspecial name, we call it the \\emph{Arnoux-Rauzy-Cassaigne algorithm} (or ARC,\nfor short). \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.9\\textwidth]{BTgasket_square.pdf}\n\t\\caption{The Bruin-Troubetzkoy gasket.}\n\t\\label{fig:BTGasket}\n\\end{figure}\n\nUsing the MCF point of view, it is natural to compare the ergodic and spectral\nproperties of our algorithm with the ones of the above mentioned algorithms, and\nthe renormalization algorithm itself is the second dynamical system we are\nlooking at in the current paper. Our main result about the ARC MCF algorithm is\nthe following:\n\n\\begin{theorem}\\label{thm:pisot}\n\tThe cocycle defined by the ARC algorithm has almost always \\emph{Pisot}\n\tLyapunov spectrum. \n\\end{theorem} \n\nTo obtain it, we follow the ideas from~\\cite{CLL}. Contrary to this result, it\nis known that self-similar Bruin-Troubetzkoy ITMs of infinite type are very\noften weakly mixing \\cite{Mercat, BruinRadinger}. Moreover, we believe that weak\nmixing for Bruin-Troubetzkoy ITMs is the typical behavior. In fact, we\nconjecture that: \n\n\\begin{conjecture}\\label{weakmixingconjecture} \n\tAlmost all (with respect to the measure $\\mu$ obtained in\n\t\\cref{thm:uniquelyergodic}) Bruin-Troubetzkoy ITMs of infinite type are\n\tweakly mixing. \n\\end{conjecture} \n\nTherefore, we have discovered an interesting phenomenon that did not appear\nbefore in the situations that can be used as a natural references in our context\n(generic IETs, Arnoux-Rauzy IETs, and so on): we have a dynamical system which\nis typically weakly mixing while the renormalization algorithm satisfies the\nPisot condition. \nThis seems to be an interesting phenomenon, which warrants further investigation. \n\n\\subsection*{Organization of the paper}\nThe paper is organized as follows: we start with the detailed description of the\nrenormalization algorithm (see \\cref{sec:induction}); in the same section we\nprove \\cref{thm:uniquelyergodic} and the upper bound in\n\\cref{thm:hdimboundaries}. \\cref{sec:Pisot} is devoted to the proof of\nthe \\cref{thm:pisot}. Finally, the lower bound in the \\cref{thm:hdimboundaries} is\nproved in the \\cref{sec:HausdorffEstimate}.\n\n\\subsection*{Acknowledgments}\nWe would like to thank Sebastien Labb\\'e for interesting conversations and some improvements for the first version of the paper; we are also grateful to Paul Mercat for fruitful\ndiscussions. We thank Juan Galvis, Yessica Trujillo and Juan Pablo Sierra for\ntheir help in preparing \\cref{fig:BTGasket,fig:BTGasket_simplex}. \n\n\\section{Renormalization}\\label{sec:induction} \n\nIn this section we first describe the induction procedure for Bruin-Troubetzkoy\nfamily of ITMs. Then, we apply it to get \\cref{thm:uniquelyergodic}.\n\n\\subsection{Notation}\nFirst, we change the notation in order to make the description of our family\nmore homogeneous. Namely, we introduce new parameters: if $\\alpha>\\beta$, we\nhave\n\\[\n\t\\begin{split}\n\t\ta &= 1 - \\alpha,\\\\\n\t\tb &=\\alpha -\\beta,\\\\\n\t\tc &= \\beta.\n\t\\end{split}\n\\]\nThus, $a + b + c = 1$ and \n\\[\n\t\\begin{split}\n\t\tT([0,a)) \t&= [1-a,1).\\\\\n\t\tT([a,a+b)) &= [1-b,1).\\\\\n\t\tT([a+b,1)) &= [0, c).\n\t\\end{split}\n\\]\nWe always assume that $a$, $b$, and $c$ are \\emph{rationally independent}. \n\nLet us also enumerate the intervals of continuity of the map $T$ from the left\nto the right; thus, the first interval is the one of the length $a$, the second\nis the one of the length $b$ and the third is the one of the length $c$. The\nvector that codes the order in which the intervals appear at the preimage of\n$T$, is given by $(1,2,3).$\n\n\\subsection{The induction $\\rauzy$}\nTo define our induction, we distinguish three cases. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[][Case 1.\\label{fig:Case1}]{\\includegraphics[width=.4\\textwidth]{RauzyCase1.pdf}}\n\t\\qquad\\qquad\n\t\\subfloat[][Case 3.\\label{fig:Case3}]{\\includegraphics[width=.4\\textwidth]{RauzyCase3.pdf}}\n\t\\caption{The two cases of the $\\rauzy$ induction not (immediately) yielding a finite type ITM.}\n\t\\label{fig:RauzyInfiniteCases}\n\\end{figure}\n\n{\\bf Case 1: $a>b+c$.}\\label{case1} We consider the first return map on the\nsubinterval $[b+c,1)$. It is an ITM in the same family with the following\nlengths of intervals: \n\\[\n\t\\begin{split}\n\t\ta' &= a-b-c,\\\\\n\t\tb' &= b,\\\\\n\t\tc' &= c,\n\t\\end{split}\n\\]\nsee \\cref{fig:Case1}.\nWe observe that the order of intervals does not change: the interval of length\n$a'$ is still the leftmost, the interval of length $b'$ is in the middle, while\nthe interval of length $c'$ is the rightmost. So, the coding is again given by\nthe vector $(1,2,3)$. \n\n{\\bf Case 2: $c<a<b+c$.}\\label{case2} One can check that in this case the ITM\ncan be reduced to the ITM on 2 intervals and thus belongs to the finite case,\nsee \\cref{fig:Case2}.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=.6\\textwidth]{RauzyCase2.pdf}\n\t\\caption{The case of the $\\rauzy$ induction inducing a finite type ITM.}\n\t\\label{fig:Case2}\n\\end{figure}\n\n{\\bf Case 3: $a<c$.}\\label{case3} We consider the first return map to the\nsubinterval $[a,1)$. As result we get the following ITM:\n\\[\n\t\\begin{split}\n\t\tT([a,a+b)) &= [1-b,1),\\\\\n\t\tT([a+b,2a+b)) &= [a-1,1),\\\\\n\t\tT([1-c+a,1)) &= [0,c-a) \n\t\\end{split}\n\\] \nThen, the lengths change in the following way: \n\\[\n\t\\begin{split}\n\t\ta' &= a,\\\\\n\t\tb' &= b,\\\\\n\t\tc' &= c-a,\n\t\\end{split}\n\\]\nsee \\cref{fig:Case3}.\n\nHowever, this case is very different from the first case, since the\n\\emph{order} of the intervals has changed: now the interval of length $b'$ is\nthe leftmost one, while the interval of length $a'$ is in the middle (the\nthird interval is always the rightmost). Note that the position of the intervals\nin the image does not change: the image of the rightmost interval is always in\nthe left part and contains $0$, while the two other intervals are on the right\nand contain the rightmost point of the support interval. So, the coding is given\nby the vector $(2,1,3)$. \n\nIt is easy to see that if we start with an ITM with the combinatorial coding\ngiven by the vector $(2,1,3)$ we get the symmetric picture. More precisely, if\nthe interval labelled by $2$ is longer than half of the support interval, after\nthe induction we still have the intervals in the order $(2,1,3)$. Whereas if the\ninterval labelled by $3$ is longer than the rightmost interval (in our case it\nis interval labelled by $1$), then the resulting ITM is coded by $(1,2,3)$ again. \n\nThus, the induction process can be seen as a Markovian multidimensional continued\nfraction algorithm that can be easily described in terms of simplicial systems\n(see~\\cite{Fougeron:Simplicial}). The diagram associated with this system is\npresented in \\cref{fig:Rauzy_graph}, where we\nignore Case~2 since it corresponds to the finite case ITMs. \n\nThe following lemma clearly holds: \n\n\\begin{lemma}\\label{infinitetype} \n\tThe set $A$ of parameters that give rise to the infinite type\n\tBruin-Troubetzkoy ITMs coincides with the set of parameters that do not\n\tenter the hole during the induction procedure. \n\\end{lemma} \n\nWe recall the terminology used for the classical Rauzy induction for IETs. In\neach iteration of the induction, the longest interval, i.e., the one that gets\ncut, will be called the \\emph{winner} and the shortest one is called the\n\\emph{loser}. As with IETs, we name the intervals using the corresponding\nletter. With this convention, in Case 1 the $a$-interval is the winner and the\n$c$-interval is the loser, whereas in Case 3 it is the opposite. We will\nsometimes simply say that the letter (and not the corresponding interval) is the\nwinner or the loser. The following statement follows from \\cref{infinitetype}: \n\n\\begin{lemma}\\label{winlose} \n\tA Bruin-Troubetzkoy ITM is of infinite type if and only if each letter wins\n\tand loses infinite number of times. \n\\end{lemma} \n\nThus, given that $\\lambda = (a,b,c)$ is a vector of lengths of the intervals\nand $\\lambda' = (a', b', c')$ is the vector of lengths of the intervals after\nthe application of the induction, we have $\\lambda = \\mathcal{R} \\lambda',$ with\n$\\mathcal{R} =\\mathcal{R}(k_1,k_2,k_3,\\cdots) =\nA^{k_1}C_{A}B^{k_2}C_{B}A^{k_3}\\cdots,$ where $k_1, k_2, \\cdots \\in\\mathbb{N},$ \n\\[\n\tA = \n\t\\begin{pmatrix}\n\t1 & 1 & 1\\\\\n\t0 & 1& 0 \\\\\n\t0 & 0& 1\n\t\\end{pmatrix}\n\t\\qquad\n\t\\text{and}\n\t\\qquad\n\tB = \n\t\\begin{pmatrix}\n\t1 & 0 & 0\\\\\n\t1 & 1& 1 \\\\\n\t0 & 0& 1\n\t\\end{pmatrix},\n\\]\nwhile\n\\[\n\tC_A = \n\t\\begin{pmatrix}\n\t1 & 0 & 0\\\\\n\t0 & 1& 0 \\\\\n\t1 & 0& 1\n\t\\end{pmatrix}\n\t\\qquad\n\t\\text{and}\n\t\\qquad\n\tC_B = \n\t\\begin{pmatrix}\n\t1 & 0 & 0\\\\\n\t0 & 1& 0 \\\\\n\t0 & 1& 1\n\t\\end{pmatrix}.\n\\]\n\nWe remark that the matrices $A$ and $B$ of the induction coincide with the one\ndefined by P. Arnoux and G. Rauzy in~\\cite{AR}.\n\n\\begin{figure}[bt]\n\t\\begin{tikzpicture}[>=stealth]\n\t\t\\node (left) [left=2cm]\t{$(1, 2, 3)$};\n\t\t\\node (right) [right=2cm] {$(2, 1, 3)$};\n\t\t\\draw[->,>=latex] (left) to[bend left]\n\t\tnode[above] {$C_A$}\n\t\t(right);\n\t\t\\draw[->,>=latex] (right) to[bend left]\n\t\tnode[above] {$C_B$}\n\t\t(left);\n\t\t\\draw[->,>=latex] (right) to [out=30, in=330, looseness=4]\n\t\tnode[right] {$B$}\n\t\t(right);\n\t\t\\draw[->,>=latex] (left) to [out=150, in=210, looseness=4]\n\t\tnode[left] {$A$}\n\t\t(left);\n\t\\end{tikzpicture}\n\t\\caption{The Rauzy graph $\\rauzygraph$ of the induction $\\rauzy$.}\n\t\\label{fig:Rauzy_graph}\n\\end{figure}\n\nThe graph if the induction $\\rauzy$ is shown on \\cref{fig:Rauzy_graph}. We\nstress that the coefficients $k_i$ can be equal to $0$. However, since applying\nthe matrix $A$ implies cutting of $a$-intervals (and, similarly the matrix $B$\nimplies cutting the $b$-interval), an ITM is of infinite type if and only if we\nhave infinitely often that even and odd $k_i$'s are \\emph{strictly} positive.\n\nNow one can check that any $\\rauzy$ that contains $A$ and $B$ in positive\npowers together with $C_A$ and $C_B$ has strictly positive entries. Therefore,\nthe following lemma holds: \n\n\\begin{lemma}\nThere exists a special acceleration of the induction described above. \n\\end{lemma} \n\nThe definition of \\emph{special acceleration} can be seen\nin~\\cite[Remark~1]{FS}. Morally, it means a first return map to some subsimplex\ncompactly contained in the parameter space. Exploiting the machinery of\nsimplicial systems introduced in~\\cite{Fougeron:Simplicial} and related results\nfrom~\\cite{FS}, we easily obtain \\cref{thm:uniquelyergodic}.\n\n\\begin{proof}[Proof of \\cref{thm:uniquelyergodic}]\n\tThe simplicial system associate to $\\rauzy$ is \\emph{uniformly expanding}\n\tby~\\cite[Proposition~4.1]{Fougeron:Simplicial} and therefore ergodic thanks\n\tto~\\cite[Corollary~4.4]{Fougeron:Simplicial}. It is obvious that the\n\tsimplicial system is \\emph{quickly escaping} in the sense of\n\t\\cite{Fougeron:Simplicial} and thus, by Theorem~1.1 in\n\t\\cite{Fougeron:Simplicial}, we obtain the natural measure $\\mu$ that induces\n\tthe measure of maximal entropy on the natural suspension. Therefore the set\n\tof parameters which follow the same path (generic for $\\mu$) is a single\n\tpoint, and so by the standard argument originated by Veech we conclude that\n\tthe original ITM is uniquely ergodic. This completes the proof of\n\t\\cref{thm:uniquelyergodic}.\n\\end{proof}\n\nAs a corollary of the previous result, we obtain an upper estimate on the\nHausdorff dimension of the parameters yielding Bruin-Troubetzkoy ITMs of\ninfinite type.\n\n\\begin{corollary}\n\tThe set $A$ of parameters that give rise to the infinite type\n\tBruin-Troubetzkoy ITMs has Hausdorff dimension strictly smaller than $2$.\n\\end{corollary}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.9\\textwidth]{BT_Gasket.pdf}\n\t\\caption{The Bruin-Troubetzkoy gasket using the simplicial coordinates $(a, b, c)$.}\n\t\\label{fig:BTGasket_simplex}\n\\end{figure}\n\n\nWe will obtain a lower bound, using thermodynamical formalism, in\n\\cref{sec:HausdorffEstimate}. In \\cref{fig:BTGasket_simplex}, we represent the\nBruin-Troubetzkoy gasket using the parameters $(a, b, c)$ instead of $(\\alpha,\n\\beta)$ as in \\cref{fig:BTGasket}.\n\n\\subsection{Recovering Bruin and Troubetzkoy's Gauss map} \nWe now show that our induction can be accelerated to recover the Gauss map of\nBruin and Troubetzkoy.\n\n\\begin{proposition}\\label{prop:RauzyGauss}\n\tThere exists an acceleration of $\\rauzy$ such that, after rescaling the\n\toriginal interval, the induced transformation is the one obtained via the\n\tGauss map of Bruin and Troubetzkoy.\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by recalling the definition of the Gauss map. If $T_{\\alpha,\n\t\\beta}$ is a Bruin-Troubetzkoy ITM, then we define the ITM $T_{\\alpha',\n\t\\beta'}$ where\n\t\\begin{equation}\\label{eq:GaussMap}\n\t\t(\\alpha', \\beta') = \\biggl(\\frac{\\beta}{\\alpha}, \n\t\t\t\\frac{\\beta-1}{\\alpha} + \\biggl\\lfloor \\frac{1}{\\alpha} \\biggr\\rfloor\\biggr),\n\t\\end{equation}\n\twhere $\\lfloor \\cdot \\rfloor$ is the (lower) integer part. Let us recall\n\tthat we can recover the parameter $\\alpha$ and $\\beta$ from the length ones\n\tusing that $\\alpha = b + c$ and $\\beta = c$, if the intervals are in the\n\torder $(1,2,3)$, and similarly, replacing $b$ by $a$ in the other case.\n\t\n\tWe observe that, by definition of the $\\rauzy$ induction, the first case is\n\trepeated $n$ times, with $n \\ge 0$ given by\n\t\\[\n\t\tn \t= \\biggl\\lfloor \\frac{a}{b+c} \\biggr\\rfloor\n\t\t\t= \\biggl\\lfloor \\frac{1-\\alpha}{\\alpha} \\biggr\\rfloor \n\t\t\t= \\biggl\\lfloor \\frac{1}{\\alpha} \\biggr\\rfloor - 1.\n\t\\]\n\tThen, we are in the third case, and we change the order of the intervals.\n\n\tAfter the above steps, the three intervals of continuity are of lengths\n\t\\[\n\t\t\\begin{split}\n\t\t\ta' &= a - n(b+c),\\\\\n\t\t\tb' &= b,\\\\\n\t\t\tc' &= c - (a- n(b+c)). \n\t\t\\end{split}\n\t\\]\n\tThe total length is $a'+b'+c' = b+c = \\alpha$. So, if we renormalize by\n\tdividing the interval by $\\alpha$, rescaling it to length $1$, we see that\n\t\\[\n\t\t\\alpha' = \\frac{a'+c'}{\\alpha} = \\frac{c}{\\alpha} = \\frac{\\beta}{\\alpha}\n\t\\]\n\tMoreover, since\n\t\\[\n\t\tc' = \\beta - 1 +\\alpha + \\biggl( \\biggl\\lfloor \\frac{1}{\\alpha} \\biggr\\rfloor - 1\\biggr) \\alpha = \\beta - 1 + \\biggl\\lfloor \\frac{1}{\\alpha} \\biggr\\rfloor \\alpha,\n\t\\]\n\twe have that\n\t\\[\n\t\t\\beta' = \\frac{c'}{\\alpha} = \\frac{\\beta - 1}{\\alpha} + \\biggl\\lfloor \\frac{1}{\\alpha} \\biggr\\rfloor.\n\t\\]\n\tThe above formulas agree with~\\eqref{eq:GaussMap} and so we are done.\n\\end{proof}\n\n\n\n\n\\section{Pisot property for the ARC algorithm}\\label{sec:Pisot}\n\\subsection{The ARC multidimensional continued fraction\nalgorithm}\\label{sec:ARC_MCF} \n\nThe induction $\\rauzy$ introduced in the previous section defines a\nmultidimensional continued fraction algorithm (or MCF algorithm, for short),\nwhich we call the \\emph{Arnoux-Rauzy-Cassaigne} (or ARC) MCF algorithm, as we\nwill now explain. We can naturally act by the matrices of the $\\rauzy$ induction\nto the standard $2$-dimensional simplex $\\Delta = \\Delta^2 = \\{(x,y,z): x,y,z\\ge\n0, x+y+z=1\\}$. In formulas, we have:\n\\begin{equation}\\label{eq:ARC_IFS}\t\n\t\\begin{split}\n\t\tf_A (x,y,z) &= \\biggl(\\frac{1}{2-x}, \\frac{y}{2-y}, \\frac{z}{2-z}\\biggr),\\\\\n\t\tf_B (x,y,z) &= \\biggl(\\frac{x}{2-x}, \\frac{1}{2-y}, \\frac{z}{2-z}\\biggr),\\\\\n\t\tf_{C_A} (x,y,z) &= \\biggl(\\frac{1-y}{2-x-y}, \\frac{y}{2-x-y}, \\frac{z}{2-x-y}\\biggr),\\\\\n\t\tf_{C_B} (x,y,z) &= \\biggl(\\frac{x}{2-x-y}, \\frac{1-x}{2-x-y}, \\frac{z}{2-x-y}\\biggr).\n\t\\end{split}\t\n\\end{equation}\n\nWe will now show that the cocycle defined by the ARC MCF algorithm has negative\nsecond Lyapunov exponent (see below for the relevant definitions),\nfollowing~\\cite{CLL}; in the terminology of~\\cite{Lagarias}, the MCF algorithm\nis \\emph{strongly convergent}. This will imply that it satisfies the Pisot\ncondition from~\\cite{BST}. \n\nWe remark that the measure $\\mu$ obtained in the proof of\n\\cref{thm:uniquelyergodic} induces a measure, which, slightly abusing the\nnotation, we still denote $\\mu$, on set of walks along the graph $\\rauzygraph$.\nWe will call $1$ the state corresponding to the permutation $(1, 2, 3)$ and $2$\nthe permutation $(2, 1, 3)$. Thus, for instance, the path $112$ corresponds to\nthe application of the matrices $AC_A$ of the induction. As usual, let $[a_1,\na_2, \\dotsc, a_n]$ be the cylinder formed by all the words in $\\{1, 2\\}^\\NN$\nthat begin with the letters $a_1a_2\\cdots a_n$. \n\nWe now recall the definition of Lyapunov exponents in the present context (along\nwith the relevant notation). Given an infinite word $w\\in\\{1,2\\}^\\NN$, we\nconsider the product of matrices corresponding to the path on the graph\n$\\rauzygraph$ described by $w$:\n\\[\n\tX_n(w) = X_{w_0w_1} X_{w_1w_2} \\cdots X_{w_{n-2}w_{n-1}},\n\\]\nfor $n\\ge 1$. This forms a cocycle, as\n\\[\n\tX_{m+n}(w) = X_m(w) X_n(\\sigma^m w),\n\\]\nwhere $\\sigma\\colon\\{1,2\\}^\\NN\\to\\{1,2\\}^\\NN$ is the shift map. Since all the\nmatrices of the $\\rauzy$ induction are invertible, this cocycle is\nlog-integrable with respect to the measure $\\mu$ constructed in\n\\cref{thm:uniquelyergodic}:\n\\[\n\t\\int_{\\{1,2\\}^\\NN} \\log \\max\\bigl\\{\\|X_1(w)\\|, \\|X_1(w)^{-1}\\|\\bigr\\} \\, d\\mu(w) < \\infty.\n\\]\nHence there are ($\\mu$-almost everywhere) well defined Lyapunov exponents:\n\\[\n\t\\lambda_1 = \\lim_{n\\to\\infty} \\frac{\\log{\\|X_n(w)\\|}}{n},\n\t\\qquad\n\t\\lambda_1 + \\lambda_2 = \\lim_{n\\to\\infty} \\frac{\\log{\\|\\wedge^2 X_n(w)\\|}}{n},\n\t\\qquad\n\t\\lambda_3 = -(\\lambda_1 + \\lambda_2)\n\\]\nwhere $\\wedge$ is, as usual, the exterior product of matrices, and the last\nequality follows from the fact that the matrices of the induction have\ndeterminant $1$. Moreover, since, by unique ergodicity, the nested cones $X_1\nX_2 \\cdots X_n = X_{[1,n)}\\RR^3_{\\ge 0}$, where we have suppressed the\ndependence on $w$, converge for $\\mu$-almost all words $w$, to a line $\\RR_{0}\nf$, for some $f\\RR^3$, we have the following characterization of the second\nLyapunov exponent will be useful later on:\n\\[\n\t\\lambda_2 = \\lim_{n\\to\\infty} \\frac{\\log{\\Bigl\\|\\transpose{X_n(w)}|_{f^\\perp}\\Bigr\\|}}{n},\n\\]\nwhere $\\transpose{X}$ denotes the transpose matrix.\n\n\\subsection{Some technical lemmas}\nWe begin with a general lemma about matrices and norms, whose proof can be found\nin~\\cite[Lemma~3.5]{CLL}.\n\n\\begin{lemma}\\label{lemma:submultiplicative}\n\n\tLet $X$ and $Y$ non negative $d\\times d$ real matrices, such that $XY\\neq\n\t0$. Let $\\|\\cdot\\|$ denote any seminorm on $\\RR^s$ which is a norm on every\n\t$f^\\perp$, with $f\\in (X\\RR^d_{\\ge 0} \\cup Y\\RR^d_{\\ge 0}) \\setminus \\{0\\}$.\n\tWe have\n\t\\[\n\t\t\\Bigl\\| \\transpose{XY} \\Bigr\\|^{XY\\RR^d_{\\ge 0}} \\le \n\t\t\\Bigl\\| \\transpose{X} \\Bigr\\|^{X\\RR^d_{\\ge 0}} \\Bigl\\| \\transpose{Y} \\Bigr\\|^{Y\\RR^d_{\\ge 0}}.\n\t\\]\n\\end{lemma}\n\nFrom the definitions, we obtain:\n\n\\begin{lemma}\\label{lemma:products_of_M}\n\tLet $(M_n)_{n \\in \\NN} \\in \\{A, B, C_A, C_B\\}^\\NN$ be a sequence of\n\tmatrices. If $(M_n)$ contains infinitely many occurrences of $A$, $B$, $C_A$\n\tand $C_B$, then there exists an increasing sequence of integers $(n_m)_{m\n\t\\in \\NN}$, such that $n_0 = 0$ and\n\t\\[\n\t\tM_{[n_m, n_{m+1})} \\in \\{A^k C_A, B^k C_B: k\\in\\NN\\}.\n\t\\]\n\\end{lemma}\n\nFollowing~\\cite{CLL}, we introduce the seminorm $\\|\\cdot\\|_D$ on $\\RR^3$ given\nby\n\\[\n\t\\|v\\|_D = \\max v -\\min v = \\max_{i=1,2,3} v_i - \\min_{i=1,2,3} v_i.\n\\]\nWe remark that the seminorm is invariant under addition of constant vectors.\nMoreover, for any $f\\in\\RR^3_{\\ge 0}$, the restriction of the seminorm to\n$f^\\perp$ is a genuine norm, see~\\cite[Lemma~3.7]{CLL}. The same result implies\nthat, when restricted to $f^\\perp$, for a given $f\\in\\RR^3_{\\ge 0}$, we obtain a\nnorm on $3\\times 3$ matrices, which is invariant under addition of constant\nvectors, and comparable with the infinity norm. More precisely,\nin~\\cite[Lemma~3.7]{CLL} it is proven that: \n\\begin{equation}\\label{eq:Dnorm_vs_infinitynorm}\n\t\\frac{1}{2}\\bigl\\| M|_{f^\\perp} \\bigr\\|_D \\le \\bigl\\| M|_{f^\\perp} \\bigr\\|_\\infty \n\t\t\\le 2 \\, \\bigl\\| M|_{f^\\perp} \\bigr\\|_D\n\\end{equation}\n\nWith this notation, we have\n\n\\begin{lemma}[\\protect{\\cite[Lemma~3.8]{CLL}}]\\label{lemma:hyperplanes}\n\tLet $M$ be a $3\\times 3$ positive and invertible real matrix. Consider the\n\tset $\\cH$ of hyperplanes orthogonal to some vector in\n\t\\[\n\t\tS = (M\\cE \\cup (\\cE - \\cE) \\cup M(\\cE - \\cE)) \\setminus \\{0\\},\n\t\\]\n\twhere $\\cE = \\{e_1, e_2, e_3\\}$ and $\\cD$ be the finite union of\n\tone-dimensional intersections of two hyperplanes of $\\cH$:\n\t\\[\n\t\t\\cD = \\bigcup_{h_1, h_2 \\in \\cH} h_1 \\cap h_2.\n\t\\]\n\tThen, the maximal value of the norm $\\|\\cdot\\|_D$, for all restrictions\n\torthogonal to a positive vector in the cone $M\\RR^3_{\\ge 0}$, is attained at\n\tsome vector $v$ in $\\Bigl(\\cD \\setminus \\pm \\transpose{M^{-1}} \\RR^3_{>0}\\Bigr)\n\t\\setminus \\{0\\}$, i.e.,\n\t\\[\n\t\t\\Bigl\\|\\transpose{M}\\Bigr\\|_D^{M\\RR^3_{\\ge 0}} \n\t\t:= \\sup_{f\\in M\\RR^3_{\\ge 0} \\setminus \\{0\\}} \\Bigl\\|\\transpose{M}|_{f^\\perp}\\Bigr\\|_D\n\t\t = \\max_{v\\in \\bigl(\\cD \\setminus \\pm \\transpose{M^{-1}} \\RR^3_{>0}\\bigr) \\setminus \\{0\\}} \n\t\t \t\\frac{\\Bigl\\|\\transpose{M}v\\Bigr\\|_D}{\\|v\\|_D}.\n\t\\]\n\\end{lemma}\n\nWe will use the next lemma to control the norm introduced above for specific\nproducts of the matrices of our MCF.\n\n\\begin{lemma}\\label{lemma:norm_one}\n\tFor every $k\\ge 0$ we have that \n\t\\[\n\t\t\\Bigl\\|\\transpose{(A^kC_A)}\\Bigr\\|_D^{\\RR_{\\ge 0}^3} = 1, \\qquad\n\t\t\\Bigl\\|\\transpose{(B^kC_B)}\\Bigr\\|_D^{\\RR_{\\ge 0}^3} = 1.\n\t\\]\n\\end{lemma}\n\n\\begin{proof}\n\tWe will prove the lemma only for $A^kC_A$, the other case being similar.\n\tChoose a vector $f \\in \\RR^3_{\\ge 0} \\setminus \\{0\\}$ and let $v = (v_1,\n\tv_2, v_3) \\in f^\\perp$. Using the invariance of the norm $\\|\\cdot\\|_D$ under\n\taddition by constant vectors, we obtain, for every $k\\ge 0$\n\t\\[\n\t\t\\Bigl\\|\\transpose{(A^kC_A)}\\Bigr\\|_D^{\\RR_{\\ge 0}^3} = \n\t\t\\left\\| \n\t\t\t\\begin{pmatrix}\n\t\t\t\tk+1 & 0 & 1 \\\\\n\t\t\t\tk \t& 1 & 0 \\\\\n\t\t\t\tk\t& 0 & 1\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tv_1 \\\\ \n\t\t\t\tv_2 \\\\\n\t\t\t\tv_3\n\t\t\t\\end{pmatrix}\n\t\t\\right\\|_D\n\t\t=\n\t\t\\left\\|\n\t\t\t\\begin{pmatrix}\n\t\t\t\t1 & 0 & 0 \t\\\\\n\t\t\t\t0 & 1 & -1 \t\\\\\n\t\t\t\t0 & 0 & 0\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tv_1 \\\\\n\t\t\t\tv_2 \\\\\n\t\t\t\tv_3\n\t\t\t\\end{pmatrix}\n\t\t\\right\\|_D.\n\t\\]\n\n\tLet $v' = (v_1, v_2 - v_3, 0)$, then \n\t\\[\n\t\t\t\\min v \\le \\min v' \\le 0 \\le \\max v' \\le \\max v.\n\t\\]\n\tWe remark that, since we work in $3$ dimensions, we have that $\\|(v_1, v_2,\n\tv_3)\\|_D = \\max\\{|v_1 - v_2|, |v_1 - v_3|, |v_2 - v_3|\\}$, so the previous\n\tequation implies that $\\|v'\\|_D \\le \\|v\\|_D$. Hence\n\t\\[\n\t\t\t\\frac{\\Bigl\\|\\transpose{(A^k C_A)}v\\Bigr\\|_D}{\\|v\\|_D} \\le 1,\n\t\\] \n\tfor every $v\\in f^\\perp$ and $k\\ge 0$. To obtain the equality, we observe\n\tthat, since $f$ is non zero and non negative, we can take a vector $v = (a,\n\t-b, 0)$, for $a$, $b>0$ inside $f^\\perp$. For this vector, the direct\n\tcomputation yields\n\t\\[\n\t\t\\Bigl\\|\\transpose{(A^k C_A)}(a, -b, 0)\\Bigr\\|_D = \\|(a, -b, 0)\\|_D,\n\t\\]\n\tas we wanted.\n\\end{proof}\n\nWe need to take care of a ``base case'' before we can do the general one.\n\n\\begin{lemma}\\label{lemma:AC_ABC_B}\n\tLet $M = AC_A BC_B$ or $M = BC_B AC_A$. Then\n\t\\[\n\t\t\\Bigl\\| \\transpose{M} \\Bigr\\|_D^{M\\RR^3_{\\ge 0}} \\le \\frac{4}{5}.\n\t\\]\n\\end{lemma}\n\n\\begin{proof}\n\tWe will prove the result in the case $M = AC_A BC_B$, the other case is\n\tsymmetric. By direct computation:\n\t\\[\n\t\tM = AC_ABC_B =\n\t\t\\begin{pmatrix}\n\t\t\t3 & 3 & 2 \\\\\n\t\t\t1 & 2 & 1 \\\\\n\t\t\t1 & 1 & 1\n\t\t\\end{pmatrix},\n\t\t\\qquad\n\t\tM^{-1} =\n\t\t\\begin{pmatrix}\n\t\t\t1 & -1 & -1 \\\\\n\t\t\t0 & 1 & -1 \\\\\n\t\t\t-1 & 0 & 3\n\t\t\\end{pmatrix}.\n\t\\]\n\tGiven $z = (a, b, c)$ we can compute\n\t\\[\n\t\t\\Bigl\\|\\transpose{M}z\\Bigr\\|_D = \\Bigl\\|\\transpose{(3a+b+c, 3a+2b+c, 2a+b+c)}\\Bigr\\|_D \n\t\t\t= \\Bigl\\|\\transpose{(0, b, a+b+c)}\\Bigr\\|_D.\n\t\\]\n\tWe now construct the set $\\cH$ as in \\cref{lemma:hyperplanes}. By a direct\n\tcomputation:\n\t\\begin{align*}\n\t\tMe_1 &= (3, 1, 1), & e_1 - e_3 &= (1, 0, -1), & M(e_1 - e_3) &= (1, 0, 0), \\\\\n\t\tMe_2 &= (3, 2, 1), & e_1 - e_2 &= (1, -1, 0), & M(e_1 - e_2) &= (0, -1, 0), \\\\\n\t\tMe_3 &= (2, 1, 1), & e_2 - e_3 &= (0, 1, -1), & M(e_2 - e_3) &= (1, 1, 0).\n\t\\end{align*}\n\tHence, $\\cH$ is made of nine hyperplanes. Again by \\cref{lemma:hyperplanes},\n\twe need to consider vectors $z \\in \\Bigl(\\cD \\setminus \\pm \\transpose{M^{-1}}\n\t\\RR^3_{>0}\\Bigr) \\setminus \\{0\\}$. The relevant computations are in\n\t\\cref{table:CLL}, from which the result follows.\n\\end{proof}\n\n\n\\begin{table}[tp]\n\t\\[\n\t\t\\begin{array}{ccccccc}\n\t\t\t\\toprule\n\t\t\tu & v & z = u \\wedge v & \\transpose{M}z &\\|z\\|_D & \\|\\transpose{M}z\\|_D \\\\\n\t\t\t\\midrule\n\t\t\tMe_1 & Me_2 \t\t& (-1, 0, 3) \t& (0, 0, 1) \t& 4 & 1 \\\\\n\t\t\tMe_1 & Me_3 \t\t& (0, -1, -1) \t& (0, -1, 0) \t& 2 & 1 \\\\\n\t\t\tMe_1 & e_1 - e_3 \t& (-1, 4, -1) \t& (0, 4, 1) \t& 5 & 4 \\\\\n\t\t\tMe_1 & e_1 - e_2 \t& (1, 1, -4) \t& (0, 1, -1) \t& 5 & 2 \\\\\n\t\t\tMe_1 & e_2 - e_3 \t& (-2, 3, 3) \t& (0, 3, 2) \t& 5 & 3 \\\\\n\t\t\tMe_1 & M(e_1 - e_2) & (1, 0, -3) \t& (0, 0, -1) \t& 4 & 1 \\\\\n\t\t\tMe_1 & M(e_2 - e_3) & (-1, 1, 2) \t& (0, 1, 1) \t& 3 & 1 \\\\\n\t\t\tMe_1 & M(e_1 - e_3) & (0, 1, -1) \t& (0, 1, 0) \t& 2 & 1 \\\\\n\t\t\tMe_2 & Me_3 \t\t& (1, -1, -1) \t& (1, 0, 0) \t& 2 & 1 \\\\\n\t\t\tMe_2 & e_1 - e_3 \t& (-2, 4, -2) \t& (-4, 0, 2) \t& 6 & 4 \\\\\n\t\t\tMe_2 & e_1 - e_2 \t& (1, 1, -5) \t& (-1, 0, -2) \t& 6 & 2 \\\\\n\t\t\tMe_2 & e_2 - e_3 \t& (-3, 3, 3) \t& (-3, 0, 0) \t& 6 & 3 \\\\\n\t\t\tMe_2 & M(e_1 - e_2) & (1, 0, -3)\t& (0, 0, -1) \t& 4 & 1 \\\\\n\t\t\tMe_2 & M(e_2 - e_3) & (-1, 1, 1) \t& (-1, 0, 0) \t& 2 & 1 \\\\\n\t\t\tMe_2 & M(e_1 - e_3) & (0, 1, -2) \t& (-1, 0, -1) \t& 3 & 1 \\\\\n\t\t\tMe_3 & e_1 - e_3 \t& (-1, 3, -1) \t& (-1, 2, 0) \t& 4 & 3 \\\\\n\t\t\tMe_3 & e_1 - e_2 \t& (1, 1, -3) \t& (1, 2, 0) \t& 4 & 2 \\\\\n\t\t\tMe_3 & e_2 - e_3 \t& (-2, 2, 2) \t& (-2, 0, 0) \t& 4 & 2 \\\\\n\t\t\tMe_3 & M(e_1 - e_2)\t& (1, 0, -2) \t& (1, 1, 0) \t& 3 & 1 \\\\\n\t\t\tMe_3 & M(e_2 - e_3) & (-1, 1, 1) \t& (-1, 0, 0) \t& 2 & 1 \\\\\n\t\t\tMe_3 & M(e_1 - e_3) & (0, 1, -1) \t& (0, 1, 0) \t& 2 & 1 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{array}\n\t\\]\n\t\\caption{The computations involved in \\cref{lemma:AC_ABC_B}. We only wrote\n\tthe values of $u$ and $v$ which yield a $z = u \\wedge v$ in\n\t$\\RR^3\\setminus\\transpose{M}^{-1}\\RR^3_{>0}$.}\\label{table:CLL}\n\\end{table}\n\nThe main technical result is the following\n\n\\begin{lemma}\\label{lemma:measure_cylinder}\n\tLet $\\mu$ the measure on $\\{1,2\\}^\\NN$ obtained in\n\t\\cref{thm:uniquelyergodic}. For every $\\epsilon > 0$, there exists an $N$\n\tsuch that, for every $n > N$ and $\\mu$-almost every sequences $(M_n)_{n \\in\n\t\\NN} \\in \\{A, B, C_A, C_B\\}^\\NN$, we have\n\t\\[\n\t\t\\Bigl\\| \\transpose{M}_{[0,n)}|_{f^\\perp}\\Bigr\\|_\\infty \n\t\t\\le (n+1) \\biggl(\\frac{4}{5}\\biggr)^{\\frac{1}{8}n(\\mu([112211221])-\\epsilon)-\\frac{1}{8}},\n\t\\]\n\twhere $\\bigcap_{n\\in\\NN} M_{[0,n)}\\RR^3_{\\ge 0} = \\RR_{\\ge 0} f$.\n\\end{lemma}\n\n\\begin{proof}\n\tWe begin by recalling that, by its construction, the measure $\\mu$ assigns\n\tpositive measure to every cylinder. In particular, $\\mu([1])$, $\\mu([2])>0$.\n\tBy ergodicity of $\\mu$, $\\mu$-almost every sequence of matrices\n\t$(M_n)_{n\\in\\NN}$ contains infinitely often each one of the matrices $A$,\n\t$B$, $C_A$ and $C_B$. Then, by \\cref{lemma:products_of_M}, there is an\n\tincreasing sequence $(n_i)_{i\\in\\NN}$ such that $n_0 = 0$ and\n\t\\[\n\t\tN_i = M_{[n_i, n_{i+1})} \\in \\{A^k C_A, B^k C_B: k\\in\\NN\\}\n\t\\]\n\tfor all $i$. For all positive $n$, there exists a unique $m\\in\\NN$ such that\n\t$n_m \\le n-1 < n_{m+1}$. Let $g = M_{[0,n)}^{-1} f$, then by\n\t\\cref{lemma:submultiplicative}, we obtain\n\t\\[\n\t\t\\begin{split}\n\t\t\t\\Bigl\\| \\transpose{M}_{[0,n)}|_{f^\\perp}\\Bigr\\|_\\infty &\\le \\Bigl\\| \\transpose{M}_{[0,n)}\\Bigr\\|_\\infty^{M_{[0,n)} \\RR^3_{\\ge 0}} \\\\\n\t\t\t\t&\\le \\Bigl\\| \\transpose{M}_{[n_m,n)}\\Bigr\\|_\\infty^{M_{[n_m, n)} \\RR^3_{\\ge 0}} \\cdot \\Bigl\\| \\transpose{M}_{[0,n_m)}\\Bigr\\|_\\infty^{M_{[0, n_m)} \\RR^3_{\\ge 0}}\\\\\n\t\t\t\t&\\le \\Bigl\\| \\transpose{M}_{[n_m,n)}\\Bigr\\|_\\infty \\cdot \\Bigl\\| \\transpose{M}_{[0,n_m)}\\Bigr\\|_\\infty^{M_{[0, n_m)} \\RR^3_{\\ge 0}}\n\t\t\\end{split}\n\t\\]\n\tSince $M_{[n_m,n)}$ is of the form\n\t\\[\n\t\tA^k =\n\t\t\\begin{pmatrix}\n\t\t\t1 & k & k \\\\\n\t\t\t0 & 1 & 0 \\\\\n\t\t\t0 & 0 & 1\n\t\t\\end{pmatrix},\n\t\t\\qquad\n\t\tB^k =\n\t\t\\begin{pmatrix}\n\t\t\t1 & 0 & 0 \\\\\n\t\t\tk & 1 & k \\\\\n\t\t\t0 & 0 & 1\n\t\t\\end{pmatrix},\n\t\\]\n\tfor some $k\\in\\NN$, and $\\Bigl\\| \\transpose{A^k}\\Bigr\\|_\\infty = \\Bigl\\|\n\t\\transpose{B^k}\\Bigr\\|_\\infty = k+1$, we have\n\t\\[\n\t\t\\Bigl\\| \\transpose{M}_{[n_m,n)}|_{f^\\perp}\\Bigr\\|_\\infty \\le n - n_m + 1 \\le n + 1.\n\t\\]\n\n\tNow, we deal with the second term: $\\Bigl\\|\n\t\\transpose{M}_{[0,n_m)}\\Bigr\\|_\\infty^{M_{[0, n_m)} \\RR^3_{\\ge\n\t0}}$, with $M_{[0,n_m)} = \\prod_{i=0}^{m-1} N_i = N_{[0,m)}$. Let $J_m$ be\n\tthe set of indices $j\\in \\{0, 1, \\dotsc, n_m - 8\\}$ such that $M_{[j,j+8)} =(A C_A B C_B)^2$.\n\tCall $J'_m \\subseteq J_m$ a subset of maximal cardinality such that\n\t\\begin{equation}\\label{eq:estimate_on_Jm}\n\t\t\\min ((J'_m - J'_m) \\cap \\NN_{>0}) \\ge 8.\n\t\\end{equation}\n\tWe remark that $\\# J'_m \\ge \\frac{1}{8} \\# J_m$. Now, if $j\\in J'_m$ there\n\texists a unique $i = i(j)\\in\\NN$ such that $n_{i} \\in\\{j, j+1, j+2\\}$ and\n\t$N_i N_{i+1} \\in \\{AC_A, BC_B\\}$. In particular, if $j, j' \\in J'_m$ with\n\t$j\\neq j'$, then $|i(j') - i(j)| \\ge 2$, thanks\n\tto~\\eqref{eq:estimate_on_Jm}. Denote $I_m = \\{i(j), j\\in J'_m\\}$, so $\\# I_m = \\# J'_m$.\n\n\tUsing \\cref{lemma:submultiplicative} recursively\n\ttogether with~\\eqref{eq:Dnorm_vs_infinitynorm}, \\cref{lemma:norm_one} and\n\t\\cref{lemma:AC_ABC_B} we obtain\n\t\\[\n\t\t\\begin{split}\n\t\t\t\\Bigl\\|\\transpose{M}_{[0,n_m)}\\Bigr\\|_\\infty^{M_{[0, n_m)} \\RR^3_{\\ge 0}} &=\n\t\t\t\t\\Bigl\\|\\transpose{N}_{[0,m)}\\Bigr\\|_\\infty^{N_{[0, m)} \\RR^3_{\\ge 0}} \n\t\t\t\t\\le 2\t\\Bigl\\|\\transpose{N}_{[0,m)}\\Bigr\\|_D^{N_{[0, m)} \\RR^3_{\\ge 0}} \\\\\n\t\t\t\t&\\le 2 \\prod_{i\\in I_m} \\Bigl\\|\\transpose{(N_i N_{i+1})}\\Bigr\\|_D^{N_i N_{i+1} \\RR^3_{\\ge 0}}\n\t\t\t\t\\cdot \\prod_{\\substack{i\\in{0,1,\\dotsc, m-1} \\\\ i\\notin I_m, i\\notin I_m+1}} \\Bigl\\|\\transpose{N_i}\\Bigr\\|_D^{N_i \\RR^3_{\\ge 0}}\\\\\n\t\t\t\t&\\le 2 \\biggl(\\frac{4}{5}\\biggr)^{\\# I_m}.\n\t\t\\end{split}\n\t\\]\n\n\tWe can now conclude the proof. Using Birkhoff ergodic theorem, for\n\t$\\mu$-almost every $x\\in\\{1,2\\}^\\NN$, we have\n\t\\[\n\t\t\\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{k=0}^{n-8} \\chi_{[112211221]} \\circ S^k (x) = \n\t\t\\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{k=0}^{n-1} \\chi_{[112211221]} \\circ S^k (x) = \n\t\t\\mu([112211221]).\n\t\\]\n\tHence, for $\\mu$-almost every $x\\in\\{1,2\\}^\\NN$, and for all $\\epsilon>0$,\n\tthere exists an $N$ such that, if $n>N$, then\n\t\\[\n\t\t\\Biggl| \\frac{1}{n} \\sum_{k=0}^{n-1} \\chi_{[112211221]} \\circ S^k (x) - \\mu([112211221])\\Biggr| < \\epsilon,\n\t\\]\n\twhich implies that\n\t\\[\n\t\t\\begin{split}\n\t\t\t\\# J_m &= \\sum_{k=0}^{n_m-8} \\chi_{[112211221]} \\circ S^k (x) \\\\\n\t\t\t &\\ge \\sum_{k=0}^{n-8} \\chi_{[112211221]} \\circ S^k (x) -1 \\\\\n\t\t\t &\\ge n(\\mu([112211221]) -\\epsilon) - 1,\n\t\t\\end{split}\n\t\\]\n\tand the proof is complete (we used the definition of the cylinder $[112211221]$ and the coefficients $n_m$ for the first ineuquality).\n\\end{proof}\n\nThe previous work allows us to prove the negativity of the second Lyapunov\nexponent for the MCF algorithm. \n\n\\begin{theorem}\\label{thm:no_weak_mixing}\t\n\tThe MCF algorithm defined by the renormalization algorithm has $\\mu$-almost\n\teverywhere \\emph{negative} second Lyapunov exponent. \n\\end{theorem}\n\n\\begin{proof}\n\tFrom the above discussion, using \\cref{lemma:measure_cylinder}, for\n\t$\\mu$-almost every word $w$ and every $\\epsilon>0$, we have that\n\t\\[\n\t\\begin{split}\n\t\t\\lambda_2 &= \\lim_{n\\to\\infty} \\frac{\\log{\\Bigl\\|\\transpose{X_n(w)}|_{f^\\perp}\\Bigr\\|}}{n} \\\\\n\t\t&\\le \\lim_{n\\to\\infty} \\frac{\\log{ \\Bigl( (n+1) \\bigl(\\frac{4}{5}\\bigr)^{\\frac{1}{8}n(\\mu([112211221])-\\epsilon)-\\frac{1}{8}}\\Bigr)}}{n} \\\\\n\t\t&= \\lim_{n\\to\\infty} \\frac{\\log{(n+1)} + \\bigl(\\frac{1}{8}n(\\mu([112211221])-\\epsilon)-\\frac{1}{8}\\bigr)\\log{\\bigl(\\frac{4}{5}\\bigr)}}{n} \\\\\n\t\t&< \\frac{1}{8}(\\mu([112211221])-\\epsilon)\\log{\\biggl(\\frac{4}{5}\\biggr)}\\\\\n\t\t&< 0,\n\t\\end{split}\n\t\\]\n\twhich proves the statement.\n\n\t\n\\end{proof}\n\n\n\n\\section{Hausdorff dimension estimates}\\label{sec:HausdorffEstimate}\nIn this section, we show that the Hausdorff dimension of the gasket defined by\nour MCF algorithm is the same as its affinity dimension, and prove estimates for\nit. Our approach follows the very recent papers~\\cite{Jurga:Gasket,\nJiaoLiPanXu:Applications}.\n\n\\subsection{Thermodynamic formalism}\\label{sec:thermodynamic_formalism}\nIn this subsection we recall some general definitions and results\nfrom~\\cite{Jurga:Gasket} that we will use. \n\nLet $\\bX$ be a set of matrices in $\\SL(3,\\RR)$. We say that $\\bX$ is\n\\emph{irreducible} if no proper linear subspace of $\\RR^3$ is preserved by all\nthe matrices in $\\bX$.\n\nGiven a matrix $X\\in\\SL(3,\\RR)$, let $\\alpha_1(X) \\ge \\alpha_2(X) \\ge\n\\alpha_3(X)$ be its singular values. Then, for $s\\ge 0$, the singular value\nfunction $\\phi^s\\colon\\SL(3,\\RR)\\to\\RR_+$ is defined as\n\\begin{equation}\\label{eq:singular_value_f}\n\t\\phi^s =\n\t\\begin{cases}\n\t\t\\Bigl(\\frac{\\alpha_2(X)}{\\alpha_1(X)}\\Bigr)^s, & \\text{if $0\\le s \\le 1$},\\\\\t\n\t\t\\frac{\\alpha_2(X)}{\\alpha_1(X)}\\Bigl(\\frac{\\alpha_3(X)}{\\alpha_1(X)}\\Bigr)^{s-1}, & \\text{if $1\\le s \\le 2$},\\\\\n\t\t\\Bigl(\\frac{\\alpha_2(X)\\alpha_3(X)}{\\alpha_1^2(X)}\\Bigr)^{s-1}, & \\text{if $s \\ge 2$}.\n\t\\end{cases}\n\\end{equation}\n\nThe irreducibility of $\\bX$ implies that the function $\\phi$ is\nquasimultiplicative on $\\bX$, see~\\cite[Section~2.2]{Jurga:Gasket} for details.\n\nWe now define the zeta function $\\zeta_\\bX\\colon [0,\\infty)\\to[0,\\infty]$ by\n\\[\n\t\\zeta_\\bX (s) = \\sum_{n=1}^{\\infty} \\sum_{X\\in\\bX^n} \\phi^s(X).\n\\]\nFinally, the \\emph{affinity dimension} $s_\\bX$ is the critical exponent of the above\nseries:\n\\begin{equation}\\label{eq:affinity_dim}\n\ts_\\bX = \\inf \\{s\\ge 0: \\zeta_\\bX(s)<\\infty\\}.\n\\end{equation}\n\nWe will use the following result, which uses the strong open set condition\n(SOSC), see, e.g.,~\\cite[Definition~2.5]{Jurga:Gasket}.\n\n\\begin{theorem}[\\protect{\\cite[Theorem~1.3]{Jurga:Gasket}}]\\label{thm:Jurga} \n\tSuppose a finite set $\\bX$ of positive matrices in $\\SL(3,\\RR)$ generates a \n\tsemigroup $S_\\bX$ which is Zariski dense in $\\SL(3,\\RR)$ and satisfies the\n\tSOSC. Then $\\dim_H K_\\bX = \\min \\{s_\\bX, 2\\}$.\n\\end{theorem}\n\nWe say that a set of positive matrices $\\bX = \\{X_1, \\dotsc, X_n\\}$ is\n\\emph{balanced} if there exists a $c>0$ such that, for all $i=1,\\dotsc,n$,\n\\[\n\t\\frac{\\min(X_i)_{j,k}}{\\max (X_i)_{j,k}} \\ge c.\n\\]\nThis implies that the singular value function $\\phi$ is almost-submultiplicative\non the set $\\bX$, see~\\cite[Proposition~2.1]{Jurga:Gasket}.\n\nFinally, we recall the definition of the pressure $P_\\bX\\colon[0,\\infty)\\to\\RR$,\ngiven by\n\t\\[\n\t\tP_\\bX(s) = \\lim_{n\\to\\infty} \\frac{1}{n} \\log {\\Biggl(\\sum_{X\\in\\bX^n} \\phi^s(X)\\Biggr)},\n\t\\]\nwhere the limit exists since the function $\\phi^s$ is almost-submultiplicative.\nOne can see that $P_\\bX$ is a continuous, strictly decreasing, convex function\nand that its unique root is exactly the affinity dimension $s_\\bX$.\n\n\\subsection{The Bruin-Troubetzkoy gasket}\n\nThe graph $\\rauzygraph$ in \\cref{fig:Rauzy_graph} can be\nencoded, using the alphabet $\\alphabet = \\{A, C_A, B, C_B\\}$ by the matrix\n\\[\n\tT = (t_{ij}) = \\begin{pmatrix}\n\t\t1 & 1 & 0 & 0 \\\\\n\t\t0 & 0 & 1 & 1 \\\\\n\t\t0 & 0 & 1 & 1 \\\\\n\t\t1 & 1 & 0 & 0\n\t\\end{pmatrix}.\n\\]\nConsidering the paths on the graph $\\rauzygraph$ we define a topological Markov\nshift, which naturally we call the ARC topological Markov shift. Then, the set\nof (one-sided) infinite paths, starting from the vertex $1 = (1, 2, 3)$, on the\ngraph corresponds to the set\n\\[\n\t\\sW = \\{w = (w_i)_{i=1}^\\infty: w_1 = A, C_A, w_i \\in\\alphabet, t_{w_i w_{i+1}} = 1, \\text{ for all } i\\ge 1\\}.\n\\]\nA word of length $n$, $w = w_1 w_2 \\cdots w_n$, is \\emph{admissible} if $w_1\n= A, C_A$ and \n\\[\n\tt_{w_1w_2} t_{w_2w_3} \\cdots t_{w_{n-1}w_n} = 1.\n\\]\nThe set of admissible words of length $n$, which corresponds to the set of $n$\nlength path on the graph $\\rauzygraph$, is denoted $\\sW_n$. \n\nWe recall that in \\cref{sec:ARC_MCF} we defined the Arnoux-Rauzy-Cassaigne MCF\nalgorithm. Bearing in mind this algorithm, the Bruin-Troubetzkoy gasket\n$\\gasket$ is\n\\[\n\t\\gasket = \\bigcup_{w\\in\\sW} \\bigcap_{n=1}^\\infty f_{w_1} \\circ \\cdots \\circ f_{w_n} (\\Delta).\n\\]\nIn other words, it is the \\emph{attractor} of the iterated function system\ndriven by the paths on the graph $\\rauzygraph$, using the transformations\nin~\\eqref{eq:ARC_IFS}.\n\nIn the following, we will identity the alphabet $\\alphabet$ with the set of\nmatrices bearing the same names. It is easy to see that the set of matrices\n$\\alphabet$ is irreducible. Then, we can use the results of the previous\nsection.\n\n\n\\subsection{Equality between Hausdorff dimension and affinity dimension}\nThe main result of this section is the following\n\n\\begin{theorem}\\label{thm:dimH_eq_affdim}\n\tThe Hausdorff dimension of the Bruin-Troubetzkoy gasket $\\gasket$ is\n\tequal to its affinity dimension. That is\n\t\\[\n\t\t\\dim_H \\gasket = s_{\\alphabet} = \\inf \\Biggl\\{\n\t\ts\\ge 0: \\sum_{n=1}^{\\infty} \\sum_{w\\in\\sW_n} \\frac{\\alpha_2(w)}{\\alpha_1(w)}\n\t\t\t\\left(\\frac{\\alpha_3(w)}{\\alpha_1(w)}\\right)^{s-1}< \\infty\n\t\t\\Biggr\\}.\n\t\\]\n\\end{theorem}\n\nWe will closely follow the strategy used for the analogous result for the Rauzy\ngasket in the paper~\\cite{Jurga:Gasket}, see her Theorem~1.1.\n\nLet\n\\[\n\t\\Gamma = \\{A^n C_A C_B, C_A B^n C_B, (C_A C_B)^n A\\}_{n \\ge 1} \\subset \\SL(3,\\RR)\n\\]\nand $\\Gamma_N$ the $N$-th truncation of the set:\n\\[\n\t\\Gamma_N = \\{A^n C_A C_B, C_A B^n C_B, (C_A C_B)^n A\\}_{1 \\le n \\le N}.\n\\]\nWe will denote by $S_\\Gamma$ and $S_{\\Gamma_N}$ the semigroups generated by\n$\\Gamma$ and $\\Gamma_N$ respectively. The results recalled in\n\\cref{sec:thermodynamic_formalism}, applied to these semigroups allow us to\ndefine their affinity dimension as in~\\eqref{eq:affinity_dim}. We will denote by\n$s_\\Gamma$ and $s_{\\Gamma_N}$ the affinity dimensions of the semigroups $\\Gamma$\nand $\\Gamma_N$ respectively.\n\n\\begin{lemma}\\label{lemma:excluding_constants}\n\tIf $K_\\Gamma$ denotes the projective limit set of $S_\\Gamma$, then\n\t$\\gasket\\setminus K_\\Gamma$ is countable.\n\\end{lemma}\n\n\\begin{proof}\n\tThe only words on $\\alphabet$ that appear in $\\sW$ but do not appear as\n\tcombinations of elements in $\\Gamma$ are the ones which are eventually\n\tconstantly equal to either $A$, $B$ or $C_A C_B$. Since this set is\n\tcountable, we are done.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:simultaneous_conjugacy}\n\tThe matrices in $\\Gamma$ can be simultaneously conjugated to a set of\n\tbalanced matrices.\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by observing that the matrices in $\\Gamma$ are non-negative. \n\tConsider the matrix\n\t\\[\n\t\tM_\\epsilon = \\begin{pmatrix}\n\t\t\t1 \t\t\t& -\\epsilon & -\\epsilon \\\\\n\t\t\t-\\epsilon \t& 1 \t\t& -\\epsilon \\\\\n\t\t\t-\\epsilon \t& -\\epsilon & 1\n\t\t\\end{pmatrix},\n\t\\]\n\tfor some sufficiently small $\\epsilon$. For instance, $\\epsilon \\le\n\t\\frac{1}{5}$ is enough.\n\n\tA direct computation shows that the entries grow linearly with $n$. This\n\timplies that we can find two constants $0< c_1 < c_2 < \\infty$, which depend\n\tonly on $\\Gamma$ and $\\epsilon$, such that, for all $X\\in\\Gamma$ we have\n\tthat $M_\\epsilon^{-1} X M_\\epsilon = X'$ satisfies that $c_1 \\le\n\t\\frac{X'}{n} \\le c_2$. \n\t\n\tFor concreteness, let us compute $A^n C_A C_B M_\\epsilon$, we have\n\t\\[\n\t\tA^n C_A C_B M_\\epsilon = \\begin{pmatrix}\n\t\t\tn+1-3n\\epsilon & 2n-(2n+1)\\epsilon & n-(3n+1)\\epsilon \\\\\n\t\t\t-\\epsilon \t\t& 1\t\t\t\t\t& -\\epsilon \\\\\n\t\t\t1-2\\epsilon\t\t& 1-2\\epsilon\t\t& 1-2\\epsilon\n\t\t\\end{pmatrix}.\n\t\\]\n\tMultiplying by the matrix $M_\\epsilon^{-1}$ we see that all the entries,\n\tonce we divided by $n$, are bounded from above and below. Repeating the\n\tcomputation for the matrices $C_A B^n C_B$ and $(C_A C_B)^n A$, we find the\n\tconstants $c_1$ and $c_2$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:sup_affinity_exp}\n\tWe have that $\\sup_N s_{\\Gamma_N} = s_\\Gamma$.\n\\end{corollary}\n\n\n\\begin{proof}\n\tSince $\\Gamma_N \\subseteq \\Gamma$, the exponents satisfy $s_{\\Gamma_N} \\le\n\ts_\\Gamma$. Let $s < s_\\Gamma$. By \\cref{prop:simultaneous_conjugacy}, the\n\tfunction $\\phi^s$ is almost-submultiplicative on $S_\\Gamma$ and $S_N$. Then,\n\t$s_\\Gamma$ and $s_{\\Gamma_N}$ are the unique zeros of the respective\n\tpressure functions $P_\\Gamma$ and $P_{\\Gamma_N}$. \n\t\n\tSince $\\phi^s$ is\n\tquasimultiplicative,~\\cite[Proposition~3.2]{KaenmakiReeve} ensures that\n\t$0<P_\\Gamma (s) = \\sup_N P_{\\Gamma_N}$, which implies that $P_{\\Gamma_N}>0$\n\tfor some $N$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:equality_affinity_exp}\n\tWe have that $s_\\alphabet = s_\\Gamma$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $S_\\alphabet$ be the semigroup generated by $\\sW$. Since $\\Gamma\n\t\\subseteq S_\\alphabet$ we have that $s_\\Gamma \\le s_\\alphabet$. We will now\n\tshow the other direction.\n\t\n\tWe have\n\t\\[\n\t\t\\begin{split}\n\t\t\t\\zeta_\\alphabet (s) &\\le \\zeta_\\Gamma (s) + \n\t\t\t\t\\sum_{n=1}^{\\infty}\\sum_{w\\in\\sW_n} \\sum_{k=1}^{\\infty} \\phi^s(w A^k) + \\phi^s(w B^k) + \\phi^s(w (C_AC_B)^k) \\\\\n\t\t\t\t\t\t\t\t&\\le C\\zeta_\\Gamma (s) + \n\t\t\t\t\\sum_{n=1}^{\\infty}\\sum_{w\\in\\sW_n} \\sum_{k=1}^{\\infty} \\phi^s(w A^k C_A C_B) + \\phi^s(w B^kC_B) + \\phi^s(w (C_AC_B)^k A) \\\\\n\t\t\t\t\t\t\t\t&\\le 2C\\zeta_\\Gamma(s),\n\t\t\\end{split}\t\n\t\\]\n\twhere we used that we can find a constant $C < \\infty$, that only depend on\n\tthe matrices in $\\alphabet$ and $s$, such that for all matrices\n\t$X\\in\\alphabet$ and $Y\\in\\SL(3,\\RR)$, we have $\\phi^s(Y) \\le C\\phi^s(YX)$,\n\tsee, e.g.,~\\cite[Lemma~1]{BochiGourmelon}. The above inequalities imply\n\t$s_\\alphabet \\le s_\\Gamma$ and we are done.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Zariski_density}\n\tFor all sufficiently large $N$, the subgroup $S_N$ generated by $\\Gamma_N$\n\tis Zariski dense in $\\SL(3,\\RR)$.\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by recalling that the Zariski closure of any subgroup is an\n\talgebraic group. Let $G$ be the Zariski closure of $S_\\Gamma$, and\n\t$\\mathfrak{g}$ its Lie algebra, which corresponds to the tangent space to\n\tthe identity. It is clear that $\\mathfrak{g} \\subseteq \\sl(3, \\RR)$, where\n\t$\\sl(3, \\RR)$ is the Lie algebra of the Lie group $\\SL(3,\\RR)$ is the\n\t$8$-dimensional algebra of the $3\\times 3$ matrices with zero trace and the\n\tusual matrix conmutador as Lie bracket: $[X,Y] = XY - YX$. We will show that\n\t$\\mathfrak{g} = \\sl(3, \\RR)$, by finding $8$ linearly independent matrices\n\tin $\\mathfrak{g}$.\n\n\tLet us consider the matrices\n\t\\[\n\t\tA^n (C_A C_B) = \\begin{pmatrix}\n\t\t\tn+1 & 2n & n \\\\\n\t\t\t0 & 1 & 1 \\\\\n\t\t\t1 & 1 & 1\n\t\t\\end{pmatrix},\n\t\\]\n\tfor any $n\\in\\NN$. Let $P$ be a polynomial that is zero on all the points of\n\t$S_\\Gamma$. Then we can form the real polynomial\n\t\\[\n\t\tq(x) = P \\left(\\begin{pmatrix}\n\t\t\tx+1 & 2x & x \\\\\n\t\t\t0 & 1 & 1 \\\\\n\t\t\t1 & 1 & 1\n\t\t\\end{pmatrix}\\right).\n\t\\]\n\tSince $q(n)=0$ for all $n\\in\\NN$, $q\\equiv 0$, which implies that the\n\tmatrices\n\t\\[\n\t\t\\gamma(x) = \n\t\t\\begin{pmatrix}\n\t\t\tx+1 & 2x & x \\\\\n\t\t\t0 & 1 & 1 \\\\\n\t\t\t1 & 1 & 1\n\t\t\\end{pmatrix}\n\t\\]\n\tform a curve inside the real algebraic group $G$. Then\n\t\\[\n\t\tX_1 = \\frac{d}{dx} \\gamma(0)^{-1}\\gamma(x)\\biggr|_{x=0} = \\begin{pmatrix}\n\t\t\t1 & 2 & 1 \\\\\n\t\t\t0 & 0 & 0 \\\\\n\t\t\t-1 & -2 & -1\n\t\t\\end{pmatrix}\\in\\mathfrak{g}.\n\t\\]\n\tSimilarly, by considering $C_A B^n C_B$ and $(C_A C_B)^n$ respectively, we\n\tobtain that\n\t\\[\n\t\tX_2 = \\begin{pmatrix}\n\t\t\t0 & 0 & 0 \\\\\n\t\t\t1 & 1 & 1 \\\\\n\t\t\t-1 & -1 & -1\n\t\t\\end{pmatrix}\n\t\t\\qquad\n\t\t\\text{and}\n\t\t\\qquad\n\t\tX_3 = \\begin{pmatrix}\n\t\t\t0 & 0 & 0 \\\\\n\t\t\t0 & 0 & 0 \\\\\n\t\t\t1 & 1 & 0\n\t\t\\end{pmatrix}\n\t\\]\n\tare in $\\mathfrak{g}$.\n\tWe now consider the conmutators\n\t\\[\n\t\t\\begin{split}\n\t\t\tX_4 &= [X_1, X_2] = \\begin{pmatrix}\n\t\t\t\t1 & 1 & 1 \\\\\n\t\t\t\t0 & 0 & 0 \\\\\n\t\t\t\t-1 & -1 & -1\n\t\t\t\\end{pmatrix},\n\t\t\t\\qquad\n\t\t\tX_5 = [X_1, X_3] = \\begin{pmatrix}\n\t\t\t\t1 & 1 & 0 \\\\\n\t\t\t\t0 & 0 & 0 \\\\\n\t\t\t\t-2 & -3 & -1\n\t\t\t\\end{pmatrix},\\\\\n\t\t\tX_6 &= [X_2, X_3] = \\begin{pmatrix}\n\t\t\t\t0 & 0 & 0 \\\\\n\t\t\t\t1 & 1 & 0 \\\\\n\t\t\t\t-2 & -2 & -1\n\t\t\t\\end{pmatrix},\n\t\t\t\\qquad\n\t\t\tX_7 = [X_3, X_4] = \\begin{pmatrix}\n\t\t\t\t-1 & -1 & 0 \\\\\n\t\t\t\t0 & 0 & 0 \\\\\n\t\t\t\t-2 & -2 & 1\n\t\t\t\\end{pmatrix},\\\\\n\t\t\tX_8 &= [X_2, X_5] = \\begin{pmatrix}\n\t\t\t\t-1 & -1 & -1 \\\\\n\t\t\t\t-1 & -2 & 0 \\\\\n\t\t\t\t4 & 5 & 3\n\t\t\t\\end{pmatrix}.\n\t\t\\end{split}\n\t\\]\n\n\tIt can be checked that the set $\\{X_i\\}_{i=1}^8$ is a linearly independent\n\tsubset of $\\mathfrak{g}$, and hence $\\mathfrak{g}=\\sl(3,\\RR)$. Thus,\n\t$S_\\Gamma$ is Zariski dense inside $\\SL(3,\\RR)$.\n\n\tTo conclude the proof, we remark that, since $S_\\Gamma$ is a subsemigroup of\n\t$S_\\alphabet$, the latter is also Zariski dense inside $\\SL(3,\\RR)$. Since\n\t$\\SL(3,\\RR)$ is a (Zariski) closed and connected subgroup of $\\GL(3,\\RR)$,\n\tdensity of $\\Gamma_N$ for sufficiently large $N$ follows\n\tfrom~\\cite[Lemma~3.7]{MorrisSert:Variational}.\n\\end{proof}\n\nWe can now prove the main result of this section.\n\n\\begin{proof}[Proof of \\cref{thm:dimH_eq_affdim}]\n\tFrom \\cref{prop:simultaneous_conjugacy} and \\cref{prop:Zariski_density}, for\n\tsufficiently large $N$ we have can simultaneously conjugate every $\\Gamma_N$\n\tto a subset of \\emph{positive} matrices in $\\SL(3,\\RR)$ which satisfies the\n\tSOSC and that generate a Zariski dense subgroup of $\\SL(3,\\RR)$. Then,\n\t\\cref{thm:Jurga}, together with \\cref{cor:sup_affinity_exp} and\n\t\\cref{lemma:equality_affinity_exp} yield\n\t\\[\n\t\t\\dim_H \\gasket \\ge \\sup_N \\dim_H K_{\\Gamma_N} = \\sup_N s_{\\Gamma_N} = s_\\Gamma = s_\\alphabet.\n\t\\]\n\n\tLet us show the reverse inequality. Since $\\gasket \\setminus K_\\Gamma$ is\n\tcountable, we have that $\\dim_H \\gasket = \\dim_H K_\\Gamma$. By\n\t\\cref{prop:simultaneous_conjugacy} we can simultaneously conjugate $\\Gamma$\n\tto $\\Gamma_\\epsilon$, a set of positive matrices in $\\SL(3,\\RR)$. These\n\tmatrices send the positive cone into a compact subset of itself, so one can\n\treason as in~\\cite[Section~6.1.2]{Jurga:Gasket} to show that $\\dim_H\n\tK_\\Gamma \\le s_\\Gamma$. Finally, by \\cref{lemma:equality_affinity_exp}, we\n\thave $s_\\Gamma = s_\\alphabet$. Then, $\\dim_H \\gasket \\le s_\\alphabet$.\n\\end{proof}\n\n\\subsection{A lower bound on $\\dim_H \\gasket$.}\nWe recall that we denote by $\\cE = \\{e_1, e_2, e_3\\}$ the standard base of\n$\\RR^3$. The corresponding elements in $\\PP(\\RR^3)$ will be denoted by $E_i =\n\\RR e_i$. In this section, it will be more convenient to use a different set of\ngenerators for the semigroup $S_\\Gamma$. Let \n\\[\n\tD_1 = A, \\quad D_2^n = C_A B^n C_B, \\quad D_3 = C_A C_B = \\begin{pmatrix}\n\t\t1 & 0 & 0 \\\\\n\t\t0 & 1 & 0 \\\\\n\t\t1 & 1 & 1 \n\t\\end{pmatrix},\n\\]\nfor any $n \\ge 1$. Then, $\\{D_1, D_2^n, D_3, n\\in\\NN\\}$ still generates the\nsemigroup $s_\\Gamma$. Since we will also use their transpose, we list them,\nto help the reader:\n\\[\n\t\\transpose{D_1} = \\begin{pmatrix}\n\t\t1 & 0 & 0 \\\\\n\t\t1 & 1 & 0 \\\\\n\t\t1 & 0 & 1\n\t\\end{pmatrix},\n\t\\quad\n\t\\transpose{D_2^n} = \\begin{pmatrix}\n\t\t1 & n & 1 \\\\\n\t\t0 & n+1 & 1 \\\\\n\t\t0 & n & 1\n\t\\end{pmatrix},\n\t\\quad\n\t\\transpose{D_3} = \\begin{pmatrix}\n\t\t1 & 0 & 1 \\\\\n\t\t0 & 1 & 1 \\\\\n\t\t0 & 0 & 1\n\t\\end{pmatrix}.\n\\]\n\nThe following result follows from direct computations and will be left to the\nreader.\n\n\\begin{lemma}\\label{lemma:deltaprime}\n\tThe matrices $\\{D_1, D_2^n, D_3\\}$ and their transpose preserve the simplex\n\t$\\Delta$. Moreover, $\\Bigl\\{\\transpose{D_1}, \\transpose{D_2^n},\n\t\\transpose{D_3}\\Bigr\\}$ also preserves the open sub-simplex $\\Delta'$ with\n\tvertices $(0:1:1)$, $(1:0:1)$ and $(1:1:0)$.\n\\end{lemma}\n\nFrom this, we obtain the following useful corollary. Let us introduce some\nnotation we will need. Let $\\gamma = \\gamma_1 \\gamma_2 \\cdots \\gamma_k \\in\nS_\\Gamma$, then $\\transpose{\\gamma} = \\transpose{\\gamma_k} \\cdots\n\\transpose{\\gamma_2} \\transpose{\\gamma_1}$. Moreover, given $1\\le m \\le k$, we\ndenote $\\transpose{\\gamma}_{[1,m)} = \\transpose{\\gamma_k} \\cdots\n\\transpose{\\gamma_{k-m+1}}$. We stress that, in the previous notation, we first\ntake the transpose and then cut the product after $m$ terms. \n\n\\begin{corollary}\\label{cor:insidedeltaprime}\n\tFor any $i=1,2,3$, for any $k \\ge 1$ and any $\\gamma\\in S_\\Gamma$ of length \n\t$k$, if $\\gamma_j = i$ for some $j$, then $\\transpose{\\gamma} E_i \\in \\Delta'$.\n\\end{corollary}\n\n\\begin{proof}\n\tA direct computation shows that\n\t\\[\n\t\t\\transpose{D_1} E_1 = \\transpose{D_3} E_3 = (1:1:1)\\in\\Delta',\n\t\\]\n\tand\n\t\\[\n\t\t\\transpose{D_2^n} E_2 = (n:n+1:n)\\in\\Delta',\n\t\\]\n\tas we wanted.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:gammainDelta}\n\tLet $\\gamma\\in S_\\Gamma$, and assume that its last $m$ letters are\n\tnot the same. Then, $\\transpose{\\gamma}\\gamma \\Delta \\subset \\transpose{\\gamma}_{[1,m)}\n\t\\Delta \\cap \\overline{\\Delta'}$.\n\\end{proposition}\n\n\\begin{proof}\n\tThe inclusion $\\transpose{\\gamma}\\gamma \\Delta \\subset \\transpose{\\gamma}_{[1,m)} \\Delta$\n\tholds trivially. Hence, we need to show that $\\transpose{\\gamma}\\gamma \\Delta \\subset\n\t\\Delta'$. If all the letters appear in $\\gamma$, then we can conclude by\n\t\\cref{cor:insidedeltaprime}.\n\t\n\tSince not all the last $m$ letters of $\\gamma$ are the same, we must have\n\tthat two among $\\{D_1, D_2^n, D_3\\}$ appear in the last $m$ digits of\n\t$\\gamma$. We consider each case separately,\n\n\t\\textbf{Case 1: only $D_1$ and $D_2^n$ appear.}\n\tIn this case, we actually have to distinguish whether $D_1$ or $D_2^n$\n\toccurs first. If we have $D_1 D_2^n$ inside $\\gamma$, then\n\t$\\transpose{\\gamma}$ contains $\\transpose{D_2^n} \\transpose{D_1}$. Since\n\t\\[\n\t\t\\transpose{D_2^n} \\transpose{D_1} = \\begin{pmatrix}\n\t\t\t1 & n & 1 \\\\\n\t\t\t0 & n+1 & 1 \\\\\n\t\t\t0 & n & 1\n\t\t\\end{pmatrix}\n\t\t\\begin{pmatrix}\n\t\t\t1 & 0 & 0 \\\\\n\t\t\t1 & 1 & 0 \\\\\n\t\t\t1 & 0 & 1\n\t\t\\end{pmatrix}\n\t\t=\n\t\t\\begin{pmatrix}\n\t\t\tn+2 & n & 1 \\\\\n\t\t\tn+2 & n+1 & 1 \\\\\n\t\t\tn+1 & n & 1\n\t\t\\end{pmatrix},\n\t\\]\n\twe have that $\\transpose{D_2^n} \\transpose{D_1} E_i \\in \\Delta'$ for\n\t$i=1,2,3$, as we wanted.\n\n\tSimilarly, if $D_2 D_1^n$ is contained inside $\\gamma$, then\n\t$\\transpose{\\gamma}$ contains $\\transpose{D_1} \\transpose{D_2^n}$. Since\n\t\\[\n\t\t\\transpose{D_1} \\transpose{D_2^n} = \\begin{pmatrix}\n\t\t\t1 & n & 1 \\\\\n\t\t\t1 & 2n+1 & 2 \\\\\n\t\t\t1 & 2n & 2\n\t\t\\end{pmatrix},\n\t\\]\n\twe have that $\\transpose{D_1} \\transpose{D_2^n} E_i \\in \\Delta'$ for\n\t$i=1,2,3$, and we are done with this case.\n\n\t\\textbf{Case 2: only $D_3$ and $D_2^n$ appear.}\n\tLet us define the set\n\t\\[\n\t\t\\nabla_x = \\{(x:y:z) \\in \\Delta: x \\le y+z\\}.\n\t\\]\n\tIt can be checked that this set is invariant under the action of $D_2^n$,\n\t$D_3$ and their transposes. Moreover, $D_2^n \\Delta \\subset \\nabla_x$ and\n\t$D_3 \\Delta \\subset \\nabla_x$. Hence, $\\transpose{\\gamma}\\gamma \\Delta\n\t\\subset \\nabla_x$. Now we compute\n\t\\[\n\t\t\\transpose{D_3} \\transpose{D_2^n} = \\begin{pmatrix}\n\t\t\t1 & 2n & 2 \\\\\n\t\t\t0 & 2n+1 & 2 \\\\\n\t\t\t0 & n & 1\n\t\t\\end{pmatrix}\n\t\t\\qquad\n\t\t\\text{and}\n\t\t\\qquad\n\t\t\\transpose{D_2^n} \\transpose{D_3} = \\begin{pmatrix}\n\t\t\t1 & n & n+2 \\\\\n\t\t\t0 & n+1 & n+2 \\\\\n\t\t\t0 & n & n+1\n\t\t\\end{pmatrix}.\n\t\\]\n\tWe observe that, in both cases, the last two columns belong to $\\Delta'$,\n\twhile the first coordinate is invariant. In other words,\n\t$\\transpose{\\gamma}_{[1,m)} E_2$, $\\transpose{\\gamma}_{[1,m)} E_3 \\in\n\t\\Delta'$, whereas $\\transpose{\\gamma}_{[1,m)} E_1 = E_1$. Finally we have\n\t$\\transpose{\\gamma}\\gamma \\Delta \\subset \\transpose{\\gamma}_{[1,m)} \\Delta\n\t\\cap \\nabla_x = \\transpose{\\gamma}_{[1,m)} \\Delta \\cap \\overline{\\Delta'}$,\n\tand we are done.\n\n\t\\textbf{Case 3: only $D_1$ and $D_2$ appear.}\n\tThis case can be treated as the previous one, replacing $D_2^n$ by $D_1$ and\n\t$\\nabla_x$ by $\\nabla_z$, which is defined analogously.\n\\end{proof}\n\nThe following is the key technical result of this section.\n\n\\begin{lemma}\\label{lemma:lastdigitsgamma}\n\tFor every $m\\in\\NN$, there exists an $\\epsilon_m>0$ such that, for all\n\t$\\gamma\\in\\Gamma$, if the last $m$ letters of $\\gamma$ are not the same,\n\tthen we have\n\t\\[\n\t\t\\| \\gamma e_i \\| \\ge \\epsilon_m \\alpha_1(\\gamma),\n\t\\] \n\tfor $i=1,2,3$.\n\\end{lemma}\n\n\\begin{proof}\n\tUsing the $KA^+K$ Cartan decomposition of $\\SL(3,\\RR)$, we can write every\n\tmatrix $X\\in\\SL(3,\\RR)$ as $\\tilde{k}_X a_X k_X$ where $\\tilde{k}_X$,\n\t$k_X\\in\\SO(3,\\RR)$ and $a_X$ is the diagonal matrix made by the singular\n\tvalues $\\alpha_1(X)\\ge\\alpha_2(X)\\ge\\alpha_3(X)$.\n\tBy~\\cite[Lemma~14.2]{BenoistQuint:RandomBook}, one has\n\t\\[\n\t\t\\|\\gamma e_i \\| \\ge \\|\\gamma\\| d(E_i, H_\\gamma),\n\t\\]\n\twhere $H_\\gamma = k_\\gamma^{-1} (E_i^\\perp)$ is a repelling\n\thyperplane for $\\gamma$ and\n\t\\[\n\t\td(\\RR v, \\RR w) = \\frac{\\|v \\wedge w \\|}{\\|v\\| \\|w\\|},\n\t\\]\n\twith $\\|\\cdot\\|$ the standard Euclidean norm on $\\RR^3$ and the induced one\n\ton $\\wedge^2 \\RR^3$.\n\n\tOne can check that $(\\tilde{k}_\\gamma E_i)^\\perp =\n\t(V_{\\transpose{\\gamma}})^\\perp = H_\\gamma$. Hence, it is enough to\n\tcheck that the angle between $E_i$ and $V_{\\transpose{\\gamma}}$ is\n\tbounded away from $\\frac{\\pi}{2}$. Since $E_i^\\perp$ is the span of $E_j$,\n\tfor $j\\neq i$, which is an edge of the simplex $\\Delta$, it is enough to\n\tshow that $d(V_{\\transpose{\\gamma}}, \\partial\\Delta)$ is bounded from\n\tbelow by a constant that only depends on $m$, not on $\\gamma$. By\n\tdefinition, $V_{\\transpose{\\gamma}}$, is the attracting fixed point of\n\t$\\transpose{\\gamma}\\gamma$. In particular,\n\t$V_{\\transpose{\\gamma}}\\in\\transpose{\\gamma}\\gamma\\Delta$.\n\n\tBy \\cref{prop:gammainDelta}, we have that $\\transpose{\\gamma}\\gamma \\Delta\n\t\\subset \\transpose{\\gamma}_{[1,m)} \\Delta \\cap \\overline{\\Delta'}$. We\n\tremark that $\\transpose{\\gamma}_{[1,m)} \\Delta \\cap \\overline{\\Delta'}$ is a\n\tquadrilateral which does not intersect the boundary of the simplex. Thus,\n\t$d(\\partial\\Delta, \\transpose{\\gamma}_{[1,m)} \\Delta \\cap\n\t\\overline{\\Delta'}) > 0$. Since, for any given $m$, there exists only\n\tfinitely many $\\gamma$ of length $m$, we can find a $d_m>0$ such that\n\t\\[\n\t\td(V_{\\transpose{\\gamma}}, \\partial\\Delta) \\ge d\\Bigl(\\transpose{\\gamma}\\gamma\\Delta, \\partial\\Delta\\Bigr)\n\t\t\t\\ge d\\Bigl(\\transpose{\\gamma}_{[1,m)} \\Delta \\cap \\overline{\\Delta'}, \\partial\\Delta\\Bigr) > d_m,\n\t\\]\n\twhich completes the proof.\n\\end{proof}\n\nWe will need an estimation of the distortion of the simplex $\\Delta$ by an\nelement $\\gamma\\in\\Gamma$.\n\n\\begin{lemma}\\label{lemma:Distortion}\n\tAssume that the last two letters of $\\gamma$ are not the same. Then, there\n\texists a constant $C_2 > 1$ such that:\n\t\\begin{enumerate}\n\t\t\\item $\\diam(\\gamma\\Delta) \\le C_2\n\t\t\\frac{\\alpha_2(\\gamma)}{\\alpha_1(\\gamma)}$.\n\t\t\\item $\\area(\\gamma\\Delta) \\le C_2 \\alpha_1(\\gamma)^{-3}$.\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\tWe begin with the first point. It is enough to show that $d(\\gamma E_i,\n\t\\gamma E_j) \\le C_2 \\frac{\\alpha_2(\\gamma)}{\\alpha_1(\\gamma)}$, for\n\t$i, j = 1, 2, 3$.\n\tWe have that\n\t\\[\n\t\t\\begin{split}\n\t\t\td(\\gamma E_i, \\gamma E_j) &= \\frac{\\|\\gamma e_i \\wedge \\gamma e_j\\|}{\\|\\gamma e_i\\| \\|\\gamma e_j\\|} \\\\\n\t\t\t\t&\\le \\frac{\\alpha_1(\\gamma)\\alpha_2(\\gamma) \\|e_i \\wedge e_j \\|}{\\|\\gamma e_i\\| \\|\\gamma e_j\\|} \\\\\n\t\t\t\t&\\le \\frac{\\alpha_1(\\gamma)\\alpha_2(\\gamma)}{\\epsilon_2^2 (\\alpha_1(\\gamma))^2} \\\\\n\t\t\t\t&= \\epsilon_2^{-2}\\frac{\\alpha_2(\\gamma)}{\\alpha_1(\\gamma)},\n\t\t\\end{split}\n\t\\]\n\twhere we used \\cref{lemma:lastdigitsgamma} in the second inequality.\n\n\tTo prove the second point, we begin with the following elementary\n\tgeometrical fact. Let $x,y,z\\in\\RR^3\\setminus\\{0\\}$, then the area of the\n\ttriangle $\\stackrel{\\triangle}{xyz}$ with vertices $x$, $y$ and $z$ is given\n\tby\n\t\\[\n\t\t\\area(\\stackrel{\\triangle}{xyz}) = \\frac{\\|x \\wedge y \\wedge z\\|}{2d_E(0,\\stackrel{\\triangle}{xyz})},\n\t\\]\n\twhere $d_E$ is the distance from the origin to the plane containing the\n\tthree points. Slightly abusing the notation, we identify the projective\n\tsimplex $\\Delta$ with the ordinary $3$-simplex in $\\RR^3$. Let $x_\\gamma =\n\t\\gamma e_1$, $y_\\gamma = \\gamma e_2$ and $z_\\gamma = \\gamma e_3$ the three\n\tvertices of $\\gamma\\Delta$. By definition,\n\t\\[\n\t\tx_\\gamma = \\frac{\\gamma e_1}{\\|\\gamma e_1\\|_1},\n\t\\]\n\twhere $\\|\\cdot\\|_1$ is the $\\ell^1$-norm on $\\RR^3$, and similarly for the\n\tother points. Hence\n\t\\[\n\t\t\\|x_\\gamma \\wedge y_\\gamma \\wedge z_\\gamma \\| = \\frac{\\|\\gamma e_1 \\wedge \\gamma e_2 \\wedge \\gamma e_3\\|}{\\prod_{i=1}^3\\|\\gamma e_i \\|_1}\n\t\\]\n\tSince $\\|\\gamma e_1 \\wedge \\gamma e_2 \\wedge \\gamma e_3\\| = \\|e_1 \\wedge e_2\n\t\\wedge e_3\\| = 1$, and the entries of $\\gamma$ are non-negative, then\n\t$\\|\\gamma e_i \\|_1 \\ge \\|\\gamma e_i\\|$, so by \\cref{lemma:lastdigitsgamma}\n\twe are done.\n\\end{proof}\n\nWe will need the following geometrical result, which follows from the definition\nof Hausdorff dimension and was proven in~\\cite[Lemma~4.1]{PollicottSewell}.\n\n\\begin{lemma}\\label{lemma:PS} For every $\\delta > 0$, there exists a $C_\\delta >\n\t0$ such that, for all $\\gamma\\in\\Gamma$, there exists a finite open cover\n\t$\\{B_i(\\gamma)\\}_{i=1,\\dotsc,k}$ of $\\gamma\\Delta$ with\n\t$\\diam(B_i(\\gamma))\\le\\diam(\\gamma\\Delta)$ such that\n\t\\[\n\t\t\\sum_{i=1}^k \\diam^{1+\\delta} B_i(\\gamma) \\le \n\t\t\tc_\\delta \\cdot \\diam^{1-\\delta} \\gamma\\Delta \\cdot \\area^\\delta \\gamma\\Delta.\n\t\\]\n\\end{lemma}\n\nExactly as in \\cref{lemma:excluding_constants}, we can decompose the $\\gasket$\nas a set of \\emph{nice} points $\\gasket_{\\text{nice}}$ whose coding is not\neventually constant and a countable set. However, we remark that\n$\\gasket_{\\text{nice}} \\neq K_\\Gamma$, since we have switched the generating\nset.\n\nLet\n\\[\n\t\\Gamma^m = \\{\\gamma\\in\\Gamma: \\text{ the last two digits of $\\gamma$ are different and } \\diam\\gamma\\Delta \\le 1/m\\},\n\\]\nand consider the two families of coverings:\n\\[\n\t\\mathcal{U}_m = \\{B_i(\\gamma), \\gamma\\in\\Gamma^m\\}, \n\t\\quad \\text{and} \\quad\n\t\\mathcal{U}'_m = \\{\\gamma\\Delta, \\gamma\\in\\Gamma^m\\},\n\\]\nwith $B_i(\\gamma)$ defined by \\cref{lemma:PS}. We define\n\\[\n\tY = \\bigcap_{m=1}^\\infty \\bigcup_{U\\in\\mathcal{U}_m} U,\n\t\\quad \\text{and} \\quad\n\tY' = \\bigcap_{m=1}^\\infty \\bigcup_{U\\in\\mathcal{U}'_m} U.\n\\]\t\nThen $\\mathcal{U}_m$ is a Vitali cover of $Y$: for every $y\\in Y$ and every\n$\\delta>0$, there exists some $U\\in\\mathcal{U}_m$ such that $\\diam U < \\delta$\nand $y\\in U$. Similarly, $\\mathcal{U}'_m$ is a Vitali cover of $Y'$. Moreover,\nby construction $Y\\supset Y'$.\n\nWe have\n\n\\begin{lemma}\\label{lemma:nice_points}\n\tWe have the inclusion $\\gasket_{\\text{nice}} \\subset Y$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $x$ be a point in $\\gasket_{\\text{nice}}$. Then, its coding with respect\n\tto $\\{D_1, D_2, D_3\\}$ is not eventually constant. In particular, there are\n\tinfinitely many pairs of adjacent letters in its coding which are different\n\tone from the other. By diving the coding into subwords after each of these\n\tpairs appears, we form infinitely many words $\\gamma\\in\\Gamma$ whose two\n\tlast letters are different. Thus $x\\in\\gamma\\Delta$. Moreover, since\n\t$\\diam\\gamma\\Delta\\to 0$ as the length of $\\gamma$ increases, we see that\n\t$x\\in Y' \\subset Y$.\n\\end{proof}\n\nLet $\\Gamma_0 = \\langle D_1, D_3 \\rangle$ be the semigroup generated by the\nmatrices $D_1 = A$ and $D_3 = C_A C_B$. Then, the arc $I = I(E_1, E_3) = \\{\\RR(s\ne_1 + t e_3), s,t\\in\\RR_{\\ge 0}\\}$ is preserved by $\\Gamma_0$.\n\n\\begin{lemma}\n\tThere exists an $\\epsilon > 0$ such that, for all $\\gamma\\in\\Gamma_0$ having\n\tthe last two digits different from each other, we have\n\t\\[\n\t\t\\alpha_2(\\gamma) \\ge \\epsilon, \n\t\t\\qquad \\text{and} \\qquad\n\t\t\\epsilon |\\gamma I| \\le \\frac{1}{\\alpha_1(\\gamma)^2},\n\t\\]\n\twhere $|\\gamma I|$ is the length of the arc $\\gamma I$.\n\\end{lemma}\n\n\\begin{proof}\n\tSince the matrices $D_1$ and $D_3$ preserve $I$ and their restriction to the\n\tsubspace generated by $E_1$ and $E_3$ has determinant one, by\n\t\\cref{lemma:lastdigitsgamma}, we have\n\t\\[\n\t\t|\\gamma I| = d(\\gamma E_1, \\gamma E_3) = \n\t\t\t\\frac{\\|\\gamma e_1 \\wedge \\gamma e_3\\|}{\\|\\gamma e_1\\| \\|\\gamma e_3\\|} \n\t\t\t= \\frac{1}{\\|\\gamma e_1\\| \\|\\gamma e_3\\|} \\le \\frac{1}{\\alpha_1(\\gamma)^2}.\n\t\\]\n\tWe recall that the top singular value gives the operator norm of a matrix.\n\tHence, $\\alpha_1 (\\gamma) \\ge \\|\\gamma e_i \\|_1$ for $i=1$, $2$, $3$. So, as\n\tin the proof of \\cref{lemma:Distortion}, we obtain that\n\t\\[\n\t\t\\area(\\gamma\\Delta) = C_2 \\frac{\\|\\gamma e_1 \\wedge \\gamma e_2 \\wedge \\gamma e_3\\|}{\\prod_{i=1}^{3}\\|\\gamma e_i\\|_1} \n\t\t\t\\ge C_2 \\frac{1}{\\alpha_1(\\gamma)^3}.\n\t\\]\n\tCombining the last two inequalities, we obtain\n\t\\[\n\t\t\\max\\{|\\gamma I(E_1, E_2)|, |\\gamma I(E_2, E_3)|\\} \\ge \\frac{\\area(\\gamma\\Delta)}{|\\gamma I|} \\ge C_2 \\frac{1}{\\alpha_1(\\gamma)}.\n\t\\]\n\tFinally, using the first part of \\cref{lemma:Distortion}, we have\n\t\\[\n\t\t\\alpha_2(\\gamma) \\ge C_2 \\alpha_1 (\\gamma )\\diam(\\gamma\\Delta) \n\t\t\t\\ge C_2 \\alpha_1 (\\gamma ) \\max\\{|\\gamma I(E_1, E_2)|, |\\gamma I(E_2, E_3)|\\}\n\t\t\t\\ge C_2^2 \\frac{1}{\\alpha_1(\\gamma)}\n\t\\]\n\tand we are done.\n\\end{proof}\n\nWe are now ready to conclude and prove the main result of this section.\n\n\\begin{theorem}\n\tThe Hausdorff dimension of the Bruin-Troubetzkoy gasket $\\gasket$ is\n\tgreater than $3/2$:\n\t\\[\n\t\t\\dim_H \\gasket = s_\\alphabet \\ge \\frac{3}{2}.\n\t\\]\n\\end{theorem}\n\n\\begin{proof}\n\tLet $\\gamma\\in\\Gamma_0$ be an element whose last two digits are different,\n\tthe singular value function~\\eqref{eq:singular_value_f} we have\n\t\\[\n\t\t\\phi^{3/2} (\\gamma) = \\frac{\\alpha_2(\\gamma)}{\\alpha_1(\\gamma)} \\biggl(\\frac{\\alpha_3(\\gamma)}{\\alpha_1(\\gamma)}\\biggr)^{1/2}\n\t\t= \\frac{\\alpha_2(\\gamma)^{1/2}}{\\alpha_1(\\gamma)^2} \\ge \\frac{\\epsilon^{1/2}}{\\alpha_1(\\gamma)^2} \\ge \\epsilon^2 |\\gamma I|.\n\t\\]\n\tThen\n\t\\begin{multline*}\n\t\t\\sum_{n=1}^{\\infty} \\sum_{w\\in\\sW_n} \\phi^{3/2}(w) \n\t\t\\ge \\sum_{\\gamma\\in\\Gamma_0} \\phi^{3/2}\n\t\t\\ge \\sum_{\\substack{\\gamma\\in\\Gamma_0 \\\\ \\text{last two digits different}}} \\phi^{3/2} \\\\\n\t\t\\ge \\epsilon^2 \\sum_{\\substack{\\gamma\\in\\Gamma_0 \\\\ \\text{last two digits different}}} |\\gamma I|.\n\t\\end{multline*}\n\n\tWe remark that, by \\cref{lemma:nice_points}, every point inside $\\gamma \\cap\n\t\\gasket_{\\text{nice}}$ is covered infinitely many times by\n\t\\[\n\t\t\\{\\gamma I, \\gamma\\in\\Gamma_0 \\text{ with the last two digits different from each other}\\}.\n\t\\]\n\tSince $I \\setminus (I\\cap \\gasket_{\\text{nice}})$ is countable, the series\n\t$\\ge \\sum_{\\gamma\\in\\Gamma_0} \\phi^{3/2}$ diverges and $\\dim_H \\gasket \\ge\n\t\\frac{3}{2}$, as we wanted.\n\\end{proof}\n\n\\printbibliography\n\\end{document}\n\n\\typeout{get arXiv to do 4 passes: Label(s) may have changed. 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Email: {\\tt twchao@mit.edu}}\n\n\\thanks{\\thankssymb{2}Department of Mathematics, Princeton University, Princeton, NJ 08544\\@. Email: {\\tt hansonyu@princeton.edu}}\n\n\n\\title{When entropy meets Tur\\'an: \\linebreak new proofs and hypergraph Tur\\'an results}\n\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nIn this paper, we provide a new proof of a density version of Tur\\'an's theorem.\nWe also rephrase both the theorem and the proof using entropy.\nWith the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius.\nIn addition, we also determine the Tur\\'an density for a new family of hypergraphs, which we call tents.\nOur result can be seen as a new generalization of Mubayi's result on the extended cliques.\n\\end{abstract}\n\n\\section{Introduction}\nFor any $k$-graph (i.e. $k$-uniform hypergraph) $F$, its \\emph{Tur\\'an number} $\\ex(n,F)$ is the maximum number of edges in an $F$-free $k$-graph $G$ on $n$ vertices.\nHere, $G$ is $F$-free if it contains no subgraph (not necessarily induced) isomorphic to $F$.\nThe study of Tur\\'an numbers was initiated by Tur\\'an \\cite{Turan85}, who first considered the case where $k=2$ and $F$ is the complete graph $K_{r+1}$ on $(r+1)$ vertices.\nThere, Tur\\'an showed that $\\ex(n,F)$ is maximized by the balanced complete $r$-partite graph $T_{n,r}$, which we now refer to as the Tur\\'an graph.\nTur\\'an's foundational work has motivated subsequent works on related problems, driving continuing research in extremal graph theory.\n\nThe general Tur\\'an problem is fairly understood when $k=2$.\nAlthough the exact value of $\\ex(n,F)$ is not known for general graphs $F$, the celebrated Erd\\H{o}s--Stone theorem asserts that $\\ex(n,F) = \\left(1-\\frac{1}{r}+o(1)\\right)\\binom{n}{2}$ if $\\chi(F) = r+1$, where $T_{n,r}$ is an asymptotic extremizer.\nIf we define the \\emph{Tur\\'an density} to be\n\\[\\pi(F) = \\lim_{n\\to\\infty}\\frac{\\ex(n,F)}{\\binom{n}{k}}\\]\nfor a $k$-graph $F$, then the Erd\\H{o}s--Stone theorem can be rephrased as $\\pi(F) = 1-\\frac{1}{\\chi(F)-1}$ when $F$ is a graph.\nIt is worth pointing out that when $\\chi(F)=2$, Erd\\H{o}s--Stone gives that $\\pi(F)=0$, showing that $\\ex(n,F)$ is subquadratic but does not determine the asymptotic behavior of $\\ex(n,F)$.\nDespite lots of effort, there are still many interesting open problems regarding the asymptotic behavior of $\\ex(n,F)$ when $F$ is bipartite.\nHowever, in this paper, we will focus on the non-degenerate case where $\\pi(F)>0$.\n\nGiven how much we know about Tur\\'an numbers and Tur\\'an densities of graphs, it might be surprising how little we know about hypergraph Tur\\'an problems.\nIn fact, the exact value of $\\pi(F)$ is still unknown even for $F=K_4^{(3)}$, the $3$-uniform clique on $4$ vertices.\nTur\\'an showed that $\\pi(K_4^{(3)})\\geq \\frac{5}{9}$ and conjectured that it is actually an equality.\nHowever, proving this conjecture still seems hard to date, and the current best upper bound $\\pi(F)\\leq 0.561666$ was obtained by Razborov \\cite{Raz10} using flag-algebraic computation, which was later verified by \\cite{BT11} and \\cite{F-RV13}.\nThe difficulty comes from the fact that hypergraph Tur\\'an problems have drastically different behaviors from the graph case.\nFor example, there is a large family of constructions all showing $\\pi(K_4^{(3)})\\geq \\frac{5}{9}$ given in \\cite{Kos82} (also see \\cite{F-D-F88}).\nIn comparison, the Erd\\H{o}s--Simonovits theorem states that any asymptotic extremizer of $\\pi(K_{r+1})$ should be close to $T_{n,r}$.\nWe will discuss other interesting phenomena for hypergraph Tur\\'an problems in \\cref{subsec:hypergraph-turan-density}.\n\nThe aim of this paper is to find inspiration for new ways to approach hypergraph Tur\\'an problems by examining our new proof of the density Tur\\'an theorem, i.e. $\\pi(K_{r+1}) = 1-\\frac{1}{r}$.\nThis leads to new hypergraph Tur\\'an results regarding hypergraphs that we call ``tents'', which generalizes Mubayi's result \\cite{Mub06} on the extended cliques.\nWe will introduce our results and related work in more detail in \\cref{subsec:hypergraph-turan-density}.\n\nBefore diving into hypergraph Tur\\'an problems, we will first give a quick overview of known proofs of Tur\\'an's theorem.\nWe will then introduce the entropy method, which we use to rephrase both the theorem statement and our proof.\nThen we will mention our hypergraph Tur\\'an results that can be obtained using the new perspective, which can be thought of as one of our main results.\n\n\\subsection{Proofs of Tur\\'an's theorem}\nTur\\'an's original proof \\cite{Turan85} works by a clever induction on the number of vertices by removing a $K_r$ from the graph.\nErd\\H{o}s \\cite{Erdos70} later provided another proof that modified the graph step by step, maintaining the $K_{r+1}$-freeness and making the graph complete multipartite at the end.\nThis method has the benefit that it is easier to see that the Tur\\'an graph $T_{n,r}$ is the extremizer.\nA proof of the same spirit is a folklore proof that proceeds with symmetrization (also known now as Zykov Symmetrization as this trick was used by Zykov \\cite{Zyk49,Zyk52} in his work).\nThe proof modifies the graph by taking two non-adjacent vertices, and replacing one with another (see \\cite[Chapter 41]{AZ18}).\nUnfortunately, all those proofs do not easily generalize to hypergraphs as they all use properties of graphs crucially.\n\nOne proof that looks entirely different from the previous proofs is by applying the Caro--Wei theorem, which is due to Alon and Spencer \\cite{AS00}.\nThe Caro--Wei theorem, independently proven by Caro \\cite{Caro79} and Wei \\cite{Wei81}, gives a lower bound on the independence number of a graph $G$ based on its degree sequence.\nThe standard proof of the Caro--Wei theorem is a nice probabilistic argument, which can be found in \\cite{AS00}.\nBy taking the complement and an application of Cauchy--Schwarz, the density Tur\\'an theorem immediately follows from Caro--Wei.\nHowever, this argument does not generalize well to higher uniformities---although the Caro--Wei theorem can be extended to hypergraphs (see \\cite{CT91}), applying the inequality on the complement no longer gives tight hypergraph Tur\\'an results.\n\nAnother proof that is seemingly different from all the above is a proof due to Motzkin and Straus \\cite{MS65}.\nTheir proof relies crucially on a quantity called \\emph{Lagrangian}.\nThe Lagrangian $L(G)$ of a graph $G=(V,E)$ is defined as\n\\[\\max \\sum_{\\{u,v\\}\\in E}x_ux_v \\textup{ subj. to } x_v\\geq 0\\quad\\forall v\\in V\\textup{ and }\\sum_{v\\in V}x_v=1.\\]\nDespite its somewhat long definition, it is a natural quantity to consider in the context of Tur\\'an problems.\nTo see this, let $N$ be some large positive integers.\nConsider the \\emph{blowup} of $G$ obtained by putting in $(x_v+o(1))N$ copies of each vertex $v\\in V$ so that there are $N$ vertices in total, where $(x_v)_{v\\in V}$ is the extremizer for the Lagrangian.\nThen there are $(L(G)+o(1))N^{2}$ edges in the blowup.\nOn the other hand, it is clear that $\\abs{E}\\leq L(G)\\abs{V}^2$, which shows that the density Tur\\'an theorem is equivalent to that $L(G)\\leq \\frac{1}{2}\\left(1-\\frac{1}{r}\\right)$ for every $K_{r+1}$-free graph $G$.\nMotzkin and Straus' idea is that if $u$ and $v$ are not adjacent, then there is an extremizer with either $x_u=0$ or $x_v=0$ for $L(G)$.\nTherefore if $G$ is $K_{r+1}$-free, then there is an extremizer with support of size at most $r$.\nA simple application of Cauchy--Schwarz then concludes the proof.\nDespite its algebraic look, this proof is actually similar to Zykov Symmetrization in spirit.\n\nIt is natural to generalize graph Lagrangian to hypergraph Lagrangian.\nFor any $k$-graph $G=(V,E)$, its \\emph{hypergraph Lagrangian} $L(G)$ is defined as the maximum of $\\sum_{\\{u_1,\\ldots, u_k\\}\\in E}x_{u_1}\\cdots x_{u_v}$ under the same condition.\nAs before, when each $v\\in V$ is blown-up to $(x_v+o(1))N$ vertices where $(x_v)_{v\\in V}$ is the extremizer for the Lagrangian, there are $(L(G)+o(1))N^k$ edges in the blowup.\nAs we will mostly talk about the density of a hypergraph rather than the number of edges, it is convenient to define $b(G)=k!L(G)$ to be the \\emph{blowup density} of $G$.\nIntuitively, it is the largest edge density of the blowups of $G$.\nAs it turns out, hypergraph Lagrangian is indeed useful for some hypergraph Tur\\'an problems, and we will discuss some of those later in \\cref{subsec:hypergraph-turan-density} and \\cref{sec:known}.\n\nA lesser-known but nonetheless interesting algebraic argument was discovered by Li and Li \\cite{LL81}. \nThere, they considered the polynomial \n\\[f\\left((x_v)_{v\\in V(G)}\\right) = \\prod_{uv\\not\\in E}(x_u-x_v)\\]\nfor any graph $G$.\nThe key observation is that if $G$ is $K_{r+1}$-free, then $f$ vanishes whenever $r+1$ of the variables $(x_v)_{v\\in V(G)}$ are equal to one another.\nIn light of this, let $I$ be the ideal of polynomials that vanish whenever $r+1$ of the variables are equal.\nThen $f\\in I$, and Tur\\'an's theorem follows from an explicit description of the generators of $I$ that Li and Li worked out.\n\nOur proof looks different from all the proofs mentioned above.\nFor graphs, our proof can be seen as a double-counting argument that, peculiarly, counts infinitely many objects.\nIn particular, we will lower bound the number of stars of each size, and show that $K_{r+1}$-freeness actually imposes an upper bound on the numbers.\nAn interesting feature our proof has is that in order to get the tight bound on the Tur\\'an density, it is necessary to take stars of any size into account.\nDespite the distinctive look of our proof, our proof is closely related to the standard probabilistic proof of the Caro--Wei theorem.\nIn fact, if one runs the standard proof on the blowup of the graph, and take the size of the blowup to infinity, then the limit of the argument becomes our argument (we thank Maya Sankar for pointing this out to us).\n\n\nIn spite of the similarity to the proof of the Caro--Wei theorem, our counting argument has the advantage that it can be easily rephrased in terms of entropy.\nThis will be crucial as it will inform us how we should adapt the proof for hypergraphs.\nWe will therefore give an introduction to the entropy method in the next subsection.\n\n\\subsection{The entropy method}\nThe concept of entropy in the context of information theory was first formulated by Shannon in his seminal work in 1948 on the noisy-channel coding theorem \\cite{Sha48}.\nRoughly speaking, the entropy of a random variable measures how much information the random variable carries.\nUsing entropy, Shannon determined the best efficiency of a code transmitted through a noisy channel that can be corrected with high probability.\nThis has become the foundation of information theory, and many other definitions of entropy have been made as well.\nHowever, in this paper, we will only use Shannon's definition of entropy.\n\nThe adaptation of Shannon entropy in combinatorics and outside the context of information theory came much later in comparison.\nSome early examples include Chung, Frankl, Graham and Shearer's work on triangle-intersecting family of graphs \\cite{CGFS86} (where Shearer's inequality was introduced), Radhakrishnan's entropic proof of the Br\\'egman's theorem \\cite{Rad97}, and Friedgut and Kahn's theorem on the number of copies of a fixed hypergraph in another hypergraph with a given number of edges \\cite{FK98}.\nThere is nonetheless a significant growth in work using the entropy method in the past decade or two.\nTwo recent exciting, and perhaps unexpected, examples are Gilmer's breakthrough on the union-closed set conjecture \\cite{Gil22} and the work of Gowers, Green, Manners and Tao resolving Marton's conjecture (also known as the polynomial Freimann--Ruzsa conjecture over $\\FF_2$) \\cite{GGMT24}.\n\nIn the context of extremal graph theory, the entropy method is particularly useful when dealing with counts of homomorphisms or homomorphism densities.\nHere, for any $F,G$ that are graphs or general $k$-graphs, a \\emph{homomorphism} from $F$ to $G$ is a function $f:V(F)\\to V(G)$ that sends edges of $F$ to edges of $G$. \nIn particular, $f$ must be injective on any edge of $F$.\nThe \\emph{homomorphism density} $t(F,G)$ is the probability that a uniformly random chosen function from $V(F)\\to V(G)$ is actually a homomorphism.\nIn this terminology, a corollary of the Kruskal--Katona theorem says that $t(K_3, G)\\leq t(K_2, G)^{\\frac{3}{2}}$, which follows immediately from Shearer's inequality (see also \\cite{CY24} for an entropic proof of a slightly stronger result).\nIn the last decade, the entropy method has been applied to show that various bipartite graphs $F$ are \\emph{Sidorenko}, i.e. $t(F,G)\\geq t(K_2,G)^{e(F)}$.\nThis was first formalized by Szegedy \\cite{Sze15} building on a previous work \\cite{LS11}, and this was later adapted to attack Sidorenko's conjecture \\cite{Par14, CL17, CKLL18-1, CKLL18-2} and related problems \\cite{Fitch18, Lee21, GLLV22, BMN24}.\nIn fact, we will also prove some Sidorenko-type result using arguments similar to Szegedy's in our entropic proofs.\n\nGiven how much the entropy method has been utilized to understand relations between homomorphism densities, it should be surprising that no entropic proof for Tur\\'an's theorem was known.\nIndeed, an equivalent formulation of the density Tur\\'an theorem is that if $t(K_{r+1},G)=0$ then $t(K_2, G)\\leq 1-\\frac{1}{r}$.\nIn this paper, we give the first entropic proof of the density Tur\\'an theorem.\nTo do so, we rephrase the density Tur\\'an theorem in the following way, and we will later show the equivalence between the two formulations.\nBelow, and throughout the paper, we use $\\HH(X)$ to denote the Shannon entropy of a random variable $X$ (see \\cref{sec:prelim} for definitions and basic properties).\n\n\\begin{theorem}[Entropic Tur\\'an theorem]\\label{thm:entropic-turan}\n Let $r$ be a positive integer, and let $G$ be a $K_{r+1}$-free graph.\n Let $X,Y$ be random variables distributed on $V(G)$ so that $\\{X,Y\\}$ is always an edge in $G$. Assume $X,Y$ are symmetric, i.e. the distribution of $(X,Y)$ and the one of $(Y,X)$ are the same.\n Then\n \\[\\HH(X,Y) \\leq 2\\HH(X)+\\log_2\\left(1-\\frac{1}{r}\\right).\\]\n\\end{theorem}\n\nWe make a brief remark that the equivalence is shown via an entropic reinterpretation of blowup density and Langrangian.\nIndeed, it turns out that for a given graph $G$, the maximum of the quantity $\\HH(X,Y)-2\\HH(X)$ for symmetric $V(G)$-valued random variables $X,Y$ with $\\{X,Y\\}\\in E(G)$ is related to the blowup density $b(G)$ of $G$.\nMore surprisingly, the maximum of $\\HH(X,Y)-\\HH(X)$ is related to the spectral radius $\\rho(G)$ of $G$.\nThose connections will be made precise and proven in \\cref{sec:connection}, where we also generalize the connections to hypergraphs.\nOne benefit is that as an immediate corollary of our entropic Tur\\'an theorem, we can generalize spectral Tur\\'an theorems established by Wilf \\cite{Wil86} and Nikiforov \\cite{Nik02,Nik06}.\n\n\\begin{theorem}\\label{thm:spectral-Turan-tree}\n Let $r\\geq 2$ and $T$ be a tree with $\\ell\\geq 1$ vertices. For any $K_{r+1}$-free graph $G$, we have\n \\[\\rho(G)^\\ell\\leq \\left(1-\\frac{1}{r}\\right)\\#\\{\\text{homomorphisms from $T$ to $G$}\\}.\\]\n\\end{theorem}\nTo see that this is indeed a generalization of Wilf's and Nikiforov's results, we can take $T$ to be the path $P_{\\ell}$ on $\\ell$ vertices.\nWilf's result corresponds to $\\ell=1$, whereas Nikiforov's results correspond to $\\ell=2$ and general $\\ell$. \n\\begin{theorem}[\\cite{Wil86,Nik02,Nik06}]\\label{thm:spectral-Turan}\n Let $r\\geq 2$. For any $K_{r+1}$-free graph $G$ with $n$ vertices and $m$ edges, we have\n \\[\\rho(G)\\leq \\left(1-\\frac{1}{r}\\right)n,\\]\n \\[\\rho(G)^2\\leq \\left(1-\\frac{1}{r}\\right)\\cdot 2m,\\]\n and\n \\[\\rho(G)^\\ell\\leq \\left(1-\\frac{1}{r}\\right)w_\\ell(G),\\]\n where $w_\\ell(G)$ denotes the number of $\\ell$-walks in $G$.\n\\end{theorem}\n\n\\subsection{Hypergraph Tur\\'an densities}\\label{subsec:hypergraph-turan-density}\nUsing the idea from our entropic proof of the density Tur\\'an theorem, we can determine the Tur\\'an densities for some new family of hypergraphs.\nBefore presenting our results, let us first introduce some definitions and previous work that are relevant.\n\nFor any family of $k$-graphs $\\cF$, its Tur\\'an number $\\textup{ex}(n,\\cF)$ is defined to be the maximum number of edges in a $k$-graph $G$ that is $F$-free for every $F\\in \\cF$.\nThe Tur\\'an density is defined analogously by $\\pi(\\cF) = \\lim_{n\\to\\infty}\\textup{ex}(n,\\cF)/\\binom{n}{k}$.\nFor any family of $k$-graphs $\\cF$ and a $k$-graph $G$, we say that $G$ is \\emph{$\\cF$-hom-free} if there does not exist any homomorphism $F\\to G$ for every $F\\in \\cF$.\nA $F$-hom-free $k$-graph is simply a $k$-graph that is $\\{F\\}$-hom-free.\n\nIt is a standard result in the field that $\\pi(\\cF)$ is the supremum of $b(G)$ where $G$ runs through all $\\cF$-hom-free $k$-graphs (see \\cite[Section 2]{Kee11} or \\cite[Lemma 2.2]{San24} for example).\nNotice that a single edge has blowup density $k!/k^k$, showing that $b(G)\\geq k!/k^k$ if $G$ is not empty.\nThis immediately shows that either $\\pi(\\cF)=0$ or $\\pi(\\cF)\\geq k!/k^k$ for any family of $k$-graphs $\\cF$.\nWe see that among the possible values of Tur\\'an density, there is a ``jump'' going from $0$ to $k!/k^k$.\nWhen $k=2$, this is indeed the behavior of Tur\\'an densities: the Erd\\H{o}s--Stone theorem shows that all possible values are $0, \\frac{1}{2}, \\frac{2}{3}, \\frac{3}{4},\\ldots$, showing that there are only jumps in the case of graphs.\nHowever, for hypergraphs, the set of possible Tur\\'an densities has a different behavior.\nIt was first discovered by Frankl and R\\\"odl \\cite{FR84} that for each $k\\geq 3$, there are infinitely many \\emph{non-jumps} $\\delta$, where for every $\\varepsilon>0$ there exists a family $\\cF$ of $k$-graphs with $\\pi(\\cF)\\in (\\delta,\\delta+\\varepsilon)$.\nOn the other hand, Baber and Talbot \\cite{BT11} showed that jumps do exist above $k!/k^k$ when $k=3$.\nHowever, our understanding in jumps and non-jumps is still limited, and we do not even know whether $k!/k^k$ is a jump.\n\nA standard argument shows that $k!/k^k$ is a jump if and only if there exists a finite family $\\cF$ of $k$-graph with $\\pi(\\cF)=k!/k^k$ and $b(F)>k!/k^k$ for each $F\\in \\cF$ (see \\cite{FR84}).\nThe fact that we do not know whether $k!/k^k$ is a jump can thus be seen as a result of not having sufficient understanding in the families $\\cF$ with $\\pi(\\cF)=k!/k^k$.\nIndeed, known families with Tur\\'an densities equal to $k!/k^k$ are so few that we can list them here.\nFor general $k$, Mubayi \\cite{Mub06} showed that the $k$-uniform extended clique $E^{(k)}_{k+1}$ of size $k+1$ has Tur\\'an density $k!/k^k$.\nHere, the \\emph{extension} of a hypergraph is another hypergraph with higher uniformity obtained by adding different vertices into the edges, and an \\emph{extended clique} is an extension of a complete graph. \nIn particular, $E^{(k)}_{k+1}$ is obtained by adding $k-2$ extra vertices to each edge of $K_{k+1}$, where no two edges share any extra vertices.\nThis was later generalized by Mubayi and Pikhurko \\cite{MP07}, who showed that the hypergraph $\\Delta_{(1,1,\\ldots, 1)}$ with edges\n\\[\\left\\{v_1,\\ldots, v_k\\right\\}\\text{ and }\\{w,v_i,u^{(i)}_1,\\ldots, u^{(i)}_{k-2}\\}\\text{ for }i\\in [k]\\]\nalso has Tur\\'an density $k!/k^k$.\nHere, and later whenever the vertex set is not explicitly described, the vertex set consists of vertices that appear in the description of the edges.\nMubayi and Pikhurko's result is indeed an improvement as $E^{(k)}_{k+1}$ is homomorphic to $\\Delta_{(1,1,\\ldots, 1)}$, showing that $E^{(k)}_{k+1}$-hom-free graphs are also $\\Delta_{(1,1,\\ldots,1)}$-hom-free and so $\\pi(E^{(k)}_{k+1})\\leq \\pi(\\Delta_{(1,1,\\ldots,1)})$.\n\nWe remark that both Mubayi's \\cite{Mub06} and Mubayi and Pikhurko's \\cite{MP07} results are stronger---the exact Tur\\'an numbers were determined for sufficiently many vertices.\nIf we only care about the Tur\\'an density, then an argument of Sidorenko \\cite{Sid89} based on hypergraph Lagrangian can be modified to show that $\\pi(\\Delta_{(1,\\ldots,1)})=k!/k^k$ as well---this is an observation by Keevash \\cite[Theorem 3.1]{Kee11}.\n\nFor smaller $k$'s, slightly more is known.\nWhen $k=3$, Bollob\\'as \\cite{Bol74} showed that $\\pi(\\{K_4^{-},F_5\\}) = \\frac{2}{9}$ where $K_4^{-} = \\{123,124,134\\}$ and $F_5=\\{123,124,345\\}$.\nThis was improved by Frankl and F\\\"uredi \\cite{FF83}, who showed that $\\pi(F_5)$ is already equal to $\\frac{2}{9}$.\nUsing flag algebra, Baber and Talbot \\cite{BT12} improved this further by showing that $\\pi(\\{123,124,345,156\\}) = \\frac{2}{9}$.\nFinally, when $k=4$, Pikhurko \\cite{Pik08} showed that $\\pi(\\{1234, 1235, 4567\\}) = \\frac{3}{32}$.\n\nAs shown above, not a lot is known about families $\\cF$ of $k$-graphs with $\\pi(\\cF)=k!/k^k$.\nAs an application of our entropic proof of the density Tur\\'an theorem, we will generalize our argument to show $\\pi(\\cF)=k!/k^k$ for a new family $\\cF$ of $k$-graphs.\nOur method has a benefit that we may first come up with an argument and then see what family of $k$-graphs need to be forbidden in order for the argument to work.\nWe believe that this advantage can help discovering more families $\\cF$ with minimum positive Tur\\'an densities.\n\n\n\\begin{figure}[h]\\centering\\label{fig:Tent}\n\\begin{tikzpicture}[scale=0.8]\n\n \\coordinate (A) at (0,0);\n \\coordinate (B) at (1,0);\n \\coordinate (C) at (2,0);\n \\coordinate (D) at (5,0);\n \\coordinate (E) at (6,0);\n \\coordinate (F) at (4.5,1.732/2);\n \\coordinate (G) at (2.75,1.732*3/4);\n \\coordinate (H) at (4,1.732);\n \\coordinate (I) at (3.5,1.732*3/2);\n\n \\draw [fill] (A) circle (1.6pt);\n \\draw [fill] (B) circle (1.6pt);\n \\draw [fill] (C) circle (1.6pt);\n \\draw [fill] (D) circle (1.6pt);\n \\draw [fill] (E) circle (1.6pt);\n \\draw [fill] (F) circle (1.6pt);\n \\draw [fill] (G) circle (1.6pt);\n \\draw [fill] (H) circle (1.6pt);\n \\draw [fill] (I) circle (1.6pt);\n\\draw[rounded corners=8pt,black,line width=2pt] \n (0-0.2,0.5)--(-0.5-0.2,0)--(0-0.2,-0.5)--(6+0.2,-0.5)--(6.5+0.2,0)--(6+0.2,0.5)--cycle;\n \n\\draw[rounded corners=6pt,black,line width=2pt] \n(0-0.2,0.3)--(-0.3-0.2,0)--(0-0.2,-0.3)--(2+0.3/1.732,-0.3)--(3.5+0.1+0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(3.5+0.1+0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(3.5+0.1-0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(2-0.3/1.732,0.3)--cycle;\n\n\\draw[rounded corners=6pt,black,line width=2pt] \n(6+0.2,0.3)--(6+0.3+0.2,0)--(6+0.2,-0.3)--(5-0.3/1.732,-0.3)--(3.5-0.1-0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(3.5-0.1-0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(3.5-0.1+0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(5+0.3/1.732,0.3)--cycle;\n\n\\node at (3.5,0) {Base};\n\\node at (4.7,1.732*3/2) {Apex};\n\\end{tikzpicture}\n\\caption{$(3,2)$-tent}\n\\end{figure}\n\n\n\n\nTo state our result, for any partition $\\lambda$ of $k$, let $\\lambda = (\\lambda_1,\\ldots, \\lambda_{\\ell})$ where $\\ell = \\ell(\\lambda)$ is the length of $\\lambda$, and $\\lambda_1\\geq \\cdots\\geq \\lambda_{\\ell}$.\nWe also denote $\\sum_{i=1}^{\\ell}\\lambda_i$ by $\\abs{\\lambda}$ (which is equal to $k$ by definition).\nFor any $\\lambda$ with $\\ell(\\lambda)\\geq 2$, we define the \\emph{$\\lambda$-tent}, denoted by $\\Delta_{\\lambda}$, to be the following $k$-graph.\nThe $\\lambda$-tent comes with an edge $e$ that is the \\emph{base} and a vertex $v$ that is the \\emph{apex}.\nSetting $\\ell=\\ell(\\lambda)$ to be the length of $\\lambda$, for each $i\\in[\\ell]$ we also have an edge $e_i$ containing $v$ such that $\\abs{e_i\\cap e}=\\lambda_i$.\nMoreover, we require that $e_i\\cap e_j = \\{v\\}$ for any $i\\neq j\\in [\\ell]$.\nIt is clear that this determines $\\Delta_{\\lambda}$ uniquely up to isomorphism---in fact, we must have $e\\cap e_1,\\ldots, e\\cap e_{\\ell}$ partition $e$.\nIt is easy to check that this definition matches the definition of $\\Delta_{(1,1,\\ldots,1)}$ above, $F_5 = \\Delta_{(2,1)}$ (with base $123$ and $4$ being the apex) and Pikhurko's result can be rephrased as $\\pi(\\Delta_{(3,1)})=\\frac{3}{32}$.\nOur result can now be stated as follows.\n\n\\begin{theorem}\\label{thm:main-tent}\n Let $k\\geq 2$ be a positive integer, and let $\\cF_k$ be the family of $\\lambda$-tents with $\\abs{\\lambda}=k$ and $\\ell(\\lambda)=2$.\n Then $\\pi(\\cF_k) = k!/k^k$.\n\\end{theorem}\n\nNote that this is a stronger statement than Mubayi's and Mubayi and Pikhurko's results.\nIn fact, $\\Delta_{(1,1,\\ldots, 1)}$ admits a homomorphism to $\\Delta_{\\lambda}$ for every $\\abs{\\lambda}=k$ and $\\ell(\\lambda)=2$, which shows that $\\pi(\\Delta_{(1,1,\\ldots,1)})\\leq \\pi(\\cF_k)$. \nUsing the same argument, we can transform \\cref{thm:main-tent} into a Tur\\'an result of a single $k$-graph.\n\n\\begin{theorem}\\label{thm:one-tent}\n Let $k\\geq 2$ be a positive integer, and let $\\lambda$ be a partition of $k$ such that $\\lambda_1\\leq \\lceil k/2\\rceil$ and $\\lambda_i=1$ for all $1<i\\leq \\ell(\\lambda)$.\n Then $\\pi(\\Delta_\\lambda) = k!/k^k$.\n\\end{theorem}\n\nAlthough when $k=3$ and $4$, \\cref{thm:one-tent} is subsumed by the known results mentioned above, this gives a new Tur\\'an result for larger $k$'s. \nTo show that this should be a nontrivial result for larger $k$'s, we prove the following result in the opposite direction.\n\n\\begin{theorem}\\label{thm:not-Turan}\n There exists a constant $\\alpha<1$ so that for all sufficiently large $k\\in\\NN$ and any partition $\\lambda$ of $k$ with $\\ell(\\lambda)\\geq 2$, if $\\lambda_1 >\\alpha k$ then $\\pi(\\Delta_{\\lambda})>k!/k^k$.\n\\end{theorem}\n\n\\cref{thm:one-tent} shows that the constant in \\cref{thm:not-Turan} cannot be smaller than $1/2$, and it seems like an interesting question to determine the best possible value of $\\alpha$.\nIt might help us understand the $k$-graphs $F$ with $\\pi(F)=k!/k^k$ as well.\nWe leave this as a future direction for interested readers.\n\nBeyond showing $\\pi(\\cF) = k!/k^k$ for various families $\\cF$ of $k$-graphs, our method also applies to some other scenarios where the extremizers are blowups of complete hypergraphs.\nUnfortunately, we have not been able to find an argument that proves a new and clean statement in those settings.\nWe nonetheless include the arguments later in \\cref{sec:known} in the hope that they will be enlightening for readers interested in adapting our arguments.\nThe relevant background will also be introduced there.\n\\subsection{Structure of the paper}\nWe will first present our new proof of the density Tur\\'an theorem in \\cref{sec:counting}.\nWe will then introduce the necessary entropic tools in \\cref{sec:prelim}, which will set us up for \\cref{sec:entropy}, where we rephrase our proof in terms of entropy.\nIn \\cref{sec:connection}, we will show how our entropic formulation captures quantities such as hypergrpah Lagrangian and spectral radius.\nWe will use the connection to prove the spectral Tur\\'an theorems and the equivalence between the entropic Tur\\'an theorem and the density Tur\\'an theorem.\nIn \\cref{sec:partial-hypergraph}, we set up some notations and propositions that will be useful in the later sections.\nIn \\cref{sec:main-proof}, we will apply the entropic argument in \\cref{sec:entropy} to show \\cref{thm:main-tent} in two different ways, and we will also prove \\cref{thm:one-tent,thm:not-Turan}.\nSome further generalization of our arguments is included in \\cref{sec:known}, where we also introduce some related known results.\nFinally, we will end with some concluding remarks in \\cref{sec:conclusion}.\n\n\n\n\\section{A new proof of the density Tur\\'an theorem}\\label{sec:counting}\nIn this section, we give a new proof to the density Tur\\'an theorem. The key idea is to lower bound the density of stars of each size in terms of edge density by their Sidorenko property. If the densities are large, then we shall find a large clique. The main difference of this proof from all the previous ones is that we consider stars of all sizes at once.\n\n\\begin{proof}[Proof of the density Tur\\'an theorem]\n For any two graphs $H,G$, let $t(H,G)$ be the homomorphism density of $H$ in $G$. That is, $t(H,G)$ is the probability that a function $f:V(H)\\rightarrow V(G)$ chosen uniformly at random is a homomorphism from $H$ to $G$. We will need the following lemma about lower bounding the homomorphism density of stars in terms of edge density, which is a special case of Sidorenko's conjecture. We include the proof here since the proof is short.\n \\begin{lemma}\n For $i\\geq 0$, let $S_i=K_{1,i}$ be the star with $i+1$ vertices. Then\n \\[t(S_i,G)\\geq t(K_2,G)^i\\]\n holds for any graph $G$.\n \\end{lemma}\n \\begin{proof}\n Assume $n=\\abs{V(G)}$ and $m=\\abs{E(G)}$. Note that $S_i$ has $i+1$ vertices, and hence \n \\[t(S_i,G)=\\frac{\\sum_{v\\in V(G)}\\deg(v)^i}{n^{i+1}}\\geq \\frac{1}{n^i}\\left(\\frac{\\sum_{v\\in V(G)}\\deg(v)}{n}\\right)^i=\\frac{(2m)^i}{n^{2i}}=t(K_2,G)^i,\\]\n where the inequality follows from the convexity of $x^i$.\n \\end{proof}\n Now we assume the graph $G$ is $K_{r+1}$-free. We sample a sequence of i.i.d.\\@ random vertices $v_0,v_1,\\dots$ from $V(G)$ uniformly at random. For $i\\geq 0$, let $A_i$ be the event that the induced graph on vertices $v_0,\\dots,v_{i-1},v_i$ contains $S_i$ as a subgraph centered at $v_i$. In particular, $A_0$ is the true event. Note that there can only be at most $r$ events happening at the same time. Otherwise, assume $A_{i_0},A_{i_1},\\dots,A_{i_r}$ are all true for some $0=i_0<i_1<\\dots<i_r$. Then $v_{i_0},\\dots,v_{i_r}$ form an $(r+1)$-clique in $G$. Therefore, by double counting, we may conclude that\n \\[\\PP(A_0)+\\PP(A_1)+\\dots\\leq r.\\]\n \n On the other hand, we know that $\\PP(A_i)=t(S_i,G)\\geq t(K_2,G)^i$ for all $i$. Thus, we have\n \\[\\frac{1}{1-t(K_2,G)}\\leq \\PP(A_0)+\\PP(A_1)+\\dots\\leq r.\\]\n After rearranging, we get\n \\[\\frac{2m}{n^2}=t(K_2,G)\\leq 1-\\frac{1}{r},\\]\n and we are done.\n\\end{proof}\n\n\n\n\n\\section{Shannon entropy}\\label{sec:prelim}\nIn this section, we introduce the definition of Shannon entropy and some of the properties we will use from the literature. We refer the readers to \\cite[Section 14.6]{AS00} for a more detailed introduction.\nWe will also prove a lemma which upper bounds the entropies of random variables by the entropy of their mixture. This lemma will be one of the key ingredients of many of the proofs in the rest of this paper.\n\n\\subsection{Preliminaries}\n\nFor any discrete random variable $X$, we write $p_X(x)\\eqdef\\PP(X=x)$. Also, we denote by $\\supp(X)$ the support of $X$, i.e. the set of all $x$ such that $p_X(x)>0$.\nThrough out this paper, the random variables we will consider are always discrete with finite support, i.e. $\\abs{\\supp(X)}<\\infty$. For any such random variable, we may define its Shannon entropy.\n\n\\begin{definition}\n For any random variable $X$, we define its Shannon entropy\n \\[\\HH(X)\\eqdef\\sum_{x\\in\\supp(X)}-p_X(x)\\log_2p_X(x).\\]\n For any sequence of random variables $X_1,\\dots,X_n$, we use $\\HH(X_1,\\dots,X_n)$ to denote the entropy of the random tuple $(X_1,\\dots,X_n)$.\n\\end{definition}\n\nWe also define the conditional entropy of $X$ given $Y$.\n\\begin{definition}\n For any two random variables $X,Y$, the conditional entropy of $X$ given $Y$ is given by\n \\[\\HH(X\\mid Y)\\eqdef\\HH(X,Y)-\\HH(Y).\\]\n Equivalently, we have\n \\begin{align*}\n \\HH(X\\mid Y)=&\\quad\\smashoperator{\\sum_{y\\in\\supp(Y)}}p_Y(y)\\HH(X\\mid Y=y)\\\\\n =&\\quad\\smashoperator{\\sum_{(x,y)\\in \\supp(X,Y)}}-p_{X,Y}(x,y)\\log_2\\left(\\frac{p_{X,Y}(x,y)}{p_Y(y)}\\right).\n \\end{align*}\n\\end{definition}\n\nUsing the definition of conditional entropy, we have the following chain rule.\n\\begin{proposition}[Chain rule]\n For any random variables $X_1,\\dots,X_n$, we have\n \\[\\HH(X_1,\\dots,X_n)=\\HH(X_1)+\\HH(X_2\\mid X_1)+\\dots+\\HH(X_n\\mid X_1,\\dots,X_{n-1}).\\]\n\\end{proposition}\n\nThe following proposition says that on a fixed support, the entropy is maximized by the uniform distribution on that support.\n\\begin{proposition}[Uniform bound]\n For any random variable $X$, we have\n \\[\\HH(X)\\leq \\log_2\\abs{\\supp(X)},\\]\n where the equality holds if and only if $X$ is uniform.\n\\end{proposition}\n\nWe will also need the following two propositions about entropy.\n\\begin{proposition}[Subadditivity]\nFor any three random variables $X,Y,Z$, we have\n\\[\\HH(X,Y)\\leq \\HH(X)+\\HH(Y),\\]\n\\[\\HH(X,Y\\mid Z)\\leq \\HH(X\\mid Z)+\\HH(Y\\mid Z).\\]\n\\end{proposition}\n\\begin{proposition}[Dropping condition]\nFor any three random variables $X,Y,Z$, we have\n\\[\\HH(X\\mid Y)\\leq \\HH(X),\\]\n\\[\\HH(X\\mid Y,Z)\\leq \\HH(X\\mid Z).\\]\n\\end{proposition}\n\n\\subsection{Mixture and the mixture bound}\nIn this subsection, the concern is what is called the \\emph{mixture} of random variables.\n\\begin{definition}\n For random variables $X_1,\\dots,X_n$ and weights $w_1,\\dots,w_n\\geq 0$ with $\\sum_{i=1}^n w_i=1$, we say that $Z$ is the \\emph{mixture of $X_1,\\dots,X_n$ with weight $w_1,\\dots, w_n$} if $Z$ is obtained from the following procedure. We first pick an independent random index $\\bi$ with probability $\\PP(\\bi =i)=w_i$. Then we set $Z=X_{\\bi}$.\n\\end{definition}\nIn our applications, we will consider mixtures of random variables whose supports do not overlap too much.\n\\begin{definition}\n Let $a$ be a positive integer.\n We say that the random variables $X_1,\\dots,X_n$ have \\emph{$(a+1)$-wise disjoint supports} if for any element $x\\in \\cup_{i=1}^n \\supp (X_i)$, there are at most $a$ many indices $i$ such that $x\\in\\supp (X_i)$.\n\\end{definition}\n\nWith the definitions above, we may state our lemma about an upper bound on the entropies of random variables with $(a+1)$-wise disjoint supports, in terms of the entropy of their mixture.\n\\begin{lemma}[Mixture bound]\\label{lemma:mix}\n Let $X_1,\\dots,X_n$ be random variables with $(a+1)$-wise disjoint supports. Then there exists a mixture of $X_1,\\dots,X_n$, say $Z$, such that\n \\[\\sum_{i=1}^n 2^{\\HH(X_i)}\\leq a2^{\\HH(Z)}.\\]\n\\end{lemma}\n\\begin{proof}\n Let $s_i=2^{\\HH(X_i)}$ and we define\n \\[w_i=\\frac{s_i}{\\sum_{j=1}^n s_j}.\\]\n Let $\\bi$ be an independent random index with probability $\\PP(\\bi =i)=w_i$ and let $Z=X_{\\bi}$ be the mixture. By chain rule, we have $\\HH(Z,\\bi)=\\HH(\\bi)+\\HH(Z\\mid \\bi)=\\HH(Z)+\\HH(\\bi\\mid Z)$. Therefore,\n \\[\\HH(Z)=\\HH(\\bi)+\\HH(Z\\mid \\bi)-\\HH(\\bi\\mid Z).\\]\n By the definition of entropy and conditional entropy, we have\n \\[\\HH(\\bi)=\\sum_{i=1}^n -w_i\\log_2 w_i=\\frac{-s_i}{\\sum_{i=1}^n s_i}\\log_2\\bigl(\\frac{s_i}{\\sum_{i=1}^n s_i}\\bigr)\\]\n and\n \\[\\HH(Z\\mid \\bi)=\\sum_{i=1}^n w_i\\HH(X_i)=\\frac{s_i\\log_2 s_i}{\\sum_{i=1}^n s_i}.\\]\n We may upper bound $\\HH(i\\mid Z)$ by uniform bound. For any $x\\in\\cup_{i=1}^n\\supp(X_i)$, when conditioning on $Z=x$, there are at most $a$ possible indices as an outcome of $\\bi$. Thus, we have\n \\[\\HH(\\bi\\mid Z)\\leq \\log_2 a.\\]\n Combining all above, we get\n \\begin{align*}\n \\HH(Z)\\geq &\\frac{-s_i}{\\sum_{i=1}^n s_i}\\log_2\\bigl(\\frac{s_i}{\\sum_{i=1}^n s_i}\\bigr)+\\frac{s_i\\log_2 s_i}{\\sum_{i=1}^n s_i}-\\log_2 a\\\\\n =&\\log_2\\left(\\sum_{i=1}^n s_i\\right)-\\log_2 a,\n \\end{align*}\n and we are done after rearranging.\n\\end{proof}\n\nThe following example shows that \\cref{lemma:mix} resembles a double counting on $(a+1)$-wise disjoint sets. Thus, the mixture bound can be viewed as an entropic version of this double counting.\n\\begin{example}\n Let $a$ be an integer and let $S_1,\\dots,S_n$ be some sets that are $(a+1)$-wise disjoint. Assume $X_i$ is a random element chosen from $S_i$ uniform at random for each $i\\in [n]$, and let $Z$ be the mixture of $X_1,\\ldots, X_n$ provided by \\cref{lemma:mix}. We have $2^{\\HH(X_i)}=\\abs{S_i}$, and by uniform bound we have $2^{\\HH(Z)}\\leq \\abs{\\cup_{i=1}^nS_i}$.\n Hence, \\cref{lemma:mix} implies that\n \\[\\sum_{i=1}^n\\abs{S_i}\\leq a2^{\\HH(Z)}\\leq a\\abs{\\bigcup_{i=1}^n S_i},\\]\n which gives the same bound as the double counting argument on pairs $(x,i)$ with $x\\in S_i$.\n\\end{example}\n\n\\section{Reformulation using the entropy method}\\label{sec:entropy}\nIn this subsection, we reformulate the proof in \\cref{sec:counting} using entropy to prove \\cref{thm:entropic-turan}. As expected, we shall sample the stars in the same way as Szegedy did \\cite{Sze15}, and we will use \\cref{lemma:mix} to replace the double counting argument.\n\\begin{proof}[Proof of \\cref{thm:entropic-turan}]\n Recall that we have a $K_{r+1}$-free graph $G$ and symmetric random variables $X,Y$ distributed on $V(G)$ with $\\{X,Y\\}\\in E(G)$ always holding.\n We first fix an integer $N\\in\\NN$, and we will take $N$ goes to infinity later. \n \\begin{claim}\\label{claim:StarSido}\n For each $i=0,1,\\dots, N$, there exists a random tuple $T_i=(v_0^{(i)},\\dots,v_N^{(i)})\\in V(G)^{N+1}$ such that \n \\begin{enumerate}\n \\item there is always an edge between $v_j^{(i)},v_i^{(i)}$ for all $j=0,\\dots,i-1$,\n \\item the marginal distributions of $v_j^{(i)}$ and $X$ are the same for all $j=0,1\\dots,N$, and\n \\item $\\HH(T_i)=i\\HH(Y\\mid X)+(N+1-i)\\HH(X)$.\n \\end{enumerate}\n \\end{claim}\n \\begin{proof}\n For $i=0$, it is easy to check that $N+1$ i.i.d. random vertices $v_0^{(0)},\\dots,v_N^{(0)}$ with the law of $X$ satisfy the condition.\n\n For $i\\geq 1$, we first sample an edge $(v_0^{(i)},v_i^{(i)})$ using the law of $(X,Y)$. Next, we condition on $v_i^{(i)}$ and resample $v_0^{(i)}$ $(i-1)$ times conditionally independently to get $v_1^{(i)},\\dots,v_{i-1}^{(i)}$. Finally, we sample $v_{i+1}^{(i)},\\dots, v_N^{(i)}$ independently using the law of $X$. \n\n Note that the first two conditions are true from the way we sample the random variables. It remains to compute $\\HH(T_i)$. Note that $\\HH(T_i)=\\HH(v_0^{(i)},\\dots,v_i^{(i)})+(N-i)\\HH(X)$ since we sampled $v_{i+1}^{(i)},\\dots, v_N^{(i)}$ independently. By chain rule, we have \n \\begin{align*} \\HH(v_0^{(i)},\\dots,v_i^{(i)})=&\\HH(v_0^{(i)},\\dots,v_{i-1}^{(i)}\\mid v_i^{(i)})+\\HH(v_i^{(i)})\\\\\n =&i\\HH(v_0^{(i)}\\mid v_i^{(i)})+\\HH(v_i^{(i)})\\\\\n =&i\\HH(Y\\mid X)+\\HH(X).\n \\end{align*}\n Therefore, $\\HH(T_i)=i\\HH(Y\\mid X)+(N+1-i)\\HH(X)$.\n \\end{proof}\n Now, we may apply \\cref{lemma:mix} to the random tuples $T_0,\\dots,T_N$ in \\cref{claim:StarSido}. Since $G$ is $K_{r+1}$-free, similar to the proof in \\Cref{sec:counting}, any tuple of $N+1$ vertices is in at most $r$ supports $\\supp(T_i)$. Therefore, the supports of $T_0,\\dots,T_N$ are $(r+1)$-wise disjoint. Thus, there is a mixture $T=(v_0,\\dots,v_N)$ of $T_0,\\dots,T_N$ such that\n \\[\\sum_{i=0}^N 2^{\\HH(T_i)}\\leq r2^{\\HH(T)}.\\]\n Note that the marginal distribution of $v_i$ is also the same as the marginal distribution of $X$, so we may upper bound $\\HH(T)$ by $(N+1)\\HH(X)$ by subadditivity. By using $\\HH(T_i)=i\\HH(Y\\mid X)+(N+1-i)\\HH(X)$, we get\n \\[\\sum_{i=0}^N x^i\\leq r,\\]\n where $x\\eqdef 2^{\\HH(Y\\mid X)-\\HH(X)}$. By taking $N$ to infinity, we conclude that $1/(1-x)\\leq r$. Therefore,\n \\[\\HH(Y\\mid X)-\\HH(X)=\\log_2x\\leq \\log_2\\left(1-\\frac{1}{r}\\right).\\qedhere\\]\n\\end{proof}\n\n Let $\\abs{V(G)}=n$ and $\\abs{E(G)}=m$. \n If we pick $(X,Y)$ uniformly at random from all the oriented edges, \\cref{thm:entropic-turan} and the uniform bound give\n \\[\\log_2(2m)=\\HH(X,Y)\\leq 2\\HH(X)+\\log_2\\left(1-\\frac{1}{r}\\right)\\leq 2\\log_2 n+\\log_2\\left(1-\\frac{1}{r}\\right).\\]\n That is, $m\\leq \\left(1-\\frac{1}{r}\\right)\\frac{n^2}{2}$, which recovers the density Tur\\'an theorem.\n In the next section, we will see that \\cref{thm:entropic-turan} is in fact equivalent to the density Tur\\'an theorem by relating entropy to blowup densities.\n \n \n\n\n\\section{Connecting entropy to Lagrangian and spectral radius}\\label{sec:connection}\n\nIn this section, we will show that \\cref{thm:entropic-turan} is equivalent to the density Tur\\'an theorem.\nWe will actually generalize this equivalence in many ways: we will show it for hypergraphs, and we will also go much beyond Lagrangian and blowup densities.\nThis will be useful later to draw connection to the spectral radius of graphs.\n\nWe first observe that in \\cref{thm:entropic-turan}, the quantity that we care about is actually the maximum of $\\HH(X,Y)-2\\HH(X)$ when $(X,Y)$ ranges over all possible symmetric distributions on the oriented edges of $G$.\nThis quantity turns out to be related to the blowup density $b(G)$.\nTo extend this to hypergraphs, we make the following definitions.\n\n\\begin{definition}[Random edge with uniform ordering]\n Let $G$ be a $k$-graph, we say that a tuple of random vertices $(X_1,\\dots,X_k)\\in V(G)^k$ is a \\emph{random edge with uniform ordering on $G$} if $(X_1,\\dots,X_k)$ is symmetric and $\\{X_1,\\dots,X_k\\}$ is always an edge of $G$. Here, $(X_1,\\dots,X_k)$ being symmetric means the distribution of $(X_{\\sigma(1)},\\dots,X_{\\sigma(k)})$ is always the same for any permutation $\\sigma$ of $[n]$.\n\\end{definition}\n\n\\begin{definition}[Entropic density]\n \n For any $k$-graph $G$, define its \\emph{entropic density} $b_{\\textup{entropy}}(G)$ to be the largest possible value of $2^{\\HH(X_1,\\ldots, X_k)-k\\HH(X_1)}$ for any random edge with uniform ordering $(X_1,\\ldots, X_k)$.\n\\end{definition}\n\nNote that $b_{\\textup{entropy}}(G)$ exists as the space of random edge with uniform ordering is compact.\nWe will show that $b_{\\textup{entropy}}(G)$ is equal to $b(G)$, which immediately shows that \\cref{thm:entropic-turan} is equivalent to the density Tur\\'an theorem.\nWe will actually show a stronger statement.\nTo that end, we make the following notations.\nFor any $k$-graph $G$, let $\\vec{E}(G)$ be the set of oriented edges.\nFor each $p>0$, let $b_p(G)$ be the maximum of $\\prod_{(v_1,\\ldots, v_k)\\in\\vec{E}(G)}x_{v_1}\\cdots x_{v_k}$ for $(x_v)_{v\\in V(G)}$ subject to $\\norm{x_v}_{\\ell^p}=1$ (the same definition was made by Keevash, Lenz and Mubayi \\cite{KLM14} where they called the quantity the \\emph{$p$-spectral radius}).\nAlso let $b_{p,\\textup{entropy}}(G)$ be the largest possible value of $2^{\\HH(X_1,\\ldots, X_k)-\\frac{k}{p}\\HH(X_1)}$ for any random edge with uniform ordering $(X_1,\\ldots, X_k)$.\nNote that $b_p(G)$ and $b_{p,\\textup{entropy}}(G)$ both exist by compactness.\n\n\\begin{example}\\label{ex:entropic-lagrangian}\n When $p=1$, we clearly have $b_p(G)=b(G)$ and $b_{p,\\textup{entropy}}(G) = b_{\\textup{entropy}}(G)$.\n When $G$ is a graph and $p=2$, it is not hard to see that $b_p(G)$ is the maximum \n \\[\\max\\vec{x}^{\\intercal}A_G\\vec{x}\\textup{ subject to }\\norm{(x_v)_{v\\in V(G)}}_{\\ell^2}=1\\]\n where $A_G$ is the adjacency matrix of $G$.\n It is a standard fact that this is exactly the spectral radius of $G$.\n In this case, $b_{2,\\textup{entropy}}(G)$ is the largest possible value of $2^{\\HH(X,Y)-\\HH(X)} = 2^{\\HH(Y\\mid X)}$ for any random edge with uniform ordering $(X,Y)$. \n\n For general $k$, if $p=k$, then $b_p(G)$ corresponds to the spectral radius of the adjacency \\linebreak$k$-tensor of $G$, which was proven in \\cite{Qi13}.\n The quantity $b_{k,\\textup{entropy}(G)}$ is the largest possible value of $2^{\\HH(X_1,\\ldots, X_k)-\\HH(X_1)} = 2^{\\HH(X_2,\\ldots, X_k\\mid X_1)}$. Once we prove $b_k(G) = b_{k,\\textup{entropy}(G)}$, this would provide a nice alternative interpretation of the spectral radius for hypergraphs.\n\\end{example}\n\nNow we will show that $b_p(G)$ and $b_{p,\\textup{entropy}}(G)$ are equal to each other.\nThe proof uses Lagrange multiplier in a crucial way.\n\n\\begin{proposition}\\label{prop:entropic-lagrangian}\n For any $k$-graph $G$ and any $p>0$, $b_{p,\\textup{entropy}}(G) = b_p(G)$.\n \n\\end{proposition}\n\n\n\\begin{proof}\n For any $v\\in V(G)$, let $\\vec{L}_v(G)$ be the oriented link of $v$, i.e. the set $(v_2,\\ldots, v_k)$ such that $(v,v_2,\\ldots, v_k)\\in \\vec{E}(G)$.\n \n We start with the following claim that helps us simplify $\\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)$ when $(X_1,\\dots,X_k)$ is in a certain form.\n \\begin{claim}\\label{claim:entropy-lag}\n For any tuple $(x_v)_{v\\in V(G)}\\in \\RR_{\\geq 0}^{V(G)}$, we consider a random edge with uniform ordering $(X_1,\\dots,X_k)$ on $G$ given by\n \\[\\PP((X_1,\\dots,X_k)=(v_1,\\dots,v_k))=\\frac{1}{\\beta}\\prod_{i=1}^kx_{v_i},\n \\text{ where }\\beta\\eqdef\\sum_{(v_1,\\dots,v_k)\\in \\vec{E}(G)}\\prod_{i=1}^kx_{v_i}.\\]\n We also define\n \\[y_v\\eqdef \\PP(X_1=v)=\\frac{x_v}{\\beta}\\sum_{(v_2,\\dots,v_k)\\in \\vec{L}_v(G)}\\prod_{i=2}^kx_{v_i}.\\]\n Then we have\n \\[\\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)=\\log_2\\beta-\\frac{k}{p}\\sum_{v\\in V(G)}y_v\\log_2 \\left(\\frac{x_v^p}{y_v}\\right).\\]\n \\end{claim}\n \\begin{proof}\n First, we have\n \\begin{align*}\n \\HH(X_1,\\dots,X_k)=&\\sum_{(v_1,\\dots,v_k)\\in \\vec{E}(G)}-\\frac{1}{\\beta}\\prod_{i=1}^kx_{v_i}\\log_2\\left(\\frac{1}{\\beta}\\prod_{i=1}^kx_{v_i}\\right)\\\\\n =&\\sum_{(v_1,\\dots,v_k)\\in \\vec{E}(G)}\\frac{1}{\\beta}\\prod_{i=1}^kx_{v_i}\\left(\\log_2\\beta-\\sum_{i=1}^k\\log_2x_{v_i}\\right)\\\\\n =&\\log_2\\beta-k\\sum_{v\\in V(G)}y_v\\log_2 x_v\n \\end{align*}\n Combining this with $\\HH(X_1)=\\sum_{v\\in V(G)}-y_v\\log_2y_v$, we get\n \\begin{align*}\n \\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)=&\\log_2\\beta-\\frac{k}{p}\\sum_{v\\in V(G)}\\left(py_v\\log_2 x_v-y_v\\log_2y_v\\right)\\\\\n =&\\log_2\\beta-\\frac{k}{p}\\sum_{v\\in V(G)}y_v\\log_2 \\left(\\frac{x_v^p}{y_v}\\right).\\qedhere\n \\end{align*}\n \\end{proof}\n Now, we may prove the proposition. We first show that $b_{p,\\textup{entropy}}(G)\\geq b_p(G)$. \n \n Let $(x_v)_{v\\in V(G)}\\in \\RR_{\\geq 0}^{V(G)}$ be the tuple that achieves the maximum in the definition of $b_p(G)$. Define $(X_1,\\dots,X_k)$, $\\beta$, and $(y_v)_{v\\in V(G)}$ in the same way as in \\cref{claim:entropy-lag}. Note that $\\beta=b_p(G)$ and $\\sum_{v\\in V(G)}x_v^p=1$. From \\cref{claim:entropy-lag}, we have\n \\begin{align*}\n \\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)=&\\log_2\\beta-\\frac{k}{p}\\sum_{v\\in V(G)}y_v\\log_2 \\left(\\frac{x_v^p}{y_v}\\right)\\\\\n \\geq& \\log_2\\beta-\\frac{k}{p}\\log_2 \\left(\\sum_{v\\in V(G)}x_v^p\\right)=\\log_2\\beta,\n \\end{align*}\n where the inequality follows from the Jensen's inequality and the concavity of $\\log_2x$.\n Therefore $b_{p,\\textup{entropy}}(G)\\geq b_p(G)$.\n\n For the opposite direction, let $(X_1,\\ldots, X_k)$ be a random edge with uniform ordering achieving the maximum of $b_{p,\\textup{entropy}}(G)$.\n For any unoriented edge $e\\in E(G)$, let $q_e$ be the probability $\\PP(\\{X_1,\\ldots, X_k\\}=e)$.\n Also let $x_v = \\left(\\frac{1}{k}\\sum_{e\\ni v} q_e\\right)^{1/p}$.\n Then\n \\[\\HH(X_1,\\ldots, X_k) = \\HH(X_1,\\ldots, X_k\\mid \\{X_1,\\ldots, X_k\\})+\\HH(\\{X_1,\\ldots, X_k\\}) = \\log_2 k! -\\sum_{e\\in E(G)}q_e\\log_2 q_e\\]\n and\n \\[\\HH(X_1) = \\sum_{v\\in V}-x_v^p\\log_2 x_v^p.\\]\n Therefore, $(q_e)_{e\\in E(G)}$ is a maximizer of \n \\[-\\sum_{e\\in E(G)}q_e\\log_2 q_e+\\frac{k}{p}\\sum_{v\\in V(G)}x_v^p\\log_2 x_v^p\\]\n subject to $q_e \\geq 0$ for all $e\\in E(G)$ and $\\sum_{e\\in E(G)}q_e=1$.\n Note that $\\partial x_v^p/\\partial q_e$ is nonzero only if $v\\in e$, and if that is the case we have $\\partial x_v^p/\\partial q_e=1/k$.\n By Lagrange multiplier, we know that \n \\[-\\log_2 q_e-1 + \\frac{1}{p}\\sum_{v\\in e}\\left(1+\\log_2 x_v^p\\right)\\]\n is constant for all $e\\in E(G)$ with $q_e>0$.\n Therefore\n \\[\\alpha \\eqdef \\frac{q_e}{\\prod_{v\\in e}x_v}\\]\n is the same for all $e\\in E(G)$ with $q_e>0$.\n Notice that $\\PP(X_1 = v) = x_v^p$ for any $v\\in V(G)$, and for any $(v_1,\\ldots, v_k)\\in \\vec{E}(G)$, we have \n \\[\\PP((X_1,\\ldots, X_k)=(v_1,\\ldots, v_k)) = \\frac{q_e}{k!}=\\frac{\\alpha}{k!} \\prod_{i=1}^{k}x_{v_i}.\\]\n Therefore, using \\Cref{claim:entropy-lag} with $\\beta=k!/\\alpha$, we see that\n \\begin{align*}\n \\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)=&\\log_2\\beta-\\frac{k}{p}\\sum_{v\\in V(G)}y_v\\log_2 \\left(\\frac{x_v^p}{y_v}\\right),\n \\end{align*}\n where, in this case, $y_v=x_v^p$. Thus, $\\HH(X_1,\\dots,X_k)-\\frac{k}{p}\\HH(X_1)=\\log_2 \\beta$.\n Note that $\\sum_{v\\in V(G)}x_v^p=1$. \n Therefore by the fact that \n \\[\\beta = \\sum_{(v_1,\\ldots, v_k)\\in \\vec{E}(G)}\\prod_{i=1}^{k}x_{v_i},\\]\n we have $b_{p,\\textup{entropy}}(G)\\leq b_p(G)$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:equiv-to-entropy}\n For any family $\\cF$ of $k$-graphs, $\\pi(\\cF)$ is the supremum of $2^{\\HH(X_1,\\ldots, X_k)-k\\HH(X_1)}$ for any random edge with uniform ordering $(X_1,\\ldots, X_k)$ on any $\\cF$-hom-free $k$-graph $G$.\n\\end{corollary}\n\\begin{proof}\n Since $\\pi(\\cF)$ is the supremum of $b(G)$ for all $\\cF$-hom-free $k$-graphs $G$, we know that $\\pi(\\cF)$ is the supremum of $b_{\\textup{entropy}}(G)$ for all $\\cF$-hom-free $k$-graphs $G$ as well.\n The statement follows from the definition of entropic density $b_{\\textup{entropy}}(G)$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:entropy-equiv-to-Turan}\n The entropic Tur\\'an theorem (\\cref{thm:entropic-turan}) is equivalent to the density Tur\\'an theorem.\n\\end{corollary}\n\\begin{proof}\n By \\cref{cor:equiv-to-entropy}, it suffices to show that if $G$ is $K_{r+1}$-free, then $G$ is $K_{r+1}$-hom-free.\n This is clear as any homomorphic image of $K_{r+1}$ is $K_{r+1}$.\n\\end{proof}\n\\begin{remark}\n In the previous section, we showed that \\cref{thm:entropic-turan} implies the density Tur\\'an theorem using a simpler argument.\n This turns out to be the direction we care about in this paper.\n For all the Tur\\'an-type results proven later in this paper using entropy and \\cref{prop:entropic-lagrangian}, we may also avoid the use of \\cref{prop:entropic-lagrangian} by a similar simpler argument.\n However, we think \\cref{prop:entropic-lagrangian} is interesting on its own, so we establish the proposition here and will freely use it from now on.\n\\end{remark}\n\n Setting $p=2$, we can now prove \\cref{thm:spectral-Turan-tree} by combining \\cref{thm:entropic-turan} and Szegedy's method of sampling a random homomorphic image of the tree $T$.\n \\begin{proof}[Proof of \\cref{thm:spectral-Turan-tree}]\n From \\cref{prop:entropic-lagrangian} and the observation in \\cref{ex:entropic-lagrangian}, there exists a random edge with uniform ordering $(X,Y)$ on $G$ such that $\\log_2\\rho(G)=\\HH(Y\\mid X)$. By \\cref{thm:entropic-turan}, we have\n \\[\\ell\\log_2\\rho(G)=\\ell\\HH(Y\\mid X)\\leq \\HH(X)+(\\ell-1)\\HH(Y\\mid X)+\\log_2\\left(1-\\frac{1}{r}\\right).\\]\n \n Let $v_1,\\ldots, v_{\\ell}$ be an ordering of the vertices of $T$ where for every $i\\in\\{2,\\ldots,\\ell\\}$, the vertex $v_i$ is adjacent to exactly one $v_j$ with $j<i$.\n Now, we sample random vertices $X_1,\\dots,X_\\ell$ in $G$ as follows. Let $X_1$ be a random vertex sampled using the law of $X$. Assume we have already sampled $X_1,\\dots,X_{i-1}$, and assume $v_j$ is the neighbor of $v_i$ with $j<i$. We sample $X_i$ conditionally independently such that $X_i\\mid X_j\\sim Y\\mid X$. It follows that $X_1,\\dots,X_\\ell$ is always a homomorphic image of $T$ in $G$. Also, from the way we sample, we know that $\\HH(X_1,\\dots,X_\\ell)=\\HH(X)+(\\ell-1)\\HH(Y\\mid X)$. Thus, we have\n \\[\\HH(X)+(\\ell-1)\\HH(Y\\mid X)=\\HH(X_1,\\dots,X_\\ell)\\leq \\log_2\\#\\{\\text{homomorphisms from $T$ to $G$}\\},\\]\n and we are done by combining this with the previous inequality and rearranging.\n \\end{proof}\n\n For general $p$, recall that our definition of $b_p(G)$ matches the definition of $p$-spectral radius given by Keevash, Lenz and Mubayi. Thus, by combining \\cref{prop:entropic-lagrangian} with \\cref{thm:entropic-turan}, we recover the following theorem for graphs by Kang and Nikiforov \\cite{KN14}.\n \\begin{theorem}[\\cite{KN14}]\\label{thm:p-spectral-Turan}\n Let $r\\geq 2$ be a positive integer and $p\\geq 1$ be a real number. For any $K_{r+1}$-free graph $G$ with $n$ vertices and $m$ edges, we have\n \\[b_p(G)\\leq \\left(1-\\frac{1}{r}\\right)n^{2-2/p},\\]\n and\n \\[b_p(G)\\leq \\left(1-\\frac{1}{r}\\right)^{1/p}(2m)^{1-1/p}.\\]\n \\end{theorem}\n \\begin{proof}\n From \\cref{prop:entropic-lagrangian}, there exists a random edge with uniform ordering $(X,Y)$ on $G$ such that $\\log_2b_p(G)=\\HH(X,Y)-\\frac{2}{p}\\HH(X)$. We have\n \\[\\HH(X,Y)-\\frac{2}{p}\\HH(X)\\leq \\left(2-\\frac{2}{p}\\right)\\HH(X)+\\log_2\\left(1-\\frac{1}{r}\\right)\\leq \\left(2-\\frac{2}{p}\\right)\\log_2 n+\\left(1-\\frac{1}{r}\\right),\\]\n and\n \\[\\HH(X,Y)-\\frac{2}{p}\\HH(X)\\leq \\left(1-\\frac{1}{p}\\right)\\HH(X,Y)+\\frac{1}{p}\\log_2\\left(1-\\frac{1}{r}\\right)\\leq \\left(1-\\frac{1}{p}\\right)\\log_2 (2m)+\\frac{1}{p}\\left(1-\\frac{1}{r}\\right).\\qedhere\\]\n \\end{proof}\n \n We also remark that, by utilizing \\cref{prop:entropic-lagrangian}, we can translate \\cref{thm:main-entropy-ver} and also results in \\cref{sec:known} into spectral results using arguments in the proofs of \\cref{thm:spectral-Turan-tree} and \\cref{thm:p-spectral-Turan}.\n\n\n\\section{Partial hypergraphs}\\label{sec:partial-hypergraph}\nIn this section, we introduce some notations and an entropic lemma that will be useful in the later sections.\nThose notations are non-standard and are set for our own notational convenience when describing hypergraphs and homomorphisms.\n\nA \\emph{partial $k$-graph} $F$ is a simplicial complex whose faces have size at most $k$.\nIts set of vertices is denoted by $V(F)$, and its set of faces, or partial edges, is denoted by $E(F)$.\nA \\emph{homomorphism} from a partial $k$-graph $F$ to a $k$-graph $G$ is a map $f:V(F)\\to V(G)$ such that for any partial edge $e\\in E(F)$, $f$ is injective on $e$ and $f(e)$ is contained in some edge in $E(G)$.\nNow for any partial $k$-graph $F$, its \\emph{extension} $\\Tilde{F}$ is the $k$-graph obtained as follows: first let $E'$ be the set of maximal partial edges in $E(F)$, and then extend each partial edge in $E'$ to a $k$-edge by adding in extra vertices, where two different edges do not share any extra vertices.\nNotice that if $F$ is a simplicial complex generated by edges of some $k'$-graph $F'$ with $k'<k$, then $\\Tilde{F}$ is the extension of $F'$ as defined in the introduction.\n\n\\begin{example}[Definition of partial tents]\n In \\cref{sec:main-proof}, the partial $k$-graphs and the corresponding extensions of concern would be the following.\nFor any partition $\\lambda$ of $k$ with $\\ell\\eqdef \\ell(\\lambda)\\geq 2$, the \\emph{partial $\\lambda$-tent} $\\Delta^p_\\lambda$ is the partial $k$-graph obtained by taking the simplicial complex generated by $\\Delta_\\lambda$, and then restricting it to $e\\cup \\{v\\}$ where $e$ is the base and $v$ is the apex.\nIt is easy to verify that $\\Delta_\\lambda$ is the extension of the partial $k$-graph $\\Delta^p_{\\lambda}$.\n\\end{example}\n\n\n\n\\begin{figure}[h]\\centering\\label{fig:PartialTent}\n\\begin{tikzpicture}[scale=0.8]\n\n \\coordinate (A) at (0,0);\n \\coordinate (B) at (1,0);\n \\coordinate (C) at (2,0);\n \\coordinate (D) at (5,0);\n \\coordinate (E) at (6,0);\n \\coordinate (F) at (4.5,1.732/2);\n \\coordinate (G) at (2.75,1.732*3/4);\n \\coordinate (H) at (4,1.732);\n \\coordinate (I) at (3.5,1.732*3/2);\n\n \\draw [fill] (A) circle (1.6pt);\n \\draw [fill] (B) circle (1.6pt);\n \\draw [fill] (C) circle (1.6pt);\n \\draw [fill] (D) circle (1.6pt);\n \\draw [fill] (E) circle (1.6pt);\n \\draw [fill] (I) circle (1.6pt);\n\\draw[rounded corners=8pt,black,line width=2pt] \n (0-0.2,0.5)--(-0.5-0.2,0)--(0-0.2,-0.5)--(6+0.2,-0.5)--(6.5+0.2,0)--(6+0.2,0.5)--cycle;\n \n\\draw[rounded corners=6pt,black,line width=2pt] \n(0-0.2,0.3)--(-0.3-0.2,0)--(0-0.2,-0.3)--(2+0.3/1.732,-0.3)--(3.5+0.1+0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(3.5+0.1+0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(3.5+0.1-0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(2-0.3/1.732,0.3)--cycle;\n\n\\draw[rounded corners=6pt,black,line width=2pt] \n(6+0.2,0.3)--(6+0.3+0.2,0)--(6+0.2,-0.3)--(5-0.3/1.732,-0.3)--(3.5-0.1-0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(3.5-0.1-0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(3.5-0.1+0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(5+0.3/1.732,0.3)--cycle;\n\n\\draw[black,line width=2pt]\n(7,1.4)--(9,1.4);\n\n\\draw[black,line width=2pt]\n(8.7,1.7)--(9,1.4);\n\n\\draw[black,line width=2pt]\n(8.7,1.1)--(9,1.4);\n\n\\node at (8,2) {extension};\n\n\n \\draw [fill] (9,1.4) circle (0.9pt);\n\n \\draw [fill] (A)+(10,0) circle (1.6pt);\n \\draw [fill] (B)+(10,0) circle (1.6pt);\n \\draw [fill] (C)+(10,0) circle (1.6pt);\n \\draw [fill] (D)+(10,0) circle (1.6pt);\n \\draw [fill] (E)+(10,0) circle (1.6pt);\n \\draw [fill,red] (F)+(10,0) circle (1.6pt);\n \\draw [fill,red] (G)+(10,0) circle (1.6pt);\n \\draw [fill,red] (H)+(10,0) circle (1.6pt);\n \\draw [fill] (I)+(10,0) circle (1.6pt);\n\\draw[rounded corners=8pt,black,line width=2pt] \n (10+0-0.2,0.5)--(10-0.5-0.2,0)--(10+0-0.2,-0.5)--(10+6+0.2,-0.5)--(10+6.5+0.2,0)--(10+6+0.2,0.5)--cycle;\n \n\\draw[rounded corners=6pt,black,line width=2pt] \n(10+0-0.2,0.3)--(10-0.3-0.2,0)--(10+0-0.2,-0.3)--(10+2+0.3/1.732,-0.3)--(10+3.5+0.1+0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(10+3.5+0.1+0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(10+3.5+0.1-0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(10+2-0.3/1.732,0.3)--cycle;\n\n\\draw[rounded corners=6pt,black,line width=2pt] \n(10+6+0.2,0.3)--(10+6+0.3+0.2,0)--(10+6+0.2,-0.3)--(10+5-0.3/1.732,-0.3)--(10+3.5-0.1-0.15*1.732,1.732*3/2+0.1*1.732-0.15)--(10+3.5-0.1-0.15,1.732*3/2+0.1*1.732+0.15*1.732)--(10+3.5-0.1+0.15*1.732,1.732*3/2+0.1*1.732+0.15)--(10+5+0.3/1.732,0.3)--cycle;\n\n\n\n\n\\end{tikzpicture}\n\\caption{Partial $(3,2)$-tent and its extension. Note that for the partial tent, only the maximal edges are shown.}\n\\end{figure}\n\n\n\nThose definitions are useful as for any partial $k$-graph $F$, a homomorphism $F\\to G$ is essentially the same as a homomorphism $\\Tilde{F}\\to G$.\nThis would be helpful later as instead of considering homomorphisms from $\\Delta_{\\lambda}$, we can consider homomorphisms from $\\Delta^p_{\\lambda}$, which are easier to describe.\n\n\n\n\n\\begin{proposition}\\label{prop:partial-tent-hom}\n Let $F$ be a partial $k$-graph, and let $G$ be a $k$-graph.\n Then there is a homomorphic image of $F$ in $G$ if and only if there is a homomorphic image of $\\Tilde{F}$ in $G$.\n\\end{proposition}\n\\begin{proof}\n Let $(\\Tilde{F})_{\\textup{cpx}}$ be the simplicial complex generated by the edges in $\\Tilde{F}$, which we will think of as a partial $k$-graph.\n Then $F$ is the restriction of $(\\Tilde{F})_{\\textup{cpx}}$ on $V(F)$.\n For any homomorphism $f:V(\\Tilde{F})\\to V(G)$ from $\\Tilde{F}$ to $G$, we also have that it is an homomorphism from $(\\Tilde{F})_{\\textup{cpx}}$ to $G$.\n It is then easy to check that $f|_{V(F)}$ is a homomorphism from $F$ to $G$.\n\n Conversely, suppose that $g:V(F)\\to V(G)$ is a homomorphism from $F$ to $G$.\n Note that for each $e\\in E(\\Tilde{F})$, we know that $e\\cap V(F)\\in E((\\Tilde{F})_{\\textup{cpx}})$ and so $e\\cap V(F)$ is in $E(F)$ as well.\n By the definition, this implies that for every $e\\in E(\\Tilde{F})$, we have that $g$ is injective on $e\\cap V(F)$ and $g(e\\cap V(F))$ is contained in some edge in $G$.\n As any vertex in $V(\\Tilde{F})\\backslash V(F)$ is in exactly one edge in $E(\\Tilde{F})$, it is possible to extend $g$ to $\\Tilde{g}:V(\\Tilde{F})\\to V(G)$ so that $g(e)$ is an edge in $G$ for each $e\\in E(\\Tilde{F})$.\n The extended map $\\Tilde{g}$ is indeed a homomorphism from $\\Tilde{F}$ to $G$.\n\\end{proof}\n\nLater on, as in the proof in \\cref{sec:entropy}, we will need to show that we can sample random homomorphisms from some tree-like structures with high entropy.\nBefore we can do so, we need to first describe what the tree-like structures are.\n\n\\begin{definition}[Partial forest and forest sequence]\n \nFor any partial $k$-graph $F$, any linear order $<$ on $V(F)$, and any vertex $v\\in V(F)$, let $M_{F,<}(v)$ be the set of partial edges whose maximum vertex is $v$.\nA partial $k$-graph $F$ is a \\emph{partial forest} with respect to a linear order $<$ on $V(F)$ if for every $v\\in V(F)$, there is exactly one maximal partial edge $e_v$ in $M_{F,<}(v)$.\nIn this case, the \\emph{forest sequence} of $(F,<)$ is a sequence $(n_{1},\\ldots, n_{k})$ where for each $i\\in[k]$, $n_{i}$ is the number of vertices $v\\in V(F)$ with $\\abs{e_v}=i$. \n\\end{definition}\n\nWe also define quantities that are analogs of the quantity $2^{\\HH(Y\\mid X)-\\HH(X)}$ we used in \\cref{sec:entropy}.\n\n\\begin{definition}[Ratio sequence]\n Let $(X_1,\\dots,X_k)\\in V(G)^k$ be a random edge with uniform ordering on a $k$-graph $G$. We define the \\emph{ratio sequence} $0< x_1\\leq\\dots\\leq x_k= 1$ of $(X_1,\\dots,X_k)$ by $x_i=2^{\\HH(X_i\\mid X_{i+1},\\dots,X_k)-\\HH(X_i)}$ for each $i\\in [k]$.\n\\end{definition}\n\nWe are now ready to apply Szegedy's argument to sample homomorphisms from partial forests with high entropy.\n\n\\begin{lemma}\\label{lemma:sample-tree}\n Let $(X_1,\\ldots, X_k)$ be a random edge with uniform ordering on a $k$-graph $G$ and let $x_1,\\ldots, x_k$ be its ratio sequence.\n For any partial forest $F$ with a linear order $<$, if $(n_1,\\ldots, n_k)$ is its forest sequence, then one can sample a random homomorphism $(Y_v)_{v\\in V(F)}$ from $F$ to $G$ with entropy equal to\n \\[v(F)H(X_1)+\\log_2\\left(\\prod_{i=1}^{k}x_i^{n_{k+1-i}}\\right).\\]\n Moreover, the random homomorphism can be sampled such that for any partial edge $e\\in E(F)$, the distribution of $(Y_v)_{v\\in e}$ is the same as $(X_i)_{k-\\abs{e}+1\\leq i\\leq k}$.\n\\end{lemma}\n\\begin{proof}\n We will induct on $v(F)$.\n The case $v(F)=0$ is vacuously true.\n Now suppose that it holds for partial forest of size $v(F)-1$.\n Let $v_{\\max}$ be the maximum vertex in $V(F)$.\n Then $F\\backslash \\{v_{\\max}\\}$ is also a partial forest, and so we may sample a random homomorphism $(Y_v)_{v\\in V(F)\\backslash \\{v_{\\max}\\}}$ with the prescribed properties.\n Let $e$ be the maximal partial edge in $M_{F,<}(v_{\\textup{max}})$, and let $j=k+1-\\abs{e}$.\n By the inductive hypothesis, $(Y_v)_{v\\in e\\backslash v_{\\max}}$ is identically distributed as $(X_i)_{j+1\\leq i\\leq k}$.\n Therefore, we may sample $Y_{v_{\\max}}$ given $(Y_v)_{v\\in e\\backslash v_{\\max}}$ conditionally independently so that $(Y_v)_{v\\in e}$ is identically distributed as $(X_i)_{j\\leq i\\leq k}$.\n This way,\n \\begin{align*}\n \\HH\\left((Y_v)_{v\\in V(F)}\\right)=&\\HH\\left((Y_v)_{v\\in V(F)\\backslash \\{v_{\\max}\\}}\\right)+\\HH\\left(Y_{v_{\\max}}\\mid (Y_v)_{v\\in e\\backslash \\{v_{\\max}\\}}\\right)\\\\\n =&\\left(v(F)-1\\right)H(X_1)+\\log_2\\left(x_j^{-1}\\prod_{i=1}^{k}x_i^{n_{k+1-i}}\\right)+H(X_j\\mid X_{j+1},\\ldots, X_k)\\\\\n =&v(F)H(X_1)+\\log_2\\left(\\prod_{i=1}^{k}x_i^{n_{k+1-i}}\\right)\n \\end{align*}\n where we use that $\\HH(X_i)= \\HH(X_1)$ for any $i\\in[k]$.\n It remains to show that for any partial edge $e'$ containing $v_{\\max}$, the distribution of $(Y_v)_{v\\in e'}$ is the same as $(X_i)_{k-\\abs{e'}+1\\leq i\\leq k}$.\n This is true as $e'\\subseteq e$ by the definition of $e$ and $v_{\\max}$, and the distribution $(X_1,\\ldots, X_k)$ is symmetric.\n\\end{proof}\n\n\n\\section{Proof of \\cref{thm:main-tent,thm:one-tent}}\\label{sec:main-proof}\nIn this section, we will first give two proofs of \\cref{thm:main-tent}.\nWe will then show how \\cref{thm:main-tent} implies \\cref{thm:one-tent}.\nFinally, we will conclude this section with a proof of \\cref{thm:not-Turan}.\n\nThroughout this section, we will fix a $k$-graph $G$ and a random edge with uniform ordering $(X_1,\\ldots, X_k)$ on $G$.\nWe will also set $0< x_1\\leq \\cdots\\leq x_k=1$ to be its ratio sequence.\nWe make an observation that to upper bound $b(G)=b_{\\textup{entropy}}(G)$, it suffices to upper bound\\linebreak $2^{\\HH(X_1,\\ldots, X_k)-\\HH(X_k)}=x_1\\cdots x_{k-1}$ by the chain rule. \nTherefore, the upper bound of \\cref{thm:main-tent} follows from the following statement.\n\n\\begin{theorem}\\label{thm:main-entropy-ver}\n If $G$ is $\\lambda$-tent-hom-free for every $\\abs{\\lambda} = k$ and $\\ell(\\lambda)=2$, then we have\n \\[\\HH(X_1,\\dots,X_k)- k\\HH(X_1)=\\log_2(x_1\\cdots x_k)\\leq \\log_2\\frac{k!}{k^k}.\\]\n\\end{theorem}\n\nWe first show that \\cref{thm:main-tent} indeed follows from \\cref{thm:main-entropy-ver}.\n\\begin{proof}[Proof of \\cref{thm:main-tent} using \\cref{thm:main-entropy-ver}]\n First, it is clear that $\\pi(\\cF_k)\\geq k!/k^k$ as a single edge does not contain any homomorphic image of any tents, and it has blowup density $k!/k^k$.\n To show the reverse inequality, if $G$ is $\\cF_k$-hom-free, then by \\cref{thm:main-entropy-ver}, we have $b(G)=b_{\\textup{entropy}}(G)\\leq k!/k^k$.\n This shows that $\\pi(\\cF_k)\\leq k!/k^k$.\n\\end{proof}\n\n\\subsection{First proof of \\cref{thm:main-entropy-ver}}\n\nTo prove \\cref{thm:main-entropy-ver}, we will apply \\cref{lemma:sample-tree} and \\cref{lemma:mix} to obtain several inequalities involving $x_1,\\ldots,x_{k}$.\nThen we will solve for the maximum of $x_1\\cdots x_{k-1}$ subject to the inequalities.\n\n\n\\begin{lemma}\\label{lemma:alt-tent-ineq}\nIf $G$ is $\\lambda$-tent-hom-free for every $\\abs{\\lambda} = k$ and $\\ell(\\lambda)=2$, then for any $i,j\\in[k]$ with $i+j\\leq k$, we have $x_i+x_j\\leq x_{i+j}$.\n \n\\end{lemma}\n\\begin{proof}\n We will consider two partial forests $F^{(1)}$ and $F^{(2)}$ both on $V=\\{v_1,\\ldots, v_k,w\\}$.\n Let $F^{(1)}$ be spanned by the two partial edges $\\{v_1,\\ldots, v_k\\}$ and $\\{v_{i+1},\\ldots,v_k, w\\}$.\n Let $F^{(2)}$ be spanned by the two partial edges $\\{v_1,\\ldots, v_k\\}$ and $\\{v_1,\\ldots, v_{k-j}, w\\}$.\n Then both partial $k$-graphs are indeed partial forests with respect to the linear order $v_1<\\cdots <v_k<w$.\n It is clear that in $F^{(1)}$ with the forest sequence $(n_1,\\ldots, n_k)$, the vertices $v_1,\\ldots, v_k$ contribute one to $n_1,\\ldots, n_k$ and $w$ contributes to $n_{k-i+1}$.\n Similarly, the forest sequence of $F^{(2)}$ is all-one except for $n_{k-j+1}=2$.\n\n Let $(Y^{(1)}_v)_{v\\in V},(Y^{(2)}_v)_{v\\in V}$ be the random homomorphism from $F^{(1)},F^{(2)}$ given by \\cref{lemma:sample-tree}, respectively.\n Note that if some tuple of vertices is in the supports of both $(Y^{(1)}_v)_{v\\in V}$ and $(Y^{(2)}_v)_{v\\in V}$, then this tuple corresponds to a homomorphism from $F^{(1)}\\cup F^{(2)}$ to $G$.\n As $F^{(1)}\\cup F^{(2)}$ clearly contains a partial $(i,k-i)$-tent with base $\\{v_1,\\ldots, v_k\\}$ and apex $w$, we know that the two random homomorphisms have disjoint support.\n Suppose that $(Z_v)_{v\\in V}$ is the mixture given by \\cref{lemma:mix}, then by \\cref{lemma:mix,lemma:sample-tree} we know\n \\[\\left(x_1\\cdots x_{k-1}\\cdot x_i+x_1\\cdots x_{k-1}\\cdot x_j\\right)2^{(k+1)\\HH(X_1)}\\leq 2^{\\HH((Z_v)_{v\\in V})}.\\]\n Observe that both $F^{(1)}$ and $F^{(2)}$ contains the partial edges $\\{v_1,\\ldots, v_k\\}$ and $\\{v_{i+1},\\ldots, v_{k-j},w\\}$.\n Therefore $(Y^{(1)}_{v_1},\\ldots, Y^{(1)}_{v_k})$ and $(Y^{(2)}_{v_1},\\ldots, Y^{(2)}_{v_k})$ both have the same distributions as $(X_1,\\ldots, X_k)$ by \\cref{lemma:sample-tree}, which shows that $(Z_{v_1},\\ldots, Z_{v_k})$ has the same distribution as $(X_1,\\ldots, X_k)$ as well.\n Using a similar argument, we can show that $(Z_w, Z_{v_{i+1}},\\ldots, Z_{v_{k-j}})$ has the same distribution as $(X_{i+j}, \\ldots, X_k)$.\n As a consequence,\n \\begin{align*}\n \\HH\\left((Z_v)_{v\\in V}\\right)\\leq& \\HH(Z_{v_1},\\ldots, Z_{v_k})+\\HH(Z_w\\mid Z_{v_{i+1}},\\ldots, Z_{v_{k-j}})\\\\\n =&\\HH(X_1,\\ldots, X_{k})+\\HH(X_{i+j}\\mid X_{i+j+1},\\ldots, X_{k})\\\\\n =& (k+1)\\HH(X_1)+\\log_2(x_1\\cdots x_{k-1}\\cdot x_{i+j}).\n \\end{align*}\n This shows that \n \\[x_1\\cdots x_{k-1}2^{(k+1)\\HH(X_1)}(x_i+x_j)\\leq x_1\\cdots x_{k-1}2^{(k+1)\\HH(X_1)}\\cdot x_{i+j}\\]\n and so the desired statement follows.\n\\end{proof}\n\nOur next goal is to upper bound $ x_1\\cdots x_{k-1}$.\nTo upper bound the product, we prove the following auxiliary inequality.\n\n\\begin{lemma}\\label{lemma:aux-ineq}\n Suppose that $y_1,\\ldots, y_k$ are some non-negative real numbers with $y_i+y_j\\leq y_{i+j}$ for any $i,j\\in[k]$ with $i+j\\leq k$.\n Then\n \\[y_1\\cdots y_k\\leq k!\\left(\\frac{y_1+\\cdots +y_k}{\\binom{k+1}{2}}\\right)^k.\\]\n\\end{lemma}\n\\begin{proof}\n We will prove this by induction.\n It clearly holds when $k=1$.\n Now suppose that $k\\geq 2$ and the statement holds for $k-1$.\n Then by the inductive hypothesis,\n \\[y_1\\cdots y_k\\leq (k-1)!\\left(\\frac{y_1+\\cdots +y_{k-1}}{\\binom{k}{2}}\\right)^{k-1}y_k\\leq k!\\left(\\frac{(k-1)\\cdot\\frac{y_1+\\cdots+y_{k-1}}{\\binom{k}{2}}+\\frac{y_k}{k}}{k}\\right)^k\\]\n by AM-GM.\n Since\n \\[y_1+\\dots+y_{k-1} = \\frac{1}{2}\\sum_{i=1}^{k-1}(y_i+y_{k-i})\\leq \\frac{k-1}{2}y_k,\\]\n we know \n \\[(k-1)\\cdot\\frac{y_1+\\cdots+y_{k-1}}{\\binom{k}{2}}+\\frac{y_k}{k} = \\frac{2}{k}\\left(y_1+\\cdots +y_{k-1}+\\frac{y_k}{2}\\right)\\leq \\frac{2}{k}\\cdot \\frac{k}{k+1}\\left(y_1+\\cdots +y_k\\right)\\]\n and so\n \\[y_1\\cdots y_k\\leq k!\\left(\\frac{\\frac{2}{k+1}(y_1+\\cdots+y_k)}{k}\\right)^k =k!\\left(\\frac{y_1+\\cdots +y_k}{\\binom{k+1}{2}}\\right)^k, \\]\n as desired.\n\\end{proof}\n\nCombining \\cref{lemma:alt-tent-ineq} and \\cref{lemma:aux-ineq}, we are now ready to prove \\cref{thm:main-entropy-ver}.\n\n\\begin{proof}[Proof of \\cref{thm:main-entropy-ver}]\n By \\cref{lemma:alt-tent-ineq}, $x_1,\\ldots, x_k$ are non-negative reals satisfying the condition of \\cref{lemma:aux-ineq}.\n We also know that $x_k=1$, so $x_1+\\cdots +x_k\\leq \\frac{k-1}{2}+1 = \\frac{k+1}{2}$.\n Thus by \\cref{lemma:aux-ineq},\n \\[x_1\\cdots x_{k-1} = x_1\\cdots x_k\\leq k!\\left(\\frac{\\frac{k+1}{2}}{\\binom{k+1}{2}}\\right)^k = \\frac{k!}{k^k},\\]\n which is the desired statement\n\\end{proof}\n\\subsection{Second proof of \\cref{thm:main-entropy-ver}}\nHere, we give an alternative proof using much more complicated partial forests.\nAlthough the proof is more involved, this proof would be the one we generalize later in \\cref{sec:known}.\n\n\\begin{lemma}\\label{lemma:tent-ineq}\n If $G$ is $\\lambda$-tent-hom-free for every $\\abs{\\lambda}=k$ and $\\ell(\\lambda)=2$, then for every $i\\in[k-1]$, we have $x_j<x_{i+1}$ for each $j\\leq i$ and\n \\[\\prod_{j=1}^{i}\\frac{x_j}{x_{i+1}-x_j}\\leq 1.\\]\n\\end{lemma}\n\\begin{proof}\n\n We will fix $i$ throughout this proof.\n As in what we did in \\cref{sec:entropy}, we will temporarily fix an integer $N\\in\\NN$ that will later be taken to infinity.\n For any $1=t_0<t_1<t_2<\\cdots <t_i\\leq N$, we will define a partial forest $F^{(\\vec{t})}$ on $V=\\{v_1,\\ldots, v_{k-i-1}, w_1,\\ldots, w_N\\}.$\n The partial forest $F^{(\\vec{t})}$ is spanned by the partial edges $\\{w_m, w_{t_{j+1}},\\ldots, w_{t_i}\\}\\cup \\{v_1,\\ldots, v_{k-i-1}\\}$ for every $t_j\\leq m<t_{j+1}$, where $t_{i+1}$ is set to be $N+1$.\n This is indeed a partial forest with respect to the linear order $<$ with $v_1<\\cdots<v_{k-i-1}<w_N<\\cdots <w_1$.\n We can compute the forest sequence with respect to the linear order as follows: each $v_j$ contributes one to $n_j$ for each $j\\leq k-i-1$, and each $w_m$ with $t_j\\leq m<t_{j+1}$ contributes $1$ to $n_{k-j}$.\n Therefore the forest sequence $(n_1,\\ldots, n_k)$ is $(t_1-t_0,\\ldots, t_{i+1}-t_i, 1,\\ldots, 1)$.\n Now let $(Y^{(\\vec{t})}_v)_{v\\in V}$ be the random homomorphism produced by \\cref{lemma:sample-tree}.\n This gives\n \\begin{align}\\label{eq:tent-2ndproof-eq1}\n \\HH\\left((Y^{(\\vec{t})}_v)_{v\\in V}\\right) = (N+k-i-1)\\HH(X_1)+\\log_2\\left(x_{i+2}\\cdots x_{k}\\cdot \\prod_{j\\leq i+1}x_j^{t_{j}-t_{j-1}}\\right).\n \\end{align}\n\n We will now show that the supports of $(Y^{(\\vec{t})}_v)_{v\\in V}$ are disjoint for different choices of $\\vec{t}$.\n Suppose for the sake of contradiction that for some $\\vec{t}\\neq \\vec{t}'$ there is a tuple of vertices from $V(G)$ lying in the supports of $(Y^{(\\vec{t})}_v)_{v\\in V}$ and $(Y^{(\\vec{t}')}_v)_{v\\in V}.$\n Then this tuple witnesses a homomorphism sending $F^{(\\vec{t})}\\cup F^{(\\vec{t}')}$ to $G$.\n We will show a contradiction by demonstrating that $F^{(\\vec{t})}\\cup F^{(\\vec{t}')}$ contains a homomorphic image of some partial $\\lambda$-tent with $\\ell(\\lambda)=2$.\n\n Let $j\\geq 1$ be the minimum index in which $\\vec{t}$ and $\\vec{t}'$ differ, and without loss of generality, suppose that $t_j'<t_j$.\n Then we can find partial edges $e=\\{v_1,\\ldots,v_{k-i-1},w_{t_0},w_{t_1},\\ldots, w_{t_i}\\}$,\\linebreak $e_1=\\{v_1,\\ldots, v_{k-i-1},w_{t_j'}, w_{t_j},\\ldots, w_{t_i}\\}$ in $F^{(\\vec{t})}$ and $e_2 = \\{w_{t_0'},\\ldots, w_{t_j'}\\}$ in $F^{( \\vec{t}')}$.\n By the minimality of $j$, we know $e_2=\\{w_{t_0},\\ldots, w_{t_{j-1}}, w_{t_j'}\\}.$\n Note that $e,e_1,e_2$ form a partial $(k-j,j)$-tent with base $e$ and apex $w_{t_j'}$, showing that $F^{(\\vec{t})}\\cup F^{(\\vec{t}')}$ contains a partial $(k-j,j)$-tent, which is a contradiction.\n \n Therefore we may now apply \\cref{lemma:mix} with $a=1$.\n Suppose that $(Z_v)_{v\\in V}$ is the resulting mixture of $(Y_v^{(\\vec{t})})_{v\\in V}$ for all possible $\\vec{t}$.\n By \\cref{lemma:sample-tree} and the fact that $\\{v_1,\\ldots, v_{k-i-1},w_m\\}$ is present in all partial forests we take for any $m\\in[N]$, we know that $(Z_{v_1},\\ldots, Z_{v_{k-i-1}},Z_{w_m})$ has the same distribution as $(X_{i+1},\\ldots, X_k)$ for each $m\\in[N]$.\n Hence\n \\begin{align}\n \\HH\\left((Z_v)_{v\\in V}\\right)\\leq& \\HH(Z_{v_1},\\ldots, Z_{v_{k-i-1}})+\\sum_{m=1}^{N}\\HH(Z_{w_m}\\mid Z_{v_1},\\ldots, Z_{v_{k-i-1}}) \\nonumber\\\\\n =& \\HH(X_{i+2},\\ldots,X_k)+N\\HH(X_{i+1}\\mid X_{i+2},\\ldots,X_k) \\nonumber\\\\\n =& (N+k-i-1)\\HH(X_1)+\\log_2(x_{i+2}\\cdots x_k\\cdot x_{i+1}^N).\\label{eq:tent-2ndproof-eq2}\n \\end{align}\n Thus \\cref{lemma:mix,eq:tent-2ndproof-eq1,eq:tent-2ndproof-eq2} now gives\n \\[2^{(N+k-i-1)\\HH(X_1)}\\sum_{1=t_0<t_1<\\cdots <t_{i+1}= N+1}x_{i+2}\\cdots x_k\\cdot \\prod_{j\\leq i+1}x_j^{t_{j}-t_{j-1}}\\leq x_{i+2}\\cdots x_k\\cdot x_{i+1}^{N}\\cdot 2^{(N+k-i-1)\\HH(X_1)},\\]\n and so\n \\[\\sum_{1=t_0<t_1<\\cdots <t_{i+1}= N+1}\\prod_{j\\leq i+1}\\left(\\frac{x_j}{x_{i+1}}\\right)^{t_j-t_{j-1}}\\leq 1.\\]\n Note that we may replace $j\\leq i+1$ by $j<i+1$ in the product.\n This way, when we take $N$ to approach infinity, we must have $x_j<x_{i+1}$ for each $j\\in[i]$ in order for the left hand side to converge.\n Moreover, the left hand side becomes\n \\[\\sum_{\\delta_1,\\ldots, \\delta_{i}\\in \\NN}\\prod_{j\\leq i}\\left(\\frac{x_j}{x_{i+1}}\\right)^{\\delta_j} =\\prod_{j\\leq i}\\frac{x_j}{x_{i+1}-x_j},\\]\n as desired.\n\\end{proof}\n\nOnce again, to prove \\cref{thm:main-entropy-ver}, we need to upper bound $x_1\\cdots x_{k-1}$ given the inequalities in \\cref{lemma:tent-ineq}.\nWe will prove a slightly stronger statement, which will also be useful in the next section.\n\n\\begin{lemma}\\label{lemma:another-aux-ineq}\n Let $k$ be a positive integer.\n Fix real numbers $0<z_1<\\cdots <z_k$.\n Let $0< y_1< \\ldots< y_k$ be real numbers with\n \\[\\prod_{j\\leq i}\\frac{y_j}{y_{i+1}-y_j}\\leq \\prod_{j\\leq i}\\frac{z_j}{z_{i+1}-z_j}\\]\n for any $i=1,\\ldots, k-1$.\n Then \n \\[y_1\\cdots y_{k-1}\\leq \\frac{z_1\\cdots z_{k-1}}{z_k^{k-1}}y_k^{k-1}.\\]\n\\end{lemma}\n\\begin{proof}\n We will prove by induction on $k$.\n When $k=1$ this is clearly true.\n Now suppose that $k\\geq 2$ and the statement is true for all smaller $k$.\n Then we have\n \\[\\frac{y_1\\cdots y_i}{z_1\\cdots z_i}\\leq \\frac{y_{i+1}^i}{z_{i+1}^{i}}\\]\n for all $i<k-1$ by the inductive hypothesis.\n Now let \n \\[\\alpha_i =\\frac{1}{i}\\sum_{j\\leq i} \\frac{z_j}{z_k-z_j}\\] for any $i\\leq k-1$.\n Note that for any $i<k-1$, we have\n \\[\\left(\\frac{y_1\\cdots y_{i+1}}{z_1\\cdots z_{i+1}}\\right)^{\\alpha_{i+1}}\\leq \\left(\\frac{y_1\\cdots y_i}{z_1\\cdots z_i}\\right)^{\\alpha_{i}}\\left(\\frac{y_{i+1}^i}{z_{i+1}^i}\\right)^{(\\alpha_{i+1}-\\alpha_i)}\\left(\\frac{y_{i+1}}{z_{i+1}}\\right)^{\\alpha_{i+1}}=\\left(\\frac{y_1\\cdots y_i}{z_1\\cdots z_i}\\right)^{\\alpha_{i}}\\left(\\frac{y_{i+1}}{z_{i+1}}\\right)^{\\frac{z_{i+1}}{z_k-z_{i+1}}}.\\]\n Here, we are using that $\\alpha_{i+1}-\\alpha_i\\geq 0$ as $\\frac{z_1}{z_k-z_1}<\\cdots <\\frac{z_{i+1}}{z_k-z_{i+1}}$.\n Multiplying these up for $i=1,\\ldots, k-2$, and we get\n \\[\\left(\\frac{y_1\\cdots y_{k-1}}{z_1\\cdots z_{k-1}}\\right)^{\\alpha_{k-1}}\\leq \\left(\\frac{y_1}{z_1}\\right)^{\\frac{z_1}{z_k-z_1}}\\cdots\\left(\\frac{y_{k-1}}{z_{k-1}}\\right)^{\\frac{z_{k-1}}{z_k-z_{k-1}}}.\\]\n Thus\n \\begin{align*}\n \\left(\\frac{y_1\\cdots y_{k-1}}{z_1\\cdots z_{k-1}}\\right)^{\\alpha_{k-1}+1}\\leq& \\prod_{i=1}^{k-1}\\left(\\frac{y_i}{z_i}\\right)^{\\frac{z_i}{z_k-z_i}}\\left(\\frac{y_k-y_i}{z_k-z_i}\\right)\\\\\n =&\\prod_{i=1}^{k-1}\\left[\\left(\\frac{y_i}{z_i}\\right)^{\\frac{z_i}{z_k}}\\left(\\frac{y_k-y_i}{z_k-z_i}\\right)^{\\frac{z_k-z_i}{z_k}}\\right]^{\\frac{z_k}{z_k-z_i}}\\\\\n \\leq &\\prod_{i=1}^{k-1}\\left(\\frac{z_i}{z_k}\\cdot \\frac{y_i}{z_i}+\\frac{z_k-z_i}{z_k}\\cdot \\frac{y_k-y_i}{z_k-z_i}\\right)^{\\frac{z_k}{z_k-z_i}}&\\textup{(weighted AM-GM)}\\\\\n = &\\prod_{i=1}^{k-1}\\left(\\frac{y_k}{z_k}\\right)^{\\frac{z_k}{z_{k}-z_i}}\n =\\left(\\frac{y_k}{z_k}\\right)^{(k-1)(\\alpha_{k-1}+1)},\n \\end{align*}\n completing the inductive step.\n \n\\end{proof}\n\n\\begin{proof}[Alternative Proof of \\cref{thm:main-entropy-ver}]\n Suppose $G$ is $\\cF_k$-hom-free.\n Set $z_i=i$ for each $i\\in[k]$.\n By \\cref{lemma:tent-ineq}, we know that\n \\[\\prod_{j\\leq i}\\frac{x_j}{x_{i+1}-x_j}\\leq 1 = \\prod_{j\\leq i}\\frac{j}{(i+1)-j}=\\prod_{j\\leq i}\\frac{z_j}{z_{i+1}-z_j}.\\]\n Therefore by \\cref{lemma:another-aux-ineq} and the fact that $x_k=1$,\n \\[x_1\\cdots x_{k-1}\\leq \\frac{(k-1)!}{k^{k-1}}=\\frac{k!}{k^k},\\]\n as desired.\n\\end{proof}\n\n\\subsection{Proof of \\cref{thm:one-tent,thm:not-Turan}}\nAs mentioned in the introduction, \\cref{thm:one-tent} is an immediate corollary of \\cref{thm:main-tent}.\nWe give a detailed argument of how \\cref{thm:one-tent} follows from \\cref{thm:main-tent} below.\n\n\\begin{proof}[Proof of \\cref{thm:one-tent}]\n Let $\\lambda$ be a partition of $k$ with $\\lambda_1\\leq \\lceil k/2\\rceil$ and $\\lambda_i = 1$ for all $1<i\\leq \\ell(\\lambda)$.\n Again, it is clear that $\\pi(\\Delta_\\lambda)\\geq k!/k^k$, so it suffices to show that $\\pi(\\Delta_\\lambda)\\leq k!/k^k$.\n To show this, it suffices to show that any $\\Delta_\\lambda$-hom-free $k$-graph $G$ is also $\\Delta_{\\lambda'}$-hom-free for any $\\lambda'$ with $\\abs{\\lambda'}=k$ and $\\ell(\\lambda')=2$.\n This will follow immediately if we show that $\\Delta_{\\lambda}$ admits a homomorphism to $\\Delta_{\\lambda'}$ for any such $\\lambda'$.\n By \\cref{prop:partial-tent-hom}, it is sufficient to show that $\\Delta_{\\lambda'}$ admits a homomorphism from $\\Delta^p_{\\lambda}$ for any $\\lambda'$ with $\\abs{\\lambda'}=k$ and $\\ell(\\lambda')=2$.\n This is now simple: suppose that $\\Delta_{\\lambda'}$ has base $e'$ and apex $v'$, and $e_1', e_2'$ are two edges such that $\\abs{e_i'\\cap e'}=\\lambda_i'$ for $i\\in[2]$.\n We also suppose that $\\Delta^p_{\\lambda}$ has base $e$ and apex $v$, and $e_1,\\ldots, e_{\\ell}$ are partial edges such that $\\abs{e_i\\cap e}=\\lambda_i$ for $i\\in[\\ell]$.\n As $\\lambda_1'\\geq \\lceil k/2\\rceil \\geq \\lambda_1$, we can take $f:e\\cup \\{v\\}\\to V(\\Delta_{\\lambda'})$ so that $f(v)=v'$, $f(e)=e'$ and $f(e\\cap e_1)\\subseteq e'\\cap e_1'$.\n This is a homomorphism from $\\Delta^p_{\\lambda}$ to $\\Delta_{\\lambda'}$ as any vertex in $e'$ shares an edge with $v'$ in $\\Delta_{\\lambda'}$.\n\\end{proof}\n\nFinally, we give a proof of \\cref{thm:not-Turan} by demonstrating a $k$-graph $G$ that has $b(G)>k!/k^k$ and is $\\Delta_\\lambda$-free for large $\\lambda_1$.\nSimilar to an earlier lower-bound construction by Frankl and F\\\"uredi \\cite{FF89} for $\\Delta_{(k-1,1)}$, we will do so by constructing a $k$-graph $G$ so that the intersection of any two edges is small.\n\n\\begin{proof}\n Let $\\alpha<1$ be some constant that is close to $1$.\n In particular, assume that $\\alpha > 1/2$.\n Let $G_{\\textup{aux}}$ be an auxiliary graph with vertices $\\binom{[2k]}{k}$, and two vertices are connected if the corresponding subsets have intersection at least $\\alpha k$.\n Then $G_{\\textup{aux}}$ is a regular graph with degree\n \\[\\sum_{i\\leq (1-\\alpha)k}\\binom{k}{i}^2< k\\binom{k}{\\lfloor (1-\\alpha)k\\rfloor}^2 = 2^{(2h(\\alpha)+o(1))k},\\]\n where $h(\\alpha) = -\\alpha\\log_2\\alpha-(1-\\alpha)\\log_2\\alpha$ and we use that \n \\[\\binom{k}{(1-\\alpha+o(1))k}=2^{(h(\\alpha)+o(1))k}\\]\n when $\\alpha>1/2$.\n \n By the Caro--Wei theorem, there exists an independent set of size\n \\[\\frac{\\binom{2k}{k}}{2^{(2h(\\alpha)+o(1))k}} = 2^{(2-2h(\\alpha)+o(1))k}.\\]\n This corresponds to a $k$-graph $G$ on $[2k]$ with $2^{(2-2h(\\alpha)+o(1))k}$ edges so that any two edges have intersection less than $\\alpha k$.\n\n Now if $G$ contains a homomorphic image of $\\Delta_\\lambda$ where $\\lambda_1>\\alpha k$, let $e$ be its base and let $e_1$ be the edge with $\\abs{e\\cap e_1}=\\lambda_1$.\n Also let $f$ be a homomorphism from $\\Delta_\\lambda$ to $G$.\n Then $\\abs{f(e)\\cap f(e_1)}>\\alpha k$, and so $f(e)=f(e_1)$.\n This shows if $v$ is the apex of $\\Delta_\\lambda$, then $f(v)=f(u)$ for some $u\\in e$.\n However, $\\{uv\\}$ is contained in some edge in $\\Delta_\\lambda$, which is a contradiction.\n Thus $\\pi(\\Delta_\\lambda)$ is at least $b(G)$, which is at least the density of $G$.\n The density of $G$ is\n \\[\\frac{k! \\cdot 2^{(2-2h(\\alpha)+o(1))k} }{(2k)^k} = 2^{(1-2h(\\alpha)+o(1))k}\\cdot \\frac{k!}{k^k},\\]\n which is strictly greater than $k!/k^k$ for sufficiently large $k$ as long as $h(\\alpha)<1/2$.\n As $h$ is continuous on $[1/2, 1]$ and $h(1)=0$, this is true for $\\alpha$ sufficiently close to $1$.\n\\end{proof}\nThe proof roughly gives $\\alpha \\approx 0.89$.\nAlthough our proof is not fully optimized, we believe that it would not give the correct upper bound for $\\alpha$ even after being fully optimized.\nTherefore we do not pursue in this direction.\n\n\\section{Other applications of our method}\\label{sec:known}\n\nRecall from the introduction that Mubayi \\cite{Mub06} showed $\\pi(E^{(k)}_{k+1})=k!/k^k$ where $E^{(k)}_{k+1}$ is the extended clique of size $k+1$, and Mubayi and Pikhurko \\cite{MP07} strengthened it to $\\pi(\\Delta_{(1,1,\\ldots,1)}) = k!/k^k$.\nIn fact they both proved more general results than this: Mubayi showed that for each $r\\geq k$,\n\\[\\pi(E^{(k)}_{r+1})=b(K^{(k)}_r) = \\prod_{i=1}^{k-1}\\left(1-\\frac{i}{r}\\right)\\]\nand Mubayi and Pikhurko strengthened it as follows: consider the partial $k$-graph $F$ on $r+1$ vertices generated by $[k]$ and all the $2$-subsets of $[r+1]$, and then take its extension $\\Tilde{F}$.\nThen $\\pi(\\Tilde{F})=b(K^{(k)}_r)$ as well.\nNote that $E^{(k)}_{r+1}$ is the extension of $K_{r+1}$ as a partial $k$-graph, and there is a homomorphism from $K_{r+1}$ to $\\Tilde{F}$.\nTherefore $\\pi(E^{(k)}_{r+1})\\leq \\pi(\\Tilde{F})$, and so $\\pi(\\Tilde{F}) = b(K^{(k)}_r)$ is indeed a stronger statement.\nWe remark that Keevash's adaptation \\cite[Theorem 3.1]{Kee11} of Sidorenko's argument \\cite{Sid89} gives a much more general result than Mubayi and Pikhurko's result in this case, and we refer the readers to Keevash's survey for the statement.\n\nWe are able to prove $\\pi(\\Tilde{F}) = b(K^{(k)}_r)$ as well, though our proof is considerably more complicated, and it seems hard to produce a clean stronger statement.\nWe nonetheless outline the argument here for readers interested in improving our argument.\n\n\\begin{theorem}\\label{thm:fix-k-big-r-full-edge}\n Let $k,r$ be positive integers with $r\\geq k$.\n Let $\\cF$ be a family of $k$-graphs such that the following holds.\n For any $i=1,\\ldots, k-1$, if we take the union of any $\\binom{r-k+i}{i}+1$ different partial forests $F^{(\\vec{t})}$ as in the proof of \\cref{lemma:tent-ineq}, then its extension is not $\\cF$-hom-free.\n \n Then $\\pi(\\cF) \\leq b(K^{(k)}_r)$.\n\\end{theorem}\n\\begin{proof}\n Suppose that $G$ is $\\cF$-hom-free.\n Let $(X_1,\\ldots, X_k)$ be any random edge with uniform ordering on $G$ and let $x_1,\\dots,x_k$ be its ratio sequence.\n We first fix some $i\\in[k-1]$.\n Temporarily fix some large positive integer $N$ as in the proof of \\cref{lemma:tent-ineq}.\n For any $1=t_0<t_1<\\cdots <t_i\\leq N$, let $(Y_v^{(\\vec{t})})_{v\\in V}$ be the random homomorphism from $F^{(\\vec{t})}$ to $G$ sampled via \\cref{lemma:sample-tree} as in the proof of \\cref{lemma:tent-ineq}.\n Then by the assumption on $\\cF$ and that $G$ is $\\cF$-hom-free, we know that the supports of the random homomorphisms $(Y_v^{(\\vec{t})})_{v\\in V}$ are $\\left(\\binom{r-k+i}{i}+1\\right)$-wise disjoint.\n Therefore, if $(Z_v)_{v\\in V}$ is the mixture of the $(Y_v^{(\\vec{t})})_{v\\in V}$'s provided by \\cref{lemma:mix}, we have\n \\[ \\sum_{1=t_0<t_1<\\cdots <t_i\\leq N}2^{\\HH\\left((Y_v^{(\\vec{t})})_{v\\in V}\\right)}\\leq\\binom{r-k+i}{i}2^{\\HH\\left((Z_v)_{v\\in V}\\right)}.\\]\n Using what we have computed in the proof of \\cref{lemma:tent-ineq}, when $N$ is taken to infinity, we get\n \\[\\prod_{j\\leq i}\\frac{x_j}{x_{i+1}-x_j}\\leq \\binom{r-k+i}{i}.\\]\n\n Now let $z_i = r-k+i$ for each $i=1,\\ldots, k$.\n Then it is easy to verify that\n \\[\\binom{r-k+i}{i}= \\prod_{j\\leq i}\\frac{z_j}{z_{i+1}-z_j}\\]\n for each $i\\in[k-1]$.\n Therefore, by \\cref{lemma:another-aux-ineq}, we get that\n \\[x_1\\cdots x_{k-1}\\leq \\frac{z_1\\cdots z_{k-1}}{z_k^{k-1}} = \\frac{(r-k+1)\\cdots (r-1)}{r^{k-1}} = \\prod_{i=1}^{k-1}\\left(1-\\frac{i}{r}\\right)=b(K_r^{(k)}).\\]\n This shows that $b(G) = b_{\\textup{entropy}}(G)\\leq b(K_r^{(k)})$ for any $\\cF$-hom-free $k$-graph $G$, and so we have $\\pi(\\cF)\\leq b(K_r^{(k)})$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:fix-k-big-r-full-edge}\n Let $F$ be the partial $k$-graph on $r+1$ vertices generated by $[k]$ and all the $2$-subsets of $[r+1]$.\n Let $\\Tilde{F}$ be its extension.\n Then $\\pi(\\Tilde{F}) = b(K_r^{(k)})$.\n\\end{corollary}\n\\begin{proof}\n First of all, it is clear that $K_r^{(k)}$ is $F$-hom-free.\n Therefore, by \\cref{prop:partial-tent-hom}, $K_r^{(k)}$ is also $\\Tilde{F}$-hom-free, and so $\\pi(\\Tilde{F})\\geq b(K_r^{(k)})$.\n\n To show that $\\pi(\\Tilde{F})\\leq b(K_r^{(k)})$, it now suffices to show that the assumption of \\cref{thm:fix-k-big-r-full-edge} holds for any $i\\in[k-1]$.\n Indeed, for any collection $T$ of $\\binom{r-k+i}{i}+1$ different possible $\\vec{t}$'s, we may construct $S\\subseteq\\NN$ with size $r-k+i+1$ that satisfies the following: for each $s\\in S$ there exists $\\vec{t}\\in T$ such that $s\\in\\{t_1,\\ldots, t_i\\}$, and there exists a $\\vec{t}\\in T$ with $\\{t_1,\\ldots, t_i\\}\\subseteq S$.\n Indeed, set $S' =\\bigcup_{\\vec{t}\\in T}\\{t_1,\\ldots, t_i\\}$.\n Then $\\abs{T}\\leq \\binom{\\abs{S'}}{i}$, which shows that $\\abs{S'}\\geq r-k+i+1$.\n Now simply take $S\\subseteq S'$ of size $r-k+i+1$ while containing some $\\{t_1,\\ldots, t_i\\}$ for some $\\vec{t}\\in T$.\n Label this $\\vec{t}$ as $\\vec{t^*}$.\n\n Now we need to show that there is a homomorphic image of $\\Tilde{F}$ in the extension of $\\bigcup_{\\vec{t}\\in T}F^{(\\vec{t})}$.\n By \\cref{prop:partial-tent-hom}, it suffices to construct a homomorphism from $F$ to $\\bigcup_{\\vec{t}\\in T}F^{(\\vec{t})}$.\n To do so, we will simply map $1,\\ldots,k-i-1$ to $v_1,\\ldots, v_{k-i-1}$, map $k-i,\\ldots, k$ to $w_{t^*_0},\\ldots, w_{t^*_i}$, and then map the rest of the vertices into $S\\backslash \\{t^*_1,\\ldots, t^*_i\\}$ bijectively.\n To show that this is indeed a homomorphism, notice first that $\\{v_1,\\ldots, v_{k-i-1},w_{t_0^*},\\ldots, w_{t_i^*}\\}$ is a partial edge in $F^{(\\vec{t^*})}$.\n Therefore it remains to check that $\\{w_{s_1},w_{s_2}\\}$ and $\\{v_{m},w_{s_1}\\}$ are both in $\\bigcup_{\\vec{t}\\in T}F^{(\\vec{t})}$ for any $s_1\\neq s_2\\in S$ and $m\\in[k-i-1]$.\n Indeed, if $s_1<s_2$ and $s_2 = t_j$ for some $\\vec{t}\\in T$, then $\\{v_m, w_{s_1}, w_{s_2}\\}$ is indeed a partial edge in $F^{(\\vec{t})}$, which shows that both $\\{w_{s_1},w_{s_2}\\}$ and $\\{v_{m},w_{s_1}\\}$ are partial edges in $F^{(\\vec{t})}$ as well.\n\\end{proof}\n\nWe remark that \\cref{thm:fix-k-big-r-full-edge} seems much stronger than \\cref{cor:fix-k-big-r-full-edge}, though we do not see a clean way to extract a stronger statement from \\cref{thm:fix-k-big-r-full-edge}.\nWe leave this as a potential future direction for interested readers.\n\nWith a completely different method, we can improve Mubayi's result in a slightly different way, and this is closer to what Sideorenko actually did in his paper \\cite{Sid89} using hypergraph Lagrangian.\nIn that paper, Sidorenko showed that many extensions of partial $k$-graphs on $r+1$ vertices have Tur\\'an density equal to $b(K_r^{(k)})$, as long as $r$ is at least some threshold $M_k$ that depends on $k$.\nOne special case related to our result is the $k$-graph $F_{r+1}^{(k,k-1)}$ that can be obtained as follows: consider the partial $k$-graph on $[r+1]$ spanned by the edges $\\{[k-1]\\cup i:i=k,\\ldots, r+1\\}$ and all the $2$-subsets of $[r+1]$, and then take the extension of the partial $k$-graph.\nFor example, $F_{k+1}^{(k,k-1)}$ is the tent $\\Delta_{(k-1,1)}.$\nSidorenko's result is more general and relies on trees $T$ that satisfy the Erd\\H{o}s--S\\'os conjecture $\\textup{ex}(T,n) \\leq \\frac{1}{2}(v(T)-2)n$, and we refer the readers to Sidorenko's original paper \\cite{Sid89} for more details (also see \\cite[Section 2]{Ste20} or \\cite{TT22} for some families of trees where the Erd\\H{o}s--S\\'os conjecture is known to hold).\n\nWith a slightly different choice of partial forests, we can also prove that $\\pi(F^{(k,k-1)}_{r+1}) = b(K^{(k)}_r)$ for sufficiently large $r$ with respect to $k$.\nOur argument actually gives a more general statement: for any $s<k\\leq r$, let $F^{(k,s)}_{r+1}$ be the extension of the partial $k$-graph spanned by $\\{[s]\\cup i:i=s+1,\\ldots, r+1\\}$ and all the $2$-subsets of $[r+1]$.\nThen we obtain a sufficient condition for $\\pi(F^{(k,s)}_{r+1}) = b(K_r^{(k)})$.\n\n\\begin{theorem}\\label{thm:Sidorenko-type}\n Let $k,r,s$ be positive integers with $k\\leq r$ and \n \\begin{equation}\\label{eq:krs-relation}\n k-s\\geq \\sum_{i=1}^{s-1}\\frac{i}{r-i}.\n \\end{equation}\n Then $\\pi(F^{(k,s)}_{r+1}) = b(K_r^{(k)})$.\n\\end{theorem}\n\\begin{proof}\n It is clear that $K_r^{(k)}$ is $F^{(k,s)}_{r+1}$-hom-free. Therefore, $\\pi(F^{(k,s)}_{r+1})\\geq b(K_r^{(k)})$.\n \n To prove the other direction $\\pi(F^{(k,s)}_{r+1})\\leq b(K_r^{(k)})$,\n we may fix a $F^{(k,s)}_{r+1}$-hom-free $k$-graph $G$ and a random with uniform ordering $(X_1,\\dots,X_k)$ on $G$.\n Let $x_1,\\dots,x_k$ be the ratio sequence of $(X_1,\\dots,X_k)$. We will solve for the maximum of $x_1\\dots x_{k-1}$ under the constraints given by the following lemma.\n \\begin{lemma}\\label{lemma:sido-type-forest}\n For any integers $i,j$ with $i\\in [k-s], i\\leq j< k$, we have\n \\[\\frac{x_i}{r-k+i}\\leq x_{j+1}-x_j.\\]\n \\end{lemma}\n \\begin{proof}\n\n We will fix $i,j$ throughout this proof.\n As in what we did in \\cref{sec:entropy}, we will temporarily fix an integer $N\\in\\NN$ that will later be taken to infinity.\n For any $t\\in [N]$, we will define a partial forest $F^{(t)}$ on $V=\\{v_1,\\dots,v_{k-i},w_1,\\dots,w_N\\}$.\n The partial forest $F^{(t)}$ is spanned by the partial edges $\\{v_1,\\dots,v_{k-i},w_t\\}$, $\\{v_1,\\dots,v_{k-j-1},w_m,w_t\\}$ for every $m<t$, and $\\{v_1,\\dots,v_{k-j-1},w_m\\}$ for every $m>t$.\n With the linear order $<$ given by $v_1<\\dots<v_{k-i}<w_N<\\cdots <w_1$, we know that $F^{(t)}$ is indeed a partial forest.\n We can compute the forest sequence with respect to the linear order as follows: each $v_m$ contributes one to $n_m$ for each $m\\leq k-i$. For the contribution of $w_m$, if $m> t$ it contributes one to $n_{k-j}$; if $m=t$ it contributes one to $n_{k-i+1}$; otherwise it contributes one to $n_{k-j+1}$.\n Therefore the forest sequence $(n_1,\\ldots, n_k)$ is $\\vec{e}_1+\\dots+\\vec{e}_{k-i}+(N-t)\\vec{e}_{k-j}+\\vec{e}_{k-i+1}+(t-1)\\vec{e}_{k-j+1}$, where $\\vec{e}_1,\\dots,\\vec{e}_k$ are the vectors in the standard basis.\n Now let $(Y^{(t)}_v)_{v\\in V}$ be the random homomorphism produced by \\cref{lemma:sample-tree}.\n This gives\n \\begin{align}\\label{eq:sido-type-tree-eq1}\n \\HH\\left((Y^{(t)}_v)_{v\\in V}\\right) = (N+k-i)\\HH(X_1)+\\log_2\\left(x_{i}\\cdots x_{k}\\cdot x_j^{t-1}x_{j+1}^{N-t}\\right).\n \\end{align}\n\n Now, we show that the random tuples $(Y^{(1)}_v)_{v\\in V},\\dots,(Y^{(N)}_v)_{v\\in V}$ have $(r-k+i+1)$-wise disjoint supports.\n Note that, for any $t_1<\\dots<t_{r-k+i+1}$, the extension of the union $\\cup_{\\ell=1}^{r-k+i+1} F^{(t_{\\ell})}$ contains a homomorphic image of $F^{(k,k-i)}_{r+1}$, given by the partial edges $\\{v_1,\\dots,v_{k-i},w_{t_{\\ell}}\\}$ for $\\ell\\in [r-k+i+1]$ and $\\{w_{t_{\\ell'}},w_{t_{\\ell}}\\}$ for $1\\leq \\ell'<\\ell\\leq r-k+i+1$. Since $k-i\\leq s$, this is also a homomorphic image of $F^{(k,s)}_{r+1}$. Thus, no sequence of vertices is in $\\cap_{\\ell=1}^{r-k+i+1}\\supp ((Y^{(t_\\ell)}_v)_{v\\in V})$.\n \n Therefore we may now apply \\cref{lemma:mix} with $a=r-k+i$.\n Suppose that $(Z_v)_{v\\in V}$ is the resulting mixture of $(Y_v^{(t)})_{v\\in V}$ for all $t\\in [N]$. \n Note that the partial edge $\\{v_1,\\dots,v_{k-i}\\}$ is present in all partial forests, so by \\cref{lemma:sample-tree} we know that $(Z_{v_1},\\ldots, Z_{v_{k-i}})$ has the same distribution as $(X_{i+1},\\ldots, X_k)$. Similarly, for each $m\\in[N]$, since the partial edge $\\{v_1,\\dots,v_{k-j-1},w_m\\}$ is present in all partial forests, we know that $(Z_{v_1},\\ldots, Z_{v_{k-j-1}},Z_{w_m})$ has the same distribution as $(X_{j+1},\\ldots, X_k)$.\n Hence\n \\begin{align}\n \\HH\\left((Z_v)_{v\\in V}\\right)\\leq& \\HH(Z_{v_1},\\ldots, Z_{v_{k-i}})+\\sum_{m=1}^{N}\\HH(Z_{w_m}\\mid Z_{v_1},\\ldots, Z_{v_{k-j-1}}) \\nonumber\\\\\n =& \\HH(X_{i+1},\\ldots,X_k)+N\\HH(X_{j+1}\\mid X_{j+2},\\ldots,X_k)\\nonumber \\\\\n =& (N+k-i)\\HH(X_1)+\\log_2(x_{i+1}\\cdots x_k\\cdot x_{j+1}^N).\\label{eq:sido-type-tree-eq2}\n \\end{align}\n Thus \\cref{lemma:mix,eq:sido-type-tree-eq1,eq:sido-type-tree-eq2} now give\n \\[\\sum_{t=1}^N x_{i}\\cdots x_k\\cdot x_j^{t-1}x_{j+1}^{N-t}\\cdot 2^{(N+k-i)\\HH(X_1)}\\leq (r-k+i)x_{i+1}\\cdots x_k\\cdot x_{j+1}^{N}\\cdot 2^{(N+k-i)\\HH(X_1)},\\]\n and so\n \\[\\sum_{t=1}^Nx_ix_j^{t-1}x_{j+1}^{-t}\\leq r-k+i.\\]\n By rearranging and taking $N$ goes to infinity, we obtain\n \\[\\frac{x_i}{x_{j+1}}\\cdot \\frac{1}{1-\\frac{x_j}{x_{j+1}}}=\\sum_{t=1}^\\infty \\frac{x_i}{x_{j+1}}\\left(\\frac{x_j}{x_{j+1}}\\right)^{t-1}\\leq r-k+i,\\]\n and the lemma follows.\n \\end{proof}\n\n \nOnce again, to prove \\cref{thm:Sidorenko-type}, we need to upper bound $x_1\\cdots x_{k-1}$ given the inequalities in \\cref{lemma:sido-type-forest}. We start with the following inequality similar to \\cref{lemma:aux-ineq}.\n\n\\begin{lemma}\\label{lemma:sido-type-aux-ineq}\n Suppose that $y_1,\\ldots, y_t$ and $z$ are some non-negative real numbers.\n Then\n \\[y_1\\cdots y_t\\leq \\left(\\sum_{i=1}^{t}\\frac{y_i}{z+i}\\right)^t\\frac{(z+1)\\cdots(z+t)}{t^t}.\\]\n\\end{lemma}\n\\begin{proof}\n We will prove this by inducting on $t$. For $t=1$, the inequality is trivial.\n \n Assume the statement is true for $t-1$. From the inductive hypothesis and AM-GM inequality, we have\n \\begin{align*}\n y_1\\cdots y_t\\leq& y_t\\left(\\sum_{i=1}^{t-1}\\frac{y_i}{z+i}\\right)^{t-1}\\frac{(z+1)\\cdots(z+t-1)}{(t-1)^{t-1}}\\\\\n = &\\left(\\frac{t-1}{z+t}y_t\\right)\\left(\\sum_{i=1}^{t-1}\\frac{y_i}{z+i}\\right)^{t-1}\\frac{(z+1)\\cdots(z+t)}{(t-1)^{t}}\\\\\n \\leq &\\left(\\frac{t-1}{t}\\sum_{i=1}^{t}\\frac{y_i}{z+i}\\right)^{t}\\frac{(z+1)\\cdots(z+t)}{(t-1)^{t}}\\\\\n = &\\left(\\sum_{i=1}^{t}\\frac{y_i}{z+i}\\right)^t\\frac{(z+1)\\cdots(z+t)}{t^t}.\\qedhere\n \\end{align*}\n\\end{proof}\n\nNow, by using this lemma with $t=k-1, y_i=x_i$ and $z=r-k$, it is sufficient to upper bound right hand side using the conditions from \\cref{lemma:sido-type-forest}.\n\\begin{claim}\\label{claim:sido-type-xk}\n We have\n \\[\\frac{x_1}{r-k+1}+\\dots+\\frac{x_{k-1}}{r-1}\\leq \\frac{k-1}{r}x_k.\\]\n\\end{claim}\n\\begin{proof}\n Let $s'$ be the largest integer such that\n \\[k-s'\\geq \\sum_{i=1}^{s'-1}\\frac{i}{r-i}\\]\n holds. In particular, we have $s\\leq s'<k$.\n Set $c$ to be the real number such that\n \\[\\frac{k-1}{r}=(1-c)\\frac{1}{r-s'}+\\frac{1}{r-s'+1}+\\dots+\\frac{1}{r-1}.\\]\n From the definition of $s'$, we have\n \\[k-1\\geq s'-1+\\frac{s'-1}{r-s'+1}+\\dots+\\frac{1}{r-1}=\\frac{r}{r-s'+1}+\\dots+\\frac{r}{r-1}\\]\n and\n \\[k-1< s'+\\frac{s'}{r-s'}+\\dots+\\frac{1}{r-1}=\\frac{r}{r-s'}+\\dots+\\frac{r}{r-1}.\\]\n Therefore, $c\\in (0,1]$. By replacing the coefficient of $x_k$ using the definition of $c$ and rearranging, we may rewrite the inequality we want to show as the following.\n \\begin{align}\n &\\frac{x_1}{r-k+1}+\\dots+\\frac{x_{k-s-1}}{r-s-1}+c\\frac{x_{k-s}}{r-s}\\nonumber\\\\\n \\leq &(1-c)\\frac{s'}{r-s'}\\frac{x_k-x_{k-s'}}{s'}+\\frac{s'-1}{r-s'+1}\\frac{x_k-x_{k-s'+1}}{s-1}+\\dots+\\frac{1}{r-1}\\frac{x_k-x_{k-1}}{1}.\\label{eq:sido-type-wts}\n \\end{align}\n\n Note that \\cref{lemma:sido-type-forest} implies that\n \\[\\frac{x_i}{r-k+i}\\leq \\frac{x_k-x_{k-j}}{j}\\]\n holds for all $i\\leq k-s'\\leq j$. Thus, to prove \\cref{eq:sido-type-wts}, it is sufficient to check\n \\[k-s-1+c\\leq (1-c)\\frac{s'}{r-s'}+\\frac{s'-1}{r-s'+1}+\\dots+\\frac{1}{r-1}.\\]\n Actually, the equality holds because, by the choice of $c$, we have\n \\begin{align*}\n k-s-1+c=&(1-c)\\frac{r}{r-s'}+\\frac{r}{r-s'+1}+\\frac{r}{r-s'+2}+\\dots+\\frac{r}{r-1}-s+c\\\\\n =&(1-c)\\frac{s'}{r-s'}+\\frac{s'-1}{r-s'+1}+\\frac{s'-2}{r-s'+2}+\\dots+\\frac{1}{r-1}.\\qedhere\n \\end{align*}\n\\end{proof}\nBy combining \\cref{lemma:sido-type-aux-ineq,claim:sido-type-xk}, we get\n\\[x_1\\dots x_{k-1}\\leq \\left(\\frac{k-1}{r}x_k\\right)^{k-1}\\frac{(r-k+1)\\cdots(r-1)}{(k-1)^{k-1}}=\\frac{(r-k+1)\\cdots(r-1)}{r^{k-1}}=b(K_r^{(k)}).\\qedhere\\]\n\\end{proof}\n\nTo give a sense of what the inequality in \\cref{thm:Sidorenko-type} means, with some standard computation, we can show the following. \nIf $r,k$ are growing positive integers such that $r = (C+o_{k\\to\\infty}(1))k$ for some $C\\geq 1$, then the largest positive integer $s$ satisfying \\cref{eq:krs-relation} is $(C(1-\\exp(-C^{-1}))+o_{k\\to\\infty}(1))k$.\nIn a different regime where $s=k-d$ for some positive integers $d$, we can get that the smallest positive integer $r$ satisfying the inequality is $((2d)^{-1}+o_{k\\to\\infty}(1))k^2$.\nWe include those computations in the appendix (\\cref{prop:linear-r,prop:quadratic-r}).\n\nWe briefly remark that the threshold $M_k$ Sidorenko deduced on $r$ is the same as ours when $s=k-1$.\nHowever, Sidorenko's argument works for a more general family of hypergraphs.\nIt is also possible that by modifying Sidorenko's argument appropriately, we may get a statement analogous to \\cref{thm:Sidorenko-type} with the extra parameter $s$.\n\n\\section{Concluding remarks}\\label{sec:conclusion}\n\n\\subsection{Exact result and stability}\nIn this paper, we mostly focus on the Tur\\'an density rather than the Tur\\'an number. \nHowever, we believe that with more work, it is possible to extract the exact Tur\\'an number for sufficiently many vertices from our density Tur\\'an theorems \\cref{thm:main-tent,thm:Sidorenko-type} at least when we also forbid all homomorphic images.\nMore specifically, we believe that there is a corresponding stability result for \\cref{thm:main-tent,thm:Sidorenko-type}, which is usually helpful to deduce the exact Tur\\'an number for sufficiently many vertices.\nIndeed, many exact results were deduced using stability results in a crucial way.\nFor some examples, we refer the readers to \\cite{KS05,MP07, Pik08, Pik13, BIJ17, NY17, NY18, LMR23, San24}.\n\n\\subsection{Other extremizers}\nAll the Tur\\'an results we are able to prove in this paper have blowups of $K^{(k)}_r$ as their asymptotic extremizers, and this is not a coincidence.\nWe find it much easier to construct partial forests that would give tight inequalities on the ratio sequences $x_1,\\ldots, x_{k}$ with equality holding when $(X_1,\\ldots,X_k)$ is a uniform oriented edge in $K^{(k)}_r$.\nHowever, as mentioned in the introduction, many difficulties of hypergraph Tur\\'an problems come from the potential complicated structures in the extremizers.\nIt would thus be more exciting if our method can be applied to problems with extremizers not as simple as $K^{(k)}_r$.\n\nThe first step would probably be to extend this to other Tur\\'an problems where the extremizers are blowups of some other hypergraphs.\nTwo candidates are the complete bipartite $3$-graph $(A\\sqcup B, E)$ where $E = \\binom{A}{2}\\times B\\cup A\\times\\binom{B}{2}$, and the complete oddly bipartite $k$-graph $(A\\sqcup B,E)$ where $k$ is even, and $E$ is the $k$-edges $e$ such that $\\abs{A\\cap e}$ is odd.\nAlthough they are not formally blowups of some smaller hypergraphs, one can think of the complete bipartite $3$-graphs as the blowups of $(\\{1,2\\},\\{\\{1,1,2\\},\\{1,2,2\\}\\})$, and the completely oddly bipartite $k$-graphs are the blowups of some $2$-vertex ``degenerate'' hypergraphs as well.\n\nThere are many known Tur\\'an results where the two hypergraphs are (asymptotic) extremizers.\nFor example, a classical result of De Caen and F\\\"uredi \\cite{DCF00} shows that the complete bipartite $3$-graph is an asymptotic extremizer for the Fano plane.\nThis was later extended by Mubayi--R\\\"odl \\cite{MR02} and Baber--Talbot \\cite{BT12}.\nOn the other hand, Keevash and Sudakov \\cite{KS05} showed that the complete oddly bipartite $k$-graph is the extremizer for expanded triangle.\nA very recent breakthrough of Sankar \\cite{San24} showed that the complete oddly bipartite $4$-graph is an asymptotic extremizer for tight cycles of sufficiently large length not divisible by $4$.\n\nWe are unable to construct any partial forests that give tight inequalities when $G$ is the complete bipartite $3$-graph.\nFor $G$ being complete oddly $k$-graphs, it is possible to construct such partial forests following the argument in \\cref{thm:entropic-turan} and Sidorenko's \\cite{Sid92} and Frankl's \\cite{Fra90} ideas, which used auxiliary $2$-graphs to show that the Tur\\'an densities of expanded triangles are $1/2$.\nHowever, we have not found any other partial forests that use essentially different ideas.\nIt would be interesting to see if there are ways to obtain tight inequalities for those two candidates of $G$ in the hope that they would give rise to new Tur\\'an results.\n\nLet us close this discussion by mentioning that our method seems to capture a little structure in the conjectured extremizer for $K_4^{(3)-}$, the $3$-graph on $4$ vertices with $3$ edges.\nLet $G_1$ be a $3$-graph on $6$ vertices with $10$ edges so that any $2$-subset is in exactly $2$ edges---it turns out that $G_1$ does exist and is unique up to isomorphism.\nThe \\emph{iterated blowup} $G_m$ of $G_1$ is constructed inductively by replacing each vertex in $G_1$ with $G_{m-1}$.\nThen $G_m$ is $K_4^{(3)-}$-free, and by taking $m$ to infinity, we get that $\\pi(K_4^{(3)-})\\geq \\frac{2}{7}$.\nThis is a construction of Frankl and F\\\"uredi \\cite{FF84}, and the construction is conjectured to be optimal.\nThe current best upper bound $\\pi(K_4^{(3)-})\\leq 0.2871$ is obtained by Baber and Talbot \\cite{BT11} using flag algebra.\nThough we cannot say anything new about the Tur\\'an problem of $K_4^{(3)-}$ itself, our method seems to capture some structure in $G_1$.\nIndeed, by the partial forests $F^{(i)}=([4],\\{[3],[4]\\backslash \\{i\\}\\})$ for $i=1,2,3$, we can show that if $G$ is $K_4^{(3)-}$-free and $(X_1,X_2,X_3)$ is a random edge with uniform ordering on $G$, then\n\\[x_1 \\eqdef 2^{\\HH(X_1\\mid X_2,X_3)-\\HH(X_1)} \\leq \\frac{1}{3}.\\]\nThis is indeed achieved when $(X_1,X_2,X_3)$ is a uniformly chosen oriented edge in $G_1$.\n\n\\subsection{Entropic spectral radius}\nIn \\cref{sec:connection}, we showed that for any $k$-graph $G$, its spectral radius is related to the maximum of $\\HH(X_2,\\ldots, X_k\\mid X_1)$ for symmetric distribution $(X_1,\\ldots,X_k)$ on the oriented edges of $G$.\nIt would be interesting if this connection can be utilized to deduce some properties of spectral radius.\nOne possible candidate is a result of Kang, Liu and Shan \\cite{KLS18} that showed that\n\\[\\rho(G)\\geq \\left(\\frac{1}{v(G)}\\sum_{v\\in V(G)}\\deg(v)^{\\frac{k}{k-1}}\\right)^{\\frac{k-1}{k}}\\]\nfor any $k$-graph $G$, where $\\rho(G)$ is the spectral radius of $G$.\n\n\\subsection{Entropic flag algebra}\nAs one may have observed, many upper bounds on Tur\\'an densities, especially for those that are still open, were obtained using flag algebra.\nSuch upper bounds using flag algebra, roughly speaking, are obtained via carefully chosen sum-of-squares inequalities, enumeration of possible small configurations, and numerical computationg of positive semidefinite programs.\nSee \\cite{Raz13} for a more detailed discussion of the method.\n\nThe inequalities obtained using our argument seem to be really different from the inequalities obtained by sum-of-squares.\nThis suggests a possibility that maybe the flag algebra bounds can be improved with this new idea and some enumeration of possible partial forests to use in the argument.\nHowever, aside from the time complexity enumerating through the possible partial forests, there seem to be several technicalities to overcome for this to work.\nThe first is that in most of our proofs, we need to look at infinitely many partial forests in order to get a tight bound.\nIn addition, the inequalities we get, unlike the ones in flag-algebraic arguments, are highly non-linear.\nHowever, if we are just aiming for some numerical upper bound that is close to the truth, then hopefully finite but sufficiently many partial forests together with an approximation of the supremum of $x_1\\cdots x_{k-1}$ subject to the inequalities would be enough.\n\nThe most serious issue is probably that there has not been a framework for automated entropic computation.\nSo far, the flag-algebraic tools are developed to keep track of the homomorphism densities of labeled graphs.\nUnfortunately, it seems that all our arguments for hypergrpah Tur\\'an problems cannot be rephrased using homomorphism densities as we also crucially use the marginal distributions of the random homomorphisms sampled by \\cref{lemma:sample-tree}.\nIt would thus be necessary to come up with an ``entropic flag algebra'' framework and implement corresponding software to execute the idea in this subsection.\nWe refer the readers to \\cite{CY24-2} for another entropic argument that motivates this idea of ``entropic flag algebra''.\n\\section*{Acknowledgement}\nThe project was motivated when the first author was visiting Hong Liu at Institute for Basic Science, and the first author would like to thank his hospitality.\nWe would also like to thank Ryan Alweiss and Freddie Manners for discussions during the early stage of this project, Dhruv Mubayi and Maya Sankar for pointing us to references for hypergraph Tur\\'an problems, Yongtao Li for pointing us to references for spectral Tur\\'an problems, and Noga Alon for pointing us to other useful references.\nLast but not least, we would like to thank Zeev Dvir, Xiaoyu He, Cosmin Pohoata and Maya Sankar for helpful comments on an earlier draft.\n\n\\bibliographystyle{amsplain0}\n\\bibliography{ref_joints}\n\n\n\\appendix\n\\section{Explicit relation between $r,s$ and $k$ in \\cref{thm:Sidorenko-type}}\nIn this appendix, we will relate positive integers $k,r,s$ with $k\\leq r$ satisfying the inequality\n\\begin{equation}\\label{eq:appendix-rks}\n k-s\\geq \\sum_{i=1}^{s-1}\\frac{i}{r-i}.\n\\end{equation}\nWe first compute the right hand side.\n\\begin{lemma}\\label{lemma:integral-approx}\n Suppose that $k,r,s$ are positive integers satisfying \\cref{eq:appendix-rks}.\n Then $r(k-s) = \\Omega(s^2)$, $r-s = \\Omega(r)$ and\n \\[\\sum_{i=1}^{s-1}\\frac{i}{r-i} = r\\log\\left(\\frac{r-1}{r-s}\\right)-(s-1)+O\\left(\\frac{s}{r}\\right).\\]\n\\end{lemma}\n\\begin{proof}\n We first show $r(k-s)=\\Omega(s^2)$.\n This is clear as \n \\[k-s\\geq \\sum_{i=1}^{s-1}\\frac{i}{r-i}\\geq \\left\\lfloor \\frac{s-1}{2}\\right\\rfloor \\frac{\\left\\lceil \\frac{s-1}{2}\\right\\rceil}{r-\\left\\lceil \\frac{s-1}{2}\\right\\rceil}=\\Omega(s^2r^{-1}).\\]\n \n Now we show that $r-s = \\Omega(r)$.\n This is clear when $r\\geq 2k$, so it suffices to check the case when $r<2k$.\n In this case, we have $2k(k-s)>r(k-s)\\geq \\Omega(s^2)$.\n This forces $s \\leq ck$ for some constant $c<1$, and so $r-s=\\Omega(r)$ as $s<k\\leq r$.\n\n Now let $\\cE$ be the error term defined by\n \\[\\cE =\\sum_{i=1}^{s-1}\\frac{i}{r-i}-\\int_{1}^{s}\\frac{x}{r-x}\\textup{ d}x = \\int_1^s\\left(f(\\lfloor x\\rfloor)-f(x)\\right)\\textup{ d}x\\]\n where we set $f(x) = x/(r-x)$.\n Note that $f'(x) = r(r-x)^{-2}$ is positive and increasing in $x$ when $x\\in [1,s]\\subseteq [1,r-1]$.\n Therefore\n \\[0\\geq f(\\lfloor x\\rfloor)-f(x)\\geq (\\lfloor x\\rfloor -x)f'(x)>-\\frac{r}{(r-x)^2}\\]\n for any $x\\in [1,s]$.\n This shows that \n \\[0\\geq \\cE \\geq -\\frac{(s-1)r}{(r-s)^2},\\]\n which shows that $\\cE = O(s/r)$.\n Therefore\n \\[\\sum_{i=1}^{s-1}\\frac{i}{r-i} =\\int_{1}^{s}\\frac{x}{r-x}\\textup{ d}x+O\\left(\\frac{s}{r}\\right)= r\\log\\left(\\frac{r-1}{r-s}\\right)-(s-1)+O\\left(\\frac{s}{r}\\right).\\qedhere\\]\n \n\\end{proof}\n\n\\begin{proposition}\\label{prop:linear-r}\n Let $r\\geq k$ be a positive integer growing with $k$ so that $r = (C+o_{k\\to\\infty}(1))k$ for some constant $C\\geq 1$.\n Then the largest positive integer $s$ satisfying \\cref{eq:appendix-rks} also satisfies \\linebreak$s=C(1-\\exp(-C^{-1})+o_{k\\to\\infty}(1))k$. \n\\end{proposition}\n\\begin{proof}\n By the choice of $s$, we know\n \\[k-s\\geq \\sum_{i=1}^{s-1}\\frac{i}{r-i}\\]\n and\n \\[k-(s-1)< \\sum_{i=1}^{s-2}\\frac{i}{r-i}.\\]\n Therefore \n \\[k-s+O(1) = \\sum_{i=1}^{s-1}\\frac{i}{r-i}+O\\left(\\frac{s}{r-s}\\right).\\]\n By \\cref{lemma:integral-approx}, we know that this implies\n \\[k-s +O(1) = r\\log\\left(\\frac{r-1}{r-s}\\right)-(s-1)+O\\left(\\frac{s}{r}\\right).\\]\n Rearranging, we get\n \\[\\frac{r-1}{r-s} = \\exp\\left(\\frac{k-1}{r}+O(r^{-1})\\right),\\]\n and so\n \\begin{align*}\n s =& 1+(r-1)\\left(1-\\exp\\left(-\\frac{k-1}{r}+O(r^{-1})\\right)\\right)\\\\\n =& 1+(C+o_{k\\to\\infty}(1))k \\cdot \\left(1-\\exp\\left(-C^{-1}+o_{k\\to\\infty}(1)\\right)\\right),\n \\end{align*}\n where we use the fact that $r^{-1} = O(k^{-1})$.\n The desired statement thus follows.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:quadratic-r}\n Let $k,d$ be positive integers with $d<k$, and let $s=k-d$.\n Then the smallest positive integer $r$ satisfying \\cref{eq:appendix-rks} also satisfies $r=(\\frac{1}{2d}+o_{k\\to\\infty}(1))k^2$.\n\\end{proposition}\n\\begin{proof}\n\n \n By the choice of $r$, we also have\n \\[d\\geq \\sum_{i=1}^{s-1}\\frac{i}{r-i}\\]\n and\n \\[d<\\sum_{i=1}^{s-1}\\frac{i}{r-1-i}.\\]\n Note that\n \\[\\frac{i}{r-i}-\\frac{i}{r-1-i} = O(ir^{-2})\\]\n for every $i\\leq s-1$ as we know that $r-s = \\Omega(r)$ by \\cref{lemma:integral-approx}.\n Therefore\n \\[d = \\sum_{i=1}^{s-1}\\frac{i}{r-i}+O(s^2r^{-2}).\\]\n By \\cref{lemma:integral-approx}, we know that $r = \\Omega(d^{-1}s^2) = \\Omega(s^2)$.\n Therefore by \\cref{lemma:integral-approx},\n \\[d = r\\log\\left(\\frac{r-1}{r-s}\\right)-(s-1)+O\\left(\\frac{s}{r}\\right)=r\\log\\left(\\frac{r-1}{r-s}\\right)-(s-1)+o_{k\\to\\infty}(1).\\]\n \n Note that\n \\begin{align*}\n r\\log\\left(\\frac{r-1}{r-s}\\right)=&r\\log\\left(1+\\frac{s-1}{r}+\\frac{s(s-1)}{r^2}+O\\left(\\frac{s^3}{r^3}\\right)\\right)\\\\\n =&r\\left(\\frac{s-1}{r}+\\frac{(s+1)(s-1)}{2r^2}+O\\left(\\frac{s^3}{r^3}\\right)\\right)\\\\\n =&s-1+\\frac{s^2}{2r}+o_{k\\to\\infty}(1).\n \\end{align*}\n Therefore we get that\n \\[\\frac{r}{k^2} = \\left(1+o_{k\\to\\infty}(1)\\right)\\frac{r}{s^2}= \\frac{1}{2d}+o_{k\\to\\infty}(1),\\]\n as desired.\n \n\\end{proof}\n\\end{document}\n\n"}}},{"rowIdx":93,"cells":{"id":{"kind":"string","value":"2412.08183v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08183v1"},"title":{"kind":"string","value":"States and IR divergences in factorization algebras"},"full_text":{"kind":"string","value":"\n\\documentclass[preprint]{ptephy_v1}\n\\preprintnumber{YITP-24-168, KUNS-3030} \\usepackage{hyperref}\n\n\\usepackage{bm}\n\\usepackage[all]{xy}\n\\usepackage{tcolorbox}\n\\usepackage{tcolorbox}\n\\tcbuselibrary{breakable, skins, theorems}\n\\usepackage{mathrsfs}\n\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{condition}{Condition}\n\\newtheorem{definition}{Definition}\n\\newtheorem{remark}{Remark}\n\\newtheorem*{theorem*}{Theorem}\n\\newtheorem{conjecture}{Conjecture}\n\n\\numberwithin{equation}{subsection}\n\\numberwithin{definition}{subsection}\n\\numberwithin{theorem}{subsection}\n\\numberwithin{remark}{subsection}\n\\numberwithin{conjecture}{subsection}\n\n\\begin{document}\n\n\\title{States and IR divergences in factorization algebras}\n\n\n\\author{Masashi Kawahira}\n\\affil{Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502 Japan \\email{masashi.kawahira@yukawa.kyoto-u.ac.jp}}\n\n\\author[2]{Tomohiro Shigemura}\n\\affil{Department of Physics, Kyoto University, Kyoto 606-8502, Japan \\email{shigemura@gauge.scphys.kyoto-u.ac.jp}}\n\n\n\n\n\n\n\n\n\n\\begin{abstract}There are several choices of states.\nIn factorization algebras, we often use a natural augmentation state $\\langle-\\rangle_{\\rm aug}$. \nIn physics, we use a state given by compactification of spacetime $\\langle-\\rangle_{\\rm cptf}$, or a state corresponding Schwartz boundary condition $\\langle-\\rangle_{\\rm Sch}$.\nAt first glance, the relation between these three states is not so clear.\n\nIn this paper, propose a definition of compactification method in factorization algebras and give a way to treat IR divergences in massless theory.\nAs a result, we see the three sates are equivalent both in massive theory and massless theory.\n\\end{abstract}\n\n\\subjectindex{A12, A13, B05, B33 B34, B39}\n\\maketitle\n\\section{Introduction}\nQuantum field theory (QFT) is a central topic in theoretical physics. \nThis is useful to explain various kinds of phenomena.\nAnd also, people have discovered the various mathematical conjectures by using QFTs.\nNevertheless we have not obtain the complete understandings of quantum field theories.\n\nIn the last ten years, Kevin Costello and Owen Gwilliam have developed a new formulation based on factorization algebras\\cite{Costello:2016vjw}\\cite{Costello:2021jvx}.\nThis formulation works well in perturbative field theories.\nThe advantage is that it make clear a lot of concepts of QFTs like path-integral or operator products.\n\nIn this paper, we aim to make clear the concepts of IR divergences in massless free scalar theories.\nSince long ago, physicists have been studying IR divergences \\cite{Bloch:1937pw}.\nAnd recently some physicists pay attention to it again\\cite{Strominger:2017zoo}.\nInterestingly by using factorization algebras, we can treat the concepts of IR divergences {\\it without any divergences}.\n\nIn order to discuss IR divergences in a mathematical way, we sort the constructions of states in factorization algebras.\nEspecially, we consider three states. We name them as {\\it a natural augmentation state, compactification state and Schwartz state}. \n\nThis paper is organized as follows. \nIn section 2, we will see the goal of this paper.\nIn section 3, we will review the formulation by Costello and Gwilliam.\nIn section 4, we will review a natural augmentation state.\nIn section 5, we will give a compactification state and a treatment of IR divergences.\nIn section 6, we will review a Schwartz state for massive case and give a Schwartz state for massless case and a treatment of IR divergences.\nIn section 7, we will prove the equivalence of the above sates. \nSection 8 is devoted for conclusion and discussion.\n\n\n\\section{What will we see ?}\nIn this paper, we consider observables in some region $U$ of $\\mathbb{R}^d$, which we denote $\\mathcal{O}_U\\in{\\rm Obs}(U):={\\rm Sym}(C_{\\rm c}^\\infty(U))$. In factorization algebras, by taking the cohomology and states we obtain expectation values\\footnote{In later section, we will give definitions of the cohomology and the state.}.\n\\begin{align}\n H^0{\\rm Obs}^{\\rm cl}(U)\\ni [\\mathcal{O}_U]_U\\mapsto \\langle [\\mathcal{O}_U]_U\\rangle \\in\\mathbb{C}\n\\end{align}\n\nWe will explain the following three states.\n\\begin{itemize}\n \\item Natural augmentation state \n $\\langle-\\rangle_{\\rm aug}$\n \\item Schwartz state\n $\\langle-\\rangle_{\\rm Sch}$\n \\item Compactification state\n $\\langle-\\rangle_{\\rm cptf}$\n\\end{itemize}\n\n\\noindent{}$\\langle-\\rangle_{\\rm aug}$ and $\\langle-\\rangle_{\\rm Sch}$\nare introduced by Costtelo-Gwilliam\\cite{Costello:2016vjw}, and they showed that these are equivalent.\nNote that natural augmentation state $\\langle-\\rangle_{\\rm aug}$ is well-defined in massive and massless case, while Schwartz state is defined only for massive case. This is essentially from IR divergence.\nIn this paper, we will see how to get rid of IR divergence and define Schwartz state in massless case.\n\nAnd we will define compactification state $\\langle-\\rangle_{\\rm cptf}$ which represents compactification-notion of quantum field theories.\nThis is well-defined in massive case. \nIn massless case, we need to remove IR divergence in order to define $\\langle-\\rangle_{\\rm cptf}$.\n\nIn addition, We will prove that three states are equivalent both in massive case and massless case.\n\n\\section{Brief review of factorization algebra of free real scalar theory}\n\\subsection{Observable algebra}\nIn this paper, we would like to discuss path-integral-formulation in mathematical manner.\nIn contrast to operator formulation, obeservables are not operators but ``functionals with compact support.\"\nIn order to consider it, we define observable algebra as follows.\\footnote{Precisely, we need to take a completion in order to make ${\\rm Obs}(U)$ a topological vector space. However we omit the discussion. If you are interested in it, check \\cite{Costello:2016vjw}.\\label{ref:completion}}\n\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\nObservable algebra ${\\rm Obs}(U)$ is defined as \n\\begin{align}\n {\\rm Obs}(U):={\\rm Sym}(C_{\\rm c}^\\infty(U,\\mathbb{C}))\n\\end{align} and the elements are called observables on $U$.\n\\end{definition}\n \\end{tcolorbox}\n\nMotivation of the above definition is that observables can be regarded as a polynomial of functions with compact supports:\n$\\mathcal{O}=c+f+f_1*f_2+\\cdots$,\nwhere $*$ is a formal symmetric product, $c\\in\\mathbb{C}$ and $f,f_1,f_2,\\cdots\\in C_{\\rm c}^\\infty(U,\\mathbb{C})$. And $\\mathcal{O}$ acts $\\Phi$ as\n\\begin{align}\n \\mathcal{O}:\\Phi\\mapsto\n c\n +\\int_M f\\Phi\n +\\int_M f_1\\Phi \\int_M f_2\\Phi\n +\\cdots\n\\end{align}\n\n \n\\subsection{Classical derived observable space}\nIn order to perform ``path-integral,\" we introduce derived observable algebras.\nRoughly, \\textit{derived} means that we will add a concept of \\textit{degree} to the observable space ${\\rm Obs}(U)$.\nThis formulation is originally given by Batalin and Vilkovisky\\cite{Batalin:1981jr}\\cite{Batalin:1983ggl}.\n\\begin{align}\n &C_{\\rm c}^\\infty(U)^{0}:=(C_{\\rm c}^\\infty(U,\\mathbb{C}),0),\\\\\n &C_{\\rm c}^\\infty(U)^{-1}:=(C_{\\rm c}^\\infty(U,\\mathbb{C}),-1).\n\\end{align}\nThe first one is a \\textit{degree-0 linear observable}, and the second one is a \\textit{degree-$(-1)$ linear observable} which corresponds to \\textit{anti-field} in physics literature.\nFor simplicity, we denote $(f,0)\\in C_{\\rm c}^\\infty(U)^{0}$ and $(f,-1)\\in C_{\\rm c}^\\infty(U)^{-1}$ as $f$ and $f^\\star$.\nThe symmetric product $*$ is defined for them as\n\\begin{align}\n a*b=(-1)^{|a||b|}b*a\\label{eq:*}.\n\\end{align}\nAnd when we have degree $+1$ operator $\\Delta^{\\rm cl}$, we can define a classical derived observable algebra.\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n Classical derived observable algebra ${\\rm Obs}^{\\rm cl}(U)$ is defined as\n \\begin{align}\n {\\rm Obs}^{\\rm cl}(U)\n :=\n {\\rm Sym}\\left(C_{\\rm c}^\\infty(U)^{-1}\\xrightarrow{\\Delta^{\\rm cl}} C_{\\rm c}^\\infty(U)^{0}\\right)\n \\end{align}\n where $\\Delta^{\\rm cl}$ is a classical Batalin-Vilkovisky operator which is defined in Definition \\ref{def:cl_BV_op}.\n\\end{definition} \n\\end{tcolorbox}\n\n\\noindent{}The concrete form of $\\Delta^{\\rm cl}$ depends on what theory we want. In this paper, we are interested in free theory, thus we will defined as follows.\n\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\label{def:cl_BV_op}\\\\\n Classical Batalin-Vilkovisky operator $\\Delta^{\\rm cl}$ is a map \n \\begin{align}\n C_{\\rm c}^\\infty(U)^{-1}\\ni f^\\star\\mapsto -(-\\Delta+m^2)f\\in C_{\\rm c}^\\infty(U)^{0}. \n \\end{align}\n Hence $\\Delta^{\\rm cl}$ has a degree $+1$.\n\\end{definition} \n\\end{tcolorbox}\n\nBy (\\ref{eq:*}), we can rewrite classical derived observable space ${\\rm Obs}^{\\rm cl}(U)$.\n\\begin{align}\n &{\\rm Obs}^{\\rm cl}(U)= \n \\notag\\\\\n &\\bigg(\\cdots \n \\xrightarrow{\\Delta^{\\rm cl}}\\bigwedge^2 C_{\\rm c}^\\infty(U)^{-1}* {\\rm Sym}\\left(C_{\\rm c}^\\infty(U)^{0}\\right)\n \\xrightarrow{\\Delta^{\\rm cl}}C_{\\rm c}^\\infty(U)^{-1}* {\\rm Sym}\\left(C_{\\rm c}^\\infty(U)^{0}\\right)\n \\xrightarrow{\\Delta^{\\rm cl}}{\\rm Sym}\\left(C_{\\rm c}^\\infty(U)^{0}\\right)\\bigg)\n\\end{align}\nThis is a chain complex. It is called \\textit{classical Batalin–Vilkovisky complex}.\nThe cohomology of classical Batalin–Vilkovisky complex\n is called \\textit{classical Batalin–Vilkovisky cohomology}:\n \\begin{align}\n H^*\\left({\\rm Obs}^{\\rm cl}(U)\\right).\n \\end{align}\n\n\nLet us consider the physical meanings of the cohomology $H^0\\left({\\rm Obs}^{\\rm cl}(U)\\right)$.\nWe take a 0-degree observables $\\mathcal{O}_1,\\mathcal{O}_2$ and assume that these are same in the cohomology, i.e., $\\exists X\\ {\\rm s.t.}$\n\\begin{align}\n \\mathcal{O}_2-\\mathcal{O}_1\n =\n \\Delta^{\\rm cl}X.\n\\end{align}\nLet $\\Phi_{\\rm cl}$ be a solution of the equation of motion $(-\\Delta+m^2)\\Phi=0$\\footnote{If you want a theory with some interactions, we need to add some terms to $\\Delta^{\\rm cl}$.}. We have\n\\begin{align}\n \\mathcal{O}_2(\\Phi_{\\rm cl})-\\mathcal{O}_1(\\Phi_{\\rm cl})\n &=\n \\Delta^{\\rm cl}X(\\Phi_{\\rm cl})\\notag\\\\\n &=\n 0.\n\\end{align}\nHence we can see $H^0\\left({\\rm Obs}^{\\rm cl}(U)\\right)$ as the on-shell evaluation of observables. However $[\\mathcal{O}_1](=[\\mathcal{O}_2])$ is not a number. Then we define a map called \\textit{state} $\\langle-\\rangle$.\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n A state $\\langle-\\rangle$ is a smooth map:\n \\begin{align}\n \\langle-\\rangle:\n H^0\\left({\\rm Obs}^{\\rm cl}(M)\\right)\n \\to \n \\mathbb{C}.\n \\end{align}\n\\end{definition}\n\\end{tcolorbox}\n\n\n\\section{Natural augmentation state}\n\\subsection{Massive and massless case}\nBy definition, ${\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)$ has a natural augmentation map. Costello-Gwilliam use it to define a state.\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n We have a projection:\n \\begin{align}\n \\begin{array}{ccc}\n {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)^{0}\n &\\longrightarrow\n &\\mathbb{C}\\\\\n \\rotatebox{90}{$\\in$}\n &\n &\\rotatebox{90}{$\\in$}\\\\\n c+f+f_1*f_2+\\cdots\n &\\longmapsto\n &c\n \\end{array}\n \\end{align}\n where $c\\in\\mathbb{R}$ and $f,f_1,f_2,\\cdots\\in C_{\\rm c}^\\infty(U)$.\n This induces\n \\begin{align}\n \\langle-\\rangle_{\\rm aug}:\n H^0({\\rm Obs}^{\\rm cl}(\\mathbb{R}^d))\n \\to\n \\mathbb{C}.\n \\end{align}\n We call it a natural augmentation state.\n\\end{definition}\n\\end{tcolorbox}\nThe cohomology is\n\\begin{align}\n H^0({\\rm Obs}^{\\rm cl}(\\mathbb{R}^d))\n =\n \\frac\n {{\\rm Obs}^{\\rm cl}(\\mathbb{R}^d))^{0}}\n {\\Delta^{\\rm cl}\n ({\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)^{-1})}.\n\\end{align}\nHence, to check the well-definedness of $\\langle-\\rangle_{\\rm aug}$, we see for any $X\\in {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)^{-1}$,\n\\begin{align}\n \\Delta^{\\rm cl}X\n \\notin\n \\mathbb{C}\n \\subset\n {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)^{0}.\n\\end{align}\nWrite $X$ as $P*f^\\star$ where $P$ is in ${\\rm Sym}(C_{\\rm c}^\\infty(\\mathbb{R}^d)^0)$ and $f^\\star$ is in $C_{\\rm c}^\\infty(\\mathbb{R}^d)^{-1}$.\n\\begin{align}\n \\Delta^{\\rm cl}X\n =\n P*(\\Delta^{\\rm cl}f^\\star)\n\\end{align}\nIn order that $\\Delta^{\\rm cl}X$ sits in $\\mathbb{C}$, \nboth $P$ and $\\Delta^{\\rm cl}f^\\star$ must be constants.\nHowever, by definition\n\\begin{align}\n \\Delta^{\\rm cl}f^\\star\n \\in\n C_{\\rm c}^\\infty(\\mathbb{R}^d)^0.\n\\end{align}\nThen it can not be a constant. Thus $\\langle-\\rangle_{\\rm aug}$ is well-defined.\n\n\\section{Compactification state}\n\\subsection{Massive case}\nIn order to consider compactification, locality plays an essential role.\nHence we assume the measure of $U$ is finite: $\\mu(U)\\ll \\infty$. We call it {\\it locality condition}.\nUnder this condition, we have an inclusion map\n\\begin{align}\n i:{\\rm Obs}^{\\rm cl}(U)\\to \n {\\rm Obs}^{\\rm cl}(T^d)\n\\end{align}\nand\n\\begin{align}\n i:H^*{\\rm Obs}^{\\rm cl}(U)\\to \n H^*{\\rm Obs}^{\\rm cl}(T^d)\n\\end{align}\nfor sufficiently large torus $T^d$. \nStrictly speaking, we need to distinguish the above two maps, but we use the same letter $i$ for simplicity.\n\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n In massive case,\n \\begin{align}\n H^n{\\rm Obs}^{\\rm cl}(T^d)\n =\n \\left\\{\n \\begin{array}{ll}\n \\mathbb{C} & (n=0) \\\\\n 0 & ({\\rm otherwise})\n \\end{array}\n \\right.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n \\begin{align}\n A:=\n \\left(C_{\\rm c}^\\infty\\left(T^d\\right)^{-1}\\xrightarrow{\\Delta^{\\rm cl}}C_{\\rm c}^\\infty\\left(T^d\\right)^0\\right)\n \\end{align}\n This is an isomorphism. Thus $H^*(A)=0$, then $H^0({\\rm Sym}(A))=\\mathbb{C}$ and $H^{n\\le -1}({\\rm Sym}(A))=0$\n\\end{proof}\n\nThen $i$ gives a state $\\langle-\\rangle_{\\rm cptf}$.\n\\footnote{${\\rm cptf}$ means a ``compactification.\"}\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\\ \\\\\n A compactification state $\\langle-\\rangle_{\\rm cptf}$ is a smooth map:\n \\begin{align}\n \\langle-\\rangle_{\\rm cptf}:\n H^0{\\rm Obs}^{\\rm cl}(U)\n \\to\n H^0{\\rm Obs}^{\\rm cl}(T^d)=\\mathbb{C}.\n \\end{align}\n\\end{definition}\n\\end{tcolorbox}\n\n\\subsection{Massless case}\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n In massless case,\n \\begin{align}\n H^*{\\rm Obs}^{\\rm cl}(T^d)\n =\n \\mathbb{C}[q,r]/r^2\n \\end{align}\n where $q$ is a generator with degree $0$ and $r$ is a generator with degree $-1$.\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n We show a quasi isomorphism:\n \\begin{align}\n A\\xrightarrow{\\pi} B\n \\end{align}\n where\n \\begin{align}\n A=\\left(C_{\\rm c}^\\infty\\left(T^d\\right)^{-1}\\xrightarrow{\\Delta^{\\rm BV}}C_{\\rm c}^\\infty\\left(T^d\\right)^0\\right),\\ \n B=(\\mathbb{C}\\xrightarrow{0}\\mathbb{C}).\n \\end{align}\n We denote the basis of $\\mathbb{R}$ as $r,q$.\n \n First of all, we will see the following commutative diagram:\n \\begin{align}\n \\xymatrix{\n C_{\\rm c}^\\infty\\left(T^d\\right)^{-1} \\ar[r]^{\\Delta^{\\rm cl}} \\ar[d]_{\\pi^{-1}} & C_{\\rm c}^\\infty\\left(T^d\\right)^0 \\ar[d]_{\\pi^0} \\\\\n \\mathbb{C} \\ar[r]^{0} & \\mathbb{C} \n }\n \\end{align}\n where $\\pi^{-1}$ is defined as \n \\begin{align}\n \\pi^{-1}(f^{\\star})\n :=\n r\\int_{T^d}{\\rm d}x_1,\\cdots,x_d\\ f(x_1,\\cdots,x_d)\n =\n r\\int_{0}^{2\\pi} {\\rm d}x_1\n \\cdots\n \\int_{0}^{2\\pi} {\\rm d}x_d\n \\ f(x_1,\\cdots,x_d)\n \\end{align}\n for $f^{\\star}\\in C_{\\rm c}^\\infty\\left(T^d\\right)^{-1}$, \n and $\\pi^{0}$ is defined as\n \\begin{align}\n \\pi^{0}(g)\n :=\n q\\int_{T^d}{\\rm d}x_1\\cdots{\\rm d}x_d\\ g(x_1,\\cdots,x_d)\n =\n q\\int_{0}^{2\\pi} {\\rm d}x_1\n \\cdots\n \\int_{0}^{2\\pi} {\\rm d}x_d\n \\ g(x_1,\\cdots,x_d)\n \\end{align}\n for $g\\in C_{\\rm c}^\\infty\\left(T^d\\right)^0$.\n We can easily check that\n $\\pi^0(g)=0$ holds if $g=\\Delta^{\\rm cl}f^{\\star}.$\n\n Second, we will see $H^0(A)=H^0(B)$. By definition,\n\\begin{align}\n H^0(A)\n =\n \\frac{C_{\\rm c}^\\infty(I)^0}\n {{\\rm im}(\\Delta^{\\rm cl})}.\n\\end{align}\nWe will show that ${\\rm im}(\\Delta^{\\rm cl})={\\rm ker}(\\pi^0)$. If it holds, $H^0(A)=H^0(B)$. Obviously,\n\\begin{align}\n {\\rm im}(\\Delta^{\\rm cl})\\subset {\\rm ker}(\\pi^0).\n\\end{align}\nThen we will check ${\\rm im}(\\Delta^{\\rm cl})\\supset {\\rm ker}(\\pi^0)$.\nTake $f\\in {\\rm ker}(\\pi^0)$. $f$ satisfies\n\\begin{align}\n \\int_{T^d} {\\rm d}x_1,\\cdots,x_d\\ f(x_1,\\cdots,x_d)=0.\\label{eq:int=0}\n\\end{align}\nThen we define functions $\\tilde{f},\\tilde{\\tilde{f}}$ as\n\\begin{align}\n &\\tilde{f}(x_1,\\cdots,x_d)\n :=\n \\int_0^{x_1} {\\rm d}y_1\\cdots\\int_0^{x_d} {\\rm d}y_d\n \\ f(y_1,\\cdots,y_d),\\\\\n &\\tilde{\\tilde{f}}(x_1,\\cdots,x_d)\n :=\n \\tilde{f}(x_1,\\cdots,x_d)-\\frac{1}{(2\\pi)^d}\\int_{T^d}\n {\\rm d}y_1\\cdots{\\rm d}y_d\\ \\tilde{f}(y_1,\\cdots,y_d)\n\\end{align}\n$\\tilde{f}$ and $\\tilde{\\tilde{f}}(x)$ are in $C_{\\rm c}^\\infty(T^d)$ because of (\\ref{eq:int=0}). And we have\n\\begin{align}\n \\int_{S^1}{\\rm d}x\\ \\tilde{\\tilde{f}}(x)=0\n \\label{eq:int_tilde=0}\n\\end{align}\nThen we define\n\\begin{align}\n F(x_1,\\cdots,x_d)\n :=\n \\int_0^{x_1} {\\rm d}y^1\n \\cdots\n \\int_0^{x_d} {\\rm d}y^d\n \\ \n \\tilde{\\tilde{f}}(y_1,\\cdots,y_d)\n\\end{align}\n$F$ is in $C_{\\rm c}^\\infty(T^d)$ because of (\\ref{eq:int=0}).\nWe can see $\\Delta^{\\rm cl}F^\\star=f$, thus $f\\in {\\rm im}(\\Delta^{\\rm cl})$ and ${\\rm im}(\\Delta^{\\rm BV})\\supset {\\rm ker}(\\pi^0)$.\n\nThird, we will see $H^{-1}(A)=H^{-1}(B)$. Clearly $H^{-1}(B)=\\mathbb{C}$. Then let us think of\n\\begin{align}\n H^{-1}(A)={\\rm ker}(\\Delta^{\\rm cl}).\n\\end{align}\nThis is $\\mathbb{C}$, since non-trivial solutions of \n\\begin{align}\n \\Delta f=0\n\\end{align}\nis $f={\\rm const}\\in C_{\\rm c}^\\infty(T^d)$.\nAnd $\\pi^{-1}$ gives just a $(2\\pi)^d$-multiplication. Then $H^{-1}(A)=H^{-1}(B)=\\mathbb{C}$.\n\\end{proof}\n\n\nHence $i$ does NOT give a state.\n\\begin{align}\n i:\n H^0{\\rm Obs}^{\\rm cl}(U)\n \\to\n H^0{\\rm Obs}^{\\rm cl}(T^d)\n =\n \\mathbb{R}[q]\n\\end{align}\nHowever by sending $q$ to $0$, we obtain a state.\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\\ \\\\\n A compactification state $\\langle-\\rangle_{\\rm cptf}$ is defined as $j\\circ i$ where $j$ sends $q$ to $0$.\n \\begin{align}\n \\langle-\\rangle_{\\rm cptf}:\n H^0{\\rm Obs}^{\\rm cl}(U)\n \\xrightarrow{i}\n \\mathbb{C}[q]\n \\xrightarrow{j}\n \\mathbb{C}\n \\end{align}\n where $\\mu(U)\\ll\\infty$.\n\\end{definition}\n\\end{tcolorbox}\n\\subsection{Why does $j$ mean removal of IR divergence ?}\\label{subsec:why_j_?}\nIn order to see physical meaning of $j$, \nwe review the following theorem in one-dimensional system.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\label{thm:M=I_cl}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n If $I\\subset \\mathbb{R}$ is an interval, \n \\begin{align}\n H^*\\left({\\rm Obs}^{\\rm cl}(I)\\right)\n =\n \\left\\{\n \\begin{array}{ll}\n \\mathbb{C}[q,p] & (n=0) \\\\\n 0 & ({\\rm otherwise})\n \\end{array}\n \\right.\n \\end{align}\n where $q,p$ has degree 0.\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n We show a quasi-isomorphism:\n \\begin{align}\n A\\sim B\n \\end{align}\n where\n \\begin{align}\n A=\\left(C_{\\rm c}^\\infty(I)^{-1}\\xrightarrow{\\Delta^{\\rm cl}}C_{\\rm c}^\\infty(I)^0\\right),\\ \n B=(0\\to\\mathbb{R}^2).\n \\end{align}\n $\\mathbb{C}^2$ sits in degree 0. And we denote the basis of $\\mathbb{C}^2$ as $q,p$.\n The cohomology of ${\\rm Sym}(A)$ and ${\\rm Sym}(B)$ are $H^*({\\rm Obs}^{\\rm cl}(I))$ and $\\mathbb{C}[q,p]$ respectively.\n \n First of all, we will see the following commutative diagram:\n \\begin{align}\n \\xymatrix{\n C_{\\rm c}^\\infty(I)^{-1} \\ar[r] \\ar[d]_{\\pi^{-1}} & C_{\\rm c}^\\infty(I)^0 \\ar[d]_{\\pi^0} \\\\\n 0 \\ar[r] & \\mathbb{C}^2\n }\n \\end{align}\n where $\\pi^0$ is defined as \n \\begin{align}\n \\pi^0(g)\n :=q\\int_I {\\rm d}x\\ g(x)\\phi_q(x)+p\\int_I {\\rm d}x\\ g(x)\\phi_p(x)\n \\end{align}\n for $g\\in C_{\\rm c}^\\infty(I)^0$.\n The definition of $\\phi_q,\\phi_p\\in C^\\infty(\\mathbb{R})$ is as follows.\n In case that $m>0$ we define $\\phi_q,\\phi_p\\in C^\\infty(\\mathbb{R})$ as\n\\begin{align}\n \\phi_q(x)=\\frac{1}{2}(e^{mx}+e^{-mx}),\\ \n \\phi_p(x)=\\frac{1}{2m}(e^{mx}-e^{-mx}).\n\\end{align}\nThey form the kernel of $-\\Delta+m^2$.\nIf $m=0$, we define\n\\begin{align}\n \\phi_q(x)=1,\\ \n \\phi_p(x)=x.\n\\end{align}\nWe note that $\\phi_p'(x)=\\phi_q(x)$.\nWe can easily check that\n $\\pi^0(g)=0$ holds if $g=\\Delta^{\\rm cl}f^{\\star}.$\n\nNext, we will see $H^0(A)=H^0(B)$. By definition,\n\\begin{align}\n H^0(A)\n =\n \\frac{C_{\\rm c}^\\infty(I)^0}\n {{\\rm im}(\\Delta^{\\rm cl})}.\n\\end{align}\nWe will show that ${\\rm im}(\\Delta^{\\rm cl})={\\rm ker}(\\pi^0)$. If it holds, $H^0(A)=H^0(B)$. Obviously,\n\\begin{align}\n {\\rm im}(\\Delta^{\\rm cl})\\subset {\\rm ker}(\\pi^0).\n\\end{align}\nThen we will check ${\\rm im}(\\Delta^{\\rm cl})\\supset {\\rm ker}(\\pi^0)$.\nTake $f\\in {\\rm ker}(\\pi^0)$. $f$ satisfies\n\\begin{align}\n &\\int_I {\\rm d}x\\ f(x)e^{mx}=0\\ {\\rm and}\n \\int_I {\\rm d}x\\ f(x)e^{-mx}=0\\ \\ \n ({\\rm for\\ massive\\ case}),\\notag\\\\\n &\\int_I {\\rm d}x\\ f(x)x=0\\ {\\rm and}\n \\int_I {\\rm d}x\\ f(x)=0\\ \\ \n ({\\rm for\\ massless\\ case}) \\label{eq:integral_of_f},\n\\end{align}\nbecause $\\int_I {\\rm d}x f(x)\\phi_p(x)=0$ and $\\int_I {\\rm d}x f(x)\\phi_q(x)=0$.\nLet $G\\in C^0(\\mathbb{R})$ be the Green function:\n\\begin{align}\n &G(x)=\\frac{1}{2m}e^{-m|x|}\\ \\ \n ({\\rm for\\ massive\\ case}),\\notag\\\\\n &G(x)=-\\frac{1}{2}|x|\\ \\ \n ({\\rm for\\ massless\\ case}).\n\\end{align}\nThen the convolution of $f$ and $G$ is\n\\begin{align}\n (G\\cdot f)(x):=\\int_I {\\rm d}y\\ G(x-y)f(y).\n\\end{align}\nThis is in $C_{\\rm c}^\\infty(I)^0$ because of (\\ref{eq:integral_of_f}). For $(G\\cdot f)^\\star\\in C_{\\rm c}^\\infty(I)^{-1}$, \n\\begin{align}\n f=\\Delta^{\\rm cl}(G\\cdot f)^\\star.\n\\end{align}\nThus $f\\in {\\rm im}(\\Delta^{\\rm cl})$, and ${\\rm im}(\\Delta^{\\rm cl})\\supset {\\rm ker}(\\pi^0)$.\n\nFinally, we will see $H^{-1}(A)=H^{-1}(B)$. Clearly $H^{-1}(B)=0$. Then let us think of\n\\begin{align}\n H^{-1}(A)={\\rm ker}(\\Delta^{\\rm cl}).\n\\end{align}\nThis is trivial since there are not non-trivial solutions of \n\\begin{align}\n (-\\Delta+m^2)f=0\n\\end{align}\nfor $f\\in C_{\\rm c}^\\infty(I)$. Then $H^{-1}(A)=H^{-1}(B)=0$.\n\\end{proof}\n\n Focus on massless case. \n The physical meaning of $j$ is to remove the IR divergences.\n The origin of the generator $q$ is a constant field configuration $\\phi_q(x)=1$ in the definition of $\\pi^0$, in other words a long wave mode. ($\\phi_p(x)$ does not live in $T^{d=1}$.)\n \n Since massive theory does not have long wave mode, we can determine the expectation values $\\langle-\\rangle_{\\rm cptf}$ directly. On the other hand, massless theory has a long wave mode, and it occurs IR divergences. \n In path-integral calculation, we need to get rid of a long wave mode to decide the expectation values. Sending $q$ to $0$ is the same procedure.\n\n\n\\section{Schwartz state}\n\\subsection{Massive case}\nAlthough the notion of compactification state is clear, we need the locality condition $\\mu(U)\\ll\\infty$ to define it.\nThis condition is well-behaved in the case of local obseravables.\nHowever extended observables might violate locality condition. Schwartz state is well-defined without locality condition then this is good to apply to gauge theory which has extended observables like Wilson loop.\n\nFunctions with compact supports are Schwartz functions. Then we have\n\\begin{align}\n i:C_{\\rm c}^\\infty(\\mathbb{R}^d)\n \\hookrightarrow\n \\mathcal{S}(\\mathbb{R}^d)\n\\end{align}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is Schwartz space.\nWe define\n\\begin{align}\n {\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d)\n :=\n {\\rm Sym}\n \\left(\n \\mathcal{S}(\\mathbb{R}^d)^{-1}\n \\xrightarrow{\\Delta^{\\rm cl}}\n \\mathcal{S}(\\mathbb{R}^d)^{0}\n \\right).\n\\end{align}\nThen we have\n\\begin{align}\n i:\n {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)\n \\to\n {\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d)\n\\end{align}\nand\n\\begin{align}\n i:\n H^*{\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)\n \\to\n H^*{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d).\n\\end{align}\nActually $i$ gives a state. To see this, we introduce the following theorem.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n In massive case,\n \\begin{align}\n H^n{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d)\n =\n \\left\\{\n \\begin{array}{ll}\n \\mathbb{C} & (n=0) \\\\\n 0 & ({\\rm otherwise})\n \\end{array}\n \\right.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n \\begin{align}\n A:=\n \\left(\\mathcal{S}\n (\\mathbb{R}^d)^{-1}\n \\xrightarrow{\\Delta^{\\rm cl}}\n \\mathcal{S}\n (\\mathbb{R}^d)^0\\right)\n \\end{align}\n This is an isomorphism. To see this, we consider Fourier transformation:\n \\begin{align}\n \\begin{array}{rccc}\n &\\mathcal{S}(\\mathbb{R}^d)&\\longrightarrow& \\mathcal{S}(\\mathbb{R}^d)\\\\\n & \\rotatebox{90}{$\\in$}& & \\rotatebox{90}{$\\in$} \\\\\n & f & \\longmapsto & \\hat{f}\n \\end{array}\n \\end{align}\n where \n \\begin{align}\n \\hat f(k_1,\\cdots,k_d)\n :=\n \\int_{\\mathbb{R}^d} f(x_1,\\cdots,x_n)e^{ik\\cdot x}\n {\\rm d}x.\n \\end{align}\n Fourier transformation gives an isomorphism \n $\\mathcal{S}(\\mathbb{R}^d)\\stackrel{\\sim}{\\longrightarrow}\\mathcal{S}(\\mathbb{R}^d)$.\n Then we have\n \\begin{align}\n \\hat{A}\n :=\n \\left(\\mathcal{S}\n (\\mathbb{R}^d)^{-1}\n \\xrightarrow{\\hat\\Delta^{\\rm cl}}\n \\mathcal{S}\n (\\mathbb{R}^d)^0\\right)\n \\end{align}\n where $(\\hat\\Delta^{\\rm cl} \\hat f)(k)=-(k^2+m^2)f(k)$.\n Since it is massive, $\\hat{\\Delta}^{\\rm cl}$ is an isomorphism. Then $\\Delta^{\\rm cl}$ is also an isomorphism.\n Thus $H^*(A)=0$, then $H^0({\\rm Sym}(A))=\\mathbb{C}$ and $H^{n\\le -1}({\\rm Sym}(A))=0$\n\\end{proof}\nThen $i$ gives a state $\\langle-\\rangle_{\\rm Sch}$.\n\\footnote{${\\rm cptf}$ means a ``compactification.\"}\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n A Schwartz state $\\langle-\\rangle_{\\rm Sch}$ is a smooth map:\n \\begin{align}\n \\langle-\\rangle_{\\rm Sch}:\n H^0{\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)\n \\to\n H^0{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d)\n =\\mathbb{C}\n \\end{align}\n\\end{definition}\n\\end{tcolorbox}\n\n\\subsection{Massless case in one-dimension}\nIn massless case, $H^0{\\rm Obs}^{\\rm cl}_\\mathcal{S}(\\mathbb{R}^d)$ is NOT isomorphic to $\\mathbb{C}$.\nThis is because\n\\begin{align}\n \\hat\\Delta^{\\rm cl}:\n \\mathcal{S}(\\mathbb{R}^d)\\ni\n \\hat{f}(k)\\mapsto -k^2 \\hat{f}(k)\n \\in\\mathcal{S}(\\mathbb{R}^d)\n\\end{align}\nis not an isomorphism since \n\\begin{align}\n ({\\hat\\Delta^{\\rm cl}})^{-1}:\n \\hat{f}(k)\\mapsto -\\frac{\\hat{f}(k)}{k^2}\n\\end{align}\nis not well-defined.\nPhysicists call it IR divergence.\n\nHowever the IR divergence can be controlled by the following theorem.\nUsing this, we can define Schwartz state in the massless case.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\label{thm:IR_one-dim}\\ \\\\\n Let $\\hat f\\in \\mathcal{S}(\\mathbb{R})$.\n If $\\hat f$ satisfies\n \\begin{align}\n \\hat f(0)=0,\\\\\n \\partial \\hat{f}(0)=0,\n \\end{align}\n then\n \\begin{align}\n \\hat h(k)\n :=\n -\n \\frac{\\hat{f}(k)}{k^2}\n \\end{align}\n is also a Schwartz function on $\\mathbb{R}$.\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n First of all, we will find $\\hat g\\in\\mathcal{S}(\\mathbb{R})$ satisfying\n \\begin{align}\n \\hat{f}(k)=\n k\\hat{g}(k).\n \\end{align}\n Since $\\hat f(0)=0$, then\n \\begin{align}\n \\hat f(k)\n =\n \\hat f(k)\n -\n \\hat f(0).\n \\end{align}\n We have a line integral\n \\begin{align}\n \\hat f(k)\n -\n \\hat f(0)\n &=\n \\int_0^k \n {\\rm d}p\\ \n (\\partial\\hat{f})(p)\n \\notag\\\\\n &=\n k\n \\int_0^1 \n {\\rm d}t\\ \n (\\partial\\hat{f})(kt).\n \\end{align}\n where $p:=kt$.\n Define \n \\begin{align}\n \\hat g(k)\n := \\int_0^1\n {\\rm d}t\\ \n (\\partial \\hat f)(kt),\n \\end{align}\n then\n \\begin{align}\n \\hat{f}(k)\n =\n k\\hat{g}(k)\\label{eq:f=kg}.\n \\end{align}\n $\\hat{g}$ is Schwartz, since we can interchange differentiation and integration, because $\\partial \\hat{f}$ is Schwartz.\n\n Now $\\hat{g}(0)=0$ by $\\partial \\hat f(0)=0$. By the same procedure as the above one, we have $\\hat{h}\\in\\mathcal{S}(\\mathbb{R})$ satisfying\n \\begin{align}\n \\hat{g}(k)=k\\hat{h}(k)\\label{eq:g=kh}.\n \\end{align}\n\n By (\\ref{eq:f=kg}) and (\\ref{eq:g=kh}),\n \\begin{align}\n \\hat{f}(k)=k^2\\hat{h}(k).\n \\end{align}\n\\end{proof}\n\n\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\label{thm:Schwatz_massless}\\ \\\\\n In massless and $d=1$ case \n \\begin{align}\n H^n{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R})\n =\n \\left\\{\n \\begin{array}{ll}\n \\mathbb{C}[q,p] & (n=0) \\\\\n 0 & ({\\rm otherwise})\n \\end{array}\n \\right.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\n\\begin{proof}\n We show a quasi-isomorphism:\n \\begin{align}\n A\\sim B\n \\end{align}\n where\n \\begin{align}\n A=\n \\left(\\mathcal{S}(\\mathbb{R})^{-1}\\xrightarrow{\\Delta^{\\rm cl}}\\mathcal{S}(\\mathbb{R})^0\\right),\\ \n B=(0\\to\\mathbb{R}^2).\n \\end{align}\n $\\mathbb{C}^2$ sits in degree 0. And we denote the basis of $\\mathbb{R}^2$ as $q,p$.\n The cohomology of ${\\rm Sym}(A)$ and ${\\rm Sym}(B)$ are $H^*{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R})$ and $\\mathbb{C}[q,p]$ respectively.\n \n First of all, we will see the following commutative diagram:\n \\begin{align}\n \\xymatrix{\n \\mathcal{S}(\\mathbb{R})^{-1} \\ar[r] \\ar[d]_{\\pi^{-1}} & \\mathcal{S}(\\mathbb{R})^0 \\ar[d]_{\\pi^0} \\\\\n 0 \\ar[r] & \\mathbb{C}^2\n }\n \\end{align}\n where $\\pi^0$ is defined as \n \\begin{align}\n \\pi^0(g)\n :=\n q\\int_{\\mathbb{R}} {\\rm d}x\n \\ g(x)\\phi_q(x)\n +\n p\\int_{\\mathbb{R}} {\\rm d}x\n \\ g(x)\\phi_{p_1}(x)\n \\label{eq:pi^0}\n \\end{align}\n for $g\\in \\mathcal{S}(\\mathbb{R})^0$.\n The definition of $\\phi_q,\\phi_{p}\\in C^\\infty(\\mathbb{R})$ is as follows.\n\\begin{align}\n\\begin{cases}\n \\phi_q(x)=1, \\\\\n \\phi_{p}(x)=x.\n \\end{cases}\n\\end{align}\nThey form the kernel of $-\\Delta$.\nWe can easily check that\n $\\pi^0(g)=0$ holds if $g=\\Delta^{\\rm cl}f^{\\star}.$\nNote that $\\pi^0$ is well-defined, in other words all integrations in (\\ref{eq:pi^0}) converge, since products of Schwartz function $g$ and polynomials $\\phi_q,\\phi_{p}$ are also Schwartz functions.\n\nNext, we will see $H^0(A)=H^0(B)$. By definition,\n\\begin{align}\n H^0(A)\n =\n \\frac{\\mathcal{S}(\\mathbb{R})^0}\n {{\\rm im}(\\Delta^{\\rm cl})}.\n\\end{align}\n$\\pi^0$ is surjective \\footnote{The proof is in the appendix.},\nthen by the isomorphism theorem\n\\begin{align}\n H^0(B)\n =\n \\frac{\\mathcal{S}(\\mathbb{R})^0}\n {{\\rm ker}(\\pi^0)}.\n\\end{align}\nWe will show that ${\\rm im}(\\Delta^{\\rm cl})={\\rm ker}(\\pi^0)$. If it holds, $H^0(A)=H^0(B)$. Obviously,\n\\begin{align}\n {\\rm im}(\\Delta^{\\rm cl})\\subset {\\rm ker}(\\pi^0).\n\\end{align}\nThen we will check ${\\rm im}(\\Delta^{\\rm cl})\\supset {\\rm ker}(\\pi^0)$.\nTake $f\\in {\\rm ker}(\\pi^0)$. $f$ satisfies\n\\begin{align}\n &\\int_{\\mathbb{R}}f(x)\\ \n {\\rm d}x\n =0,\\label{eq:pi^0_q}\\\\\n &\\int_{\\mathbb{R}}xf(x)\\ \n {\\rm d}x\n =0\\label{eq:pi^0_p}.\n\\end{align}\nIn order to make it clear, we consider Fourier transformation of $A$.\n\\begin{align}\n \\hat A\n =\n \\left(\\mathcal{S}(\\mathbb{R})^{-1}\\xrightarrow{\\hat\\Delta^{\\rm cl}}\\mathcal{S}(\\mathbb{R})^0\\right)\n\\end{align}\nwhere $\\hat\\Delta^{\\rm cl}:\\hat f(k)\\mapsto -k^2f(k)$. \nAnd the above conditions (\\ref{eq:pi^0_p}) (\\ref{eq:pi^0_q}) are the same as\n\\begin{align}\n \\hat f(0)=0,\\\\\n \\partial\\hat f(0)=0.\n\\end{align}\nUnder these conditions \n\\begin{align}\n \\hat h(k):=-\\frac{\\hat{f}(k)}{k^2}\n\\end{align}\nis in $\\mathcal{S}(\\mathbb{R})$ by Theorem \\ref{thm:IR_one-dim}.\nHence, $\\hat \\Delta^{\\rm cl} \\hat h=\\hat f$ or $\\Delta^{\\rm cl} h=f$. \nThus $f\\in {\\rm im}(\\Delta^{\\rm cl})$, and ${\\rm im}(\\Delta^{\\rm cl})\\supset {\\rm ker}(\\pi^0)$.\n\nFinally, we will see $H^{-1}(A)=H^{-1}(B)$. Clearly $H^{-1}(B)=0$. Then let us think of\n\\begin{align}\n H^{-1}(A)={\\rm ker}(\\Delta^{\\rm cl}).\n\\end{align}\nThis is trivial since the solution of \n\\begin{align}\n \\Delta f(x)=0\n \\ \n {\\rm or}\n \\ \n -k^2\\hat f(k)=0\n\\end{align}\nis $f=0$ or $\\hat f=0$.\nThen $H^{-1}(A)=H^{-1}(B)=0$.\n\\end{proof}\n\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\\ \\\\\n A Schwartz state $\\langle-\\rangle_{\\rm Sch}$ is a smooth map:\n \\begin{align}\n \\langle-\\rangle_{\\rm Sch}:\n H^0{\\rm Obs}^{\\rm cl}(\\mathbb{R})\n \\to\n H^0{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R})\n =\\mathbb{C}[q,p]\n \\xrightarrow{j} \\mathbb{C}\n \\end{align}\n where $j$ is a map: $q,p\\mapsto 0$.\n\\end{definition}\n\\end{tcolorbox}\n\nIn case of compactification state, sending $q\\mapsto 0$ means to get rid of IR divergences as explained in \\ref{subsec:why_j_?}.\nHowever in case of Schwartz state, $j$ sends not only $q$ but also $p$ to $0$.\n\nIf we translate the map $p\\mapsto 0$ to ordinary physics words, it is the same as that the state $|0\\rangle$ is assumed to be\n\\begin{align}\n P|0\\rangle=0.\n\\end{align}\nIn other words, it is translation invariance. \n\nThe compactification state automatically has translation invariance because we take periodic boundary condition. In contrast, Schwartz state need to be assumed translation invariance.\n\n\\subsection{Massless case in higher dimension }\nIn one-dimensional case, we show\n\\begin{align}\n \\left(\\mathcal{S}(\\mathbb{R})^{-1}\\xrightarrow{\\Delta^{\\rm cl}}\\mathcal{S}(\\mathbb{R})^0\\right)\n \\xrightarrow[\\ \\pi\\ ]{\\sim}{}\n (0\\to\\mathbb{C}^2).\n\\end{align}\nThe $\\mathbb{C}^2$ is the solution space of Laplace equation $\\Delta \\phi=0$ in one-dimensional space.\n\nIn order to generalize to higher dimensional case, we need to consider the following solution space\\footnote{$\\mathscr{H}$ denotes ``Harmonics.\"}:\n\\begin{align}\n \\mathscr{H}\n :=\n \\{\\phi\\in \\mathcal{S}'(\\mathbb{R}^d)\\ |\\ \\Delta_{\\mathcal{S}'}\\phi=0\\}\n\\end{align}\nwhere $\\mathcal{S}'(\\mathbb{R}^d)$ is a space of tempered distributions and $\\Delta_{\\mathcal{S}'}$ is a Laplacian in $\\mathcal{S}'(\\mathbb{R}^d)$.\n$\\mathcal{S}'(\\mathbb{R}^d)$ is a natural dual space of $\\mathcal{S}(\\mathbb{R}^d)$, therefore the product $\\langle \\phi,f\\rangle$ is well-defined for $f\\in \\mathcal{S}(\\mathbb{R}^d)$ and $\\phi\\in\\mathcal{S}'(\\mathbb{R}^d)$.\nIf $\\phi$ is a function, we can represent $\\langle \\phi,f\\rangle$ as integration:\n\\begin{align}\n \\langle \\phi,f\\rangle\n =\n \\int_{\\mathbb{R}^d} {\\rm d}x_1\\cdots{\\rm d}x_d\n \\ \n \\overline{\\phi(x_1,\\cdots,x_d)}\\ f(x_1,\\cdots,x_d).\n\\end{align}\n\nLater we will see a quasi-isomorphism: \n\\begin{align}\n \\left(\\mathcal{S}(\\mathbb{R}^d)^{-1}\\xrightarrow{\\Delta^{\\rm cl}}\\mathcal{S}(\\mathbb{R}^d)^0\\right)\n \\xrightarrow[\\ \\pi\\ ]{\\sim}{}\n (0\\to\\mathscr{H}).\n\\end{align}\nRoughly speaking, $\\pi$ is defined\\footnote{This is not well-defined because $\\mathscr{H}$ has $\\infty$-dimension. More accurately we need to consider the completion of $\\mathscr{H}$ and deform the definition of $\\pi$.} as \n\\begin{align}\n \\begin{array}{rccc}\n \\pi^0\\colon\n &\\mathcal{S}(\\mathbb{R}^d)^0&\\longrightarrow& \\mathscr{H}\\\\\n & \\rotatebox{90}{$\\in$}& & \\rotatebox{90}{$\\in$} \\\\\n & f & \\longmapsto & \n \\sum_{\\phi\\in B(\\mathscr{H})}\\phi\\langle \\phi,f\\rangle.\n \\end{array}\n\\end{align}\nwhere $B(\\mathscr{H})$ is a basis of $\\mathscr{H}$.\nThis is a natural generalization of one-dimensional case.\n\nIn order to make clear the meaning of $B(\\mathscr{H})$, \nwe introduce the following known theorem.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n Let $\\phi\\in \\mathcal{S}'(\\mathbb{R}^d)$.\n If $\\Delta_{\\mathcal{S'}}\\phi=0$, then $\\phi$ can be represented as a polynomial.\n\\end{theorem}\n\\end{tcolorbox}\n\\noindent{} In other words, $\\phi\\in\\mathcal{H}$ is a harmonic polynomial.\nFor polynomials, Fisher inner product is convenient.\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}{\\rm (Fisher inner product)} \\\\\nLet $\\phi,\\psi$ be polynomials on $\\mathbb{R}^d$.\nFisher inner product is defined as\n\\begin{align}\n (\\phi|\\psi)\n :=\\left[\n \\overline{\\phi\\left(\\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_1}\\right)}\n \\psi(x_1,\\cdots,x_d)\\right]_{x_1=\\cdots x_d=0}.\n\\end{align}\nLet $\\phi$ be $k$-degree and $\\psi$ be $l$-degree.\nIf $k\\neq l$ then $(\\phi|\\psi)=0$.\n\\end{definition}\n\\end{tcolorbox}\n\\noindent{}And we obtain an orthogonal decomposition of $\\mathscr{H}$.\n\\begin{align}\n \\mathscr{H}\n =\n \\bigoplus_{k=0}^\\infty \\mathscr{H}_k.\n\\end{align}\nThe dimension of $\\mathscr{H}_k$ is finite as follows:\n\\begin{align}\n {\\rm dim}(\\mathscr{H}_k)\n =\n \\frac{(d+2k-2)(d+k-3)!}{(d-2)!k!}.\n\\end{align}\nWe take Gram–Schmidt orthonormalization and get basis $B(\\mathscr{H})=\\bigoplus_{k=0}^\\infty B(\\mathscr{H}_k)$.\n\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\label{thm:higher_dim}\\ \\\\\n In massless and higher dimensional case we have a dense map $\\pi$:\n \\begin{align}\n H^n{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R})\n \\xrightarrow{\\pi}\n \\left\\{\n \\begin{array}{ll}\n {\\rm Sym}(\\widetilde{\\mathscr{H}}\\ ) & (n=0) \\\\\n 0 & ({\\rm otherwise})\n \\end{array}\n \\right.\n\\end{align}\nwhere $\\widetilde{\\mathscr{H}}$ is a completion of $\\mathscr{H}$ in terms of Fisher inner product.\n\\end{theorem}\n\\end{tcolorbox}\nThe proof is too long to write here, hence we place the details in the appendix.\n\n\\begin{tcolorbox}[colframe=red,colback=red!3!]\n\\begin{definition}\\ \\\\\n A Schwartz state $\\langle-\\rangle_{\\rm Sch}$ is a smooth map:\n \\begin{align}\n \\langle-\\rangle_{\\rm Sch}:\n H^0{\\rm Obs}^{\\rm cl}(\\mathbb{R})\n \\to\n H^0{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R})\n ={\\rm Sym}(\\widetilde{\\mathscr{H}}\\ )\n \\xrightarrow{j} \\mathbb{C}\n \\end{align}\n where $j$ sends all generators to $0$.\n\\end{definition}\n\\end{tcolorbox}\nIn one-dimensional case, all we assume is essentially the translation invariance:\n\\begin{align}\n P|0\\rangle=0.\n\\end{align}\nHowever, in higher-dimensional case, it looks that we need to assume more invariance.\n\n\n\\section{Equivalence}\n\\subsection{Massive case}\nAs said in the beginning, we will show the equivalence of a natural augmentation state, a compactification state and a Schwartz state in massive case. \nFirst of all, we introduce the following theorem shown by K. Costello and O. Gwilliam \\cite{Costello:2016vjw}.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\n{\\rm (K. Costello, O. Gwilliam \\cite{Costello:2016vjw})}\\\\\n In massive case,\n \\begin{align}\n \\langle-\\rangle_{\\rm aug}\n =\n \\langle-\\rangle_{\\rm Sch}.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\noindent{}In addition, we will show that $\\langle-\\rangle_{\\rm aug}$ is the same as $\\langle-\\rangle_{\\rm cptf}$ in massive cases.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n In massive case, for observables satisfies the locality condition,\n \\begin{align}\n \\langle-\\rangle_{\\rm aug}\n =\n \\langle-\\rangle_{\\rm cptf}.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\nTake $\\mathcal{O}\\in {\\rm Obs}^{\\rm cl}(U)\\subset {\\rm Obs}^{\\rm cl}(T^d)$, and calculate $\\langle\\mathcal{O}\\rangle_{\\rm cptf}$. \n\\begin{align}\n \\Delta^{\\rm cl}_{T^d}\n :\n C_{\\rm c}^\\infty(T^d)^{-1}\n \\to\n C_{\\rm c}^\\infty(T^d)^0\n\\end{align}\nis an isomorphism. Therefore\n\\begin{align}\n \\mathcal{O}\n =\n c+f+f_1*f_2+\\cdots\n =\n c+\\Delta^{\\rm cl}_{T^d}(\\cdots).\n\\end{align}\nHence,\n\\begin{align}\n \\langle\\mathcal{O}\\rangle_{\\rm cptf}=c=\\langle\\mathcal{O}\\rangle_{\\rm aug}.\n\\end{align}\n\n\\end{proof}\n\n\\subsection{Masless case}\nWe will show the equivalence of the threer states in massless case.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n In massless case, for observables satisfies the locality condition,\n \\begin{align}\n \\langle-\\rangle_{\\rm aug}\n =\n \\langle-\\rangle_{\\rm cptf}.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\nTake $\\mathcal{O}\\in {\\rm Obs}^{\\rm cl}(U)$, and calculate $\\langle\\mathcal{O}\\rangle_{\\rm cptf}$.\n\\begin{align}\n \\langle\\mathcal{O}\\rangle_{\\rm cptf}\n =j\\circ\\pi^0(\\mathcal{O})\n\\end{align}\n$\\mathcal{O}=c+f+f_1*f_2+\\cdots$, then\n\\begin{align}\n \\pi^0(c+f+f_1*f_2+\\cdots)\n &=c+\\pi^0(f)+\\pi^0(f_1)\\cdot \\pi^0(f_2)+\\cdots\\notag\\\\\n &=c+q\\times ({\\rm some\\ number}).\n\\end{align}\nHence,\n\\begin{align}\n \\langle\\mathcal{O}\\rangle_{\\rm cptf}=c=\\langle\\mathcal{O}\\rangle_{\\rm aug}.\n\\end{align}\n\\end{proof}\n\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n In massless case,\n \\begin{align}\n \\langle-\\rangle_{\\rm aug}\n =\n \\langle-\\rangle_{\\rm Sch}.\n\\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\nTake $\\mathcal{O}\\in {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)$, and calculate $\\langle\\mathcal{O}\\rangle_{\\rm Sch}$.\n\\begin{align}\n \\langle\\mathcal{O}\\rangle_{\\rm cptf}\n =j\\circ\\pi^0(\\mathcal{O})\n\\end{align}\n$\\mathcal{O}=c+f+f_1*f_2+\\cdots$, then\n\\begin{align}\n \\pi^0(c+f+f_1*f_2+\\cdots)\n &=c+\\pi^0(f)+\\pi^0(f_1)\\cdot \\pi^0(f_2)+\\cdots\\notag\\\\\n &=c+({\\rm some\\ generator})\\times ({\\rm some\\ number}).\n\\end{align}\nHence,\n\\begin{align}\n \\langle\\mathcal{O}\\rangle_{\\rm Sch}=c=\\langle\\mathcal{O}\\rangle_{\\rm aug}.\n\\end{align}\n\\end{proof}\n\n\\section{Conclusion and discussion}\nWe have seen the concrete constructions of states in factorization algebras. \nIn the case of the compactification state,\n\\begin{align}\n \\langle-\\rangle_{\\rm cptf}:\n {\\rm Obs}^{\\rm cl}(U)\n \\to{\\rm Obs}^{\\rm cl}(T^d)\n \\cong\n \\begin{cases}\n \\mathbb{C} & ({\\rm massive}) \\\\ \n \\mathbb{C}[q] \\xrightarrow{j}\\mathbb{C} &({\\rm massless}) .\n \\end{cases}\n\\end{align}\nHere the origin of $q$ is the long wave mode of scalar field, so $j:q\\mapsto 0$ means to get rid of IR divergences.\nAnd the case of the Schwartz state,\n\\begin{align}\n \\langle-\\rangle_{\\rm Sch}:\n {\\rm Obs}^{\\rm cl}(\\mathbb{R}^d)\n \\to{\\rm Obs}^{\\rm cl}_{\\mathcal{S}}(\\mathbb{R}^d)\n \\cong\n \\begin{cases}\n \\mathbb{C} & ({\\rm massive}) \\\\ \n {\\rm Sym}(\\widetilde{\\mathscr{H}}\\ ) \\xrightarrow{j}\\mathbb{C} &({\\rm massless}) .\n \\end{cases}\n\\end{align}\nEspecially, in one-dimensional case, ${\\rm Sym}(\\widetilde{\\mathscr{H}}\\ ) $ is just $\\mathbb{C}[q,p]$.\nThis space might imply the symmetry in asymptotic symmetries. For example, in one-dimensional case, $j:p\\mapsto 0$ means the assumption of translation invariance for the state: $P|0\\rangle=0$.\n\nThe number of basis of ${\\rm Sym}(\\widetilde{\\mathscr{H}}\\ )$ might be uncountablly infinite. It seems to come from the uncountable label $x\\in\\mathbb{R}^{d-1}$ of $\\Phi(x)$.\nIn other words, generally QFTs have infinitely number of observables.\n\nWe think that it is interesting to investigate ${\\rm Sym}(\\widetilde{\\mathscr{H}}\\ )$ in two-dimensional case.\nGenerally, in order to avoid Coleman-Mermin-Wagner theorem\\cite{Coleman:1973ci}\\cite{Mermin:1966fe}, we need conformal symmetry (Virasoro algebra):\n\\begin{align}\n L_m|0\\rangle=0.\n\\end{align}\nWe expect that Virasoro algebra can be derived from ${\\rm Sym}(\\widetilde{\\mathscr{H}}\\ )$.\n\n\n\\section*{Acknowledgment}\nWe would like to thank Yuma Furuta for helpful discussions.\nThe work of M. K. is supported by Grant-in-Aid for JSPS Fellows No. 22KJ1989.\nThe work of T. S. was supported by JST SPRING, Grant Number JPMJSP2110.\n\n\\appendix\n\\section{About the proof of Theorem \\ref{thm:Schwatz_massless}}\n\\subsection{$\\pi^0$ is surjective}\\label{subsec:pi^0_surjective}\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n $\\pi^0:\\mathcal{S}(\\mathbb{R})\\to\\mathbb{R}^2$ is surjective where\n $\\pi^0$ is defined as\n \\begin{align}\n \\pi^0(g)\n =\n q\\int_{\\mathbb{R}} {\\rm d}x\n \\ g(x)\\phi_q(x)\n +\n p\\int_{\\mathbb{R}} {\\rm d}x\n \\ g(x)\\phi_{p}(x)\n \\end{align}\n and $q,p$ is the basis of $\\mathbb{R}^2$.\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n It is enough to show that there are $Q,P\\in\\mathcal{S}(\\mathbb{R})$ satisfying\n \\begin{align}\n \\pi^0(Q)=q,\\ \\pi^0(P)=p.\n \\end{align}\n\n One good choice of $Q$ is a {\\it smeared $\\delta$-function}.\n \\begin{align}\n Q(x)\n :=\n \\delta_{\\rm smeared}(x)\n \\end{align}\n where we assume that $\\delta_{\\rm smeared}$ is even, has a compact support and satisfies\n \\begin{align}\n \\int{\\rm d}x\n \\ \\delta_{\\rm smeared}(x)=1.\n \\end{align}\n $\\phi_{p}$ is odd, then we have $\\pi^0(\\delta_{\\rm smeared})=q$.\n And $P_i$ is given as\n \\begin{align}\n P(x)\n :=\n -\n \\frac{\\partial}{\\partial x}\n \\delta_{\\rm smeared}(x).\n \\end{align}\n We can easily show that $\\pi^0(P)=p$.\n \n\n Another choice of $Q$ is given by\n \\begin{align}\n Q(x)\n :=\n \\frac{1}{\\sqrt{\\pi}}\\exp(x).\n \\end{align}\n And $P$ is given as\n \\begin{align}\n P(x)\n :=\n -\\frac{\\partial}{\\partial x}Q(x)\n =\n \\frac{2}{\\sqrt{\\pi}}x\\exp(x^2).\n \\end{align}\n\\end{proof}\n\\subsection{Physical meanings of $q$ and $p$}\nBy Theorem \\ref{thm:Schwatz_massless}, we have an isomorphism:\n\\begin{align}\n\\begin{array}{rccc}\n\\pi^0\\colon &H^0{\\rm Obs}^{\\rm cl}_\\mathcal{S}(\\mathbb{R}^d) &\\stackrel{\\sim}{\\longrightarrow}& \\mathbb{R}[q,p]\\\\\n& \\rotatebox{90}{$\\in$}& & \\rotatebox{90}{$\\in$} \\\\\n& [Q]_\\mathcal{S} & \\longmapsto & q\\\\\n& [P]_\\mathcal{S} & \\longmapsto & p.\n\\end{array}\n\\end{align}\nEspecially if we take $Q=\\delta_{\\rm smeared}$ and $P=-\\partial\\delta_{\\rm smeared}$, $Q$ and $P$ is in $C_{\\rm c}^\\infty(\\mathbb{R})$. Then we have an inclusion map:\n\\begin{align}\n\\begin{array}{rccc}\ni\\colon&H^0{\\rm Obs}^{\\rm cl}(\\mathbb{R}) &\\hookrightarrow& H^0{\\rm Obs}^{\\rm cl}_\\mathcal{S}(\\mathbb{R}^d)\\\\\n& \\rotatebox{90}{$\\in$}& & \\rotatebox{90}{$\\in$} \\\\\n& [Q] & \\longmapsto & [Q]_\\mathcal{S}\\\\\n& [P] & \\longmapsto & [P]_\\mathcal{S}.\n\\end{array}\n\\end{align}\nTherefore originally $q$ and $p$ are from observables\n\\begin{align}\n Q=\\delta_{\\rm smeared},\\ P=-\\partial\\delta_{\\rm smeared}.\n\\end{align}\nThe action for the field $\\Phi\\in C^\\infty(\\mathbb{R}^d)$ is\n\\begin{align}\n &Q(\\Phi)\n =\\int_{\\mathbb{R}} \n {\\rm d}x\\ \n \\delta_{\\rm smeared}(x)\n \\Phi(x)\n \\sim \\Phi(0),\\\\ \n &P(\\Phi)\n =\\int_{\\mathbb{R}} \n {\\rm d}x\\ \n (-\\partial\\delta_{\\rm smeared}(x))\n \\Phi(x)\n \\sim \\partial\\Phi(0).\n\\end{align}\nThese are the same as the position observable and the momentum observables in physics literature.\n\n\\section{Some properties of harmonic polynomials}\n\\subsection{Hecke identities and a convenient representation of Fisher inner product}\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n If $\\phi\\in\\mathscr{H}$, then we have two identities.\n \\begin{align}\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\phi(x_1,\\cdots,x_d)\n e^{-\\frac{1}{2}x^2}\n e^{ik\\cdot x}\n =\n \\phi(ik_1,\\cdots,ik_d)\n e^{-\\frac{1}{2}k^2},\\\\\n \\phi\\left(\n \\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_d}\\right)\n e^{-\\frac{1}{2}x^2}\n =\n \\phi(-x_1,\\cdots,-x_d)e^{-\\frac{1}{2}x^2}.\n \\end{align}\n We call them Hecke identities.\n\\end{theorem}\n\\end{tcolorbox}\n\\noindent{}By using Hecke identities, we have the following theorem.\n\\begin{tcolorbox}[colframe=blue,colback=blue!3!]\n\\begin{theorem}\\ \\\\\n If $\\phi,\\psi\\in\\mathscr{H}$, then\n \\begin{align}\n (\\phi|\\psi)\n &=\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\overline{\\phi(x_1,\\cdots,x_d)}\n \\psi(x_1,\\cdots,x_d)\n e^{-\\frac{1}{2}x^2}\\notag\\\\\n &=\\left\\langle \\phi,\\frac{1}{(2\\pi)^{d/2}}\\psi e^{-\\frac{1}{2}x^2}\\right\\rangle\n \\end{align}\n\\end{theorem}\n\\end{tcolorbox}\n\\begin{proof}\n $\\overline{\\phi\\left(\n \\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_d}\\right)}\n \\psi\\left(x_1,\\cdots,x_d\\right)$ is also in $\\mathscr{H}$,\n then by the first Hecke identities, we obtain\n \\begin{align}\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\overline{\\phi\\left(\n \\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_d}\\right)}\n \\psi\\left(x_1,\\cdots,x_d\\right)\n e^{-\\frac{1}{2}x^2}\n e^{ik\\cdot x}\\notag\\\\\n =\n \\overline{\\phi\\left(-i\n \\frac{\\partial}{\\partial k_1},\n \\cdots,\n -i\\frac{\\partial}{\\partial k_d}\\right)}\n \\psi\\left(k_1,\\cdots,k_d\\right)\n e^{-\\frac{1}{2}k^2}.\n \\end{align}\n Substitute $k_1=\\cdots=k_d=0$, then\n \\begin{align}\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\overline{\\phi\\left(\n \\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_d}\\right)}\n \\psi\\left(x_1,\\cdots,x_d\\right)\n e^{-\\frac{1}{2}x^2}\n =\n (\\phi|\\psi).\n \\end{align}\n Interate by part, \n \\begin{align}\n (\\phi|\\psi)\n &=\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\psi\\left(x_1,\\cdots,x_d\\right)\n \\overline{\\phi\\left(\n \\frac{\\partial}{\\partial x_1},\n \\cdots,\n \\frac{\\partial}{\\partial x_d}\\right)}\n e^{-\\frac{1}{2}x^2}\\notag\\\\\n &=\n \\frac{1}{(2\\pi)^{d/2}}\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_d\\ \n \\psi(x_1,\\cdots,x_d)\n \\overline{\\phi(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{2}x^2}.\n \\end{align}\n We used the second Hecke identity in the last line.\n\\end{proof}\n\n\\section{About the proof of Theorem \\ref{thm:higher_dim}}\n\n\\subsection{Outline of the proof}\nThe outline of the proof is the same as the one of Theorem \\ref{thm:Schwatz_massless}.\nHence naively it seem to be enough to see the following things.\n\\begin{itemize}\n \\item[(1)] The accurate definition of $\\pi^0$\n \\item[(2)] $\\pi^0$ is a surjective map.\n \\item[(3)] ${\\rm ker}(\\pi^0)\\cong{\\rm im}(\\Delta^{\\rm cl})$\n\\end{itemize}\n\\noindent{}However, since $\\mathscr{H}$ has infinitely large dimension, we rephrase (2) and (3) as\n\\begin{itemize}\n \\item[(2)'] $\\pi^0$ is a dense map.\n \\item[(3)'] ${\\rm ker}(\\pi^0)\\cong\\overline{{\\rm im}(\\Delta^{\\rm cl})}$ where $\\overline{{\\rm im}(\\Delta^{\\rm cl})}$ is a closure of ${\\rm im}(\\Delta^{\\rm cl})$.\n\\end{itemize}\nNote that\n\\begin{align}\n \\frac{\\mathcal{S}(\\mathbb{R}^2)^0}{{\\rm im(\\Delta^{\\rm cl})}}\n \\cong\n \\frac{\\mathcal{S}(\\mathbb{R}^2)^0}{{\\rm \\overline{im(\\Delta^{\\rm cl})}}},\n\\end{align}\nthen it is sufficient to see ${\\rm ker}(\\pi^0)\\cong\\overline{{\\rm im}(\\Delta^{\\rm cl})}$.\nWe will see each of them in the later sections.\n\n\\subsection{The accurate definition of $\\pi^0$}\nSo as to define $\\pi^0$, we consider the sequence\n\\begin{align}\n \\{\n \\pi^0_n:\\mathcal{S}\\to \\mathscr{H}\n \\}_{n=0,1,\\cdots}\n\\end{align}\nas\n\\begin{align}\n \\pi^0_n(f)\n =\n \\sum_{k=0}^n \n a_k\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle \\phi_k.\n\\end{align}\nHere $\\phi_k$ forms the orthogonal basis $B(\\mathscr{H})$ and the label $k$ is set in ascending order of the polynomial degree.\nThen we have $k>\\deg(\\phi_k)$.\nAnd $a_k$ will be defined as $\\frac{1}{k^{\\frac{k}{2}+1}}$ in order to make $\\pi_n^0(f)$ Cauchy sequence, in other words\n\\begin{align}\n ||\\pi_n^0(f)-\\pi^0_m(f)||\\to 0\\ {\\rm as}\\ n,m\\to\\infty\n\\end{align}\nwhere $||\\cdot ||$ is decided by Fisher inner product.\n\nFrom now, we will estimate $||\\pi_n^0(f)-\\pi^0_m(f)||$ for $n>m$.\n\\begin{align}\n ||\\pi_n^0(f)-\\pi^0_m(f)||^2\n &=\n \\left(\n \\sum_{k=m}^n \n a_k\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle \\phi_k\n \\middle|\n \\sum_{\\ell=m}^n \n a_{\\ell}\\langle \\phi_{\\ell}, e^{-\\frac{1}{4}x^2}f\\rangle \\phi_{\\ell}\n \\right)\\notag\\\\\n &=\n \\sum_{k=m}^n\\sum_{\\ell=m}^n\n \\overline{a_k}a_\\ell\n \\overline{\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle}\n \\langle \\phi_{\\ell}, e^{-\\frac{1}{4}x^2}f\\rangle\\notag\\\\\n &=\n \\sum_{k=m}^n\n |a_k|^2\n |\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle|^2\n\\end{align}\n$|a_k|$ is $\\frac{1}{k^{\\frac{k}{2}+1}}$ by definition and $|\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle|$ is\n\\begin{align}\n |\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle|\n &=\n \\left|\\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_{\\rm d}\\ \n \\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}f(x_1,\\cdots,x_d)\\right|\\notag\\\\\n &<\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_{\\rm d}\\ \n \\left|\\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}f(x_1,\\cdots,x_d)\\right|\\notag\\\\\n &<\n \\sup_{x_1,\\cdots,x_d}\n \\left|\\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}\\right|\n \\int_{\\mathbb{R}^d}\n {\\rm d}x_1\\cdots{\\rm d}x_{\\rm d}\\ \n \\left|f(x_1,\\cdots,x_d)\\right|\\notag\\\\\n &=\n \\sup_{x_1,\\cdots,x_d}\n \\left|\\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}\\right|\n ||f||_{L^1}.\n\\end{align}\n$\\phi_k(x_1,\\cdots,x_d)$ can be represented as\n\\begin{align}\n \\phi_k(x_1,\\cdots,x_d)\n =\n \\sum_{\\alpha_1+\\cdots+\\alpha_d=\\deg(\\phi_k)}\n c_{\\alpha_1\\cdots\\alpha_d}\n x_1^{\\alpha_1}\\cdots x_d^{\\alpha_d}\n\\end{align}\n$|c_{\\alpha_1\\cdots\\alpha_d}|$ can be estimated as being smaller than $1$. \nTherefore\n\\begin{align}\n \\sup_{x_1,\\cdots,x_d}\n \\left|\\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}\\right|\n &<\n 1\\cdot\n \\sup_{x_1,\\cdots,x_d}\n \\left|\n x_1^{\\deg(\\phi_k)}\n e^{-\\frac{1}{4}x^2}\\right|\\notag\\\\\n &=\n 1\\cdot\n \\left(\\frac{2\\deg(\\phi_k)}{e}\\right)^\\frac{\\deg(\\phi_k)}{2}\\notag\\\\\n &<\n \\deg(\\phi_k)^\\frac{\\deg(\\phi_k)}{2}\\notag\\\\\n &<\n k^{\\frac{k}{2}}\\ \\ \\ (\\ \\because\\ \\deg(\\phi_k)<k\\ ).\n\\end{align}\nThus\n\\begin{align}\n ||\\pi_n^0(f)-\\pi^0_m(f)||^2\n &=\n \\sum_{k=m}^n\n |a_k|^2\n |\\langle \\phi_k, e^{-\\frac{1}{4}x^2}f\\rangle|^2\\notag\\\\\n &<\n \\sum_{k=m}^n\n |a_k|^2\n \\sup_{x_1,\\cdots,x_d}\n \\left|\\overline{\\phi_k(x_1,\\cdots,x_d)}\n e^{-\\frac{1}{4}x^2}\\right|^2\n ||f||_{L^1}^2\\notag\\\\\n &<\n {\\rm const.}\\times\n \\sum_{k=m}^n\n \\frac{1}{k^{k+2}}k^k\\notag\\\\\n &=\n {\\rm const.}\\times\n \\sum_{k=m}^n\n \\frac{1}{k^{2}}\n\\end{align}\nIt implies that $\\pi_n^0(f)$ is a Cauchy sequence.\n\nWe have seen $\\pi_n^0(f)$ is a Cauchy sequence, then\nit converges in $\\mathscr{H}$.\n\\begin{align}\n \\pi_n^0(f)\\to \\phi_f\\ {\\rm as}\\ n\\to\\infty.\n\\end{align}\nWe define $\\pi^0:\\mathcal{S}^0(\\mathbb{R}^d)\\to\\mathscr{H}$ \nas $\\pi^0(f)=\\phi_f$.\n\n\\subsection{$\\pi^0$ is a dense map.}\nIn order to show it, we will see the existence of $\\mathcal{O}_\\phi\\in\\mathcal{S}(\\mathbb{R}^d)$ satisfying\n\\begin{align}\n \\pi^0(\\mathcal{O}_k)=\\phi_k.\n\\end{align}\nThe basic way to check it is the same as the section \\ref{subsec:pi^0_surjective}.\nThen we set $\\mathcal{O}_\\phi$ as\n\\begin{align}\n \\mathcal{O}_k(x_1,\\cdots,x_d)\n &=\n \\frac{1}{a_\\ell(2\\pi)^{d/2}}\n \\phi_k(x_1,\\cdots,x_d)\n e^{-\\frac{1}{4}x^2}\\notag\\\\\n &=\n \\frac{1}{a_\\ell(2\\pi)^{d/2}}\n \\phi_k\\left(-\\frac{\\partial}{\\partial x_1},\\cdots,-\\frac{\\partial }{\\partial x_d}\\right)\n e^{-\\frac{1}{4}x^2}.\n\\end{align}\nThus\n\\begin{align}\n \\pi^0\\left(\\mathcal{O}_k\\right)\n &=\n \\sum_{\\ell=0}^\\infty\n a_\\ell\\left\\langle \\phi_\\ell, \\mathcal{O}_ke^{-\\frac{1}{4}x^2}\\right\\rangle \\phi_\\ell\\notag\\\\\n &=\n \\sum_{\\ell=0}^\\infty \n \\left\\langle \\phi_\\ell, \n \\frac{1}{(2\\pi)^{d/2}}\\phi_k e^{-\\frac{1}{2}x^2}\\right\\rangle \\phi_\\ell\\notag\\\\\n &=\n \\phi_k.\n\\end{align}\n\nNote that $\\mathcal{O}_k$ is a natural generalization of $Q$ and $P$ for one-dimensional case in section \\ref{subsec:pi^0_surjective}.\n\\begin{align}\n Q(x)\n &:=\n \\frac{1}{\\sqrt{\\pi}}\\exp(x).,\\\\\n P(x)\n &:=\n -\\frac{\\partial}{\\partial x}Q(x)\n =\n \\frac{2}{\\sqrt{\\pi}}x\\exp(x^2).\n\\end{align}\nHence we can regard $\\mathcal{O}_k\\in \\mathscr{H}$ as the observables for asymptotic state in higher-dimension.\n\n\\subsection{${\\rm ker}(\\pi^0)\\cong\\overline{{\\rm im}(\\Delta^{\\rm cl})}$}\nTo show ${\\rm ker}(\\pi^0)=\\overline{{\\rm im}(\\Delta^{\\rm cl})}$, \nwe think the following two steps.\n\\begin{itemize}\n \\item \n ${\\rm ker}(\\pi^0)\n \\cong \n \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp$\n \\item \n ${\\rm ker}(\\Delta_{\\mathcal{S}'})\n \\cong\n ({\\rm im}(\\Delta^{\\rm cl}))^\\perp$\n\\end{itemize}\n where\n\\begin{align}\n \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp\n &:=\n \\{\n f\\in \\mathcal{S}'(\\mathbb{R}^d)\\ |\\ \n \\forall \\phi\\in {\\rm ker}(\\Delta_{\\mathcal{S}'}),\\ \n \\langle \\phi,e^{-\\frac{1}{4}x^2}f\\rangle=0\n \\},\\notag\\\\\n ({\\rm im}(\\Delta^{\\rm cl}))^\\perp\n &:=\n \\{\n \\phi\\in \\mathcal{S}'(\\mathbb{R}^d)\\ |\\ \n \\forall f\\in {\\rm im}(\\Delta^{\\rm cl}),\\ \n \\langle \\phi,e^{-\\frac{1}{4}x^2}f\\rangle=0\n \\}.\n\\end{align}\nBy the above equations, we have\n\\begin{align}\n {\\rm ker}(\\pi^0)\n \\cong\n (({\\rm im}(\\Delta^{\\rm cl}))^\\perp)^\\perp\n \\cong\n \\overline{{\\rm im}(\\Delta^{\\rm cl})}.\n\\end{align}\n\nFirst of all, we see ${\\rm ker}(\\pi^0)\\cong \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp$.\nTake $f\\in {\\rm ker}(\\pi^0)$, \nthen for all harmonic polynomial $\\phi$ we have\n\\begin{align}\n \\langle \\phi, e^{-\\frac{1}{4}x^2}f\\rangle=0.\n\\end{align}\nIn other words, for all $\\phi\\in {\\rm ker}(\\Delta_{\\mathcal{S}'})$\n\\begin{align}\n \\langle \\phi, e^{-\\frac{1}{4}x^2}f\\rangle=0.\n\\end{align}\nTherefore $f\\in ({\\rm ker}(\\Delta_{\\mathcal{S}'}))^\\perp$, i.e. \n${\\rm ker}(\\pi^0)\\subset \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp$.\nFollowing the above discussion in reverse, we have ${\\rm ker}(\\pi^0)\\supset \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp$.\nHence, ${\\rm ker}(\\pi^0)\\cong \\left({\\rm ker}(\\Delta_{\\mathcal{S}'})\\right)^\\perp$.\n\nNext, we see ${\\rm ker}(\\Delta_{\\mathcal{S}'})\\cong({\\rm im}(\\Delta^{\\rm cl}))^\\perp$.\nTake $\\phi\\in {\\rm im}(\\Delta^{\\rm cl}))^\\perp$, thus we have\n\\begin{align}\n &\\langle \\phi,\\Delta f \\rangle=0\n \\ \\ (\\forall f\\in\\mathcal{S}(\\mathbb{R}^d))\\notag\\\\\n &\\Leftrightarrow\n \\langle \\Delta_{\\mathcal{S}'} \\phi,f \\rangle=0\n \\ \\ (\\forall f\\in\\mathcal{S}(\\mathbb{R}^d))\\notag\\\\\n &\\Leftrightarrow \\Delta_{\\mathcal{S}'} \\phi=0.\n\\end{align}\nTherefore $\\phi\\in {\\rm ker}(\\Delta_{\\mathcal{S}'})$. Hence ${\\rm ker}(\\Delta_{\\mathcal{S}'})\\supset({\\rm im}(\\Delta^{\\rm cl}))^\\perp$.\nBy the above discussion in reverse, we have\n${\\rm ker}(\\Delta_{\\mathcal{S}'})\\subset({\\rm im}(\\Delta^{\\rm cl}))^\\perp$. Then we obtain\n\\begin{align}\n {\\rm ker}(\\Delta_{\\mathcal{S}'})\\cong({\\rm im}(\\Delta^{\\rm cl}))^\\perp.\n\\end{align}\n\n\\bibliographystyle{ptephy}\n\\bibliography{sample}\n\n\\end{document}\n\n"}}},{"rowIdx":94,"cells":{"id":{"kind":"string","value":"2412.08334v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08334v2"},"title":{"kind":"string","value":"Maker-Breaker on Galton-Watson trees"},"full_text":{"kind":"string","value":"\\pdfminorversion=5\n\\documentclass{article}\n\\usepackage{graphicx}\n\\usepackage{epstopdf}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{bm}\n\\usepackage{bbm}\n\\usepackage{enumerate}\n\\usepackage[T1]{fontenc}\n\\usepackage[latin1]{inputenc}\n\\usepackage{hyperref}\n\\usepackage{color}\n\\usepackage{curves}\n\\usepackage{tikz}\n\\usepackage[hang]{caption}\n\\usepackage{ntheorem}\n\\usepackage{array,float,caption,nicefrac}\n\\usepackage{mathrsfs}\n\n\n\n\\title{Maker-Breaker on Galton-Watson trees}\n\\author{Timo Vilkas\n\t\t\\\\\\normalsize Lunds Universitet\n}\n\n\n \n\\theoremstyle{break}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}{Corollary}[section]\n\\newtheorem{lemma}{Lemma}[section]\n\\newtheorem{proposition}{Proposition}[section]\n\\theorembodyfont{\\upshape}\n\\newtheorem{definition}{Definition}\n\\newtheorem*{remark}{Remark}\n\\newtheorem{example}{Example}\n\n\\makeatletter\n\\let\\c@proposition\\c@theorem\n\\let\\c@lemma\\c@theorem\n\\let\\c@corollary\\c@theorem\n\\makeatother\n\n\\newenvironment{proof}{\\noindent{\\sc Proof:}}{\\vspace{-1em}~\\hfill $\\square$\\vspace{2em}}\n\\newenvironment{nproof}[1]{\\noindent{\\sc Proof #1:}}{\\vspace{-1em}~\\hfill $\\square$\\vspace{2em}}\n\\newenvironment{AMS}{}{}\n\\newenvironment{keywords}{}{}\n\\newcommand\\IN{\\mathbb{N}}\n\\newcommand\\IR{\\mathbb{R}}\n\\newcommand\\IZ{\\mathbb{Z}}\n\\newcommand\\IQ{\\mathbb{Q}}\n\\newcommand\\IE{\\mathbb{E}\\,}\n\\newcommand\\Prob{\\mathbb{P}}\n\\renewcommand\\epsilon{\\varepsilon}\n\\renewcommand\\phi{\\varphi}\n\\DeclareMathOperator\\supp{supp}\n\\newcommand\\En{\\mathcal{E}}\n\\newcommand\\D{\\mathscr{D}}\n\\newcommand\\law{\\buildrel d \\over\\rightarrow}\n\n\\definecolor{darkblue}{rgb}{0,0,.5}\n\\hypersetup{colorlinks=true, breaklinks=true, linkcolor=blue, \ncitecolor=darkblue, menucolor=blue, urlcolor=blue}\n\\renewcommand{\\captionfont}{\\small}\n\n\n\n\\begin{document}\n\\newpage\n\\maketitle\n\\begin{abstract}\nWe consider the following combinatorial two-player game: On the random tree arising from a branching process, each round one player (Breaker) deletes an edge and by that removes the descendant and all its progeny, while the other (Maker) fixates an edge to permanently secure it from deletion. Breaker has won once the tree's root is contained in a finite component, otherwise Maker wins by building an infinite path starting at the root. It will be analyzed both as a positional game (the tree is known to both players at the start) and with more restrictive levels of information (the players essentially explore the tree during the game).\nReading the number of available edges for play as a random walk on $\\mathbb{Z}$ allows us to derive the\nwinning probability of Breaker via fixed point equations in three natural information regimes.\nThese results provide new insights into combinatorial game theory and random structures, with potential applications to network theory, algorithmic game design and probability theory.\n\\end{abstract}\n\n\\noindent\n\\textbf{Keywords:} \nMaker-Breaker game, Galton-Watson tree, branching process, percolation games, phase transition, random walk on $\\IZ$ with drift.\\\\[0.5em]\n\\textbf{MSC2020:} 05C57, 91A43, 60G50\n\n\\section{Introduction}\nMaker-Breaker games are a class of combinatorial games in which two players, called Maker and Breaker, compete by selecting elements from a finite or infinite structure, with opposing objectives. These games are well-studied in the context of graph theory, where Maker seeks to build a particular substructure while Breaker tries to prevent its formation. In the two most common variants of the game, either nodes or edges are played, a distinction which (depending on the objective) can become irrelevant in case the underlying graph is a tree.\nHistorically seen, a game of this kind was reportedly first formulated and formalized by C.E.\\ Shannon at mid-20th century and later coined as ``Shannon switching game'' \\cite{Shannon}.\nIn recent years, the study of Maker-Breaker games has been extended to boards given by random structures.\n\nGalton-Watson trees, initially introduced in the 19th century to study the extinction of family names, have since found wide applications in areas like probability theory, evolutionary biology, and network modeling. The intersection of these two fields, Maker-Breaker games on Galton-Watson trees, presents interesting challenges that combine random processes with adversarial strategies in combinatorial games.\n\nIn this article, we explore the dynamics of Maker-Breaker games on Galton-Watson trees, examining how the probabilistic nature of the tree structure and the available information influence the winning probabilities of both players, when the objective for Breaker is to cut off the root, in the sense that it is eventually contained in a finite component. We delve into key questions such as how the offspring distribution of the tree affects the likelihood of Breaker's success and how the available information about the tree influences their tactics to account for the randomness of the game environment. Through this analysis, we shed light on the intricate interplay between randomness and strategy in these games, offering insights that could have broader implications for fields ranging from algorithm design to network theory. \n\n\\subsection{Main results} \nThroughout the paper, we will write $g$ for the probability generating function of the offspring distribution (cf.\\ equation \\eqref{pgf} below) and $q$ for the corresponding extinction probability (cf.\\ Theorem \\ref{extinction}). The probability of Breaker succeeding with isolating the root in a finite component, when being in the favorable position to start the game, is denoted by $p$ (or else $\\bar{p}$, if Maker starts instead).\n\n{\\em Positional games} in general are usually characterized by a finite board and perfect information, which make the games deterministic, except for (potential) randomness of the game board. In contrast, we consider here a (potentially) infinite random tree as underlying graph, which is explored by the players during the game.\nBoth strategies and outcome under optimal play depend on the information available to the players.\nThree different regimes will be analyzed in detail: First, the usual setting in which the whole tree is revealed at the start (turning the game into a deterministic matter), then the other extreme when no information is available (besides the edges incident to either the root or already played edges) and finally an intermediate case in which each vertex connected to the root (by edges Maker picked) is marked according to whether or not it is the root of an infinite subtree.\n\nIn all three cases, the winning probability of Breaker $p$ is the solution to an equation involving the probability generating function $g$ and its derivative:\n\n\\newtheorem*{Thm4.3}{Theorem \\ref{formula}}\n\\begin{Thm4.3}\n\tIf the GW-tree is revealed to both players at the start, it holds\n\t\\begin{equation*}\n\t\tp=g(p)+(1-p)\\cdot g'(p).\n\t\\end{equation*}\n\\end{Thm4.3}\n\n\\newtheorem*{Thm5.3}{Theorem \\ref{i=0}}\n\\begin{Thm5.3}\nGiven there are no childless nodes and the players do not receive any further information, it holds\n\t\\begin{equation*}\n\t\tp^2=g(p).\n\t\\end{equation*}\n\\end{Thm5.3}\n\nIf the probability of not having any offspring is strictly positive in this regime, we are still able to determine $p$ in case the offspring distribution is separable (cf.\\ Subsection \\ref{separable}, in particular Theorem \\ref{septhm}).\n\n\\newtheorem*{Thm6.4}{Theorem \\ref{tv=i}}\n\\begin{Thm6.4}\n\tIf the information carried by visible nodes is whether or not they have infinite progeny, it holds\n\t\\begin{equation*}\n\t\tp^2\\,(1-q)=g\\big(p\\,(1-q)+q\\big)-q.\n\t\\end{equation*}\n\\end{Thm6.4}\n\nFor all three regimes we try to illustrate the results with help of examples (in terms of the offspring distribution: Binomial, geometric and Poisson among others) and corresponding numerical calculations. It appears that the phase transition from an almost sure Breaker win to a non-trivial game is discontinuous only in the first regime (where the tree is revealed right away). Furthermore, in all cases it holds $\\bar{p}=g(p)$, with the single exception when no additional information is given and the tree contains leaves (see Theorem \\ref{arch} and Proposition \\ref{ineq}).\nIn the last section, some general observations, a few open problems and suggestions for further research are presented.\n\n\\subsection{Related work}\nAs already mentioned, games of this kind were introduced into scientific literature more than half a century ago and an abundance of variants, differing either in the specific rules of the game, the players' objectives or the underlying graphs, have been looked at and analyzed since. One of the first results \\cite{Shannon} deals with the objective for Maker (called ``short'' there) to connect two given nodes on a general graph, while Breaker (called ``cut'') tries to prevent that. Other common objectives coined variants such as the ``connectivity game'', ``perfect matching game'', ``Hamiltonian game'' or ``clique game'', in which Maker tries to claim a spanning tree, a perfect matching, a Hamiltonian cycle or a clique of a given size respectively. Besides the rule that edges are picked one by one alternatingly, the more general (potentially biased) $(m,b)$-rule, according to which in each round first Maker plays $m$ edges at once, then Breaker plays $b$ edges, has been considered and analyzed in many different contexts. We will, however, stick to the basic $(1,1)$-rule here.\n\nChvtal and Erd\\H{o}s in their seminal paper \\cite{erds} considered among others the biased $(1,b)$ connectivity game on the complete graph $K_n$ and found that the threshold for the bias is around $b=\\frac{n}{\\log n}$ as $n$ grows large (in the sense that if each round first Breaker picks $b$ edges, then Maker picks one edge, it is a Maker's win for $b$ smaller and a Breaker's win for $b$ larger than this threshold).\nMore than 20 years later, Bednarska and \\L uczak \\cite{Luczak} analyzed the size of the largest component Maker is able to build in this context. Hefetz et al.\\ \\cite{Hefetz} later addressed the time until Maker wins the unbiased $(1,1)$-game in this setting, with different objectives (among others, perfect matching and Hamiltonian game).\n\nIn the context of random boards, the Erd\\H{o}s-Rnyi graph $G_{n,p}$ was unsurprisingly one of the first targets addressed. Stojakovi\\'c and Szab \\cite{StoSza} established for different objectives (connectivity, perfect matching, Hamiltonian and clique game) the threshold for the edge probability $p$, at which Maker wins the unbiased $(1,1)$-game with high probability as $n\\to\\infty$. In addition to that, they investigate the critical bias $b$ (asymptotically and depending on $p$) at which Maker wins the $(1,b)$-game. Ferber et al.\\ \\cite{Rndboards} later extended this line of research to further ranges of the edge probability $p$. London and Pluhr \\cite{London} addressed these questions for a slightly altered Maker-Breaker game, in which Maker can only play edges incident to edges she already claimed. Clemens et al.\\ \\cite{CKM} extended their investigation of the threshold probability $p$ to the unbiased $(2,2)$-variant. While their results reveal interesting deviations from the answers for the original game, it is not hard to see that this additional rule is in fact no restriction in our chosen setting. \n\nMaker-Breaker games have also been considered on random geometric graphs $G(n,r)$: $n$ points are placed independently and uniformly at random on the unit square and connected by an edge if their distance does not exceed $r$. Here, the crucial characteristic for whether it is a Maker's or a Breaker's win (with high probability as $n\\to\\infty$) turned out to be the minimum degree. That it is a local property of the random graph should not come as a surprise, since the objectives considered by Beveridge et al.\\ \\cite{RndGeo} (connectivity, perfect matching and Hamilton game) do involve {\\em all} vertices, in the sense that Maker attempts to build a structure including all nodes of the graph.\n\nIn contrast to that, global graph properties were found to be crucial in other (in particular tree) contexts, such as the {\\em branching number} governing both random walks and percolation on trees (cf.\\ Lyons \\cite{Lyons}) or the average number of offspring, determining whether the survival probability $1-q$ of a Galton-Watson tree is strictly positive (cf.\\ Theorem \\ref{extinction}).\nWhen determining the probability for a Galton-Watson tree to contain an infinite complete binary tree $T_2$, the offspring distribution's probability generating function occupies center stage (cf.\\ Dekking \\cite{Dek1}). We will revisit this result in our context as Theorem \\ref{formula}.\n\nAlso in the analysis of what Holroyd et al.\\ (already in an earlier publication) named {\\em percolation games} on rooted, directed Galton-Watson trees \\cite{HM} (a token is moved along the directed edges and the game ends when a sink is reached) the winning (and draw) probabilities undergo a phase transition at a fixed point of an equation involving the offspring distribution's generating function. Here, three game variants are considered: normal (the player unable to move loses), misre (the player reaching a sink wins) and escape (Stopper wins when the game ends, Escaper wins if it doesn't). A similar but qualitatively different game on undirected rooted Galton-Watson trees (where nodes also can be traps and targets, but edges traversed in both directions) was recently analyzed by Karmakar et al.\\ \\cite{rooted} and yielded a similar kind of results.\n\nIn terms of game variant and players' objectives, what is most closely related to the setting we consider here (besides the original Shannon switching game) is the work by Day and Falgas-Ravry: Both players in turns claim edges and while Maker's objective is to build a path, Breaker is trying to prevent that. In a first publication \\cite{DFR1}, the authors considered the task of crossing a finite rectangular grid from left to right, a special case of Shannon's original game but extended to the general $(m,b)$-rule. In a second publication \\cite{DFR2}, they consider an infinite underlying graph and Maker's task is to build an infinite path starting at a given vertex $v_0$ (the exact same variant we will adopt here, with $v_0$ being the root of the tree $\\mathbf{0}$). Their main results concern both the $d$-dimensional grid $\\mathbb{Z}^d$ and the infinite $d$-regular tree $T_d$. Observe, however, that in these cases the Maker-Breaker game is fully deterministic.\n\n\\section{Preliminaries}\\label{sec2}\nIn order to get started, let us properly introduce the two central concepts: the (1,1) Maker-Breaker game and the branching process generating the (random) tree which will be the underlying graph in our analysis.\n\n\\subsection{The Maker-Breaker game on the edge set of a graph}\nWithin graph theory, {\\em Maker-Breaker} is a combinatorial two-player game, taking place on a graph $G=(V,E)$, with one node marked as the origin $\\mathbf{0}$. All edges in $E$ are available for play until either fixated by Maker or removed by Breaker. The two opposing players in turns are allowed to pick one of the remaining edges in order to either fixate (Maker) or remove it (Breaker). The objective for Maker is to create as big a component containing the origin as possible, while Breaker tries to cut it off.\nWe will consider simple infinite graphs (i.e.\\ no loops or multiple edges, $|V|=\\infty$) with finite degrees and consider the game won for Breaker once the origin is contained in a finite component separated from the rest of the graph by removed edges. Maker wins if Breaker doesn't and by a simple compactness argument this results in the task of fixating an infinite path starting at the origin. We assume both Maker and Breaker to play optimally in the sense that they try to maximize their chance of winning in every move. Note that the game is deterministic on a given graph $G$ and only becomes a\ngame of chance if played on a random graph or by integrating randomness into the players' strategies. Further, the probability of Maker winning the game is monotonous in $G$ in the sense that adding nodes and/or edges to the graph can not decrease it. As default, unless otherwise stated, we consider Breaker to start the game.\n\nAs a first example, consider the two-dimensional grid $\\IZ^2$ (see Figure \\ref{MB} for an illustration). It takes little ingenuity to come up with a winning strategy for Maker in the $(1,1)$-game, i.e.\\ each player gets to choose one edge at a time:\nNote first that every node $u\\in\\IZ^2$ is adjacent to at least two edges pointing outwards, to be more precise $e=\\langle u,v\\rangle\\in E$, where $\\lVert u \\rVert<\\lVert v \\rVert$. If now Maker always plays an edge pointing outwards from $u$ after Breaker has done so (for the first time), in fact all vertices (in particular the origin) stay connected to each other. This kind of strategy is commonly referred to as {\\em pairing strategy}.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=1.3]{makerbreaker}\n\t\\caption{Illustrating example of a game position on $\\IZ^2$ after 10 moves each.\\label{MB}}\n\\end{figure}\n\nIn order to make it more interesting, one can consider the more general $(m,b)$-variant of the game (Breaker removes $b$ edges, then Maker fixates $m$ edges in turns). Obviously, a winning strategy for Maker will persist when increasing $m$ (as it does for Breaker when $b$ is raised). However, already the outcome of the $(m,b)$-game on $\\IZ^2$ for $b=2$ is far from trivial to determine.\n\n\\subsection{Bienaym branching process and Galton-Watson tree}\nHere, we want to consider as underlying (random) graph for the Maker-Breaker game the (potentially infinite) family\ntree of a branching process introduced by probabilist Irne-Jules Bienaym and later (independently) made widely popular by natural scientist Francis Galton and mathematician Henry William Watson.\n\nIn this classical branching process, each individual $v$ gives rise to a random number\n$\\xi_v$ of offspring. The sequence $\\{\\xi_v\\}_{v\\in V}$ consists of independent and identically distributed\ncopies of a random variable $\\xi$.\nEach individual is represented by a node $v$ and nodes corresponding to parent and child are linked by an edge,\nto form what is commonly called {\\em Galton-Watson family tree}. Depending on the offspring distribution (and chance), this tree can turn out to be finite or infinite.\n\nAll nodes at a given (graph) distance $n\\in\\IN_0$ of the root $\\mathbf{0}$ in the tree are\nconsidered to form a generation. Let $Z_n$ denote the number of individuals in generation $n$, i.e.\n\\[Z_0=1,\\ Z_{n+1}=\\sum_{j=1}^{Z_n}\\xi_{v_{nj}}\\quad\\text{for}\\ n\\in\\IN_0,\\]\nwhere the nodes are indexed in breadth-first-order so that $\\mathbf{0}=v_{01}$ and in general $v_{nj}$ corresponds to individual $j$ in generation $n$. We will denote the probability mass function (pmf) of the offspring distribution by $p_k=\\Prob(\\xi=k)$, for $k\\in\\IN_0$, and write \n\\begin{equation}\\label{pgf}\ng(x)=\\IE\\big(x^\\xi\\big)=\\sum_{k=0}^\\infty p_k\\, x^k\n\\end{equation}\nfor its {\\em probability generating function} (pgf). Throughout, we will consider the offspring distribution to be almost surely finite. At this point it might be worth mentioning that $\\mu:=\\IE\\xi=g'(1)$ is commonly referred to as \\emph{Malthusian parameter}.\nThe extinction probability of the corresponding Galton-Watson process, i.e.\\ $q:=\\lim_{n}\\Prob(Z_n=0)$ can be characterized as the first intersection of $g$ with the identity function in the first quadrant (cf.\\ Theorem 1 in \\cite{Athreya-Ney}\nfor instance):\n\n\\begin{theorem}[Extinction probability]\\label{extinction}\n\tThe extinction probability $q$ of the branching process $(Z_n)_{n\\in\\IN_0}$ is the smallest nonnegative root\n\tof the equation $g(x)=x$. It equals 1, if $\\mu\\leq 1$ (and $p_1<1$) and is strictly less than 1 if\n\t$\\mu> 1$ (or $p_1=1$).\n\\end{theorem}\nExcluding the degenerate case $p_1=1$, a Galton-Watson process with $\\mu\\leq 1$ is hence called {\\em subcritical}, one with $\\mu> 1$ instead {\\em supercritical}. \nThe standard proof of this classical theorem is straight-forward and based on the fact that for fixed $x\\in[0,1]$, the $n$-fold iterates $g\\circ g\\circ\\dots\\circ g(x)$ converge to $q$ as $n$ goes to infinity, see Figure \\ref{G-W} for an illustration.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=1]{G-W}\n\t\\caption{Examples of probability generating functions corresponding to a subcritical (left) and supercritical process (right).\\label{G-W}}\n\\end{figure}\n\nFor a given offspring distribution, let $T$ denote the (random, unlabelled) Galton-Watson (GW) tree and for an arbitrary node $v$ in $T$, we are going to write $T_v$ for the subtree rooted in $v$. By construction, every $T_v$ is a copy of $T$.\n\nThe (1,1)-version of Maker-Breaker on $T$ amounts to Breaker cutting off a subtree $T_v$ (by disconnecting $v\\neq\\mathbf{0}$ from its parent) and Maker fixating an edge $\\langle u,w\\rangle$, hence adding the corresponding child node to the component containing the root. Observe that $T$ being a tree (i.e.\\ not containing any cycles) makes it suboptimal for both players to play an edge linking two vertices that are not incident to any fixated edges (yet).\nWe will refer to the nodes connected to the root by fixated edges as {\\em internal nodes}. Nodes, corresponding to descendants in $T$ of currently internal nodes, which are neither internal nor cut off from the root, are referred to as {\\em external nodes}, see Figure \\ref{tree} for an illustration.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.9]{tree}\n\t\\caption{Illustrating example of a game position on $T$ with internal nodes black, external nodes gray and others omitted.\\label{tree}}\n\\end{figure}\n\n\\subsection{Random walk on \\texorpdfstring{$\\IZ$}{\\bf Z}}\nTo complete this introductory section, we want to include a few basic results for random walk on integers. This topic is closely related to GW-trees for the following reason: Imagine we explore the tree node by node and keep a list of nodes logged as offspring of explored nodes, which haven't been explored yet. Then the length of this list can be seen as random walk on $\\IZ$ with step size distribution $\\xi-1$, started at 1. The tree is infinite if and only if this walk stays strictly positive.\n\nSome conditions chosen for the increments in this section might seem a bit arbitrary for now, but will appear natural later on, once we express the number of edges which the two players consider for play as a random walk on $\\IZ$ (cf.\\ \\eqref{rw} below). Let us begin with a short proof of the well-known fact that random walk with upwards drift has a non-zero probability to never become negative if started at $0$.\n\n\\begin{lemma}\\label{drift}\n\tLet $(X_n)_{n\\in\\IN}$ be an i.i.d.\\ sequence with $\\IE(X_1)>0$.\n\tThen the infimum of the random walk $(S_n)_{n\\in\\IN_0}$, defined by $S_0=0,\\ S_{n+1}=S_n+X_{n+1}$, is attained at $S_0=0$ with positive probability.\n\\end{lemma}\n\\begin{proof}\n\tBy the law of large numbers it holds almost surely that \\[\\lim_{n\\to\\infty}\\frac{S_n}{n} =\\lim_{n\\to\\infty}\\frac1n \\sum_{k=1}^n X_k=\\mathbb{E}(X_1)>0.\\] Thus, there exists $n_0\\in\\mathbb{N}$ such that $\\mathbb{P}\\big(S_n >0\\ \\text{ for all } n\\geq n_0\\big)>0$.\n\tSince $n_0$ is finite, there must be some $n_1\\in[0,n_0)$ such that $S_{n_1}$ is the minimum value of the walk with strictly positive probability. As a consequence, the event $\\{\\sum_{k=n_1+1}^n X_k\\geq 0$ for all $n>n_1\\}$ has non-zero probability. Due to the fact that $(X_n)_{n>n_1}$ and $(X_n)_{n\\in\\mathbb{N}}$ have the same distribution, the claim follows.\n\\end{proof}\n\nRandom walk on the integers with increments bounded from below by $-1$, i.e.\\ $\\Prob(X_n\\geq -1)=1$, are commonly referred to as {\\em left-continuous} or {\\em skip-free to the left}. To avoid confusion with the offspring distribution $\\xi$, let us write $(\\pi_k)_{k\\geq -1}$ for the probability mass function and\n\\[\\gamma(x)=\\sum_{k=-1}^\\infty \\pi_k\\,x^k=\\frac{\\pi_{-1}}{x}+\\pi_0+\\pi_1\\,x+\\pi_2\\,x^2+\\dots\\]\nfor the probability generating function of the corresponding increment distribution. Note that $\\gamma$ is not defined in $x=0$ if $\\pi_{-1}>0$ and that the sum defining $\\gamma(x)$ is guaranteed to converge (absolutely) for $0<|x|\\leq 1$.\n\nIn the remainder of this section, we want to introduce the probabilities of different hitting times to be finite (and odd) for a left-continuous, non-monotonic random walk on $\\IZ$ with positive drift starting at $0$, i.e.\\ $(S_n)_{n\\in\\IN_0}$ defined by $S_0=0,\\ S_n=S_{n-1}+X_{n}$, where $(X_n)_{n\\in\\IN}$ is an i.i.d.\\ sequence, $\\IE(X_1)>0$, $\\Prob(X_1\\geq-1)=1>\\Prob(X_1\\geq 0)$. To this end, let\n\\begin{align*}\\tau_{-1}&=\\inf\\{n,\\;S_n=-1\\}\\quad\\text{and}\\\\\n\t\t\t\t\\tau_0^+&=\\inf\\{n>0,\\;S_n=0\\}\n\\end{align*}\ndenote the hitting time of state $-1$ and the return time to starting state $0$ respectively. Further we define the\nfollowing (conditional) probabilities:\n \\begin{equation}\\label{probs}\n \\begin{gathered}\n \t\\rho=\\Prob(\\tau_{-1}<\\infty),\\quad \\sigma=\\Prob(S_n>0\\text{ for all }n\\in\\IN)\\quad\\text{and}\\quad\\theta=\\Prob(\\tau_0^+<\\tau_{-1})\\\\\n\t\\text{as well as}\\\\\n\t\\rho_\\mathrm{odd}=\\Prob(\\tau_{-1}\\text{ is odd}\\,|\\,\\tau_{-1}<\\infty)\\quad\\text{and}\\quad \\theta_\\mathrm{odd}=\\Prob(\\tau_0^+\\text{ is odd}\\,|\\,\\tau_0^+<\\tau_{-1}).\n \\end{gathered}\n\\end{equation}\n\nTo put it in words, $\\rho$ is the probability that state $-1$ will eventually be visited and $\\rho\\cdot\\rho_\\mathrm{odd}$ the probability that this will happen for the first time after an odd number of steps. $\\sigma$ denotes the probability that the walk never returns to starting state $0$ and $\\theta$ is the probability that the walk first returns to $0$ before possibly visiting state $-1$. Finally, $\\theta\\cdot\\theta_\\mathrm{odd}$ is the probability that the walk performs a round trip started at $0$ without visiting $-1$ in an odd number of steps.\n\nFor left-continuous walks, the strong Markov property immediately implies that $\\rho$ equals the smallest solution in $[0,1]$ to $\\gamma(x)=1$, cf.\\ Lemma 2 in \\cite{skipfree}. As we will consider increments of the form $\\xi-2$, given $p_0=0<p_1$ and $\\IE\\xi>2$, we have a non-monotonic, left-continuous walk with positive drift and the corresponding probability is the unique solution in $(0,1)$ to $g(x)=x^2$, see Theorem \\ref{i=0} below.\n \nIn case the increments $\\xi-2$ can be split in two (see Subsection \\ref{separable}), the parity of the hitting times will play an important role, which is why we include the results of Lemma 3.2 and Thm.\\ 3.3 in \\cite{LCRW}, collected in a single lemma here:\n\n\\begin{lemma}\\label{calc}\n\tLet $(S_n)_{n\\in\\IN_0}$ be a left-continuous random walk with positive drift and $\\pi_{-1}>0$, started in $S_0=0$. \n\tThen the following hold:\n\t\\begin{enumerate}[(a)]\n\t\t\\item $\\pi_{-1}+\\theta+\\sigma=1$\n\t\t\\item $\\sigma=\\pi_{-1}\\cdot \\frac{1-\\rho}{\\rho}$ and $\\theta=1-\\frac{\\pi_{-1}}{\\rho}$\n\t\t\\item $\\gamma\\big(\\rho\\,(1-2\\rho_\\mathrm{odd})\\big)=-1$\n\t\t\\item $\\theta_\\mathrm{odd}=\\frac{\\pi_{-1}\\,(1-\\rho_\\mathrm{odd})}{\\rho\\theta\\,(2\\rho_\\mathrm{odd}-1)}$\n\t\\end{enumerate}\n\\end{lemma}\n\n\n\\section{Different information regimes}\\label{sec3}\n\nDuring the game, the tree is not necessarily fully known to the players at the start but can be gradually explored. What is visible to both Maker and Breaker -- besides the current set of internal and external nodes -- will vary, hence create different regimes in the subsequent analysis. The piece of information attached to every visible node $v$ will be denoted by $I(v)$. The strategies and winning probabilities will not only depend on the offspring distribution (as indicated above) but also the level of information, i.e.\\ $I(v),\\ v\\in V,$ revealed to the players. Let us denote the probability of Breaker winning the game (given Breaker started it) by\n\\[p:=\\Prob(\\text{Breaker succeeds to isolate the root $\\mathbf{0}$ in a finite component}).\\]\nTrivially, the probability of Breaker winning (no matter who starts) is bounded from below by $q$. In fact, writing $\\bar{p}=\\Prob(\\text{Breaker wins}\\,|\\,\\text{Maker starts})$ we can further conclude $p\\geq\\bar{p}\\geq q$ due to the monotonicity in $G$ mentioned earlier. To avoid trivialities, we will therefore consider the game played on family trees corresponding to supercritical GW-processes, in other words offspring distributions with $\\IE\\xi>1$, hence $q<1$.\n\nBefore we dig deeper into the different information regimes, let us state an overarching relationship between $p$ and $\\bar{p}$, that will appear in each of the corresponding sections below:\n\n\\begin{theorem}\\label{arch}\n\tIrrespectively of $I(v)$, it holds $\\bar{p}\\geq g(p)$.\n\tIn fact, in the regimes $I(v)=T_v$ and $I(v)=|T_v|$, as well as when $p_0=0$ in $I(v)=\\emptyset$ equality holds.\n\\end{theorem}\n\nThe inequality follows immediately, considering the potential strategy of Breaker not to cut any edge incident to the root but to play the game on $T_v$ as a new game instead (in which Breaker begins), right after Maker fixates the edge $\\langle\\mathbf{0},v\\rangle$. Then the Maker-Breaker game on $T$ (in which Maker starts) actually becomes a simultaneous play of $Z_1$ independent games (on independent copies of $T$, with Breaker starting). Given $Z_1=k$ nodes in the first generation, Breaker will win using this strategy if all $k$ subgames are lost for Maker, which by independence has probability $p^k$. Consequently, we get by conditioning on $Z_1$:\n\\[\\bar{p}\\geq \\sum_{k=0}^\\infty p_k\\, p^k=g(p).\\]\nIn the two regimes $I(v)=\\emptyset$ and $I(v)=|T_v|$, there is complete symmetry in the edges considered for play. The strategy sketched above is therefore optimal from Breaker's point of view, as long as there are always edges not incident to the root available for play. Therefore, the inequality is actually an equality in the cases $I(v)=|T_v|$ as well as $p_0=0,\\ I(v)=\\emptyset$.\nThe same holds for $I(v)=T_v$ in which the game is deterministic and Maker wins iff the root can be connected to a complete binary tree (cf.\\ Lemma \\ref{char} below). When Maker starts, Breaker wins iff for no $v$ in the first generation, $T_v$ contains a complete binary tree, hence by conditioning on $Z_1$: $\\bar{p}=\\sum_k p_kp^k$. For further details, see the corresponding results in the sections below.\n\nIt might be worth mentioning that $\\bar{p}=g(p)$ does not hold for $q>0$ (i.e.\\ $p_0>0$, which is not covered by Theorem \\ref{i=0}) in the regime $I(v)=\\emptyset$, see Proposition \\ref{ineq}.\nFinally, observe that Theorem \\ref{arch} immediately implies $\\bar{p}=1$ if $p=1$, since $1\\geq\\bar{p}\\geq g(p)=g(1)=1$ then.\n\n\n\\section{Instantly revealed tree, \\texorpdfstring{$I(v)=T_v$}{I(v)=T\\_v}}\\label{sec:all}\n\nIn the extreme case, where the whole tree $T$ is revealed to the players instantly, the game becomes deterministic\n(once $T$ is fixed). The possibility of either player winning the (1,1)-game then depends on whether or not\n$T$ contains an infinite complete binary tree rooted in $\\mathbf{0}$, i.e.\\ a family tree starting with progenitor $\\mathbf{0}$ in which each individual has exactly two descendants. We will let $T_2$ denote the infinite complete binary tree and, for arbitrary $n\\in\\IN_0$, write $T_{2,n}$ for the 2-regular tree with $n$ levels, which is the full binary tree on $2^{n+1}-1$ vertices and the induced subgraph obtained when $T_2$ is cut at generation $n$.\n\n\\begin{lemma}\\label{char}\n\tIn the Maker-Breaker game on a tree $T$ known to the players at the start, Maker wins if and only if $T$ contains a complete binary tree as (induced) subgraph rooted in $\\mathbf{0}$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe sufficiency of this condition is easy to verify using the following simple pairing strategy: Maker chooses an infinite complete binary subtree $T_2$ of $T$ rooted in $\\mathbf{0}$ and always fixates the edge connecting the (only) sibling in $T_2$ of the node Breaker in the same round disconnected from their common parent (arbitrary edges whenever Breaker picks an edge not in $T_2$ or the edge connecting the sibling to its parent is fixated already).\n\t\n\tThat the condition is also necessary, we prove by induction on $n$, where $T_{2,n}$ is the smallest full binary tree that cannot be found in $T$ rooted in $\\mathbf{0}$. If $n=1$, the root $\\mathbf{0}$ has at most one child, which Breaker can disconnect in the first move. For $n>1$, at most one individual in the first generation can be the root of a copy of $T_{2,n-1}$ (otherwise there would be a $T_{2,n}$ rooted in $\\mathbf{0}$). This direct descendant (if such exists) Breaker disconnects from the root in the first move. Any other $v$ in generation 1, that Maker can fixate is not the root of a $T_{2,n-1}$ and hence Breaker wins in $T_v$ by induction hypothesis. The general assumption that $\\mathbf{0}$ has finite degree (a.s.)\\ therefore concludes the argument.\n\\end{proof}\n\nFrom this result (and its proof) we can immediately conclude the following:\n\n\\begin{corollary}\n\tIf Maker starts the Maker-Breaker game on a tree $T$ known to the players at the start, Maker wins if and only if there exists a child $v$ of $\\mathbf{0}$ such that $T_v$ contains a complete binary tree as (induced) subgraph rooted in $v$.\n\\end{corollary}\n\nUsing this characterization, we can derive an equation for the winning probabilities of Maker/Breaker for general offspring distribution,\nwhich become particularly tidy if written in terms of $g$, the probability generating function of $\\xi$, cf.\\ \\eqref{pgf}.\n\n\\begin{theorem}\\label{formula}\n\tFor the (1,1)-game on a GW-tree $T$ arising from offspring distribution $\\xi$ and revealed to both players at the start, it holds\n\t\\begin{equation}\\label{fp2}\n\t\tp=g(p)+(1-p)\\cdot g'(p)\\quad \\text{and}\\quad\\bar{p}=g(p).\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\n\tAccording to Lemma \\ref{char}, the probability $1-p$ of Maker winning the game equals the probability of $T$ containing a copy of $T_2$ rooted in $\\mathbf{0}$. The latter can be described recursively with respect to the number of children the root has.\n\t\\begin{align*}\n\t\tp&=\\Prob(\\mathbf{0} \\text{ is not the root of a copy of }T_2)\\\\\n\t\t&=\\Prob(\\mathbf{0} \\text{ does not have two children being the root of a copy of }T_2)\\\\\n\t\t&=\\sum_{k=0}^\\infty \\Prob(\\xi_\\mathbf{0}=k)\\cdot \\Prob(X_k< 2),\n\t\\end{align*}\n\twhere $X_k\\sim\\mathrm{Bin}(k,1-p)$, as every node in generation 1 is the root of an independent copy of $T$ and the probability that a fixed such copy contains $T_2$ sharing the root is consequently again $1-p$.\n\t\n\tPlugging in the probabilities coming from the binomial distributions and rewriting the equation (mainly for aesthetic and computational convenience), we arrive at\n\t\\[p=p_0+p_1+\\sum_{k=2}^\\infty p_k\\cdot\\big(p^k+k\\,(1-p)\\,p^{k-1}\\big)=g(p)+(1-p)\\cdot g'(p).\\]\n\t\n\tIn the same vein, the probability that Breaker wins when Maker starts, can be written as\n\t\\[\\bar{p}=\\sum_{k=0}^\\infty \\Prob(\\xi_\\mathbf{0}=k)\\cdot\\Prob(X_k=0)=\\sum_{k=0}^\\infty p_k\\cdot p^k=g(p),\\]\n\twhich establishes the second claim.\n\\end{proof}\n\nThe recursive equation for $p$, in this regime the probability that the underlying GW-tree does not contain a complete binary tree sharing the root, was already established more than 30 years ago, cf.\\ Thm.\\ 1 in \\cite{Dek1}. It is further straightforward to verify that $p$ is in fact the smallest solution to \\eqref{fp2} in the interval $[0,1]$, see the above reference for details.\n\n\\subsection{A few tractable examples}\nWith this in hand, we can tackle a few simple cases. As the equation is implicit in $p$, however, the pgf $g$ makes it too hard to solve \\eqref{fp2} analytically in most cases, even for standard distributions for $\\xi$. From the monotonicity in the underlying graph, we generally get a phase transition in the parameter(s) of the offspring distribution $\\xi$ in the sense\nthat $p$ equals $1$ for the stochastically smaller end of the sprectrum, followed by a jump to values smaller than one at the corresponding critical value (denoted $s_\\mathrm{c}, \\lambda_\\mathrm{c}, r_\\mathrm{c}$ in the sequel).\n\n\\begin{proposition}[Geometric distribution]\\label{geoprop}\n\tThe probability of Breaker winning on an instantly revealed GW-tree with offspring distribution \n\t\\begin{enumerate}[(a)]\n\t\t\\item $\\xi\\sim\\mathrm{Geo}_{\\IN}(s)$, i.e.\\ $p_k=s\\,(1-s)^{k-1}$ for $k\\geq 1$, equals\n\t\t\\[p=\\frac{1-\\sqrt{1-4s}}{2\\,(1-s)}\\quad\\text{and}\\quad \\bar{p}=\\frac{1-2s-\\sqrt{1-4s}}{2\\,(1-s)}=p-\\frac{s}{1-s}\\]\n\t\trespectively (depending on who starts the game) for parameter $s\\in(0,\\frac14]$. Further it holds $p=\\bar{p}=1$, for $s\\in(\\frac14,1]$.\n\t\t\\item $\\xi\\sim\\mathrm{Geo}_{\\IN_0}(s)$, i.e.\\ $p_k=s\\,(1-s)^{k}$ for $k\\geq 0$, equals\n\t\t\\[p=\\frac{1+s-\\sqrt{(1-s)\\,(1-5s)}}{2\\,(1-s)}\\text{ and }\\ \\bar{p}=\\frac{1-s-\\sqrt{(1-s)\\,(1-5s)}}{2\\,(1-s)}=p-q\\]\n\t\trespectively (depending on who starts the game) for parameter $s\\in(0,\\frac15]$. Further it holds $p=\\bar{p}=1$, for $s\\in(\\frac15,1]$.\n\t\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\t\\begin{enumerate}[(a)]\n\t\t\\item In the case of $\\xi\\sim\\mathrm{Geo}_{\\IN}(s)$, the corresponding pgf and equation \\eqref{fp2} read\n\t\t\\[g(x)=\\frac{sx}{1-(1-s)x}\\quad\\text{and}\\quad p=\\frac{sp}{1-(1-s)p}+\\frac{s(1-p)}{(1-(1-s)p)^2}\\]\n\t\trespectively. The latter has solutions $p=1$ and if $s\\leq\\frac14$ additionally \\[p=\\frac{1\\pm\\sqrt{1-4s}}{2(1-s)}.\\]\n\t\tThis together with the fact that $p$ is the smallest solution to \\eqref{fp2} in $[0,1]$ establishes the first part of the claim. To arrive at $\\bar{p}$ we simply have to evaluate $g(p)$, cf.\\ Theorem \\ref{formula}:\n\t\t\\[p=\\frac{1-\\sqrt{1-4s}}{2\\,(1-s)}\\quad\\text{and}\\quad \\bar{p}=\\frac{1-2s-\\sqrt{1-4s}}{2\\,(1-s)}=p-\\frac{s}{1-s}\\]\n\t\t\\item In the case of $\\xi\\sim\\mathrm{Geo}_{\\IN_0}(s)$ -- number of failures until first success in a Bernoulli-trials experiment -- the pgf and \\eqref{fp2} read\n\t\t\\[g(x)=\\frac{s}{1-(1-s)x}\\quad\\text{and}\\quad p=\\frac{s}{1-(1-s)p}+\\frac{s(1-s)(1-p)}{(1-(1-s)p)^2}\\]\n\t\trespectively. The latter has solutions $p=1$ and if $s\\leq\\frac15$ additionally \\[p=\\frac{1+s\\pm\\sqrt{(1-s)(1-5s)}}{2(1-s)}.\\]\n\t\tSince the extinction probability equals $q(s)=\\frac{s}{1-s}$ in this case, we can rewrite $\\bar{p}$ given by\n\t\t$g(p)$ as $p-q$.\\vspace{-1em}\n\t\\end{enumerate}\n\\end{proof}\n\n\\vspace{-1em}\n\\noindent\nNote that the difference $p-\\bar{p}$ equals $\\frac{s}{1-s}$ in both cases, while the extinction probability is 0 in the first case, since $\\xi\\sim\\mathrm{Geo}_{\\IN}(s)$ guarantees at least one descendant for each individual (i.e.\\ $p_0=0$). Further it is worth mentioning that the phase transition at the critical values $s_\\mathrm{c}=\\frac14$ and $s_\\mathrm{c}=\\frac15$ respectively is a discontinuous one as the probabilities of Breaker winning jump to $1$, see Figure \\ref{Geo} below for an illustration. The critical parameters in both cases correspond to a mean value of $\\mu=4$ for the respective offspring distribution. The calculation of $1-p$ for $\\xi\\sim\\mathrm{Geo}_{\\IN_0}(s)$ was included as a special case in Section 3 of \\cite{Dek2}.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=1]{Geo}\n\t\\caption{Winning probabilities for Breaker on a GW-tree, which arises from a geometric offspring distribution on $\\IN$ (left) respectively $\\IN_0$ (right) and is revealed at the beginning, depending on the corresponding parameter $s\\in[0,1]$. Depicted in green, the corresponding extinction probability of the tree. \\label{Geo}}\n\\end{figure}\n\nIn case the random number of decendants per individual follows a Poisson distribution, equation\n\\eqref{fp2} is no longer analytically solvable. But one can still use Theorem\\ \\ref{formula} to numerically compute the winning probabilities.\n\n\\begin{example}[Poisson distribution]\n\tFor $\\xi\\sim\\mathrm{Poi}(\\lambda)$, the pgf reads $g(x)=\\mathrm{e}^{\\lambda(x-1)}$ and consequently, \\eqref{fp2} becomes\n\t\\[p=\\big(1+\\lambda(1-p)\\big)\\mathrm{e}^{\\lambda(p-1)}\\]\n\tor $1+\\frac{x}{\\lambda}=(1-x)\\,\\mathrm{e}^x$, where we substituted the exponent by $x$. For small $\\lambda>0$, this equation has only the trivial solution $x=0$ (which translates to $p=1$). Increasing $\\lambda$, a non-trivial solution enters the picture once the linear function on the left is a tangent to the graph of $x\\mapsto (1-x)\\,\\mathrm{e}^x$ in some $x_0<0$. Using this gradient condition we find that $x_0$ is a solution to $\\mathrm{e}^{-x}=1-x+x^2$, which leads to the critical parameter $\\lambda_\\mathrm{c}= 3.3509188715\\dots$ For this mean in $\\xi\\sim\\mathrm{Poi}(\\lambda)$, the winning probability of Breaker (using again Lemma \\ref{char} and Theorem\\ \\ref{formula}) drops from 1 to $p_\\mathrm{c}=0.46483869\\dots$, see Figure \\ref{Poi1} below for an illustration. Having evaluated $p$ numerically, we can calculate $\\bar{p}=g(p)$ as before.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[scale=1]{Poi1}\n\t\t\\caption{Winning probabilities for Breaker on a GW-tree, which arises from a Poisson offspring distribution, $\\xi\\sim\\mathrm{Poi}(\\lambda)$. \\label{Poi1}}\n\t\\end{figure}\n\t\n\tIt is worth noting that the mean in the critical case for a Poisson offspring distribution equals $\\lambda_\\mathrm{c}\\neq 4$. Consequently, the fact that the phase transition with respect to $q$ (extinction) occurs at $\\mu=1$ for all (non-constant) offspring distributions (cf.\\ Theorem \\ref{extinction}) does not have an equivalent, when it comes to the phase transition with respect to $p$ (absence of a complete binary subtree rooted at $\\mathbf{0}$). This was already established in \\cite{Dek1} by a simple counter-example, based on the fact that changing $p_0$ and $p_1$, while keeping $p_0+p_1$ constant, changes all moments $\\IE(\\xi^n),\\ n\\in\\IN,$ but not $p$.\n\\end{example}\n\n\\subsection{Conditions for Maker to have a chance of winning}\nIn \\cite{Dek2} both a necessary and a sufficient condition for the existence of a complete $N$-ary subtree rooted at the ancestor $\\mathbf{0}$ of a GW-tree for arbitrary offspring distribution has been established. For $N=2$ this refers to a binary tree, more specifically $p<1$ in our setting (cf.\\ Lemma \\ref{char}).\n\n\\begin{proposition}\\label{bound}\n\tConsider the $(1,1)$-Maker-Breaker game on an instantly revealed GW-tree arising from a general offspring distribution $\\{\\Prob(\\xi=k)=p_k,\\,k\\in\\IN_0\\}$, in which Breaker starts. Then the following conditions hold:\n\t\\begin{enumerate}[(a)]\n\t\t\\item Maker has a positive chance of winning (p<1), if\n\t\t\\begin{equation*}\n\t\t\t\\IE\\Big(\\frac{1}{\\xi+1}\\Big)\\leq\\frac14+\\frac{p_0}{2}+\\frac14\\,(p_0+p_1)^2.\n\t\t\\end{equation*}\n\t\t\\item If $g''(x)<\\frac{1}{1-x}$ for all $x\\in(0,1)$, the game is an a.s.\\ win for Breaker (p=1).\n\t\\end{enumerate} \n\\end{proposition}\nThis result follows directly from Thm.\\ 3 in \\cite{Dek2}, considering the special case $N=2$.\n\n\\begin{example}[Binomial distribution]\n\tFor $\\xi\\sim\\mathrm{Bin}(n,r)$, we have $p_0=(1-r)^n$ and $p_1=nr\\,(1-r)^{n-1}$ as well as $g(x)=(1-r+rx)^n$. Solving\n\t\\eqref{fp2} numerically, we can find the critical values $r_\\mathrm{c}(n)$ and calculate the winning probabilities for Breaker in both cases, $p(n,r)$ for Breaker starting and $\\bar{p}(n,r)$ for Maker starting the game, in the same way as before, see Figure \\ref{Bin2} and the table below for numerical results.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[scale=1]{Bin2}\n\t\t\\caption{Winning probabilities for Breaker on an instantly revealed GW-tree, which arises from a Binomial offspring distribution, $\\xi\\sim\\mathrm{Bin}(n,r)$, for $n=5$ (left) and $n=10$ (right). \\label{Bin2}}\n\t\\end{figure}\t\n\t\\begin{center}\n\t\t\\begin{tabular}{ m{0.2cm}||c|c|c|c|c|c|c|c|c|c @{}m{0cm}@{}}\n\t\t\t\\hline\n\t\t\t$n$&1&2&3&4&5&6&7&8&9&10& \\\\[0.3cm] \\hline\n\t\t\t$r_\\mathrm{c}$&\\small1&\\small1&$\\nicefrac89$&\\small0.7248&\\small0.6028&\\small0.5137&\\small0.4468&\\small0.3949&\\small0.3537&\\small0.3202& \\\\[0.1cm]\n\t\t\t$p_\\mathrm{c}$&\\small1&\\small1&$\\nicefrac{5}{32}$&\\small0.2584&\\small0.3105&\\small0.3418&\\small0.3625&\\small0.3773&\\small0.3883&\\small0.3969& \\\\[0.cm]\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\captionof{table}{Critical parameter $r_\\mathrm{c}$ and the corresponding winning probability for Breaker, when $\\xi\\sim\\mathrm{Bin}(n,r)$ and small values of $n$.}\n\t\\end{center}\n\tIn order to evaluate the bounds established in Proposition \\ref{bound}, we use Fubini to get\n\t\\[\\IE\\Big(\\frac{1}{\\xi+1}\\Big)=\\int_0^1\\IE(x^\\xi)\\,dx=\\int_0^1 (1-r+rx)^n\\,dx=\\frac{1-(1-r)^{n+1}}{r\\,(n+1)}.\\]\n\tSo according to part (a) of the proposition, $p(n,r)<1$ holds for pairs $(n,r)$ which fulfill\n\t\\[\\frac{1-(1-r)^{n+1}}{r\\,(n+1)}\\leq\\frac14+\\frac12\\,(1-r)^n+\\frac14\\,\\big((1-r)^n+nr\\,(1-r)^{n-1}\\big)^2.\\]\n\tFor fixed $n$, this can be solved numerically and leads to a value $r_>(n)$, above which Maker has a positive chance of winning the game.\n\t\n\tTurning to part (b), it is easy to see that $g''(x)=0$ for $n=1$ (hence the inequality holds trivially on $(0,1)$ for arbitrary $r$) and $g''(x)=2r^2$ for $n=2$, which guarantees an a.s.\\ Breaker win for $r\\geq\\frac{1}{\\sqrt{2}}$. For $n\\geq3$, the inequality\n\t\\[n(n-1)r^2\\,(1-r+rx)^{n-2}<\\frac{1}{1-x}\\]\n\tholds for $r\\geq0$ sufficiently small and increasing it, the function on the left hand side coincides with the right hand side in $(0,1)$ for the first time when also the derivatives of both functions coincide at this point. Using this to simplify the condition, we find that the inequality holds for all $0<x<1$ if $r$ is less than $r_<(n)=\\frac1n\\big(\\frac{n-1}{n-2}\\big)^{n-2}$. Consequently, the critical parameter $r_\\mathrm{c}(n)$, which separates the trivial regime ($p=1$) from the competitive one ($p<1$), lies in the interval $[r_<(n),r_>(n)]$. In Figure \\ref{Bin1}, the critical values and the corresponding bounds are illustrated for $n\\leq 15$.\n\t\\begin{figure}[H]\\hspace{-0.3cm}\n\t\t\\includegraphics[scale=1]{Bin1}\n\t\t\\caption{Depicted are the threshold $r_\\mathrm{c}$ (green) plus corresponding bounds $r_<$ (red) and $r_>$ (blue), coming from Proposition \\ref{bound}, for Binomial offspring distribution and small values of $n$. \\label{Bin1}}\n\t\\end{figure}\n\tNote that $\\lim_{n\\to\\infty}n\\,r_<(n)=\\mathrm{e}$, which is consistent with the corresponding lower bound for the Poisson distribution, see below.\n\\end{example}\n\nEvaluating the bounds from Proposition \\ref{bound} also for the cases examined earlier, we get for $\\xi\\sim\\mathrm{Geo}_{\\IN}(s)$ that $p<1$ for $s\\leq0.21332$ and \n$p=1$ for $s\\geq0.29629$, as opposed to the true threshold $s_\\mathrm{c}=\\frac14$ (cf.\\ Proposition \\ref{geoprop}).\nWith offspring distribution $\\xi\\sim\\mathrm{Geo}_{\\IN_0}(s)$ instead, the proposition guarantees $p<1$ for $s\\leq0.16401$ and $p=1$ for $s\\geq 0.22857$, while $s_\\mathrm{c}=\\frac15$.\nHaving Poisson distributed offspring numbers, $\\xi\\sim\\mathrm{Poi}(\\lambda)$, we find that $\\lambda\\geq3.654328$ guarantees $p<1$ and $\\lambda\\leq\\mathrm{e}$ ensures $p=1$, while the true threshold is $\\lambda_\\mathrm{c}= 3.3509188715\\dots$\n\n\n\\section{No extra information, \\texorpdfstring{$I(v)=\\emptyset$}{I(v)=0}}\\label{empty}\n\nIn the most restrictive regime, in which no further information is revealed but the internal and external nodes, the strategies are simply totally random. As there is complete symmetry among all external nodes (being the root of an independent copy of our initial branching process) both players in turn remove/fixate an edge between an internal and external node picked arbitrarily.\n\nThe number of available edges/external nodes (taken at times when it is Maker's turn) is hence a random walk on $\\IZ$ with stepsize $\\xi-2$ (observe that both Maker and Breaker use up one edge and a random number of $\\xi$ is added during Maker's turn). Revealing the root $\\mathbf{0}$, there are $\\xi_\\mathbf{0}$ external nodes available; hence the initial value of the corresponding random walk is either 1 (if Breaker starts) or 2 (if Maker starts). Let us formally define the \\emph{embedded random walk}\n\\begin{equation}\\label{rw}\n\tS_0=1,\\quad S_{n}=S_{n-1}+(\\xi_n-2)\\quad\\text{for}\\ n\\in\\IN,\n\\end{equation}\nwhere $(\\xi_n)_{n\\in\\IN}$ is an i.i.d.\\ sequence of copies of $\\xi$, the offspring distribution. Then the probability of Maker winning becomes $1-p=\\Prob(S_n>0 \\text{ for all }n\\in\\IN)$.\n\nLet us first consider offspring distributions concentrated on $\\{1,2,3\\}$, i.e.\\ $\\Prob(\\xi\\in\\{1,2,3\\})=1$, as this\ncondition makes the number of external nodes available to Maker a {\\em homogeneous discrete birth-and-death chain}.\nDue to the fact that the (lazy) simple symmetric random walk on $\\IZ$ is recurrent, Breaker will obviously win the game with probability 1 if played on a GW-tree arising from any offspring distribution with\n\\[p_1=p_3=r,\\ p_2=1-2r,\\quad\\text{for arbitrary }0<r\\leq\\tfrac12,\\]\nwhere $p_k=\\Prob(\\xi=k)$ as before. By a simple coupling argument (or monotonicity of Maker-Breaker in the underlying graph), more generally $0\\neq p_1\\geq p_3$ gives Breaker an almost sure win (even if Maker starts, i.e.\\ $\\bar{p}=p=1$). However, if $p_3>p_1$ (which makes the birth-and-death chain transient) we can conclude a first non-trivial result for this regime:\n\n\\begin{proposition}\\label{123}\n In the regime $I(v)=\\emptyset$, if $p_1+p_2+p_3=1$ and $p_1<p_3$, then the probability of Breaker winning on a GW-tree arising from offspring distribution $\\xi$, with $\\Prob(\\xi=k)=p_k$, equals\n \\[p=\\frac{p_1}{p_3}\\quad\\text{and}\\quad \\bar{p}=\\left(\\frac{p_1}{p_3}\\right)^2\\]\n respectively, depending on who starts the game.\n\\end{proposition}\n\\begin{proof}\nWe can assume $p_2=0$, hence $p_3=1-p_1$, w.l.o.g.\\ as including the possibility of loops will not change the\nprobability of ever hitting $0$.\nWriting $\\theta_k:=\\Prob(\\text{chain hits 0 after start in }k)$, we get $\\theta_0=1$ and $\\theta_2=\\theta_1^2$, by the strong Markov property applied to the first time hitting 1 after start in 2. Conditioning on the first step, this gives the equation\n\\[\\theta_1=p_1\\cdot\\theta_{0}+p_3\\cdot\\theta_{2}=p_1+(1-p_1)\\cdot\\theta_{1}^{\\,2},\\quad\\text{with solutions }\\theta_1=\\frac{1\\pm |1-2p_1|}{2\\,(1-p_1)}.\\]\nThe fact that $p_1<\\frac12$ by assumption and $\\theta_1<1$ by transience of the chain, leads to $\\theta_1=\\frac{p_1}{1-p_1}$. For further details, see e.g.\\ Chap.\\ 2.5 in \\cite{LPW}.\n\\end{proof}\n\nNote that $p_1<p_3$ coincides with the intuition that $\\mu=2+(p_3-p_1)$ has to be larger than 2 for Maker to have a chance.\nThe above line of argument can in fact be used to settle even the case of a geometric offspring distribution $\\xi\\sim\\text{Geo}_{\\IN}(s)$:\n\n\\begin{proposition}[Geometric distribution]\\label{emptygeo}\n\tWhen $I(v)=\\emptyset$, the probability of Breaker winning on a GW-tree arising from offspring distribution $\\xi\\sim\\mathrm{Geo}_{\\IN}(s)$ equals for fixed $s\\in[0,\\frac12]$\n\t\\[p=\\frac{s}{1-s}\\quad\\text{and}\\quad \\bar{p}=\\left(\\frac{s}{1-s}\\right)^2\\]\n\trespectively, depending on who starts the game. In addition to that, for $s\\in(\\frac12,1]$ it holds $p=\\bar{p}=1$.\n\\end{proposition}\n\\begin{proof}\n\tLet $(X_k)_{k\\in\\IN}$ be a sequence of i.i.d.\\ $\\mathrm{Ber}(s)$ random variables (commonly referred to as Bernoulli trials). Set $\\xi_0=0$ and $\\xi_n=\\min\\{k\\in\\IN:\\; X_{\\xi_{n-1}+k}=1\\}$ for $n\\in\\IN$. Then the sequence $(\\xi_n)_{n\\in\\IN}$ is i.i.d.\\ $\\mathrm{Geo}_{\\IN}(s)$. To recycle the idea of proof used in Proposition \\ref{123} it is crucial to notice that \n\t\\[\\xi_1=\\sum_{k=1}^{\\xi_1}1=2+\\sum_{k=1}^{\\xi_1}(1-2X_k)\\]\n\tand more generally\n\t\\[\\sum_{k=1}^{n}(\\xi_k-2)=\\sum_{k=1}^{S_n}(1-2X_k),\\]\n\twhere $S_n=\\sum_{k=1}^n\\xi_k$. As the left hand side describes a random walk on $\\IZ$ with stepsize distribution $\\xi-2$ and the right hand side a homogeneous discrete birth-and-death chain with birth probability $1-s$ and its steps grouped according to the sequence of down-steps (deaths), the probability of staying strictly positive is the same for both and the claim follows.\n\\end{proof}\n\nIn fact, with a similar reasoning as in the proof of Proposition \\ref{123}, we can derive an equation for the winning\nprobability of Breaker in this regime, that holds for general {\\em strictly positive} offspring distribution.\n\n\\begin{theorem}\\label{i=0} Consider $\\xi$ with $p_0=0$, where as before $p_k=\\Prob(\\xi=k)$, for $k\\in\\IN_0$, is the distribution's pmf and $g$ its pgf. Then the probability $p$ of Breaker winning on a GW-tree arising from offspring distribution $\\xi$, when the nodes carry no further information is a solution to\n\t\\begin{equation}\\label{fp1}\n\t\tx^2=g(x).\n\t\\end{equation}\n It is equal to $0$ if $p_1=0$, to $1$ if $\\mu\\leq 2,\\ p_1>0$ and to the (unique) solution in $(0,1)$ in all non-trivial cases ($\\mu>2,\\ p_1>0$). When Maker begins, the probability of Breaker winning becomes $\\bar{p}=g(p)=p^2$.\n\\end{theorem}\n\n\\begin{proof}\n\tNote that the condition $p_0=0$ makes the embedded random walk $(S_n)_{n\\in\\IN_0}$ described in \\eqref{rw} left-continuous, i.e.\\ its down steps are of size at most 1. Hence, we can write the probability $p=\\Prob(S_n=0 \\text{ for some }n\\in\\IN\\,|\\,S_0=1)$ by conditioning on the first step as\n\t\\begin{align*}p&=p_1+\\sum_{k=0}^\\infty \\Prob(\\xi_1-2=k)\\cdot\\Prob(S_n=0 \\text{ for some }n>1\\,|\\, S_1=k+1)\\\\\n\t\t\t\t &=p_1+\\sum_{k=2}^\\infty p_k\\cdot p^{k-1}=\\frac{g(p)}{p},\n\t\\end{align*}\n\twhere the strong Markov property was used in the second step and $p\\neq0=p_0$ in the last one. Since $g(0)=p_0=0$, both $x=0$ and $x=1$ are general solutions to \\eqref{fp1}. If $p_1=0$, the random walk $(S_n)_{n\\in\\IN_0}$ is non-decreasing and trivially Maker wins with probability 1. In case $p_1>0$, the walk $(S_n)_{n\\in\\IN_0}$ is not constant. If further $\\mu\\leq 2$, it does not have positive drift and will therefore visit state $0$ almost surely, i.e.\\ $p=1$. For the case $p_1>0,\\ \\mu>2$, let us consider the function $h(x):=g(x)-x^2$.\n\tThe sign of its derivative $h'$ can change at most twice, due to the convexity of $g'$. Since $h(0)=h(1)=0$ and\n\t$h'(0)=p_1>0,\\ h'(1)=\\mu-2>0$, it has exactly one zero in $(0,1)$. That in this case $(S_n)_{n\\in\\IN_0}$ is transient with positive drift (and therefore $0<p<1$, cf.\\ Lemma \\ref{drift}), completes the proof of the first part.\n\tIf Maker starts the game, which translates to $S_0=2$, we get by left-continuity of the walk \\begin{align*}\n\t\t\\bar{p}&=\\Prob(S_n=0 \\text{ for some }n\\in\\IN\\,|\\,S_0=2)\\\\\n\t\t&=\\Prob(S_n=0 \\text{ for some }n\\in\\IN\\,|\\,S_0=1)^2=p^2=g(p),\n\t\t\\end{align*}\n\twhich settles the second part of the claim.\n\\end{proof}\n\nNoting that for $\\xi\\sim\\mathrm{Geo}_\\IN(s)$ it holds $p_0=0$ and $g(x)=\\frac{sx}{1-(1-s)x}$, the result\nfrom Proposition \\ref{emptygeo} can easily be recovered using the above theorem.\n\n\\begin{example}\\label{oom}\n Let us consider the Maker-Breaker game, played on a GW-tree with what was colloquially called ``one-or-many'' offspring distribution in the last section of \\cite{Dek2}, i.e.\\ $p_n=1-p_1=r$ for some $r\\in(0,1)$ and $n\\geq 2$. Then the tree is infinite and $g(x)=(1-r)\\,x+rx^n$. For $r>\\frac{1}{n-1}$, the game is non-trivial ($\\mu>2$) and the winning probability of Breaker when starting the game is the (unique) $x\\in(0,1)$ solving $x=(1-r)+rx^{n-1}$. In order to make the example a little less abstract, for the choice of e.g.\\ $n=4$ this becomes \n \\[p=\\frac{\\sqrt{r\\,(4-3r)}-r}{2 r}\\quad\\text{and}\\quad\\bar{p}=\\frac{2-r-\\sqrt{r\\,(4-3r)}}{2r},\\]\n illustrated in Figure \\ref{Oom} below.\n\n\\begin{figure}[H]\n\t\\hspace{-0.3cm}\n\t\\includegraphics[scale=1]{OoM}\n\t\\caption{\\textbf{Left:} For $r(n-1)>1$, the function $x\\mapsto(1-r)+rx^{n-1}$ intersects exactly once with the identity function on (0,1). \\textbf{Right:} Winning probabilities for Breaker on a tree with one-or-many offspring distribution, where ``many'' refers to $n=4$.\\label{Oom}}\n\\end{figure}\n\\end{example}\n\nTo extend the analysis beyond the scope of Theorem \\ref{i=0}, i.e.\\ offspring distributions with $p_0>0$, we first\nverify that $\\bar{p}>g(p)$ in this case.\n\n\\begin{proposition}\\label{ineq}\n\tConsider $\\xi$ with $p_0>0$. Then the probability $\\bar{p}$ of Breaker winning the game Maker starts on a GW-tree $T$ arising from offspring distribution $\\xi$, when the nodes carry no further information is strictly bigger than $g(p)$ in the non-trivial case (i.e.\\ $p<1$).\n\\end{proposition}\n\n\\begin{proof}\n\tThe difference to the case $p_0=0$ examined in Theorem \\ref{arch} is that Maker might conclude the exploration of a subtree rooted at a vertex in the first generation by fixating an edge incident to a leaf in the tree. Then Breaker will consequently erase an edge incident to the root next. To be more precise, consider the embedded random walk $(S_n)_{n\\in\\IN_0}$ with $S_0=2$ and increments distributed as $\\xi-2$ and let $\\tau_{\\leq0}=\\inf\\{n\\in\\IN,\\,S_n\\leq0\\}$\n\tbe the first time the walk exits $\\IN$ (possibly $\\tau_{\\leq0}=\\infty$). In contrast to left-continuous random walk,\n\t$S_{\\tau_{\\leq0}}=-1$ has positive probability when $p_0>0$, so let $\\alpha=\\Prob(S_{\\tau_{\\leq0}}=-1)$.\n\t\n\tFollowing a depth-first strategy is optimal for both players, by symmetry in the external nodes. Given the root $\\mathbf{0}$ has $k\\geq1$ children, this will amount to $1+Y$ consecutive games on independent trees distributed like $T$ in which Breaker starts, where $Y\\sim\\mathrm{Bin}(k-1,1-\\alpha)$: Whenever Maker makes the last possible move in a subtree rooted in the first generation, which corresponds to the event $\\{S_{\\tau_{\\leq0}}=-1\\}$, Breaker erases the next one.\n\tSo the probability for a win of Breaker in the game started by Maker is given by\n\t\\[\\Prob(\\text{Breaker wins}\\,|\\,\\text{Maker starts, }Z_1=k)=\\IE(p^{Y+1})=p\\big(\\alpha+(1-\\alpha)p\\big)^{k-1}.\\]\n\tIf $p_0=0$ (hence $\\alpha=0$), this term equals $p^k$. Given $p<1$ and $p_0>0$, however, it is strictly bigger than $p^k$ for $k>1$. This entails\n\t\\[\\bar{p}=p_0+\\sum_{k=1}^\\infty p_k\\cdot p\\big(\\alpha+(1-\\alpha)p\\big)^{k-1}>\\sum_{k=0}^\\infty p_k\\,p^k=g(p),\\]\n\twhere $p_0+p_1<1$ (implied by $p<1$) was used.\n\\end{proof}\n\n\\subsection{Separable offspring distributions}\\label{separable}\n Let us now consider the case in which $\\xi$ can be written as the sum of two i.i.d.\\ integer-valued random variables:\n\\begin{definition}\n\tLet us call the distribution of $\\xi$ {\\em separable}, if there exist independent and identically distributed integer-valued random variables $X_1$ and $X_2$, such that \n\t\\begin{equation*}\\label{sep}\n\t\t\\xi\\stackrel{d}{=}X_1+X_2.\n\t\\end{equation*}\n\\end{definition}\n\nAt first glance, this might seem to be a fairly special case. However, both the Poisson distribution $\\mathrm{Poi}(\\lambda)$ and the negative binomial distribution $\\mathrm{NB}(r,s)$, generalized to $r$ not necessarily being integer (occasionally referred to as {\\em Plya distribution}), hence also the geometric distribution $\\mathrm{Geo}_{\\IN_0}(s)$, are even infinitely divisible (i.e.\\ can be written as the sum of arbitrarily many i.i.d.\\ random variables). Since the corresponding summands are integer-valued,\n\\[\\mathrm{Poi}(\\lambda)=\\mathrm{Poi}(\\tfrac\\lambda2)\\ast\\mathrm{Poi}(\\tfrac\\lambda2)\\quad\\text{resp.}\n\\quad\\mathrm{NB}(r,s)=\\mathrm{NB}(\\tfrac r2,s)\\ast\\mathrm{NB}(\\tfrac r2,s),\\]\nthese distributions are also separable in the sense above.\nThe binomial distribution $\\mathrm{Bin}(n,r)$ with an even number $n=2k$ of trials is trivially separable, as $\\mathrm{Bin}(k,r)\\,\\ast \\,\\mathrm{Bin}(k,r)=\\mathrm{Bin}(2k,r)$, but not infinitely divisible. For odd $n$, $\\mathrm{Bin}(n,r)$ can not even be the convolution of two i.i.d.\\ random variables: Assuming for contradiction that it was, the (independent) summands $X_1,X_2$ would need to attain the values $0$ and $\\frac n2$ with positive probability (so that both $0$ and $n$ can be attained by their sum, but neither negative numbers nor values bigger than $n$). Since $\\frac n2$ in this case is not an integer, but $\\Prob(X_1+X_2=\\frac n2)>0$, we can conclude that $X_1+X_2\\nsim\\mathrm{Bin}(n,r)$, a contradiction.\n\nImportant to observe is that if $\\xi$ is separable, we get $\\xi-2\\stackrel{d}{=}(X_1-1)+(X_2-1)$ and can consider\nan extended walk $(\\tilde{S}_n)_{n\\in\\IN_0}$ with increments distributed as $X_1-1$ and $\\tilde{S}_0=S_0$, essentially splitting the steps of the embedded random walk in two. Then the probability of the embedded walk to eventually hit $0$ (or $-1$) can be calculated as $\\Prob(\\tilde{S}_{2n}\\leq0\\text{ for some }n\\in\\IN)$ using the results collected in Lemma \\ref{calc}, as was done in \\cite{LCRW}. In this way, using Cor.\\ 4.5 in \\cite{LCRW}, we can calculate $p$ (and $\\bar{p}$) for the Maker-Breaker game in the regime $I(v)=\\emptyset$ on GW-trees based on most standard offspring distributions, with the exception of $\\xi\\sim \\mathrm{Bin}(n,r),$ where $n$ is odd.\n\n\\begin{theorem}\\label{septhm}\n If the offspring distribution is separable, i.e.\\ $\\xi\\stackrel{d}{=}X_1+X_2$ with i.i.d.\\ $(X_1,X_2)$, and $\\IE\\xi>2$ then the probabilities of Breaker winning on the corresponding GW-tree with the nodes carrying no further information is given by\n \\begin{equation}\\label{sep1}\n \tp=\\rho\\,\\bigg(1-\\frac{\\sigma\\cdot\\rho_\\mathrm{odd}}{1-\\theta\\cdot(1-\\theta_\\mathrm{odd})}\\bigg)\n \\end{equation}\nand\n\\begin{equation}\\label{sep2}\n\t\\bar{p}=\\rho^2\\,\\bigg(1-\\frac{2\\sigma\\cdot\\rho_\\mathrm{odd}\\,(1-\\rho_\\mathrm{odd})}{1-\\theta\\cdot(1-\\theta_\\mathrm{odd})}\\bigg)\n\\end{equation}\ndepending on whether Breaker or Maker starts the game.\n\n The (conditional) probabilities $\\rho$, $\\sigma$ and $\\theta$ as well as $\\rho_\\mathrm{odd}$ and $\\theta_\\mathrm{odd}$ appearing here refer to the ones introduced in \\eqref{probs}, corresponding to the left-continuous random walk on $\\IZ$ starting at $0$ with i.i.d.\\ increments distributed as $X_1-1$.\n\\end{theorem}\n\n\\begin{proof}\n\tCor.\\ 4.5 in \\cite{LCRW} applies almost perfectly to the embedded walk $(S_n)_{n\\in\\IN_0}$ here, just the walk there is shifted downwards by one (hitting $\\IZ\\setminus\\IN_0$ instead of $\\IZ\\setminus\\IN$). Thus, our winning probabilities for Breaker, $p$ and $\\bar{p}$, correspond to a start in $k=0$ and $k=1$ respectively there.\n\\end{proof}\n\n\\begin{example}[Poisson distribution]\n\tConsider the Maker-Breaker game, played on a GW-tree based on a Poisson offspring distribution $\\mathrm{Poi}(\\lambda)$. For $\\lambda> 2$, the game is non-trivial and the winning probability of Breaker in the regime $I(v)=\\emptyset$ are given by \\eqref{sep1} and \\eqref{sep2} respectively. Choosing a numerical value for the parameter, e.g.\\ $\\lambda=3$, we get $X_1\\sim\\mathrm{Poi}(\\frac32)$, hence approximately $\\rho=0.417188$, $\\sigma=0.311713,$ $\\theta=0.465157$, $\\rho_\\mathrm{odd}=0.706513$ and $\\theta_\\mathrm{odd}=0.817032$, which result in $p=0.31699$ and $\\bar{p}=0.14967>0.12886=g(p)$. The corresponding probabilities for small values of $\\lambda$ are depicted in Figure \\ref{Poi3}. \n\t\n\tIt is worth mentioning, that the phase transition in this case is continuous as opposed to the regime $I(v)=T_v$.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=1]{Poisep}\n\t\\caption{Winning probabilities for Breaker on a GW-tree, which arises from a Poisson offspring distribution, $\\xi\\sim\\mathrm{Poi}(\\lambda)$. \\label{Poi3}}\n\\end{figure}\n\t\n\\end{example}\n\n\\begin{example}[Geometric distribution]\n\tIf the game is played on a GW-tree based on a $\\mathrm{Geo}_{\\IN_0}(s)$ offspring distribution instead, it is non-trivial for $s<\\frac13$ and we get $X_1\\sim\\mathrm{NB}(\\frac12,s)$. As before we can calculate the corresponding (conditional) probabilities for the extended left-continuous walk using Lemma \\ref{calc} after selecting a numerical value for the parameter.\n\tThe choice $s=\\frac14$ for example gives: \\begin{equation*}\\begin{split}\\rho=\\tfrac16\\,(1+\\sqrt{13}),\\quad \\sigma=\\tfrac14\\,(\\sqrt{13}-3),\\quad \\theta=\\tfrac14\\,(5-\\sqrt{13}),\\\\ \\rho_\\mathrm{odd}=\\tfrac{1}{12}\\,(13-\\sqrt{13})\\quad \\text{and} \\quad \\theta_\\mathrm{odd}=\\tfrac{1}{12}\\,(5+\\sqrt{13}),\\end{split}\n\t\\end{equation*} \n which leads to $p=\\frac23$ and $\\bar{p}=\\frac59>\\frac12=g(p)$ in this case.\n\\end{example}\n\n\\subsection{Bounds by coupling}\nThe monotonicity in the underlying graph also allows us to derive bounds for the winning probabilities of Breaker by means of comparison with the game on other GW-trees with (preferably) more tractable offspring distribution.\nWhile the binomial distribution $\\mathrm{Bin}(n,r),$ where $n$ is odd, is not separable, hence eludes our analysis in the foregoing subsection, it holds\n\\[\\mathrm{Bin}(n-1,r)\\preceq_{st}\\mathrm{Bin}(n,r)\\preceq_{st}\\mathrm{Bin}(n+1,r).\\]\nHence the probabilities for Breaker to win on the corresponding trees have the reversed order (as switching to a subgraph favors Breaker).\n\\begin{example}[Binomial distribution]\n To get bounds for $p$ and $\\bar{p}$ relating to the Maker-Breaker game on the GW-tree with offspring distribution\n $\\xi\\sim\\mathrm{Bin}(13,\\frac14)$ for instance, one can calculate the corresponding probabilities for \n $\\xi\\sim\\mathrm{Bin}(12,\\frac14)$ and $\\xi\\sim\\mathrm{Bin}(14,\\frac14)$ respectively via Theorem \\ref{septhm} in order to conclude $0.2482>p>0.1367$ and $0.0957>\\bar{p}>0.0383$. Clearly, in this case the bounds are not particularly sharp, but for the Binomial distribution the winning probabilities for Breaker get a lot smaller, hence closer to each other as $n$ increases. As points of reference, $\\bar{p}=p=1$ for offspring distributions $\\mathrm{Bin}(n,\\frac14)$ with $n\\leq8$ (since then $\\mu\\leq2$), however approximately $p=0.078$ and $\\bar{p}=0.017$ for $\\xi\\sim\\mathrm{Bin}(16,\\frac14)$.\n\\end{example}\n\n\\section{Total number of progeny given, \\texorpdfstring{$I(v)=|T_v|$}{I(v)=|T\\_v|}}\n\nIf only the {\\em size} of the subtree rooted in vertex $v$ is revealed, once $v$ is an internal or external node, and not the whole structure of the tree (as in Section \\ref{sec:all}), the situation is very much like when no further information is given. In fact, for extinction probability $q=0$ both regimes coincide (see part (b) of Lemma \\ref{skewed2} and the remark after Theorem \\ref{tv=i}).\nWhile both players stay away from any edge towards a node that is the root of a finite subtree, there is complete symmetry in the nodes that are the root of an infinite subtree, i.e.\\ vertices $v$ with $|T_v|=\\infty$.\n\nConsequently, the number of edges towards external nodes considered for play when it is Maker's turn is again a random walk on $\\IZ$. The step\nsize, however, is now $\\xi'-2$, where $\\xi'$ is the offspring distribution skewed by the fact that vertices, which are the roots of finite subtrees, are disregarded:\n\n\\begin{lemma}\\label{skewed}\n\tIn the case $I(v)=|T_v|$ and given the non-trivial situation, where $|T_\\mathbf{0}|=\\infty$, the number of external nodes with infinite progeny at Maker's turns can be described as the random walk\n\t\\[S_0=c,\\quad S_{n}=S_{n-1}+(\\xi'_n-2)\\quad\\text{for}\\ n\\in\\IN,\\]\n\twhere $(\\xi'_n)_{n\\in\\IN}$ is an i.i.d.\\ sequence with marginal probability mass function\n\t\\[\\Prob(\\xi_1'=k)=\\frac{1}{1-q}\\, \\sum_{n\\geq k} p_n \\binom{n}{k}(1-q)^kq^{n-k}\\quad\\text{for }k\\in\\IN,\\]\n\twhere $(p_k)_{k\\in\\IN_0}$ is again the pmf of the offspring distribution corresponding to the underlying supercritical GW-process. As before, $c=1$ corresponds to Breaker starting, $c=2$ corresponds to Maker starting the game.\n\\end{lemma}\n\n\\begin{proof}\n\tThe only thing to prove (besides following the reasoning in Section \\ref{empty} verbatim), is the form of\n\tthe step-size distribution claimed here. Let us write $\\xi^*_v$ for the number of individuals having infinite progeny among the offspring of $v$ and note that this is just an independent thinning of $\\xi_v$ with factor $q$. Since $\\{|T_v|=\\infty\\}=\\{\\xi^*_v>0\\}$, we get $\\Prob\\big(\\xi^*_v=0\\,\\big|\\,|T_v|=\\infty\\big)=0$ and for any $k\\in\\IN$\n\t\\begin{align*}\n\t\t\\Prob\\big(\\xi^*_v=k\\,\\big|\\,|T_v|=\\infty\\big)&=\\frac{\\Prob(\\xi^*_v=k)}{\\Prob(|T_v|=\\infty)}\\\\\n\t\t&=\\frac{1}{1-q}\\,\\sum_{n=1}^\\infty p_n\\cdot\\Prob(\\xi^*_v=k\\,|\\,\\xi_v=n),\n\t\\end{align*}\n\twhich implies the claim, writing $\\xi'$ for $\\xi^*$ conditioned on $\\{\\xi^*>0\\}$.\n\\end{proof}\n\nWhen it comes to this skewed step size distribution $\\xi'$, two basic facts are fairly easy to verify:\n\\begin{lemma}\\label{skewed2}\n\tLet $(p_k)_{k\\in\\IN_0}$ be the pmf of $\\xi$ and for some $0\\leq q<1$ the pmf of $\\xi'$ be given by\n\t\\[\\Prob(\\xi'=k)=\\frac{1}{1-q}\\, \\sum_{n\\geq k} p_n \\binom{n}{k}(1-q)^kq^{n-k}\\quad\\text{for }k\\in\\IN.\\]\n\tThen the following holds:\n\t\\begin{enumerate}[(a)]\n\t\t\\item $\\IE\\xi'=\\IE\\xi=\\mu$\n\t\t\\item $\\xi'\\stackrel{d}{=}\\xi$ in case $q=0$. \n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\tWhile part (b) is both obvious from the construction of $\\xi'$ and straightforward to check (by setting $q=0$ in its pmf), the claim that $\\xi$ and $\\xi'$ have the same mean requires a short calculation:\n\t\\begin{align*}\n\t\t\\IE\\xi'&=\\frac{1}{1-q}\\,\\sum_{k=1}^\\infty k \\,\\sum_{n\\geq k} p_n \\binom{n}{k}(1-q)^kq^{n-k}\\\\\n\t\t&=\\frac{1}{1-q}\\,\\sum_{n=1}^\\infty p_n \\,\\sum_{k=1}^n k\\cdot\\binom{n}{k}(1-q)^kq^{n-k}\\\\\n\t\t&=\\frac{1}{1-q}\\,\\sum_{n=1}^\\infty p_n\\cdot n(1-q) = \\mu.\\\\[-1cm]\n\t\\end{align*}\n\\end{proof}\n\nFor the remainder of this section we can assume $q\\neq0$, since otherwise $|T_v|=\\infty$ for all vertices $v$, which\nputs us back into the regime $I(v)=\\emptyset$. But even for $q>0$, in light of Lemma \\ref{skewed}, the probability of Breaker winning equals the probability of a random walk on $\\IZ$ ever hitting $\\IZ\\setminus\\IN$, much alike the most restrictive information regime $I(v)=\\emptyset$ (cf.\\ \\eqref{rw} above). However, in this regime the stepsize is no longer distributed like $\\xi-2$, but by default almost surely bigger or equal to $-1$.\n\nFollowing the line of argument in Section \\ref{empty}, we can therfore use the theory of birth-and-death chains again to solve the most simple cases of offspring distributions, despite the change that every node now carries the number of its total progeny as information. For arbitrary offspring distribution $(p_k)_{k\\in\\IN_0}$, we write as before $g:[0,1]\\to[0,1]$ for its pgf and $q$ for the corresponding extinction probability.\n\n\\begin{proposition}\\label{0123}\n\tLet $(p_k)_{k\\in\\IN_0}$ be a supercritical offspring distribution concentrated on $\\{0,1,2,3\\}$, i.e.\\ $p_1<p_0+p_1+p_2+p_3=1<p_1+2p_2+3p_3$.\n\tIf $g'(q)<p_3\\,(1-q)^2$, then the probability of Breaker winning on a GW-tree arising from offspring distribution $\\xi$, with $\\Prob(\\xi=k)=p_k$, given the regime $I(v)=|T_v|$ equals\n\t\\[p=\\frac{g'(q)}{p_3\\,(1-q)^2}\\quad\\text{and}\\quad \\bar{p}=\\left(\\frac{g'(q)}{p_3\\,(1-q)^2}\\right)^2\\]\n\trespectively, depending on who starts the game. For $g'(q)\\geq p_3\\,(1-q)^2$, it holds $\\bar{p}=p=1$.\n\\end{proposition}\n\\begin{proof}\n\tIn order to mimick the argument of Proposition\\ \\ref{123}, we conclude from Lemma \\ref{skewed} that $\\xi'$ has pmf\n\t$\\Prob(\\xi'=3)=p_3\\,(1-q)^2,\\ \\Prob(\\xi'=2)=(p_2+3p_3q)(1-q)$ and $\\Prob(\\xi'=1)=p_1+2p_2q+3p_3q^2=g'(q)$.\n\tNote that these probabilities sum to 1, as $q$ equals $g(q)=p_0+p_1q+p_2q^2+p_3q^3$ by Theorem \\ref{extinction}.\n\tThe corresponding birth-and-death chain is transient if $\\Prob(\\xi'=3)>\\Prob(\\xi'=1)$ and the probability it hits\n\t0 given by \n \t\\[p=\\frac{\\Prob(\\xi'=1)}{\\Prob(\\xi'=3)}\\text{ if started in 1 and by }\\bar{p}=\\left(\\frac{\\Prob(\\xi'=1)}{\\Prob(\\xi'=3)}\\right)^2\\text{ if started in 2}.\\]\n \tIn case $\\Prob(\\xi'=1)=g'(q)=p_3\\,(1-q)^2=\\Prob(\\xi'=3)$, the random walk described in Lemma \\ref{skewed} is a (lazy) simple symmetric random walk on $\\IZ$ that will a.s.\\ hit state 0, irrespectively of its start value $c$. The statement\n \tfor $g'(q)< p_3\\,(1-q)^2$ then follows immediately by the monotonicity in the tree (or an appropriate coupling of the walks).\n\\end{proof}\n\n\\begin{example}[Binomial distribution]\n\tLet us consider the Maker-Breaker game on a GW-family tree with offspring distribution $\\xi\\sim\\mathrm{Bin(3,r)}$ and\n\t$I(v)=|T_v|$. The branching process is supercritical if $r>\\frac13$ and its extinction probability then given by the\n\tsmallest positive root of $x=g(x)=(1+r(x-1))^3$, which is\n\t\\[q(r)=1-\\frac{3}{2r}+\\frac{1}{2r^2}\\sqrt{r\\,(4-3r)}.\\]\n\tConsequently, $g'(q)=\\frac32\\,(2-r-\\sqrt{r\\,(4-3r)})<r^3\\,(1-q)^2$ holds when $r>\\frac23$. By Proposition \\ref{0123}, we can conclude that the probabilities for Breaker winning are given by\n\t\\[p(r)=\\begin{cases}1,& \\text{for }0\\leq r\\leq\\frac23\\\\\n\t\t\t\t\t\t\\dfrac{6r\\,(2-r-\\sqrt{r\\,(4-3r)})}{\\big(3r-\\sqrt{r\\,(4-3r)}\\big)^2},&\\text{for } \\frac23<r\\leq1\\end{cases}\\]\n\tand\t$\\bar{p}(r)=p^2(r)$ respectively, depending on who starts the game. An illustration can be found in Figure \\ref{Bin3} below.\n\t\\begin{figure}[H]\n\t\t\\includegraphics[scale=1]{Bin3}\n\t\t\\caption{Winning probabilities for Breaker on a GW-tree arising from $\\xi\\sim\\mathrm{Bin}(3,r)$, where\n\t\t\tnode $v$ carries information $I(v)=|T_v|$. \\label{Bin3}}\n\t\\end{figure}\n\\end{example}\n\n\\begin{theorem}\\label{tv=i}\n In the regime $I(v)=|T_v|$, the winning probability $p$ of Breaker is a solution to\n \\begin{equation}\\label{fp3}\n \tx^2\\,(1-q)=g\\big(x\\,(1-q)+q\\big)-q.\n \t\\end{equation}\n To be more precise: $p=1$ in case $\\mu\\leq 2,\\ p_2\\neq1$ and $p=0$ if $\\xi\\geq2$ almost surely; for $\\mu>2$ and $p_0+p_1>0$, $p$ equals the (unique) solution to \\eqref{fp3} in $(0,1)$.\n\\end{theorem}\n\n\\begin{proof} First note that both $x=0$ and $x=1$ (due to $q=g(q)$ and $g(1)=1$) are general solutions to equation \\eqref{fp3}.\n\tIn the trivial case $q=1$, the equation obviously holds and $p$ equals $1$. In case $q=0$ ($\\Leftrightarrow p_0=0$) the statement becomes Theorem \\ref{i=0}, which leaves case $0<q<1$ left to treat. Using Lemma \\ref{skewed} and conditioning on the first step of the random walk $(S_n)_{n\\in\\IN_0}$, we get\n\t\\begin{align*}\n\tp&=\\sum_{k=1}^\\infty \\Prob(\\xi'_1=k)\\cdot p^{k-1}\n\t =\\frac{1}{1-q}\\,\\sum_{k=1}^\\infty p^{k-1}\\,\\sum_{n\\geq k} p_n \\binom{n}{k}(1-q)^kq^{n-k}\\\\\n\t &=\\frac{1}{p\\,(1-q)}\\,\\sum_{n=1}^\\infty p_n\\,\\sum_{k=1}^n \\binom{n}{k}p^k(1-q)^kq^{n-k}\\\\\n\t &=\\frac{1}{p\\,(1-q)}\\, \\sum_{n=0}^\\infty p_n \\Big[\\big(p(1-q)+q\\big)^n-q^n\\Big]\\\\\n\t &=\\frac{1}{p\\,(1-q)}\\,\\Big(g\\big(p(1-q)+q\\big)-g(q)\\Big),\n\t\\end{align*}\nwhere we used that the factor of $p_0$ is 0 in the before last line. In fact we tacitly assumed $p\\neq 0$ as well, but as mentioned above, $x=0$ solves equation \\eqref{fp3}.\n\nTo derive the second part of the claim, first note that by Lemma \\ref{skewed2}, the random walk $(S_n)_{n\\in\\IN_0}$ with increments distributed like $\\xi'-2$ has positive drift iff $\\mu>2$. The case $\\Prob(\\xi\\geq2)=1$ is included in $q=0$ (as mentioned at the start, where $\\xi$ and $\\xi'$ have the same distribution) and trivially gives $p=0$. In case $\\mu\\leq 2,\\ p_2\\neq1$, the random walk is neither constant nor having positive drift, hence $p=1$.\nFinally, for $\\mu>2$ and $p_0>0$ (if $p_0=0$ we are back to $q=0$) let us consider the function \n\\[h(x):=g\\big(x\\,(1-q)+q\\big)-(1-q)\\,x^2-q.\\]\nIn complete analogy to the reasoning in the proof of Theorem \\ref{i=0}, the sign of its derivative $h'(x)=(1-q)\\,\\big(g'(x\\,(1-q)+q)-2x\\big)$ can change at most twice, due to the convexity of $g'$. Further, $\\mu>2$ forces both $q<1$ and $p_0+p_1<1$, which implies that $g$ is even strictly convex on $(0,1)$.\nTherefore we have not only $h(0)=h(1)=0$ and $h'(1)=(1-q)\\,(\\mu-2)>0$, but also $h'(0)=(1-q)\\,g'(q)>(1-q)\\,g'(0)=(1-q)\\,p_1\\geq0$, where the strict inequality comes from $q>0$ and strict convexity of $g$. Thus the function $h$ has exactly one zero in $(0,1)$. That $(S_n)_{n\\in\\IN_0}$ in this case is transient with positive drift (and therefore $0<p<1$, cf.\\ Lemma \\ref{drift}) concludes the proof.\n\\end{proof}\n\nNote that for offspring distributions concentrated on $\\IN$ (i.e.\\ $p_0=0$, hence $q=0$), as already indicated above, the statement of Theorems \\ref{tv=i} and \\ref{i=0} coincide, since then almost surely $|T_v|=\\infty$ for all $v$ which renders this piece of information irrelevant.\n\n\\begin{example}[Poisson distribution]\nIf we consider $\\xi\\sim\\mathrm{Poi}(\\lambda)$ in this regime, the game is still non-trivial for $\\lambda>2$. Calculating $q$ according to Theorem \\ref{extinction} and solving \\eqref{fp3} numerically, we can determine $p$ and calculate $\\bar{p}=g(p)$, cf.\\ Theorem \\ref{arch}. The resulting probabilities are depicted below.\n\\begin{figure}[H]\n\t\\includegraphics[scale=1]{Poi2}\n\t\\caption{Winning probabilities for Breaker on a GW-tree arising from $\\xi\\sim\\mathrm{Poi}(\\lambda)$, where\n\t\tnode $v$ carries information $I(v)=|T_v|$. \\label{Poi2}}\n\\end{figure}\n\\end{example}\n\n\\section{Further observations and open problems}\nAs mentioned earlier, the phase transition (from a.s.\\ win for Breaker to non-degenerate outcome of the game)\nturned out to be discontinuous in the winning probabilities only in the case of complete information, i.e.\\ $I(v)=T_v$.\nFurther, the three information regimes considered seem to suggest that an increase in the level of information\nis (more) beneficial for Breaker (than for Maker), see Figure \\ref{Poicomp} below illustrating a comparison of winning probabilities for $\\xi\\sim\\mathrm{Poi}(\\lambda)$ in the different regimes. This might not come as a surprise, owing to the fact that the underlying graph is a tree which attributes a more local effect to moves of Maker and a more global one to the moves of Breaker.\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=1]{Poicompare}\n\t\\caption{Winning probabilities for Breaker on a GW-tree arising from $\\xi\\sim\\mathrm{Poi}(\\lambda)$ in all three considered information regimes compared.\\label{Poicomp}}\n\\end{figure}\n\nOf course, there are many more regimes to be considered: When $v$ becomes an internal/external vertex, one could reveal the subtree $T_v$ not completely (as in Section \\ref{sec:all}), but only down to a fixed level $k\\in\\IN$.\nAlternatively the node $v$ could carry only the number of offspring in a certain generation of $T_v$ as information, such as the number of its children and/or grand children for instance (neither disclosing the tree-structure of the first few levels, nor whether or not $T_v$ is finite -- if the corresponding numbers are bigger than $0$).\nSuch regimes, however, appear to be more complex in their analysis and will be left for future work -- together with \nthe question if there are information regimes other than $I(v)=T_v$, in which the phase transition in $p$ and $\\bar{p}$ is a discontinuous one.\n\nAnother possible generalization would be to break the information symmetry in the sense that what is disclosed to Maker resp.\\ Breaker when $v$ appears as internal/external node differs. This makes the progression of the game and potential strategies certainly more complex as the players in that case can extract further information and learn from the chosen moves of their opponent during the game.\n\nIt should be mentioned that in all three regimes considered here, information about the distribution of $\\xi$ is not relevant for the players. In other regimes, for instance if $I(v)$ is the parity of $\\xi_v$ and the offspring distribution of the type one-or-many (cf.\\ Example \\ref{oom}) with $n$ large and even, knowing the underlying distribution makes a big difference.\n\nIn fact, in regimes where $I(v)$ contains the number of children $\\xi_v$, the outcome of the game is insensitive to changes of $(p_0,p_1)$ as long as the sum $p_0+p_1$ is kept unchanged: There is no point for Maker in fixating the edge connecting a node $v$ with $\\xi_v=1$ to the root, as Breaker can immediately disconnect $T_v\\setminus{v}$. In such a regime, changing the distribution of $\\xi$ from one-or-many to ``none-or-many'', i.e.\\ $p_n=1-p_0=r$, will change\nthe probability $q$ for the tree to be finite, but neither $p$ nor $\\bar{p}$.\n\n\\vspace*{1em}\nThe question concerning possible cases in which $p=\\bar{p}$, touched upon at the end of Section \\ref{sec3}, can be settled in full generality without further ado:\n\n\\begin{proposition}\nIrrespectively of offspring distribution and information regime, it holds \n\\[p=\\bar{p}\\quad\\Longleftrightarrow\\quad p\\in\\{0,1\\}.\\]\n\\end{proposition}\n\\begin{proof}\nSince $0\\leq g(p)\\leq \\bar{p}\\leq p\\leq 1$ (cf.\\ Theorem \\ref{arch}), the easy direction ``$\\Leftarrow$'' follows directly from $g(1)=1$. As regards ``$\\Rightarrow$'', $p\\notin\\{0,1\\}$ implies $p_0<1$ and $p_0+p_1>0$ (otherwise either $p=1$ or $\\xi\\geq2$ a.s.\\ forcing $p=0$). It is not difficult to convince yourself that in both cases, $0<p_0<1$ and $p_1>0$ respectively, the event that $\\xi_\\mathbf{0}>0$ and at most one individual $v$ in the first generation of the tree has offspring, i.e. \\[A:=\\{Z_1>0,\\ Z_2=\\xi_v\\text{ for some $v$ in generation 1}\\},\\]\nhas positive probability. On such a tree, Breaker wins with probability 1 if starting (disconnecting $v$) and probability $p$ if Maker starts (fixating the edge to $v$). Therefore, it follows\n\\begin{align*}\\bar{p}&=\\Prob(\\text{B wins}\\,|\\,\\text{M starts}, A^\\text{c})\\cdot\\Prob(A^\\text{c})+ \\Prob(\\text{B wins}\\,|\\,\\text{M starts}, A)\\cdot\\Prob(A)\\\\\n\t&\\leq \\Prob(\\text{B wins}\\,|\\,\\text{B starts}, A^\\text{c})\\cdot\\Prob(A^\\text{c}) + p\\cdot\\Prob(A)\\\\\n\t&<\\Prob(\\text{B wins}\\,|\\,\\text{B starts}, A^\\text{c})\\cdot\\Prob(A^\\text{c}) + \\Prob(\\text{B wins}\\,|\\,\\text{B starts}, A)\\cdot\\Prob(A)\\\\\n\t&=p,\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\nLast but not least, in cases when Breaker wins, the question concerning the number of rounds necessary to exhaust the tree becomes interesting. Given optimal play, it directly relates to the size $|C(\\mathbf{0})|$ of the component containing the root that Maker is able to fixate before there are no more edges available for play. In information regimes like $I(v)=\\emptyset$ and $I(v)=|T_v|$, where the number of available visible edges can be regarded as a random walk, the time the game lasts (counted in rounds) is given by $\\tau_{\\leq0}$ the time until this walk exits $\\IN$.\n\n\n\\begin{thebibliography}{9}\n\n\t\\bibitem{Athreya-Ney}\n\t \t {\\sc K.B.\\ Athreya} and {\\sc P.E.\\ Ney} (1972):\n\t \t {\\em ``Branching processes''},\n\t \t Springer Berlin, Heidelberg.\n\t\\bibitem{Luczak}\n\t \t {\\sc M.\\ Bednarska} and {\\sc T.\\ \\L uczak} (2001):\n\t \t {Biased positional games and the phase transition},\n\t \t {\\em Random Structures \\& Algorithms},\n\t \t Vol.\\ 18 (2), pp.\\ 141-152.\n\t\\bibitem{RndGeo}\n\t \t {\\sc A.\\ Beveridge,} {\\sc A.\\ Dudek,} {\\sc A.\\ Frieze,} {\\sc T.\\ Mller} and {\\sc M.\\ Stojakovi\\'c} (2014):\n\t \t {Maker-Breaker Games on Random Geometric Graphs},\n\t \t {\\em Random Structures \\& Algorithms},\n\t \t Vol.\\ 45 (4), pp.\\ 553-607.\n \\bibitem{skipfree}\n\t\t{\\sc M.\\ Brown,} {\\sc E.A.\\ Pekz} and {\\sc S.M.\\ Ross} (2010):\n\t\t{Some results for skip-free random walk},\n\t\t{\\em Probability in the Engineering and Informational Sciences},\n\t\tVol.\\ 24 (4), pp.\\ 491-507.\n \\bibitem{erds}\n \t{\\sc V.\\ Chvtal} and {\\sc P.\\ Erd\\H{o}s} (1978):\n \t{Biased positional games}, \n \t{\\em Annals of Discrete Mathematics},\n \tVol.\\ 2, pp.\\ 221-229.\n \t\\bibitem{CKM}\n \t{\\sc D.\\ Clemens,} {\\sc L.\\ Kirsch} and {\\sc Y.\\ Mogge} (2021):\n \t{Connector-Breaker games on random boards},\n \t{\\em The Electronic Journal of Combinatorics},\n \tVol.\\ 28 (3), pp.\\ 1-33.\n \\bibitem{DFR1}\n \t\t {\\sc A.N.\\ Day} and {\\sc V.\\ Falgas-Ravry} (2021):\n\t\t {Maker-Breaker percolation games I: Crossing grids},\n\t\t {\\em Combinatorics, Probability and Computing},\n\t \t Vol.\\ 30, pp.\\ 200-227.\n \\bibitem{DFR2}\n\t\t{\\sc A.N.\\ Day} and {\\sc V.\\ Falgas-Ravry} (2021):\n\t\t{Maker-Breaker percolation games II: Escaping to infinity},\n\t\t{\\em Journal of Combinatorial Theory (Series B)},\n\t\tVol.\\ 151, pp.\\ 482-508.\n\t\\bibitem{Dek1}\n\t\t{\\sc F.M.\\ Dekking} (1991):\n\t\t{Branching processes that grow faster than binary splitting},\n\t\t{\\em The American Mathematical Monthly},\n\t\tVol.\\ 98 (8), pp.\\ 728-731.\n\t\\bibitem{Rndboards}\n\t\t{\\sc A.\\ Ferber,} {\\sc R.\\ Glebov,} {\\sc M.\\ Krivelevich} and {\\sc A.\\ Naor} (2014):\n\t\t{Biased games on random boards},\n\t \t{\\em Random Structures \\& Algorithms},\n\t\tVol.\\ 46 (4), pp.\\ 651-676.\n\t\\bibitem{Hefetz}\n\t\t{\\sc D.\\ Hefetz,}{\\sc M.\\ Krivelevich,} {\\sc M.\\ Stojakovi\\'c} and {\\sc T.\\ Szab} (2009):\n\t\t{Fast winning strategies in Maker-Breaker games},\n\t\t{\\em Journal of Combinatorial Theory (Series B)},\n\t\tVol.\\ 99, pp.\\ 39-47.\n \\bibitem{HM}\n\t {\\sc A.E.\\ Holroyd} and {\\sc J.B.\\ Martin} (2021):\n\t {Galton-Watson games},\n\t {\\em Random Structures \\& Algorithms},\n \t Vol.\\ 59 (4), pp.\\ 495-521.\n \t\\bibitem{rooted}\n \t {\\sc S.\\ Karmakar,} {\\sc M.\\ Podder,} {\\sc S.\\ Roy} and {\\sc S.\\ Sadhukhan} (2023+):\n \t {Phase transition in percolation games on rooted Galton-Watson trees},\\\\\n \t \\url{https://arxiv.org/abs/2303.11402}.\n \t \\bibitem{Shannon}\n \t {\\sc A.\\ Lehman} (1964):\n \t {A solution of the Shannon switching game},\n \t {\\em Journal of the Society for Industrial and Applied Mathematics},\n \t Vol.\\ 12 (4), pp.\\ 687-725.\n \\bibitem{LPW}\n \t\t {\\sc D.A.\\ Levin,} {\\sc Y.\\ Peres} and {\\sc E.L.\\ Wilmer} (2017):\n \t\t {\\em ``Markov chains and mixing times (2nd edition)''},\n \t\t American Mathematical Society.\n \t\\bibitem{London}\n \t\t {\\sc A.\\ London} and {\\sc A.\\ Pluhr} (2018):\n \t\t {Spanning tree game as Prim would have played},\n \t\t {\\em Acta Cybernetica},\n \t\t Vol.\\ 23, pp.\\ 921-927.\n \t\\bibitem{Lyons}\n \t\t {\\sc R.\\ Lyons} (1990):\n \t\t {Random walks and percolation on trees},\n \t\t {\\em The Annals of Probability},\n \t\t Vol.\\ 18 (3), pp.\\ 931-958.\n \t\\bibitem{Dek2}\n \t\t {\\sc A.G.\\ Pakes} and {\\sc F.M.\\ Dekking} (1991):\n \t\t {On family trees and subtrees of simple branching processes},\n \t\t {\\em Journal of Theoretical Probability},\n \t\t Vol.\\ 4 (2), pp.\\ 353-369.\n \t\\bibitem{StoSza}\n \t\t {\\sc M.\\ Stojakovi\\'c} and {\\sc T.\\ Szab} (2005):\n \t\t {Positional games on random graphs},\n \t\t {\\em Random Structures \\& Algorithms},\n \t\t Vol.\\ 26 (1-2), pp.\\ 204-223.\n \t\\bibitem{LCRW}\n \t\t {\\sc T.\\ Vilkas} (2024+):\n \t\t {Left-continuous random walk on $\\mathbb{Z}$ and the parity of its hitting times},\n \t\t submitted to {\\em Electronic Communications in Probability},\\\\\n \t\t \\url{https://arxiv.org/abs/2407.06903}.\n \\end{thebibliography}\n\n\n\n\n\\vspace{0.5cm}\\noindent\n\t{\\sc \\small \n\tTimo Vilkas\\\\\n\tStatistiska institutionen,\\\\\n\tEkonomihgskolan vid Lunds universitet,\\\\\n\t220\\,07 Lund, Sweden.}\\\\\n\ttimo.vilkas@stat.lu.se\\\\\n\n\n\n\\end{document}\n\n"}}},{"rowIdx":95,"cells":{"id":{"kind":"string","value":"2412.08400v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08400v1"},"title":{"kind":"string","value":"Combinatorial Characterization of Exponential Families of Lumpable Stochastic Matrices"},"full_text":{"kind":"string","value":"\\documentclass[a4paper,11pt,oneside]{article}\n\\pdfoutput=1\n\\RequirePackage{amsmath, amsthm, amssymb, mathtools, mathrsfs}\n\\RequirePackage{natbib}\n\\RequirePackage{tikz,tikz-cd}\n\\RequirePackage{tabularray}\n\\RequirePackage{algorithm2e}\n\\RequirePackage{nicematrix}\n\\NiceMatrixOptions{\ncode-for-first-row = \\tiny ,\ncode-for-last-row = \\tiny ,\ncode-for-first-col = \\tiny ,\ncode-for-last-col = 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\nCombinatorial Characterization of \\\\ Exponential Families of Lumpable Stochastic Matrices}\n\\begin{document}\n\n\\author[1]{Shun Watanabe \\thanks{email: shunwata@cc.tuat.ac.jp.}}\n\\author[2]{Geoffrey Wolfer \\thanks{email: geo.wolfer@aoni.waseda.jp. \n}}\n\\affil[1]{Tokyo University of Agriculture and Technology, Tokyo}\n\\affil[2]{Waseda University, Center for Data Science, Tokyo}\n\n\\date{\\today}\n\n\\maketitle\n\n\n\\begin{abstract}\n It is known that the set of lumpable Markov chains over a finite state space, with respect to a fixed lumping function, generally does not form an exponential family of stochastic matrices. In this work, we explore efficiently verifiable necessary and sufficient combinatorial conditions for families of lumpable transition matrices to form exponential families.\n\\end{abstract}\n\n\\keywords{Information geometry; irreducible Markov chains; lumpability; exponential families}\n\n\\tableofcontents\n\n\\clearpage\n\n\\section{Introduction}\n\\label{section:introduction}\nExponential families (e-families) of distributions are of established importance in statistics due to their distinctive properties for inference problems. For instance, they uniquely provide sufficient statistics capable of condensing any amount of independent and identically distributed data into a fixed number of values \\citep{pitman1936sufficient, koopman1936distributions, darmois1935lois}. What is more, it is known that the maximum likelihood estimator achieves the Cram\\'{e}r-Rao lower bound only\\footnote{Note that this fact only holds when imposing additional regularity conditions on the family.} when the family of distributions forms an exponential family \\citep{wijsman1973attainment,joshi1976attainment,fabian1977cramer, muller1989attainment}.\nIn the language of information geometry \\citep{amari2007methods}, \npositive probability distributions are endowed with the structure of a smooth manifold with a pair of dual affine connections---the e-connection and m-connection---and statistical models are regarded as submanifolds. \nIn this framework, being an e-family geometrically corresponds to being autoparallel with respect to the e-connection.\nFurthermore, deviation from being an e-family---and the subsequent breakdown of the statistical properties---can be measured in terms of curvature; this characterizes second order efficiency of estimators \\citep{efron1975defining}.\nRecently, e-families have also been put under the spotlight in optimization since they allow for efficient natural-gradient computation \\citep{amari1998natural}, which finds application in machine learning.\n\nIt is possible to similarly construct a dually flat geometry on the space of irreducible stochastic matrices \\citep{nagaoka2005exponential} defined over a fixed strongly connected transition digraph. Independently and identically distributed (iid) processes, which can be regarded as memoryless Markov chains, are known to form an e-family in the larger family of irreducible stochastic matrices \\citep{ito1988}.\nThe Markovian framework is consistent with the divergence rate of the corresponding Markov processes, and\ninformation projections \\citep{boza1971asymptotically, csiszar1987conditional}\nwhich arise naturally from the study of large deviations \\citep{moulos2019optimal} and hypothesis testing \\citep{nakagawa1993converse, watanabe2017}.\nIn this regard the Markovian framework strictly encompasses the vanilla framework for distributions, while accommodating for processes with time dependencies.\n\nHowever, Markov processes\nexhibit a significantly richer structure than their iid counterparts, with numerous properties---such as irreducibility, aperiodicity or time-reversibility--- that are not pertinent to iid processes but are well-established for Markov chains. A recently initiated research program seeks to analyze how \nMarkov-centric properties translate into geometric features of the corresponding families of\nstochastic matrices.\nFor instance, a Markov chain having a uniform stationary distribution is equivalent to being represented by a doubly stochastic transition matrix; it is well-established that the set of doubly stochastic matrices forms a mixture family \\citep{hayashi2014information}.\nSimilarly, verifying the detailed-balance equation---indicating the time-reversibility of the stochastic process---means that the transition matrix is self-adjoint in a certain Hilbert space; the set of reversible stochastic matrices is known to form both a mixture family and an exponential family \\citep{wolfer2021information}.\nFor context trees, it is known that a tree model forms an e-family if and only if it is an FSMX model\n\\citep{takeuchi2007exponential, takeuchi2017information}.\nMore recently, \\citet{wolfer2024geometric} began analyzing lumpability of Markov chains \\citep{kemeny1983finite}.\nLumpable Markov chains allow for the reduction of the state space by merging symbols without losing Markovianity, making them highly practical.\nSpecifically, they showed that although the lumpable set with respect to a fixed lumping map typically forms neither a mixture family nor an e-family, it is still possible to endow the family with the structure of a mutually dual foliated manifold, leading to a mixed coordinate system \\citep[Chapter~3.7]{amari2007methods}.\nTheir construction is centered around the concept of a Markov embedding, defined as a right inverse of the lumping operation, and which is argued to serve a similar role to \\v{C}encov's statistical morphisms \\citep{cencov1983statistical} in the context of Markov chains.\n\nThe problem of selecting a good statistical model involves choosing one that enjoys favorable analytical properties. In this regard, both e-families and lumpable families are highly sought-after models, and practitioners may be interested in enjoying the best of both worlds.\nHowever, as previously mentioned, lumpable families do not generally form e-families. Indeed, they may or may not be e-families depending on their connection graph and the lumping map. This phenomenon contrasts with many previously analyzed classes; for instance, the set of reversible stochastic matrices, which forms an e-family for any symmetric connection graph.\nIn this paper, we initiate the problem of characterizing the conditions under which lumpable stochastic matrices do form e-families.\nSince Markov embeddings\ndemonstrably preserve e-families of stochastic matrices, they naturally generate one class of lumpable e-families.\nHowever, this approach proves to be quite restrictive.\nPerhaps surprisingly, it is possible to construct families that are not directly derived from the embedding of an e-family.\nIn this work we explore some necessary and sufficient conditions for the lumpable set to form an e-family.\n\n\\paragraph{Major contributions---}\n\nWe summarize our main results below.\n\n\\paragraph{Necessary and sufficient criteria with multi-row merging blocks.}\n\nWe obtain sufficient and necessary conditions on the lumpable family $\\calW_{\\kappa}(\\calY, \\calE)$ for being an e-family in terms of so-called multi-row merging blocks (refer to Definition~\\ref{definition:merging-block}). \n Namely, a sufficient condition (Corollary~\\ref{corollary:no-multi-row-merging-block-is-sufficient}) for $\\calW_\\kappa(\\calY, \\calE)$ to be an e-family is that it exhibits no multi-row merging block, while if it exhibits a multi-row merging block which is redundant (Definition~\\ref{definition:redundant-block}), this precludes the lumpable family from being exponential (Theorem~\\ref{theorem:redundant-merging-block-criterion}). However, neither of the above conditions fully characterizes the property of being an e-family.\n We also provide an alternative sufficient criteria (Proposition~\\ref{proposition:lazy-cycle-criterion}), which shows in particular that there could be an arbitrarily large number of multi-row merging blocks while still yielding an e-family.\n\n\\paragraph{Dimensional criterion.}\n\nWe show that when the lumpable family is exponential, the log-affine hull of lumpable functions modulo anti-shift functions\\footnote{The term modulo here is understood in the context of direct sums and quotient spaces.} (refer to Definition~\\ref{definition:anti-shift-functions}) has a well-understood dimension (refer to Theorem~\\ref{theorem:dimensional-criterion}).\nAs a consequence, a mismatch in dimension necessarily implies that the family is not exponential, and this can be verified using a polynomial-time algorithm.\nWe also specialize the above result into Corollary~\\ref{corollary:dimensional-criterion-simplified} to obtain a necessary condition purely based on elementary combinatorial properties of the connection graph and the lumping map.\n\n\\paragraph{Monotonicity and stability.}\n\nWe examine the property of e-families through basic operations on the edge set $\\calE$.\nIn particular, we exhibit a monotonicity property of e-families of lumpable stochastic matrices. Namely, \nwe show in Theorem~\\ref{theorem:monotonicity} that it is generally the case that when $\\calE \\subset \\calE'$\nif $\\calW_\\kappa(\\calY, \\calE')$ forms an e-family, then $\\calW_\\kappa(\\calY, \\calE)$ also forms an e-family.\nAdditionally, we exhibit an operation on loops under which the property of being an e-family is stable (refer to Proposition~\\ref{proposition:stability-diagonal-modification}).\n\n\n\\section{Preliminaries}\n\\label{section:preliminaries}\n\\subsection{Notation}\n\nWe let $(\\calY, \\calE)$ be a directed graph (digraph) with finite vertex set $\\calY$ and edge set $\\calE \\subset \\calY^2$. We assume that\n$(\\calY, \\calE)$ is strongly connected, that is every vertex is reachable from every other vertex by traversing edges in their proper direction.\nFor $\\{Y_t\\}_{t \\in \\bbN}$ a time-homogeneous discrete time Markov chain (DTMC) over the space space $\\calY$, we collect the transition probabilities into a row-stochastic matrix $P$. In other words, we write\\footnote{\nOur notation follows\nthe applied probability literature.\nIn the information theory literature, $P(y'|y)$ is sometimes used in lieu of $P(y, y')$.}\n$$P(y,y') = \\PR{Y_{t+1} = y' | Y_{t} = y}.$$\nWhen $P(y,y') > 0$ iff $(y,y') \\in \\calE$, we say that $(\\calY, \\calE)$ is a connection graph for $P$.\nWe denote $\\calW(\\calY, \\calE)$ the set of all irreducible row-stochastic matrices pertaining to the connection graph $(\\calY, \\calE)$.\nWe additionally define $\\calF(\\calY, \\calE) = \\bbR^{\\calE}$ the set of all real functions on the set of edges and \n$\\calF_+(\\calY, \\calE)$ its positive subset. As it allows us to conveniently write a function $F \\in \\calF(\\calY, \\calE)$ in the form of a square matrix, we will routinely identify\n\\begin{equation*}\n\\begin{split}\n \\calF(\\calY, \\calE) &\\cong \\set{ F \\in \\bbR^{\\calY^2} \\colon \\forall (y,y') \\not \\in \\calE \\implies F(y,y') = 0 }, \\\\\n \\calF_+(\\calY, \\calE) &\\cong \\set{ F \\in \\calF(\\calY, \\calE) \\colon \\forall (y,y') \\in \\calE \\implies F(y,y') > 0 }. \\\\\n\\end{split}\n\\end{equation*}\nThe Hadamard product of $A$ and $B$ in $\\calF(\\calY, \\calE)$ is denoted $A \\odot B$ and for $t \\in \\bbR$, $A^{\\odot t}$ is defined as the function such that for any $y,y' \\in \\calE$, $A^{\\odot t}(y,y') = A(y,y')^t$.\nWe overload $\\exp$ and $\\log$ as follows,\n\\begin{equation*}\n\\begin{split}\n \\exp \\colon \\calF(\\calY, \\calE) &\\to \\calF_+(\\calY, \\calE), \\\\\n \\log\\colon \\calF_+(\\calY, \\calE) &\\to \\calF(\\calY, \\calE),\n\\end{split}\n\\end{equation*}\nwhere for any $F \\in \\calF(\\calY, \\calE)$ and $(y,y') \\in \\calE$, $\\exp(F)(y,y') = \\exp(F(y,y'))$, and for any $F \\in \\calF_+(\\calY, \\calE$ and $(y,y') \\in \\calE$, $\\log(F)(y,y') = \\log(F(y,y'))$.\n\n\\subsection{Lumpability {\\citep{kemeny1983finite}}}\n\\label{section:lumpability}\n\nOne classical operation on Markov processes is lumping, which means merging symbols together and recording the observations on the reduced space. It is well known that this operation typically disrupts the Markov property \\citep{burke1958markovian, rogers1981markov}. Chains for which the Markov property is preserved are called lumpable. More formally, for a surjective symbol merging map $\\kappa \\colon \\calY \\to \\calX$, we say that the Markov chain\n $\\{Y\\}_{t \\in \\bbN}$ with transition matrix\n $P \\in \\calW(\\calY, \\calE)$ is $\\kappa$-lumpable whenever the stochastic process $\\{\\kappa(Y_t)\\}_{t \\in \\bbN}$ also forms a DTMC with transition matrix $P^\\flat \\in \\calW(\\calX, \\calD)$, where\nthe set \n\\begin{equation*}\n \\calD \\eqdef \\kappa(\\calE) \\eqdef \\set{ (\\kappa(y), \\kappa(y')) \\colon (y,y') \\in \\calE } \\subset \\calX^2\n\\end{equation*}\nis called the lumped edge set.\nDenoting $\\calW_\\kappa(\\calY, \\calE)$ the $\\kappa$-lumpable subset of $\\calW(\\calY, \\calE)$,\nobserve that $\\kappa$ induces \na push-forward $\\kappa_\\star$ on stochastic matrices,\n\\begin{equation*}\n\\begin{split}\n \\kappa_\\star \\colon \\calW_\\kappa(\\calY, \\calE) \\to \\calW(\\calX, \\calD)\n\\end{split}\n\\end{equation*}\nas well as a partition of the space $\\calY$, which we denote by\n\\begin{equation*}\n \\calY = \\biguplus_{x \\in \\calX} \\calS_x,\n\\end{equation*}\nwhere for any $x \\in \\calX$, we wrote $\\calS_x \\eqdef \\kappa^{-1}(x)$.\nThe following characterization of $\\calW_\\kappa(\\calY, \\calE)$ was provided by \\citet{kemeny1983finite}. It holds that $P \\in \\calW_{\\kappa}(\\calY, \\calE)$\nif and only if for any $(x,x') \\in \\calD$ and any $y_1, y_2 \\in \\calS_{x}$,\n\\begin{equation*}\n \\sum_{y' \\in \\calS_{x'}} P(y_1,y') = \\sum_{y' \\in \\calS_{x'}} P(y_2,y').\n\\end{equation*}\nWe will use the notation of \\citet{levin2009markov}\\footnote{We note that in differential geometry $\\sharp$ and $\\flat$ commonly denote the musical isomorphism. However, in this paper, we use these symbols differently.} to often---albeit not always---disambiguate objects which pertain to the larger space using the superscript $\\sharp$ and which pertain to the reduced space using the superscript $\\flat$.\nTwo lumpable families of stochastic matrices are said to be equivalent if they coincide upon relabeling of the state space and lumped state space.\nNamely, for $\\kappa_1 \\colon \\calY_1 \\to \\calX_1, \\kappa_2 \\colon \\calY_2 \\to \\calX_2$ two lumping maps, the lumpable families $\\calW_{\\kappa_1}(\\calY_1, \\calE_1)$ and $\\calW_{\\kappa_2}(\\calY_2, \\calE_2)$ are equivalent, which we denote\n\\begin{equation*}\n \\calW_{\\kappa_1}(\\calY_1, \\calE_1) \\cong \\calW_{\\kappa_2}(\\calY_2, \\calE_2),\n\\end{equation*}\nwhenever there exist two bijections $\\phi^\\sharp \\colon \\calY_1 \\to \\calY_2$ and $\\phi^\\flat \\colon \\calX_1 \\to \\calX_2$ such that\n\\begin{equation*}\n\\begin{split}\n\\forall (y,y') \\in \\calY^2, \n (\\phi^\\sharp(y),\\phi^\\sharp(y')) \\in \\calE_2 &\\Longleftrightarrow (y,y') \\in \\calE_1, \\\\\n \\forall (x,x') \\in \\calX^2, \n (\\phi^\\flat(x),\\phi^\\flat(x')) \\in \\calD_2 = \\kappa(\\calE_2), &\\Longleftrightarrow (x,x') \\in \\calD_1 = \\kappa(\\calE_1), \\\\\n \\forall y \\in \\calY_1, \\phi^\\flat(\\kappa_1(y)) &= \\kappa_2(\\phi^\\sharp (y)).\n\\end{split}\n\\end{equation*}\n\n\\subsection{Exponential families of stochastic matrices {\\citep{nagaoka2005exponential}}}\n\n\\begin{definition}[$\\stoch$-normalization]\n\\label{definition:s-normalization}\nWhen $(\\calY, \\calE)$ is strongly connected we define the mapping\n\\begin{equation*}\n\\begin{split}\n \\stoch \\colon \\calF_+(\\calY,\\calE) &\\to \\calW(\\calY,\\calE) \\\\\n F &\\mapsto P \\colon \\calE \\to \\bbR_+, (y,y') \\mapsto P(y,y') = \\frac{F(y,y') v_F(y')}{\\rho_F v_F(y)},\n\\end{split}\n\\end{equation*} \nwhere $\\rho_F$ and $v_F$ are respectively the Perron--Frobenius (PF) root and associated right eigenvector of $F$. Henceforth, $(\\rho_F, v_F)$ will be called the right PF eigen-pair of $F$.\n\\end{definition}\n\nThe above-defined $\\stoch$-normalization plays the role of the partition function in the distribution setting, in order to normalize an arbitrary non-negative irreducible matrix into a stochastic matrix \\citep{miller1961convexity}.\n\n\\begin{definition}[Anti-shift functions {\\citep[Section~3]{nagaoka2005exponential}}]\n\\label{definition:anti-shift-functions}\nIt will be convenient to define\n\\begin{equation*}\n\\begin{split}\n\\calN(\\calY, \\calE) \\eqdef \\bigg\\{ &N \\in \\calF(\\calY, \\calE) \\colon \\exists (c, f) \\in (\\bbR, \\bbR^\\calY), \\forall (y, y') \\in \\calE, N(y, y') = f(y') - f(y) + c \\bigg\\}.\n\\end{split}\n\\end{equation*}\nObserve that \n$\\mathcal{N}(\\calY, \\calE)$ forms a $\\abs{\\calY}$-dimensional vector subspace.\n\\end{definition}\n\n\\begin{definition}[e-family of stochastic matrices {\\citep{nagaoka2005exponential}}]\n\\label{definition:e-family}\n We say that the parametric family of irreducible stochastic matrices $$\\calV_e = \\set{P_\\theta \\colon \\theta = (\\theta^1, \\dots, \\theta^d) \\in \\bbR^d} \\subset \\calW( \\calY, \\calE),$$ \nis an exponential family (e-family) of stochastic matrices with natural parameter $\\theta$ and dimension $d$, when\nthere exist a function $K \\in \\calF(\\calY, \\calE)$ and $d$ linearly independent functions $G_1, \\dots, G_d \\in \\calG(\\calY, \\calE)$, \nsuch that\n \\begin{equation*}\n P_\\theta = \\stoch \\circ \\exp \\left(K + \\sum_{i = 1}^{d} \\theta^i G_i\\right),\n \\end{equation*}\n where $\\mathcal{G}(\\calY, \\calE)$ is the quotient space\n\\begin{equation*}\n \\calG(\\calY, \\calE) \\eqdef \\calF(\\calY, \\calE)/\\calN(\\calY, \\calE),\n\\end{equation*}\nwith $\\calN(\\calY, \\calE)$ is introduced in Definition~\\ref{definition:anti-shift-functions}\n and $\\stoch$-normalization follows from Definition~\\ref{definition:s-normalization}.\n\\end{definition}\n\nIn other words, there exists a one-to-one correspondence between linear subspaces of $\\calG(\\calY, \\calE)$ and e-families \\citep[Theorem~2]{nagaoka2005exponential} through the diffeomorphism\n\\begin{equation*}\n\\begin{split}\n \\stoch \\circ \\exp \\colon \\calG(\\calY, \\calE) &\\to \\calW(\\calY, \\calE).\n \\end{split}\n\\end{equation*}\n\nSimilarly, a mixture family (m-family) of stochastic matrices is induced from the affine hull of a collection of irreducible edge measures \\citep{nagaoka2005exponential} (refer also to \\citet[Section~4.2]{ hayashi2014information}).\nAn m-family which is also an e-family is called an em-family.\nFor instance, the set of all irreducible stochastic matrices $\\calW(\\calY, \\calE)$ \nis known to form an em-family \\citep{nagaoka2005exponential}.\nWhat is more, the reversible subset is also an em-family \\citep{wolfer2021information} while the subset of bistochastic matrices forms an m-family but does not form an e-family \\citep{hayashi2014information}.\n\n\\subsection{Foliation on the $\\kappa$-lumpable family.}\n\nAlthough the lumpable family $\\calW_{\\kappa}(\\calY, \\calE)$ was shown by \\citet{wolfer2024geometric} to generally not form an m-family or an e-family of stochastic matrices, it is always possible to decompose it in terms of simpler mathematical structures, called a foliation\\footnote{A foliation is a decomposition of a manifold into a union of connected but disjoint submanifolds, called leaves, all sharing the same dimension \\citep[Chapter~19]{lee2013smooth}.}.\nThis decomposition is facilitated by the notion of a Markov embedding \\citep[Definition~4.3]{wolfer2024geometric}, which corresponds to a right inverse of the lumping operation and satisfies additional natural structural constraints. As such, Markov embeddings are the counterparts of the statistical morphisms axiomatized by \\citet{cencov1983statistical} in the context of stochastic matrices.\nIn particular, any $P \\in \\calW_{\\kappa}(\\calY, \\calE)$ induces a canonical embedding \\citep[Lemma~4.8]{wolfer2024geometric} denoted $\\Lambda_\\star^{(P)} \\colon \\calW(\\calX, \\calD) \\to \\calW_{\\kappa}(\\calY, \\calE)$ satisfying $P = \\Lambda_\\star^{(P)} \\kappa_\\star P$.\nIt was established that for any $P^\\sharp_0 \\in \\calW(\\calY, \\calE)$, the embedding of the family $\\calW(\\calX, \\calD)$ by $\\Lambda_\\star^{(P^\\sharp_0)}$\n\\begin{equation*}\n \\calJ(P^\\sharp_{0}) \\eqdef \\set{ \\Lambda_\\star^{(P^\\sharp_{0})}P \\colon P \\in \\calW(\\calX, \\calD) },\n\\end{equation*}\nforms an e-family of stochastic matrices.\nAdditionally, for any $P_0^\\flat \\in \\calW(\\calX, \\calD)$, stochastic matrices lumping into $P^{\\flat}_{0}$,\n\\begin{equation*}\n \\calL(P^{\\flat}_{0}) \\eqdef \\set{ P \\in \\calW_\\kappa(\\calY, \\calE) \\colon \\kappa_\\star P = P^{\\flat}_{0}},\n\\end{equation*}\nform an m-family, and\nthe manifold $\\calW_{\\kappa}(\\calY, \\calE)$ can be endowed with the structure of an e-foliation.\n\n\\begin{theorem}[Foliation on $\\calW_{\\kappa}(\\calY, \\calE)$ {\\citep[Theorem~6.4]{wolfer2024geometric}}]\n\\label{theorem:foliation-of-lumpable-kernels}\nFor any fixed $P^\\flat_0 \\in \\calW(\\calX, \\calD)$,\n\\begin{equation*}\n \\calW_\\kappa(\\calY, \\calE) = \\biguplus_{P \\in \\calL(P^\\flat_0)} \\calJ(P),\n\\end{equation*}\n\\begin{equation*}\n \\dim \\calW_\\kappa(\\calY, \\calE) = \\abs{\\calE} - \\sum_{(x,x') \\in \\calD} \\abs{\\calS_x} + \\abs{\\calD} - \\abs{\\calX}.\n\\end{equation*}\n\\end{theorem}\n\nMutually dual foliations and mixed coordinate systems play a significant role in information geometry \\citep[Section~3.7]{amari2007methods}.\n\n\\paragraph{Problem statement---}\nOur goal is to obtain a full characterization of exponential families of lumpable stochastic matrices in terms of combinatorial properties of the connection graph $(\\calY, \\calE)$ and the lumping map $\\kappa$. Ideally, we wish to develop necessary and sufficient conditions which are all verifiable in polynomial time.\n\n\\paragraph{Motivation---}\n\nPrevious studies of the geometric structure of well-known families of stochastic matrices---such as reversible, bistochastic or memoryless---have mostly established geometric structure that are valid for general edge sets.\nThis is in stark contrast to lumpable stochastic matrices, where---perhaps surprisingly---the nature of the family seems to depend intricately on the structure of the edge set and its interplay with the lumping map.\nAnother exception is found in\n\\citep{takeuchi2007exponential, takeuchi2017information},\nwhere the e-family nature of the context tree\ndepends on some additional structural properties of the tree, which partially motivated our question.\n\nIn addition, exponential families of stochastic matrices enjoy distinctive properties that may offer analytical power to the practitioner.\nThe asymptotic variance of a function $G \\in \\bbR^\\calE$ with respect to some irreducible stochastic matrix $K$ can be expressed using the second derivative of the potential function of the one-parameter exponential family (Definition~\\ref{definition:e-family}) anchored at $K$ and tilted by $G$ \\citep{hayashi2014information}.\nFurthermore, when $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family, Bregman geometry yields a Pythagorean identity \\citep{hayashi2014information}.\nSpecifically, for any $P \\in \\calW(\\calY, \\calE)$ and $\\overline{P} \\in \\calW_\\kappa(\\calY, \\calE)$,\n\\begin{equation*}\n \\kl{P}{\\overline{P}} = \\kl{P}{P_m} + \\kl{P_m}{\\overline{P}}\n\\end{equation*}\nwhere \n\\begin{equation*}\n P_m \\eqdef \\argmin_{\\widetilde{P} \\in \\calW_{\\kappa}(\\calY, \\calE)} \\kl{P}{\\widetilde{P}}\n\\end{equation*}\nis the unique m-projection (reverse information projection) of $P$ onto $\\calW_{\\kappa}(\\calY, \\calE)$.\n\n\\paragraph{First approach---} As Markov embeddings are known to be e-geodesic affine \\citep[Theorem~10]{wolfer2024geometric}, an immediate sufficient condition for a family $\\calW_\\kappa(\\calY, \\calE)$ to be an e-family is to find the existence of an embedding $\\Lambda_\\star$ satisfying $\\calW_\\kappa(\\calY, \\calE) = \\Lambda_\\star \\calW(\\calX, \\calD)$. This corresponds to restricting the foliation of Theorem~\\ref{theorem:foliation-of-lumpable-kernels} to a single e-leaf $\\calJ$. As we will see in this paper, this condition is quite restrictive; there exist many more exponential families.\n\n\\section{The lumpable cone}\n\\label{section:lumpable-cone}\nIn this section, we assume that $(\\calY, \\calE)$ is strongly connected and that $\\kappa \\colon \\calY \\to \\calX$ is a surjective lumping function.\nSimilar to \\citet{wolfer2024geometric}, we define the set of lumpable functions as follows,\n\\begin{equation*}\n \\calF_\\kappa(\\calY, \\calE) \\eqdef \\set{ F \\in \\calF(\\calY, \\calE) \\colon \\forall (x,x') \\in \\calD, \\forall y_1, y_2 \\in \\calS_{x}, \\sum_{y' \\in \\calS_{x'}} P(y_1,y') = \\sum_{y' \\in \\calS_{x'}} P(y_2,y') }.\n\\end{equation*}\nThe positive subset $\\calF^{+}(\\calY, \\calE)$ of $\\calF(\\calY, \\calE)$, forms a blunt\\footnote{A convex cone is called blunt if it does not contain the null vector.} convex cone, while $\\calF^{+}_\\kappa(\\calY, \\calE)$ is a subcone of $\\calF^{+}(\\calY, \\calE)$, as depicted on Figure~\\ref{figure:cone-section}.\n\n\\begin{proposition}[Commutativity]\n\\label{proposition:stochastic-rescaling-and-lumping-commute}\n$\\stoch$-normalization preserves $\\kappa$-lumpability and \nthe following diagram commutes\n\\[ \\begin{tikzcd}\n \\calF_\\kappa^+(\\calY, \\calE) \\arrow{r}{\\kappa_\\star} \\arrow[swap]{d}{\\stoch} & \\calF^+(\\calX, \\calD) \\arrow{d}{\\stoch} \\\\\\calW_\\kappa(\\calY, \\calE) \\arrow{r}{\\kappa_\\star}& \\calW(\\calX, \\calD).\n\\end{tikzcd}\n\\]\n\\end{proposition}\n\n\\begin{proof}\n\nLet $F \\in \\calF_\\kappa^{+}(\\calY, \\calE)$. Since $(\\calY, \\calE)$ is strongly connected, $\\stoch$-normalization is well-defined over $F \\in \\calF_\\kappa^{+}(\\calY, \\calE)$ and we first show that $\\stoch(F) \\in \\calW_\\kappa(\\calY, \\calE)$.\nBy construction, $\\stoch(F)$ is irreducible and shares the same support as $F$. It remains to verify lumpability.\nWe let $F^\\flat = \\kappa_\\star F$ denote the lumping of $F$, and $\\rho^\\flat$ and $v^\\flat$ be respectively the Perron--Frobenius root and associated right eigenvector ---henceforth called \"right PF eigen-pair\"--- of $F^\\flat$.\nFor any $y \\in \\calY$, a classical argument of \\citet{barr1977eigenvector} (see also \\citet[Lemma~12.9]{levin2009markov}) yields that\n\\begin{equation*}\n \\begin{split}\n \\sum_{y' \\in \\calY} F(y,y') v^\\flat(\\kappa(y')) &= \\sum_{x' \\in \\calX} \\sum_{y' \\in \\calS_{x'}} F(y,y') v^{\\flat}(\\kappa(y')) = \\sum_{x' \\in \\calX} \\left(\\sum_{y' \\in \\calS_{x'}} F(y,y') \\right) v^{\\flat}(x') \\\\\n &= \\sum_{x' \\in \\calX} F^\\flat(\\kappa(y),x') v^{\\flat}(x') = \\rho^\\flat v^{\\flat}(\\kappa(y)), \\\\\n \\end{split}\n\\end{equation*}\nwhich implies that $(\\rho^\\flat, v^\\flat \\circ \\kappa)$ is the right PF eigen-pair of $F$.\nAs a consequence, for any $(y,y') \\in \\calY^2$,\n\\begin{equation*}\n \\stoch(F)(y,y') = F(y,y') \\frac{v^\\flat(\\kappa(y'))}{\\rho^\\flat v^\\flat(\\kappa(y))}.\n\\end{equation*}\nFor any $x \\in \\calX$ and any $y \\in \\calS_x$, it holds that\n\\begin{equation*}\n\\begin{split}\n \\sum_{y' \\in \\calS_{x'}}\\stoch(F)(y,y') &= \\sum_{y' \\in \\calS_{x'}} F(y,y') \\frac{v^\\flat(\\kappa(y'))}{\\rho^\\flat v^\\flat(\\kappa(y))}\n = \\sum_{y' \\in \\calS_{x'}} F(y,y') \\frac{v^\\flat(x')}{\\rho^\\flat v^\\flat(x)} = F^\\flat(x,x') \\frac{v^\\flat(x')}{\\rho^\\flat v^\\flat(x)},\n\\end{split}\n\\end{equation*}\nwhich does not depend on $y \\in \\calS_x$. As a result, $\\stoch(F)$ is $\\kappa$-lumpable, and $\\kappa_\\star \\stoch(F) = \\stoch(F^\\flat)$. \n\\end{proof}\n\n\\begin{figure}[H]\n\\begin{center}\n\n\n\\tikzset{every picture/.style={line width=0.75pt}} \n\\begin{tikzpicture}[x=0.52pt,y=0.52pt,yscale=-1,xscale=1]\n\n\\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (198,77) .. controls (198,59.33) and (248.37,45) .. (310.5,45) .. controls (372.63,45) and (423,59.33) .. (423,77) .. controls (423,94.67) and (372.63,109) .. (310.5,109) .. controls (248.37,109) and (198,94.67) .. (198,77) -- cycle ;\n\\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (198,79) -- (218.2,113.79) -- (310,281) ;\n\\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] (310,281) -- (422,81) ;\n\\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ] (269,204) .. controls (269,194.06) and (287.58,186) .. (310.5,186) .. controls (333.42,186) and (352,194.06) .. (352,204) .. controls (352,213.94) and (333.42,222) .. (310.5,222) .. controls (287.58,222) and (269,213.94) .. (269,204) -- cycle ;\n\\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] [dash pattern={on 0.84pt off 2.51pt}] (160,9) -- (208.1,96.4) ;\n\\draw [color={rgb, 255:red, 208; green, 2; blue, 27 } ,draw opacity=1 ] [dash pattern={on 0.84pt off 2.51pt}] (422,81) -- (461,11) ;\n\\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][dash pattern={on 4.5pt off 4.5pt}] (212.3,162.5) -- (556,162.5) -- (408.7,245.5) -- (65,245.5) -- cycle ;\n\\draw [dash pattern={on 0.84pt off 2.51pt}] (250,9) -- (310,281) ;\n\\draw [fill={rgb, 255:red, 255; green, 255; blue, 255 } ,fill opacity=1 ] (267,71) .. controls (267,69.34) and (265.66,68) .. (264,68) .. controls (262.34,68) and (261,69.34) .. (261,71) .. controls (261,72.66) and (262.34,74) .. (264,74) .. controls (265.66,74) and (267,72.66) .. (267,71) -- cycle ;\n\\draw [fill={rgb, 255:red, 255; green, 255; blue, 255 } ,fill opacity=1 ] (296,204) .. controls (296,202.34) and (294.66,201) .. (293,201) .. controls (291.34,201) and (290,202.34) .. (290,204) .. controls (290,205.66) and (291.34,207) .. (293,207) .. controls (294.66,207) and (296,205.66) .. (296,204) -- cycle ;\n\\draw [fill={rgb, 255:red, 255; green, 255; blue, 255 } ,fill opacity=1 ] (285,153) .. controls (285,151.34) and (283.66,150) .. (282,150) .. controls (280.34,150) and (279,151.34) .. (279,153) .. controls (279,154.66) and (280.34,156) .. (282,156) .. controls (283.66,156) and (285,154.66) .. (285,153) -- cycle ;\n\n\\draw (418,110) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 208; green, 2; blue, 27 } ,opacity=1 ] [align=left] {$\\calF^+_\\kappa(\\calY, \\calE)$};\n\\draw (540,170) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] [align=left] {$\\calW(\\calY, \\calE)$};\n\\draw (352,202) node [anchor=north west][inner sep=0.75pt] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,opacity=1 ] [align=left] {$\\calW_\\kappa(\\calY, \\calE)$};\n\\draw (260,142) node [anchor=north west][inner sep=0.75pt] [align=left] {$F$};\n\\draw (300,192) node [anchor=north west][inner sep=0.75pt] [align=left] {$\\stoch(F)$};\n\\draw (216,3) node [anchor=north west][inner sep=0.75pt] [align=left] {$[F]$};\n\\draw (266,63) node [anchor=north west][inner sep=0.75pt] [align=left] {$G: \\stoch(G) = \\stoch(F)$};\n\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{We can regard $\\calW_\\kappa(\\calY, \\calE)$ as a section of the cone $\\calF^+_\\kappa(\\calY, \\calE)$.}\n\\label{figure:cone-section}\n\\end{figure}\n\n\n\n\n\nFor $F \\in \\calF^+(\\calY, \\calE)$ the equivalence class\n\\begin{equation*}\n [F] \\eqdef \\set{ G \\in \\calF^{+}(\\calY, \\calE) \\colon \\stoch(G) = \\stoch(F) },\n\\end{equation*}\ncan be parametrized by the $\\abs{\\calY}$-dimensional positive orthant,\n\\begin{equation*}\n [F] = \\set{ \\rho \\diag(v) F \\diag(v)^{-1} \\colon \\rho \\in \\bbR_+, v \\in \\bbR_+^{\\calY}},\n\\end{equation*}\nand the collection of all such rays generates the entire irreducible cone\n\\begin{equation*}\n \\calF^+(\\calY, \\calE) = \\biguplus_{P \\in \\calW(\\calY, \\calE)} [P].\n\\end{equation*}\nWhile projecting onto stochastic matrices through $\\stoch$-normalization preserves $\\kappa$-lumpability, conjugation by an arbitrary diagonal matrix can disrupt the property. In fact, the following proposition holds.\n\\begin{proposition}[Closure under similarity transform]\n\\label{proposition:eigenvector-rescaling-constant-on-each-lumped-point}\n Let $v \\in \\bbR_+^{\\calY}$. The two following statements are equivalent.\n\\begin{enumerate}[$(i)$]\n \\item For any irreducible lumpable matrix $F \\in \\calF_{ \\kappa}^+(\\calY, \\calE)$, it holds that \\begin{equation*}\n\\diag(v) F \\diag(v)^{-1} \\in \\calF_{\\kappa}^{+}(\\calY, \\calE).\n \\end{equation*}\n \\item \n For any $x \\in \\calX$, it holds that for any $(y_1, y_2) \\in \\calS_{x}^2, v(y_1) = v(y_2)$. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n Let $v \\in \\bbR_+^{\\calY}, F \\in \\calF_\\kappa^{+}(\\calY, \\calE)$, write $F^\\flat = \\kappa_\\star F $, let $(x,x') \\in \\calD = \\kappa_2(\\calE)$, and assume that $v$ takes constant values $v_x$ on $\\calS_x$ and $v_{x'}$ on $\\calS_{x'}$. Then for any $y \\in \\calS_x$,\n \\begin{equation*}\n \\begin{split}\n \\sum_{y' \\in \\calS_{x'}} (\\diag(v) F \\diag(v)^{-1})(y, y') &= \\frac{v_x}{v_{x'}} \\sum_{y' \\in \\calS_{x'}} F(y, y') = \\frac{v_x}{v_{x'}} F^\\flat(x,x'),\n \\end{split}\n \\end{equation*}\n which does not depend on $y$,\n thus $\\diag(v) F \\diag(v)^{-1} \\in \\calF_\\kappa^{+}(\\calY, \\calE)$. Conversely, let us assume that $\\diag(v) F \\diag(v)^{-1} \\in \\calF_\\kappa^{+}(\\calY, \\calE)$.\n Let $\\rho \\in \\bbR, w \\in \\bbR_+^{\\calY}$ be the unique right PF eigen-pair of $F$, that is $\\stoch(F) = \\frac{1}{\\rho} \\diag(w)^{-1} F \\diag(w)$. From the proof of Proposition~\\ref{proposition:stochastic-rescaling-and-lumping-commute}, $w$ is constant on each $\\calS_x$. Observe now that\n $$\\stoch(F) = \\stoch(\\diag(v) F \\diag(v)^{-1}) = \\frac{1}{\\rho} \\diag(v \\hadamard w)^{-1} \\diag(v) F \\diag(v)^{-1} \\diag(v \\hadamard w),$$ that is $\\rho, v \\hadamard w$ is the right PF eigen-pair of $\\diag(v) F \\diag(v)^{-1}$. It follows that $v \\hadamard w$ must be constant over each $\\calS_x$, and so must $v$.\n\\end{proof}\n\nIn other words, the cone $\\calF_\\kappa^{+}(\\calY, \\calE)$ is closed with respect to similarity transform with positive vectors which are constant over elements of the partition of $\\calY$ induced by $\\kappa$.\n\n\\begin{corollary}[to Proposition~\\ref{proposition:stochastic-rescaling-and-lumping-commute}]\n\\label{corollary:geometric-intersection-interpretation}\n Let $F \\in \\calF^+(\\calY, \\calE)$. Then $\\stoch(F) \\in \\calW_\\kappa(\\calY, \\calE)$ if and only if $$[F] \\cap \\calF_\\kappa^{+}(\\calY, \\calE) \\neq \\emptyset.$$\n\\end{corollary}\n\n\\begin{proof}\n Suppose first that $\\stoch(F) \\in \\calW_\\kappa(\\calY, \\calE)$. Clearly, $F \\in [F]$, and since $\\calW_\\kappa(\\calY, \\calE) \\subset \\calF_\\kappa^{+}(\\calY, \\calE)$ it is immediate that $\\stoch(F) \\in [F] \\cap \\calF_\\kappa^{+}(\\calY, \\calE) \\neq \\emptyset$.\n Conversely, suppose that there exists $G \\in [F]$ such that $G \\in \\calF_\\kappa^+(\\calY, \\calE)$. Then, by Proposition~\\ref{proposition:stochastic-rescaling-and-lumping-commute}, $\\stoch(G) \\in \\calW_\\kappa(\\calY, \\calE)$, but $\\stoch(G) = \\stoch(F)$ since both are in $[F]$, thus also $\\stoch(F) \\in \\calW_\\kappa(\\calY, \\calE)$.\n\\end{proof}\n\nThe notion of a merging block, which we now introduce, will be instrumental to our analysis.\n\n\\begin{definition}[Merging block]\n\\label{definition:merging-block}\nLet $(x, x') \\in \\calD = \\kappa_2(\\calE)$ be such that for some $y \\in \\calS_x$ it holds that\n\\begin{equation*}\n \\abs{\\set{(y,y') \\in \\calE \\colon y' \\in \\calS_{x'}}} > 1.\n\\end{equation*}\nThen we say that $(x,x')$ is a merging block of $(\\calY, \\calE)$ with respect to $\\kappa$.\nFurthermore, when \n$\\abs{\\calS_x} \\geq 2$, we say that the merging block is multi-row.\n\\end{definition}\n\n\\begin{theorem}[Log-affinity]\n\\label{theorem:characterization-log-linearity}\nThe two following statements are equivalent.\n\n\\begin{enumerate}[$(i)$]\n \\item\n$(\\calY, \\calE)$ has no multi-row merging block with respect to $\\kappa$ (Definition~\\ref{definition:merging-block}). \n\\item $\\calF_\\kappa^{+}(\\calY, \\calE)$ is log-affine in the sense where for any $F_0, F_1 \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$ and for any $t \\in \\bbR$, \n \\begin{equation*}\n (1 - t)\\log F_0 + t \\log F_1 \\in \\calF_{\\kappa}^{+}(\\calY, \\calE),\n \\end{equation*}\n where the logarithm is here understood to be entry-wise.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n Let $F_0, F_1 \\in \\calF_\\kappa^{+}(\\calY, \\calE)$, let $t \\in \\bbR$, and write $F_0^\\flat = \\kappa_\\star F_0, F_1^\\flat = \\kappa_\\star F_1$. For $(y,y') \\in \\calY^2$, we define $F_t(y,y') = F_0(y,y')^{1 - t} F_1(y,y')^t$.\nLet $(x,x') \\in \\calD$, we need to show that for any $y_1, y_2 \\in \\calS_{x}$, $F_t(y_1, \\calS_{x'}) = F_t(y_2, \\calS_{x'})$. The case $\\abs{\\calS_x} = 1$ is trivial and it remains to inspect the case $\\abs{\\calS_x} \\geq 2$. Since $(x,x')$ is non-merging, for any $y \\in \\calS_{x}$ we denote $u'_{x'}(y)$ the unique element in $\\calS_{x'}$ such that $(y,u'_{x'}(y)) \\in \\calE$. For any $y \\in \\calS_x$, it then holds that\n\\begin{equation*}\n\\begin{split}\n \\sum_{y' \\in \\calS_{x'}} F_t(y,y') &= \\sum_{y' \\in \\calS_{x'}} F_0(y,y')^{1 -t} F_1(y,y')^{t} = F_0(y,u'_{x'}(y))^{1-t} F_1(y,u'_{x'}(y))^{t} \\\\\n &= \\left(\\sum_{y'\\in \\calS_{x'}} F_0(y,y') \\right)^{1-t} \\left( \\sum_{y' \\in \\calS_{x'}} F_1(y,y') \\right)^{t}\n = F_0^\\flat(x,x') ^{1-t} F_1^\\flat(x,x')^{t},\n\\end{split}\n\\end{equation*}\nwhich does not depend on $y$, thus $F_t \\in \\calF_\\kappa^+(\\calY, \\calE)$.\nConversely, let us now suppose that $(\\calY, \\calE)$ has a multi-row merging block with respect to $\\kappa$. We show the somewhat stronger claim that there exist two stochastic matrices $P_0, P_1 \\in \\calW_\\kappa(\\calY, \\calE)$ and $t \\in \\bbR$ such that $P_0^{\\hadamard (1 - t)} \\hadamard P_1^{\\hadamard t} \\not \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$.\n We let $(x_0,x_0') \\in \\calD$ be a merging block of $\\calW_\\kappa(\\calY, \\calE)$, and denote $y_0, y_\\star, y'_a, y'_b \\in \\calS_{x_0}^2, \\calS_{x_0'}^2$ a quadruplet such that $y_0 \\neq y_\\star$, $y'_a \\neq y'_b$ and $(y_\\star, y'_a), (y_\\star, y'_b) \\in \\calE$.\n Let $\\eta_a, \\eta_b \\in \\bbR_+$ with $\\eta_a < \\eta_b < 1$ and such that \n $$(\\eta_a + \\eta_b)^{-1} = \\frac{1}{2}\\abs{ \\set{\\overline{x} \\in \\calX \\colon (x_0, \\overline{x}) \\in \\calD }} \\abs{ \\set{ \\overline{y} \\in \\calS_{x_0'} \\colon (y_\\star,\\overline{y}) \\in \\calE }}.$$\n We construct,\n \\begin{equation*}\n P_{a,b}(y,y') = \\begin{cases} \n 0 &\\text{when } (y, y') \\not \\in \\calE \\\\\n \\eta_a &\\text{when } (y,y') = (y_\\star, y'_a) \\\\\n \\eta_b &\\text{when } (y,y') = (y_\\star, y'_b) \\\\\n \\cfrac{1}{\\abs{ \\set{\\overline{x} \\in \\calX \\colon (\\kappa(y), \\overline{x}) \\in \\calD }} \\abs{ \\set{ \\overline{y} \\in \\calS_{\\kappa(y')} \\colon (y,\\overline{y}) \\in \\calE }}} &\\text{otherwise}.\n \\end{cases}\n \\end{equation*}\n We define $P_0, P_1 \\in \\calW_\\kappa(\\calY, \\calE)$ as $P_0 = P_{a,b}$ and $P_1 = P_{b,a}$,\n and construct the combination $\\widetilde{P} = P_0^{\\hadamard 1/2} \\hadamard P_1^{\\hadamard 1/2}$.\n By the AM-GM inequality $2 \\sqrt{\\eta_a \\eta_b} < \\eta_a + \\eta_b$, and it follows that\n \\begin{equation*}\n \\sum_{\\overline{y} \\in \\calS_{x_0'}} \\widetilde{P}(y_\\star, \\overline{y}) < \\sum_{\\overline{y} \\in \\calS_{x_0'}} \\widetilde{P}(y_0, \\overline{y})\n \\end{equation*}\n hence $\\widetilde{P}_{1/2}$ is not $\\kappa$-lumpable.\n\\end{proof}\n\n\\section{Classification for small state spaces}\n\\label{section:classification}\n\\begin{remark}[Degenerate lumping function]\n\\label{remark:degenerate-lumping}\n If $\\kappa \\colon \\calY \\to \\calX$ is such that $\\abs{\\calX} \\in \\set{ \n1, \\abs{\\calY} }$, then $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family.\n\\end{remark}\n\nIt immediately follows that for $\\abs{\\calY} = 2$, every lumpable family forms an e-family.\nWe proceed to enumerate e-families for the three-state space.\n\n\\begin{theorem}[Three-state space classification]\n\\label{theorem:enumeration-three-state-space}\n\nWhen $\\abs{\\calY} = 3$,\n and assuming that $\\calW_{\\kappa}(\\calY, \\calE) \\neq \\emptyset$, the two following statements are equivalent.\n \\begin{enumerate}[$(i)$]\n \\item $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family.\n \\item Either $\\kappa$ is degenerate or $\\calW_{\\kappa}(\\calY, \\calE)$ is equivalent to one of the 12 below-listed families.\n \\end{enumerate}\n \\begin{center}\n\\begin{tblr}{cccc}\n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\zero\\\\\n \\plus & \\zero & \\zero\\\\\n\\end{tblr}\\right)$\n& \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\zero\\\\\n \\plus & \\zero & \\zero\\\\\n\\end{tblr}\\right)$ \n& \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\zero & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\plus\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n& \n $\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\zero & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n\\\\\n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\zero & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n&\n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n& \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\zero & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\plus\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n& \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n\\\\ \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\plus\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n& $\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\zero & \\plus\\\\\n \\plus & \\plus & \\zero\\\\\n\\end{tblr}\\right)$\n&\n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\zero & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\zero & \\plus\\\\\n\\end{tblr}\\right)$\n& \n$\\left(\\begin{tblr}{c|[dashed]cc}\n \\plus & \\plus & \\plus\\\\\\hline[dashed]\n \\plus & \\plus & \\zero\\\\\n \\plus & \\zero & \\plus\\\\\n\\end{tblr}\\right)$\n\\end{tblr}\n\\end{center}\n\n\\end{theorem}\n\n\\begin{proof}\n From Remark~\\ref{remark:degenerate-lumping}, we only need to consider the case where $\\abs{\\calX} = 2$.\nBeing an exponential family is a property common to the entire equivalence class of lumpable families.\nAfter removing empty lumpable families and grouping them into equivalence classes (refer to Section~\\ref{section:lumpability} for the definition of equivalence classes), we obtain 26 cases. The 12 families described in $(ii)$ can be shown to forms e-families by applying Corollary~\\ref{corollary:no-multi-row-merging-block-is-sufficient}, while the remaining 14 families can be shown to not be e-families using a dimensional argument introduced later in Theorem~\\ref{theorem:dimensional-criterion}.\n\\end{proof}\n\n\\begin{remark}\n In the three-state space setting, we observe that $\\calW_{\\kappa}(\\calY, \\calE)$ forming an e-family \n coincides with\n $\\calF_{\\kappa}^{+}(\\calY, \\calE)$ being log-affine.\n\\end{remark}\n\n\n\\section{Characterizing of e-families of lumpable stochastic matrices}\n\\label{section:towards-characterization}\nIn this section, we assume that $1 < \\abs{\\calX} < \\abs{\\calY}$ and that $\\calW_\\kappa(\\calY, \\calE) \\neq \\emptyset$.\n\n\\subsection{Sufficient conditions}\n\\label{section:sufficient-condition}\n\n\\subsubsection{No multi-row merging block criterion}\n\nThe first criterion is a natural consequence of Theorem~\\ref{theorem:characterization-log-linearity}.\n\n\\begin{corollary}[No multi-row merging block criterion]\n\\label{corollary:no-multi-row-merging-block-is-sufficient}\nIf $(\\calY, \\calE)$ has no multi-row merging block with respect to $\\kappa$, then $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family.\n\\end{corollary}\n\n\\begin{proof}\n Let $P_0, P_1 \\in \\calW_\\kappa(\\calY, \\calE)$ and $t \\in \\bbR$. Defining $\\widetilde{P}_t = P_0^{\\hadamard (1 - t)} \\hadamard P_1^{\\hadamard t}$, it follows from Theorem~\\ref{theorem:characterization-log-linearity} that $\\widetilde{P}_t \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$.\n It is then a consequence of Proposition~\\ref{proposition:stochastic-rescaling-and-lumping-commute} that $\\stoch(\\widetilde{P}_t) \\in \\calW_\\kappa(\\calY, \\calE)$. Since $P_0, P_1$ and $t$ were arbitrary, \\citet[Corollary~3]{nagaoka2005exponential} implies that $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family.\n\\end{proof}\n\n\n\\begin{example}[Hudson expansion {\\citep{kemeny1983finite}}]\nLet $(\\calX, \\calD)$ be a finite strongly connected graph. \nFor a Markov chain $X_1, X_2, \\dots$ sampled according to a transition matrix $P \\in \\calW(\\calX, \\calD)$, recall that the sliding window chain\n$$(X_1, X_2), (X_2, X_3), \\dots, (X_t, X_{t+1}), \\dots$$ is also a Markov chain with transition matrix $P^\\sharp \\in \\calW_h(\\calY, \\calE)$, with state space $\\calY = \\calD$, edge set\n\\begin{equation*}\n \\calE = \\set{ (e = (x_1, x_2),e' = (x_1', x_2')) \\in \\calD^2 \\colon x_2 = x_1' },\n\\end{equation*} \nand lumping function $h \\colon \\calY \\to \\calX, (x_1, x_2) \\mapsto x_2$.\nOne can verify that $(\\calY, \\calE)$ has no multi-row merging block with respect to $h$.\nAs a result of Corollary~\\ref{corollary:no-multi-row-merging-block-is-sufficient}, $\\calW_{h}(\\calY, \\calE)$ thus forms an e-family of lumpable Markov chains.\nWe therefore recover the known fact that the Hudson expansion of the first-order Markov chains forms an e-subfamily of second-order Markov chains.\nNote that, since the Hudson expansion is known to be a particular case of a Markov embedding \\citep{wolfer2024geometric}, and Markov embeddings are known to preserve e-families of stochastic matrices, the claim also follows from an embedding argument.\n\\end{example}\n\n\\subsubsection{Lazy-cycle criterion}\n\nHowever, for state spaces strictly larger than three, the above-stated condition is not necessary, as is demonstrated in the example below.\n\n\\begin{example}[Lumpable e-family with two multi-row merging blocks]\n\\label{example:e-family-two-merging-block}\n\\begin{equation*}\n \\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\plus \\\\\n \\zero & \\zero & \\plus & \\plus\\\\\\hline[dashed]\n \\zero & \\plus & \\plus & \\zero \\\\\n \\plus & \\plus & \\zero & \\plus \\\\\n\\end{tblr}\\right)\n\\end{equation*}\n\\end{example}\n\nThis is a consequence of the fact that constructing an e-geodesic further involves $\\stoch$-normalization which can return the curve to the lumpable set. \n\n\\begin{proposition}[Lazy cycle criterion]\n \\label{proposition:lazy-cycle-criterion}\n If any of the two following equivalent conditions are satisfied, then $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family.\n \\begin{enumerate}[$(i)$]\n \\item For any $P \\in \\calW_{\\kappa}(\\calY, \\calE)$, there exist a pair of non-negative matrices $D$ and $\\Pi$ such that\n \\begin{equation*}\n P = D + \\Pi,\n \\end{equation*}\n where $\\kappa_\\star \\Pi$ is a permutation matrix over $\\calX$ and $D$ is diagonal.\n \\item The graph $\\left( \\calX, \\set{ (\\kappa(y),\\kappa(y')) \\colon (y,y') \\in \\calE, y \\neq y'} \\right)$ is a cycle.\n \\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe proposition states that all diagonal blocks are diagonal and that there is exactly one non-zero off-diagonal block per block-line.\nWe first prove the statement in the special case where all diagonal blocks vanish, that is $\\{ (x,x) \\colon x \\in \\calX \\} \\cap \\calD = \\emptyset$, and when the off-diagonal blocks have full support, that is for any $(x,x') \\in \\calD$ with $x \\neq x'$, $\\calS_{x} \\times \\calS_{x'} \\subset \\calE$. An application of monotonicity (refer to Theorem~\\ref{theorem:monotonicity}) generalizes the result beyond full-sport off-diagonal blocks, and stability by diagonal modifications (refer to Proposition~\\ref{proposition:lazy-cycle-criterion}) yield the more general case where there are non vanishing diagonal blocks on the diagonal.\nWe further reduce the problem by observing that since the lumped matrix is irreducible, $\\Pi^\\flat = \\kappa_\\star \\Pi$ defines a cycle.\nFinally, it will be convenient to order states $\\calX = \\set{1, \\dots, \\abs{\\calX}}$ and $\\calY = \\set{1, \\dots, \\abs{\\calY}}$.\nAs a result, upon relabelling, we henceforth assume that can the family can be represented by\n\\begin{equation*}\n\\left(\\begin{tblr}{c|[dashed]c|[dashed]c|[dashed]c|[dashed]c}\n 0 & \\boxplus_{\\calS_{1} \\times \\calS_{2}} & 0 & \\hdots & 0\\\\\\hline[dashed]\n 0 & 0 & \\boxplus_{\\calS_{2} \\times \\calS_{3}} & \\ddots & \\vdots \\\\\\hline[dashed]\n \\vdots & & \\ddots & \\ddots & 0 \\\\\\hline[dashed]\n 0 & & & 0 & \\boxplus_{\\calS_{\\abs{\\calX} - 1} \\times \\calS_{\\abs{\\calX}}} \\\\\\hline[dashed]\n \\boxplus_{\\calS_{\\abs{\\calX}} \\times \\calS_{1}} & 0 & \\hdots & 0 & 0 \n\\end{tblr}\\right)\n\\end{equation*}\nwhere for $\\calS, \\calS' \\subset \\calY$, $\\boxplus_{\\calS \\times \\calS'}$ is $\\abs{\\calS} \\times \\abs{\\calS'}$ matrix defined by\n\\begin{equation*}\n \\textstyle{\\boxplus_{\\calS \\times \\calS'}} = \\begin{pmatrix}\n \\plus & \\plus & \\hdots & \\plus \\\\\n \\plus & \\plus & \\hdots & \\plus \\\\\n \\vdots & \\vdots & \\vdots & \\vdots \\\\\n \\plus & \\plus & \\hdots & \\plus \\\\\n \\end{pmatrix}.\n\\end{equation*}\nWe let $P_0, P_1 \\in \\calW_{\\kappa}(\\calY, \\calE)$, and for $t\\in \\bbR$, we denote $\\widetilde{P}_t = P_0^{\\hadamard (1- t)} \\hadamard P_1^{\\hadamard t} \\in \\calF^{+}(\\calY, \\calE)$ their log-affine combination. We define $\\rho_t$ and $v_t$ the right PF pair of $\\widetilde{P}_t$, whose existence follows by strong connectivity of $(\\calY, \\calE)$. By $\\stoch$-normalization $P_t = \\frac{1}{\\rho_t}\\diag(v)^{-1} \\widetilde{P}_t \\diag(v)$ is row-stochastic, \nand as a result, for any $(x,x') \\in \\calD$, it holds that for any $y \\in \\calS_{x}$,\n\\begin{equation*}\n \\sum_{y' \\in \\calS_{x'}} P_t(y,y') = \\sum_{y' \\in \\calY} P_t(y,y') = 1,\n\\end{equation*}\nthus $P_t$ is $\\kappa$-lumpable and the claim holds.\n\\end{proof}\n\n\\begin{corollary}\n There exist e-families of lumpable stochastic matrices with an arbitrary number of multi-row merging blocks.\n\\end{corollary}\n\nIt is instructive to observe that neither criterion implies the other.\nWhat is more, there exist e-families that remain unexplained by the two above-mentioned criteria.\nDeriving a set of necessary conditions for $\\calW_{\\kappa}(\\calY, \\calE)$ to form an e-family is the topic of our next section.\n\n\\subsection{Necessary conditions}\n\\label{section:necessary-conditions}\n\nAs we have seen in the previous section, absence of log-affinity does not always preclude the lumpable family from being exponential. However, the former will be a critical ingredient to show that at least some families of matrices with so called ``redundant blocks'' cannot form e-families.\nFirst, it will be convenient to consider the following notions of block removal.\n\n\\begin{definition}[Sub-matrix and block removal]\n\\label{definition:sub-matrix-block-removal}\nLet $F \\in \\calF_\\kappa^{+}(\\calY, \\calE)$ and a non-degenerate surjective lumping function $\\kappa \\colon \\calY \\to \\calX$.\n\\begin{description}\n \\item[Sub-matrix.] \n For $\\calT \\subset \\calX$, we write\n\\begin{equation*}\n\\begin{split}\n \\calY|_{\\calT} \\eqdef \\calY \\cap \\left( \\cup_{x \\in \\calT} \\calS_{x} \\right) , \\qquad \n \\calE|_{\\calT} \\eqdef \\calE \\cap \\left( \\bigcup_{(x,x') \\in \\calT} \\calS_{x} \\times \\calS_{x'}\\right),\n\\end{split}\n\\end{equation*}\nthe resulting sub-graph is ($\\calY|_{\\calT}, \\calE|_{\\calT}$) and the sub-matrix $F|_{\\calT} \\in \\calF_\\kappa^{+}(\\calY|_{\\calT}, \\calE|_{\\calT})$ is defined such that for any $(y,y') \\in \\calE|_{\\calT}$, \\begin{equation*}\n F|_{\\calT}(y,y') = F(y,y').\n\\end{equation*}\n \\item[Block removal.]\n The matrix $F$ where the block $(x_0, x_0') \\in \\kappa_2(\\calE)$ has been removed is defined as \n$F^{\\setminus (x_0, x_0')}\\in \\calF_\\kappa^{+}(\\calY, \\calE \\setminus (\\calS_{x_0} \\times \\calS_{x'_0}))$ where for any $(y,y') \\in \\calE \\setminus (\\calS_{x_0} \\times \\calS_{x'_0})$,\n\\begin{equation*}\n F^{\\setminus (x_0, x_0')}(y,y') = F(y,y').\n \\end{equation*}\n\\end{description}\n\\end{definition}\n \n\\subsubsection{Redundant merging block criterion}\n\nCritically, while both operations in Definition~\\ref{definition:sub-matrix-block-removal} preserve lumpability, any of the two can disrupt strong connectivity. When a block can be removed while in some sense preserving irreducibility, we say that it is redundant.\n\n\\begin{definition}[Redundant block]\n\\label{definition:redundant-block}\nLet $(x_0, x_0') \\in \\kappa_2(\\calE)$. If there exists $\\calT \\subset \\calX$ such that the following condition holds:\n \\begin{enumerate}[$(i)$]\n \\item The block $(x_0, x_0') \\in \\calT^2$.\n \\item The sub-graph\n$(\\calY|_{\\calT}, \\calE|_{\\calT} \\setminus (\\calS_{x_0} \\times \\calS_{x_0'}))$ \n is strongly connected.\n\\end{enumerate}\nThen we say that the block $(x_0, x_0')$ is redundant.\n\\end{definition}\n\n\\begin{theorem}[Redundant merging block criterion]\n\\label{theorem:redundant-merging-block-criterion}\n If $(\\calY, \\calE)$ has a multi-row merging block with respect to $\\kappa$ (Definition~\\ref{definition:merging-block}) which is redundant (Definition~\\ref{definition:redundant-block}),\n then $\\calW_{\\kappa}(\\calY, \\calE)$ does not form an e-family.\n\\end{theorem}\n\n\\begin{proof}\n We suppose that $(\\calY, \\calE)$ has a multi-row merging block $(x_0, x_0') \\in \\kappa_2(\\calE)$, which is redundant.\n Recall from the proof Theorem~\\ref{theorem:characterization-log-linearity} that there exist stochastic matrices $P_0, P_1 \\in \\calW_{\\kappa}(\\calY, \\calE)$ such that $\\widetilde{P}_{1/2} = P_0^{\\hadamard 1/2} \\hadamard P_1^{\\hadamard 1/2} \\not \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$.\n From Corollary~\\ref{corollary:geometric-intersection-interpretation}, it is enough to show that $[\\widetilde{P}_{1/2}] \\cap \\calF_\\kappa^{+}(\\calY, \\calE) = \\emptyset$ in order to prove that $\\stoch(\\widetilde{P}_{1/2}) \\not \\in \\calW_\\kappa(\\calE, \\calY)$. In other words, it is sufficient to show that for any $v \\in \\bbR_+^{\\calY}$, the rescaled $\\diag(v) \\widetilde{P}_{1/2}\\diag(v)^{-1}$ is not $\\kappa$-lumpable.\n Let us suppose for contradiction that there exists $v \\in \\bbR_+^{\\calY}$, such that $\\diag(v) \\widetilde{P}_{1/2}\\diag(v)^{-1} \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$.\n Since $(x_0,x_0')$ is redundant, there exists $\\calT \\subset \\calX$ such that\n $(x_0, x_0') \\in \\calT^2$ and \n$(\\calY|_{\\calT}, \\calE|_{\\calT} \\setminus (\\calS_{x_0} \\times \\calS_{x_0'}))$ \n is strongly connected.\n On one hand, by our assumption, it must hold that \\begin{equation*}\n \\diag(v) \\left(\\widetilde{P}_{1/2}|_{\\calT}^{\\setminus (x_0, x_0')}\\right)\\diag(v)^{-1} \\in \\calF_{\\kappa}^{+}(\\calY|_{\\calT}, \\calE|_{\\calT} \\setminus (\\calS_{x_0} \\times \\calS_{x'_0})).\n \\end{equation*}\n However, observe that by our construction in the proof of Theorem~\\ref{theorem:characterization-log-linearity}, we also have\n \\begin{equation*}\n\\widetilde{P}_{1/2}|_{\\calT}^{\\setminus (x_0, x_0')} \\in \\calF_{\\kappa}^{+}(\\calY|_{\\calT}, \\calE|_{\\calT} \\setminus (\\calS_{x_0} \\times \\calS_{x'_0})),\n \\end{equation*} \n and it follows from irreducibility of $(\\calY|_{\\calT}, \\calE|_{\\calT} \\setminus (\\calS_{x_0} \\times \\calS_{x'_0}))$ and an application of Proposition~\\ref{proposition:eigenvector-rescaling-constant-on-each-lumped-point} that\n $v$ must be constant over each $\\calS_{x}$ for $x \\in \\calT$, and in particular for $x \\in \\set{ x_0, x_0'}$.\n However, in this case, inspecting the multi-row merging block $(x_0, x_0')$,\n \\begin{equation*}\n \\begin{split}\n\\sum_{\\overline{y} \\in \\calS_{x_0'}} (\\diag(v) \\widetilde{P}_{1/2} \\diag(v)^{-1})(y_0, \\overline{y}) &= \\frac{v_{x_0}}{v_{x_0'}} \\frac{1}{\\abs{\\overline{x} \\in \\calX \\colon (x_0, \\overline{x}) \\in \\calD}},\n \\end{split}\n \\end{equation*}\n while\n \\begin{equation*}\n \\begin{split}\n \\sum_{\\overline{y} \\in \\calS_{x_0'}} (\\diag(v) \\widetilde{P}_{1/2} \\diag(v)^{-1})(y_\\star, \\overline{y}) &= \\frac{v_{x_0}}{v_{x_0'}} \\left( \\frac{1}{\\abs{\\overline{x} \\in \\calX \\colon (x_0, \\overline{x}) \\in \\calD}} + 2 \\sqrt{\\eta_a \\eta_b} - (\\eta_a + \\eta_b) \\right),\n \\end{split}\n \\end{equation*}\nwhich cannot be equal from the AM-GM inequality and the assumption that $\\eta_a \\neq \\eta_b$.\n\\end{proof}\n\n\\begin{example}\nLet $\\calY = \\set{0,1,2,3,4,5}, \\calX= \\set{a,b,c}$, the lumping map $\\kappa$ defined by the partition $\\calS_{a} = \\set{0,1}, \\calS_{b} = \\set{2,3}, \\calS_{c} = \\set{4,5}$, and \nconsider the lumpable family\n \\begin{equation*}\n \\calW_{\\kappa}(\\calY, \\calE) \\sim\n\\left(\\begin{tblr}{cc|[dashed]cc|[dashed]cc}\n \\plus & \\zero & \\plus & \\plus & \\plus & \\zero \\\\\n \\zero & \\plus & \\plus & \\plus & \\zero & \\plus \\\\\\hline[dashed]\n \\plus & \\zero & \\plus & \\zero & \\plus & \\plus \\\\\n \\zero & \\plus & \\zero & \\plus & \\plus & \\plus \\\\\\hline[dashed]\n \\plus & \\plus & \\plus & \\zero & \\plus & \\zero \\\\\n \\plus & \\plus & \\zero & \\plus & \\zero & \\plus \\\\\n\\end{tblr}\\right).\n\\end{equation*}\nHere, $(b,c) \\in \\calD$ is a merging block. When we remove edges pertaining to the block $(b,c)$---that is we remove $(2, 4), (2, 5), (3, 4), (3, 5)$---there exists a closed path \n$$1 \\rightarrow 2 \\rightarrow 0 \\rightarrow 4 \\rightarrow 2 \\rightarrow 0 \\rightarrow 3 \\rightarrow 1 \\rightarrow 5 \\rightarrow 3 \\rightarrow 1,$$ \ngoing through all the states in $\\calY$. As a result, the block $(b, c)$ is redundant and from Theorem~\\ref{theorem:redundant-merging-block-criterion}, $\\calW_{\\kappa}(\\calY, \\calE)$ does not form an e-family.\n\\end{example}\n\n\\begin{corollary}[Complete graph]\nWhen $\\abs{\\calY} \\geq 4$, unless $\\kappa$ is degenerate, $\\calW_{\\kappa}(\\calY, \\calY^2)$ does not form an e-family.\n\\end{corollary}\n\n\\begin{proof}\n When $\\kappa$ is non-degenerate, observe that\n $(\\calY, \\calY^2)$ must have a multi-row merging block. Additionally, removing this block still yields a strongly connected graph. \n From a direct application of Theorem~\\ref{theorem:redundant-merging-block-criterion}, it follows that $\\calW_{\\kappa}(\\calY, \\calY^2)$ does not form an e-family.\n\\end{proof}\n\nThe most intriguing scenarios occur in the intermediate range between having no multi-row merging blocks and all blocks being multi-row merging.\nIndeed, as the number of edges in the connection graph grows, it becomes increasingly challenging for the lumpable family to form an e-family. This intuition will be rigorously formalized in Section~\\ref{section:monotonicity}.\n\n\\subsubsection{Dimensional criterion}\n\nWe introduce the set\n\\begin{equation*}\n \\calG_\\kappa(\\calY, \\calE) \\eqdef \\set{ \\log F \\colon F \\in \\calF^+_\\kappa(\\calY, \\calE) }\n\\end{equation*}\nand its affine hull in $\\calF(\\calY, \\calE)$,\n\\begin{equation*} \\aff(\\calG_\\kappa(\\calY, \\calE)) \\eqdef \\set{ \\sum_{i = 1}^{k} \\alpha_i G_i \\colon k \\in \\bbN, \\alpha \\in \\bbR^k, \\sum_{i=1}^{k} \\alpha_i = 1, G_1, \\dots, G_k \\in \\calG_\\kappa(\\calY, \\calE) }.\n\\end{equation*}\nFrom Theorem~\\ref{theorem:characterization-log-linearity}, we know that $\\aff(\\calG_\\kappa(\\calY, \\calE)) = \\calG_\\kappa(\\calY, \\calE)$---that is $\\calG_\\kappa(\\calY, \\calE)$ is an affine space---if and only if $(\\calY, \\calE)$ has no multi-row merging block with respect to $\\kappa$.\nFor convenience, we define\n\\begin{equation}\n\\label{equation:block-type-nomenclature}\n\\begin{split}\n \\calM_{x,x'} &\\eqdef \\set{ y \\in \\calS_{x} \\colon \\exists y_1', y_2' \\in \\calS_{x'}, y_1' \\neq y_2', (y,y_1'), (y,y_2') \\in \\calE} \\qquad \\text{for any $(x,x') \\in \\calD$},\n \\\\ \\calM &\\eqdef \\set{ (x,x') \\in \\calD \\colon \\calM_{x,x'} \\neq \\emptyset }, \\\\\n \\calU &\\eqdef \\set{(x,x') \\in \\calD \\colon \\calM_{x,x'} \\neq \\calS_{x}}, \\\\\n \\calR &\\eqdef \\bigcup_{(x_0,x_0') \\in \\calM } \\set{ (y_0, y_0') \\in \\calM_{x_0, x_0'} \\times \\calS_{x_0'} \\colon (y_0, y_0') \\in \\calE}.\n\\end{split}\n\\end{equation}\nThus $\\calM$ is the collection of merging blocks---not necessarily multi-row---with respect to $(\\calY, \\calE)$ and $\\kappa$, for any block $(x,x') \\in \\calD$, $\\calM_{x,x'} \\subset \\calS_{x}$ is its set of merging rows, $\\calU$ is the collection of all blocks containing at least one non-merging row, and $\\calR$ is the set of all edges which appear in some merging row. In particular, when $(x,x') \\in \\calD \\setminus \\calM$ it holds that $\\calM_{x,x'} = \\emptyset$.\nFor each $(x,x') \\in \\calD$, we fix a set of $\\abs{\\calS_x}$ anchor edges\n\\begin{equation}\n\\label{eq:anchor-points}\n E_{x,x'} \\eqdef \\set{ \\left(\\bar{y}_1, \\bar{y}_1'\\right), \\dots, \\left(\\bar{y}_{\\abs{\\calS_x}}, \\bar{y}'_{\\abs{\\calS_x}}\\right) \\in \\calE \\cap \\left(\\calS_{x} \\times \\calS_{x'}\\right), i \\neq j \\implies \\bar{y}_i \\neq \\bar{y}_j}.\n\\end{equation}\n\n\\begin{lemma}\n\\label{lemma:affine-span-log-lumpable-cone-is-linear}\n $\\aff(\\calG_{\\kappa}(\\calY, \\calE))$ is a linear space\\footnote{In particular, it contains the null vector.}.\n\\end{lemma}\n\n\\begin{proof}\nBy definition $\\aff(\\calG_{\\kappa}(\\calY, \\calE))$ is an affine space,\n so it suffices to show that it contains the null vector.\nIt will be convenient to introduce $N \\eqdef \\max_{x \\in \\calX} \\abs{\\calS_{x}}$, and for $y \\in \\calY, x' \\in \\calX$,\n\\begin{equation}\n\\label{equation:definition-s}\n s_{y,x'} \\eqdef \\abs{ \\left(\\set{y} \\times \\calS_{x'}\\right) \\cap \\calE }.\n\\end{equation}\nFor $\\alpha \\in (0, 1/N)$,\n we let $F_\\alpha \\in \\calF(\\calY, \\calE)$ be such that for any $(y,y') \\in \\calE$,\n \\begin{equation*}\n F_\\alpha(y,y') = \\begin{cases}\n \\alpha &\\text{ when } (y,y') \\not \\in E_{\\kappa(y),\\kappa(y')}, \\\\\n 1 - \\left( s_{y,\\kappa(y')} - 1 \\right)\\alpha &\\text{ otherwise}.\n \\end{cases}\n \\end{equation*} We similarly introduce $F_\\beta$ for $\\beta \\in (0, 1/N)$ with $\\beta \\neq \\alpha$. \n By construction, $F_\\alpha, F_\\beta \\in \\calF_{\\kappa}^{+}(\\calY, \\calE)$.\n For $t \\in \\bbR$, let us inspect $G_t = t \\log F_\\alpha + (1 - t) \\log F_\\beta$.\n Setting \n \\begin{equation*}\n t = \\frac{\\log \\beta}{\\log \\left( \\frac{\\beta}{\\alpha} \\right)},\n \\end{equation*}\nwe obtain\n\\begin{equation*}\n G_t(y,y') = \\begin{cases}\n 0 \\qquad \\qquad &\\text{ when } (y,y') \\not \\in E_{\\kappa(y),\\kappa(y')}, \\\\\n \\log \\left( 1 - \\left(s_{y,\\kappa(y')} - 1\\right) \\beta \\right) + \\cfrac{\\log \\beta}{\\log \\left( \\frac{\\beta}{\\alpha} \\right)} \\log \\left( \\cfrac{1 - \\left(s_{y,\\kappa(y')} - 1\\right) \\alpha}{1 - \\left(s_{y,\\kappa(y')} - 1\\right) \\beta} \\right) &\\text{ otherwise}.\n \\end{cases}\n\\end{equation*}\nFurther setting $\\alpha = \\delta$ and $\\beta = 2 \\delta$ for $\\delta \\in (0, 1/(2N))$, we obtain\n\\begin{equation*}\n G_t(y,y') = \\begin{cases}\n 0 \\qquad \\qquad &\\text{ when } (y,y') \\not \\in E_{\\kappa(y),\\kappa(y')}, \\\\\n \\log \\left( 1 - 2 \\left(s_{y,\\kappa(y')} - 1\\right) \\delta \\right) + \\cfrac{\\log (2 \\delta)}{\\log \\left( 2 \\right)} \\log \\left( \\cfrac{1 - \\left(s_{y,\\kappa(y')} - 1\\right) \\delta}{1 - 2 \\left(s_{y,\\kappa(y')} - 1\\right) \\delta} \\right) &\\text{ otherwise}.\n \\end{cases}\n\\end{equation*}\nFor any fixed $s$, the function \n\\begin{equation*}\n h \\colon \\delta \\mapsto \\log \\left( 1 - 2 (s - 1) \\delta \\right) + \\cfrac{\\log (2 \\delta)}{\\log \\left( 2 \\right)} \\log \\left( \\cfrac{1 - (s - 1) \\delta}{1 - 2 (s - 1) \\delta} \\right)\n\\end{equation*}\nis continuous, negative, decreasing, and satisfies $\\lim_{\\delta^+ \\to 0} h(\\delta) = 0$. So for any $\\eps > 0$ there exists $\\delta_s$ such that $h(\\delta_s) < \\eps$.\nFinally, taking $s_\\star = \\min \\set{s_{y, \\kappa(y')} \\colon (y,y') \\in E_{x,x'}, (x,x') \\in \\calD}$, for $\\delta < \\delta_\\star$, we have that\n$\\nrm{G_t} < \\eps \\sqrt{\\abs{\\calE}}$. In other words $0 \\in \\calF(\\calY, \\calE)$ can be characterized as an accumulation point of a sequence of elements of $\\aff(\\calG_\\kappa(\\calY, \\calE))$.\nSince $\\aff(\\calG_{\\kappa}(\\calY, \\calE))$ is an affine subspace of $\\calF(\\calY, \\calE) = \\bbR^{\\calE}$, it is closed---it contains all its accumulation points, in particular the null function.\n\\end{proof}\n\nSince they coincide, we henceforth write $\\spann(\\calG_{\\kappa}(\\calY, \\calE)) = \\aff(\\calG_{\\kappa}(\\calY, \\calE))$.\n\n\\begin{proposition}\nFor $(x_0,x_0') \\in \\calU$, let\n$G^{\\uparrow}_{x_0,x_0'} \\in \\calF(\\calY,\\calE)$ be such that for any $(y,y') \\in \\calE$,\n \\begin{equation*}\n G^{\\uparrow}_{x_0,x_0'}(y,y') = \\pred{ (y,y') \\in E_{x_0,x_0'} },\n\\end{equation*}\nwhere $E_{x_0,x_0'}$ is introduced in \\eqref{eq:anchor-points} and the sets $\\calU, \\calR$ are introduced in Eq.~\\eqref{equation:block-type-nomenclature}.\nFor any $(y_0, y_0') \\in \\calR$, let $G^{\\uparrow}_{y_0, y_0'} \\in \\calF(\\calY, \\calE)$ be such that for any $(y,y') \\in \\calE$,\n\\begin{equation*}\n G^{\\uparrow}_{y_0, y_0'}(y,y') = \\pred{(y,y') = (y_0, y_0')}.\n\\end{equation*}\nThe system of functions\n\\begin{equation*}\n \\calB_{\\kappa}(\\calY, \\calE) \\eqdef \\set{G^{\\uparrow}_{x_0,x_0'} \\colon (x_0,x_0') \\in \\calU} \\cup \\set{G^{\\uparrow}_{y_0, y_0'} \\colon (y_0,y_0') \\in \\calR},\n\\end{equation*}\nforms a basis for $\\spann(\\calG_{\\kappa}(\\calY, \\calE))$, and\n\\begin{equation*}\n \\dim \\spann(\\calG_\\kappa(\\calY, \\calE)) = \n \\abs{\\calU} + \\abs{\\calR}.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\nFor any $(x_0,x_0') \\in \\calD$,\nand for $b \\in \\set{0, 1}$,\nwe let $F^{(b)}_{x_0,x_0'} \\in \\calF(\\calY, \\calE)$ be such that for any $(y,y') \\in \\calE$,\n\\begin{equation*}\n F^{(b)}_{x_0,x_0'}(y,y') \\eqdef \\begin{cases}\n \\frac{e^{b}}{s_{y,\\kappa(y')}} &\\text{ when } (\\kappa(y),\\kappa(y')) = (x_0,x_0'), \\\\\n \\frac{1}{s_{y,\\kappa(y')}} &\\text{ otherwise},\n \\end{cases}\n\\end{equation*}\nwhere $s_{y, \\kappa(y')}$ is defined in Eq.~\\eqref{equation:definition-s}.\nBy construction, $F^{(b)}_{x_0,x_0'} \\in \\calF_\\kappa^{+}(\\calY, \\calE)$,\n\\begin{equation*}\n \\log F^{(1)}_{x_0,x_0'} - \\log F^{(0)}_{x_0,x_0'} \\in \\spann(\\calG_{\\kappa}(\\calY, \\calE)),\n\\end{equation*}\nwhere we relied on the fact that $\\aff(\\calG_{\\kappa}(\\calY, \\calE)) = \\spann(\\calG_{\\kappa}(\\calY, \\calE))$ (Lemma~\\ref{lemma:affine-span-log-lumpable-cone-is-linear}),\nand\nwhen $(x_0, x_0') \\not \\in \\calM$, we have\n\\begin{equation*}\n \\log F^{(1)}_{x_0,x_0'} - \\log F^{(0)}_{x_0,x_0'} = G^{\\uparrow}_{x_0,x_0'}.\n\\end{equation*}\nNext, for $(y_0, y_0') \\in \\calR$, note that $s_{y_0, \\kappa(y_0')} > 1$, and let us construct\n\\begin{equation*}\n \\widetilde{F}^{\\uparrow}_{y_0, y'_0}(y,y') \\eqdef \\begin{cases}\n \\frac{1}{2} &\\text{ when } (y, y') = (y_0, y_0'), \\\\\n \\frac{1}{2 \\left(s_{y_0,\\kappa(y'_0)} - 1\\right)} &\\text{ when } y = y_0, \\kappa(y') = \\kappa(y'_0), y' \\neq y_0', \\\\\n \\frac{1}{s_{y,\\kappa(y')}} &\\text{ otherwise}.\n \\end{cases}\n\\end{equation*}\nObserve that\n\\begin{equation*}\n \\left(\\log \\widetilde{F}^{\\uparrow}_{y_0, y'_0} - \\log F^{(0)}_{\\kappa(y_0), \\kappa(y_0')} \\right)(y,y') = \\begin{cases}\n \\log\\left(\\frac{s_{y_0,\\kappa(y'_0)}}{2}\\right) &\\text{ when } (y, y') = (y_0, y_0'), \\\\\n \\log\\left(\\frac{s_{y_0,\\kappa(y'_0)}}{2\\left(s_{y_0,\\kappa(y'_0)} - 1\\right)}\\right) &\\text{ when } y = y_0, \\kappa(y') = \\kappa(y'_0), y' \\neq y_0', \\\\\n 0 &\\text{ otherwise}.\n \\end{cases}\n\\end{equation*}\nIn order to construct $G_{y_0,y_0'}^{\\uparrow}$ we first introduce\n\\begin{equation*}\n\\begin{split}\n G^{\\uparrow}_{y_0, \\kappa(y_0')} &\\eqdef \n \\frac{\\sum_{\\bar{y}_0' \\in \\calS_{\\kappa(y_0')}\\colon (y_0, \\bar{y}_0') \\in \\calE} \\left(\\log \\widetilde{F}^{\\uparrow}_{y_0, \\bar{y}'_0} - \\log F^{(0)}_{\\kappa(y_0),\\kappa(y_0')} \\right)}{\\log\\left(s_{y_0, \\kappa(y_0')}\\right) - \\log(2) + \\left(s_{y_0, \\kappa(y_0')} - 1\\right) \\left(\\log\\left(s_{y_0, \\kappa(y_0')}\\right)- \\log(2) - \\log\\left(s_{y_0, \\kappa(y_0')} - 1\\right)\\right)}.\\\\\n \\end{split}\n \\end{equation*}\n Observe that\n \\begin{equation*}\n G^{\\uparrow}_{y_0, \\kappa(y_0')}(y,y') = \\begin{cases}\n 1 &\\text{ when } y = y_0 \\text{ and } \\kappa(y') = \\kappa(y_0'), \\\\\n 0 &\\text{ otherwise}.\n \\end{cases}\n \\end{equation*}\n We then construct $G_{y_0,y_0'}^{\\uparrow}$ as\n \\begin{equation*}\n \\begin{split}\n G_{y_0,y_0'}^{\\uparrow} &= \\left(1 - \\frac{ \\log\\left(s_{y_0, \\kappa(y_0')} - 1\\right)}{\\log\\left( s_{y_0, \\kappa(y_0')}\\right) - \\log(2)} \\right) \\left( \\frac{\\log \\widetilde{F}^{\\uparrow}_{y_0, y'_0} - \\log F^{(0)}_{\\kappa(y_0),\\kappa(y_0')}}{\\log\\left(s_{y_0, \\kappa(y_0')}\\right) - \\log(2) - \\log\\left(s_{y_0, \\kappa(y_0')} - 1\\right)} - G^{\\uparrow}_{y_0, \\kappa(y_0')} \\right).\n \\end{split}\n\\end{equation*}\nWe obtain that for $(y,y') \\in \\calE$,\n\\begin{equation*}\n\\begin{split}\n G^{\\uparrow}_{y_0, y_0'}(y,y') &= \\begin{cases}\n 1 &\\text{ when } (y,y') = (y_0, y_0'), \\\\\n 0 &\\text{ otherwise}.\n \\end{cases}\n\\end{split}\n\\end{equation*}\nThe construction of $G^{\\uparrow}_{x_0,x_0'}$ for $(x_0,x_0') \\in \\calU$ follows,\n\\begin{equation*}\n G^{\\uparrow}_{x_0,x_0'} = \\log F^{(1)}_{x_0,x_0'} - \\log F^{(0)}_{x_0,x_0'} - \\sum_{(y_0, y_0') \\in \\calR \\cap \\left( \\calS_{x_0} \\times \\calS_{x_0'} \\right)} G^{\\uparrow}_{y_0, y_0'}.\n\\end{equation*}\nGenerativity and linear independence of the system of function are immediate.\n\\end{proof}\n\n\\begin{example}\n\\label{example:basis-affine-hull-log-lumpable-cone}\n Consider\n \\begin{equation*}\n \\calW_{\\kappa}(\\calY, \\calE) \\sim \\left(\\begin{tblr}{c|[dashed]ccc}\n \\oplus & \\oplus & \\plus & \\zero \\\\\\hline[dashed]\n \\oplus & \\oplus & \\plus & \\plus \\\\\n \\oplus & \\zero & \\oplus & \\zero \\\\\n \\oplus & \\zero & \\zero & \\oplus \\\\\n\\end{tblr}\\right),\n\\end{equation*}\n where $\\oplus$ indicates that the edge is fixed as an anchor--- that is it belongs to $E_{x,x'}$ for some $(x,x') \\in \\calD$.\nThen $\\spann(\\calG_\\kappa(\\calY, \\calE))$ is spanned by\n \\begin{equation*}\n \\begin{split}\n \\left(\\begin{tblr}{c|[dashed]ccc}\n 1 & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & 1 & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right),\\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & 1 & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n 1 & \\zero & \\zero & \\zero \\\\\n 1 & \\zero & \\zero & \\zero \\\\\n 1 & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right),\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n \\begin{split}\n \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & 1 & \\zero & \\zero \\\\\n \\zero & \\zero & 1 & \\zero \\\\\n \\zero & \\zero & \\zero & 1 \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & 1 & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & 1 & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & 1 \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right).\n\\end{split}\n\\end{equation*}\n\\end{example}\n\n\\begin{example}\n Consider\n \\begin{equation*}\n \\calW_{\\kappa}(\\calY, \\calE) \\sim \\left(\\begin{tblr}{cc|[dashed]cc}\n \\oplus & \\zero & \\oplus & \\zero \\\\\n \\zero & \\oplus & \\zero & \\oplus \\\\\\hline[dashed]\n \\zero & \\oplus & \\oplus & \\plus \\\\\n \\oplus & \\zero & \\zero & \\oplus \\\\\n\\end{tblr}\\right).\n\\end{equation*}\nThen $\\spann(\\calG_{\\kappa}(\\calY, \\calE))$ is spanned by\n\\begin{equation*}\n\\begin{split}\n \\left(\\begin{tblr}{cc|[dashed]cc}\n 1 & \\zero & \\zero & \\zero \\\\\n \\zero & 1 & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & 1 & \\zero \\\\\n \\zero & \\zero & \\zero & 1 \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right),\\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & 1 & \\zero & \\zero \\\\\n 1 & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right),\\\\ \n\\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & 1 & \\zero \\\\\n \\zero & \\zero & \\zero & 1 \\\\\n\\end{tblr}\\right),\\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & 1 & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & 1 \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right).\n\\end{split}\n\\end{equation*}\n\n\\end{example}\n\nWe recall the definition of the exponential hull of a family of stochastic matrices $\\calV \\subset \\calW(\\calY, \\calE)$ as the smallest exponential family which contains $\\calV$.\n\\begin{definition}[Exponential hull, {\\citealp[Definition~7]{wolfer2021information}}]\n\\label{definition:e-hull}\nLet $\\calV$ be a sub-family of $\\calW(\\calY, \\calE)$.\n\\begin{equation*}\n\\begin{split}\n\\ehull(\\mathcal{V}) \\eqdef \\Bigg\\{ \\stoch(\\widetilde{P}) &\\colon \\log \\widetilde{P} = \\sum_{i = 1}^{k} \\alpha_i \\log P_i, k \\in \\bbN, \\alpha \\in \\bbR^k, \\sum_{i = 1}^{k} \\alpha_i = 1, P_1, \\dots P_k \\in \\calV \\Bigg\\}, \n\\end{split}\n\\end{equation*}\nwhere $\\stoch$-normalization was introduced in Definition~\\ref{definition:s-normalization}.\n\n\\end{definition} \nIn particular, $\\ehull(\\calV) = \\calV$ if and only if $\\calV$ forms an e-family.\n For instance, it is known that the exponential hull of symmetric stochastic matrices yields the reversible family \\citep[Theorem~9]{wolfer2021information}.\nIn our analysis, it will be convenient to further define $\\calH_\\kappa(\\calY, \\calE)$ and $\\overline{\\calH}_\\kappa(\\calY, \\calE)$ as follows.\n\\begin{equation*}\n\\begin{split}\n\\calH_\\kappa(\\calY, \\calE) &\\eqdef \\set{ \\log P \\colon P \\in \\calW_{\\kappa}(\\calY, \\calE)} \\subset \\calG_{\\kappa}(\\calY, \\calE), \\\\\n\\overline{\\calH}_\\kappa(\\calY, \\calE) &\\eqdef \\set{ \\log P \\colon P \\in \\ehull(\\calW_{\\kappa}(\\calY, \\calE))}.\n \\end{split}\n\\end{equation*}\nObserve that $\\overline{\\calH}_\\kappa(\\calY, \\calE)$ is isomorphic to the affine hull of $\\calH_\\kappa(\\calY, \\calE)$ in $\\calF(\\calY, \\calE)/\\calN(\\calY, \\calE)$.\n\n\\begin{proposition}\n\\label{proposition:log-exponential-hull-isomorphic-to-quotient}\n \\begin{equation*}\n \\overline{\\calH}_\\kappa(\\calY, \\calE) \\cong \\left(\\spann(\\calG_\\kappa(\\calY, \\calE)) \\oplus \\calN(\\calY, \\calE) \\right) / \\calN(\\calY, \\calE).\n \\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n Let $\\overline{H} \\in \\overline{\\calH}_\\kappa(\\calY, \\calE)$. There exists $\\overline{G} \\in \\spann(\\calG_{\\kappa}(\\calY, \\calE))$ such that\n \\begin{equation*}\n \\exp\\left(\\overline{H}\\right) = \\stoch\\left( \\exp \\left(\\overline{G}\\right) \\right).\n \\end{equation*}\nIt follows that we can write\n\\begin{equation*}\n \\overline{H} = \\overline{G} + N,\n\\end{equation*}\nfor some $N \\in \\calN(\\calY, \\calE)$. As a result,\n\\begin{equation*}\n \\overline{H} \\in \\left(\\spann(\\calG_\\kappa(\\calY, \\calE)) \\oplus \\calN(\\calY, \\calE) \\right) / \\calN(\\calY, \\calE).\n\\end{equation*}\n Conversely, let $\\overline{G} \\in \\spann(\\calG_{\\kappa}(\\calY, \\calE))$ and $N \\in \\calN(\\calY, \\calE)$. It holds that\n \\begin{equation*}\n \\stoch \\left( \\overline{G} + N \\right) \\in \\ehull(\\calW_{\\kappa}(\\calY, \\calE)),\n \\end{equation*}\n hence\n \\begin{equation*}\n \\left[\\overline{G} + N \\right]_{\\calN(\\calY, \\calE)} \\in \\overline{\\calH}_\\kappa(\\calY, \\calE).\n \\end{equation*}\n\\end{proof}\n\n\\begin{theorem}[Dimensional criterion]\n\\label{theorem:dimensional-criterion}\n If \n \\begin{equation*}\n \\dim \\left(\\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\oplus \\calN(\\calY, \\calE) \\right)\n > \\abs{\\calE} + \\abs{\\calY} + \\abs{\\calD} - \\abs{\\calX} - \\sum_{(x,x') \\in \\calD} \\abs{\\calS_x},\n \\end{equation*}\n then $\\calW_{\\kappa}(\\calY, \\calE)$ does not form an e-family.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family. Then $\\ehull(\\calW_{\\kappa}(\\calY, \\calE)) = \\calW_{\\kappa}(\\calY, \\calE)$ and\n\\begin{equation*}\n \\dim \\overline{\\calH}_{\\kappa}(\\calY, \\calE) = \\dim \\calW_{\\kappa}(\\calY, \\calE).\n\\end{equation*}\nIt follows follows then from Proposition~\\ref{proposition:log-exponential-hull-isomorphic-to-quotient} that\n\\begin{equation*}\n \\dim \\left(\\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\oplus \\calN(\\calY, \\calE) / \\calN(\\calY, \\calE) \\right)\n = \\dim \\calW_{\\kappa}(\\calY, \\calE).\n \\end{equation*}\nRecall that $\\dim \\calN(\\calY, \\calE) = \\abs{\\calY}$, and from Theorem~\\ref{theorem:foliation-of-lumpable-kernels}, that\n \\begin{equation*}\n \\dim \\calW_\\kappa(\\calY, \\calE) = \\abs{\\calE} - \\sum_{(x,x') \\in \\calD} \\abs{\\calS_x} + \\abs{\\calD} - \\abs{\\calX},\n\\end{equation*}\nwhence the theorem holds.\n\\end{proof}\n\nA basis for $\\calN(\\calY, \\calE)$ can be constructed as follows.\nFor $y_0 \\in \\calY$, we define $N_{y_0} \\in \\calF(\\calY, \\calE)$ be such that for any $(y,y') \\in \\calE$,\n\\begin{equation*}\n N_{y_0}(y,y') = \\pred{y' = y_0} - \\pred{y = y_0}.\n\\end{equation*}\nLet $y_\\star \\in \\calY$ be arbitrary; then \n\\begin{equation*}\n \\set{C} \\cup \\set{ N_{y_0} \\colon y_0 \\in \\calY \\setminus \\set{y_\\star}},\n\\end{equation*}\nwhere $C \\equiv 1$ is the constant unit function over $\\calE$,\nforms a basis of $\\calN(\\calY, \\calE)$. As a consequence, the bases for $\\spann(\\calG_\\kappa(\\calY, \\calE))$ and $\\calN(\\calY, \\calE)$ can be concatenated, and determining the rank of the resulting family can be obtained algorithmically, for instance with Gaussian elimination. Before performing this somewhat costly computation, one can also proceed with the following preliminary verification which does not require computing the rank of a family of functions.\n\n\\begin{corollary}\n\\label{corollary:dimensional-criterion-simplified}\nIf \n \\begin{equation*}\n \\underbrace{\\sum_{(x,x') \\in \\calD} \\abs{\\calS_x}}_{\\# \\text{ of rows in every block}} > \\underbrace{\\left( \\abs{\\calD} - \\abs{\\calU} \\right)}_{\\# \\text{ completely merging blocks }} + \\underbrace{\\left(\\abs{\\calY} - \\abs{\\calX}\\right)}_{\\text{state space compression}} + \\underbrace{\\left(\\abs{\\calE} - \\abs{\\calR} \\right)}_{\\# \\text{ of non-merging transitions}},\n \\end{equation*}\n where $\\calU$ and $\\calR$ are defined in Eq.~\\eqref{equation:block-type-nomenclature},\n then $\\calW_{\\kappa}(\\calY, \\calE)$ does not form an e-family.\n\\end{corollary}\n\\begin{proof}\n \\begin{equation*}\n \\dim \\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\leq \\dim \\left(\\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\oplus \\calN(\\calY, \\calE) \\right).\n \\end{equation*}\n\\end{proof}\n\n\\begin{example}[Example~{\\ref{example:basis-affine-hull-log-lumpable-cone}} continued]\nRecall the family\n \\begin{equation*}\n \\calW_{\\kappa}(\\calY, \\calE) \\sim \\left(\\begin{tblr}{c|[dashed]ccc}\n \\oplus & \\oplus & \\plus & \\zero \\\\\\hline[dashed]\n \\oplus & \\oplus & \\plus & \\plus \\\\\n \\oplus & \\zero & \\oplus & \\zero \\\\\n \\oplus & \\zero & \\zero & \\oplus \\\\\n\\end{tblr}\\right).\n\\end{equation*}\nA basis of functions $\\calB_\\kappa(\\calY, \\calE)$ for $\\calG_{\\kappa}(\\calY, \\calE)$ is given in Example~\\ref{example:basis-affine-hull-log-lumpable-cone}.\nThe space $\\calN(\\calY, \\calE)$ is spanned by \n \\begin{equation*}\n \\left(\\begin{tblr}{c|[dashed]ccc}\n 1 & 1 & 1 & 0 \\\\\\hline[dashed]\n 1 & 1 & 1 & 1 \\\\\n 1 & \\zero & 1 & \\zero \\\\\n 1 & \\zero & \\zero & 1 \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & 1 & \\zero & \\zero \\\\\\hline[dashed]\n -1 & \\zero & -1 & -1 \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & 1 & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & 1 & \\zero \\\\\n -1 & \\zero & \\zero & \\zero \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right), \\left(\\begin{tblr}{c|[dashed]ccc}\n \\zero & \\zero & \\zero & \\zero \\\\\\hline[dashed]\n \\zero & \\zero & \\zero & 1 \\\\\n \\zero & \\zero & \\zero & \\zero \\\\\n -1 & \\zero & \\zero & \\zero \\\\\n\\end{tblr}\\right). \n\\end{equation*}\nOn one end, the manifold dimension of the lumpable family is given by\n\\begin{equation*}\n \\dim \\calW_{\\kappa}(\\calY, \\calE) = \\abs{\\calE} + \\abs{\\calD} - \\abs{\\calX} - \\sum_{(x,x') \\in \\calD} \\abs{\\calS_{x}} = 11 + 4 - 2 - 8 = 5.\n\\end{equation*}\nOn the other hand, concatenating the two bases, one can algorithmically verify that\n\\begin{equation*}\n\\begin{split}\n \\dim \\left(\\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\oplus \\calN(\\calY, \\calE) \\right) &= 10.\n\\end{split}\n\\end{equation*}\nConsequently,\n\\begin{equation*}\n \\dim \\left(\\spann\\left(\\calG_\\kappa(\\calY, \\calE)\\right) \\oplus \\calN(\\calY, \\calE) \\right) > \\dim \\calW_{\\kappa}(\\calY, \\calE) + \\dim \\calN(\\calY, \\calE),\n\\end{equation*}\nand from Theorem~\\ref{theorem:dimensional-criterion}, $\\calW_{\\kappa}(\\calY, \\calE)$ does not form an e-family.\n\\end{example}\n\n\n\\section{Monotonicity and stability}\n\\label{section:monotonicity}\nIn this section, we fix two state spaces $\\calX, \\calY$ and a surjective non-trivial lumping function $\\kappa \\colon \\calY \\to \\calX$. \nFor edge sets $\\calE, \\calE' \\subset \\calY^2$ such that $\\calE \\subset \\calE'$, it is clear that when $(\\calY, \\calE)$ is strongly connected, so must be $(\\calY, \\calE')$, and the graph $(\\calY, \\calE)$ can be incrementally transformed into $(\\calY, \\calE')$ by adding edges, naturally leading to the construction of a monotonous sequence of consecutive elements $\\calE = \\calE_0 \\subsetneq \\calE_1 \\subsetneq \\dots \\subsetneq \\calE_{L} = \\calE'$, where $L = \\abs{\\calE'} - \\abs{\\calE}$.\nWhen considering lumpability,\nmore care is required. Indeed, even if $\\calW_\\kappa(\\calY, \\calE)$ is non-vacuous, adding a single edge can easily disrupt lumpability, leading to $\\calW_\\kappa(\\calY, \\calE') = \\emptyset$.\nOnly by adding edges in already existing blocks, or introducing new valid blocks can we guarantee the existence of lumpable chains under the new graph.\nWe first explain how to construct a monotonous sequence of essentially consecutive families of irreducible and lumpable stochastic matrices.\n\n\\begin{lemma}[Chaining]\n\\label{lemma:chaining-edge-sets}\nLet $\\calE, \\calE' \\in \\calY^2$ be such that $\\calE \\subset \\calE'$, and $\\calW_{\\kappa}(\\calY, \\calE)$ and $\\calW_{\\kappa}(\\calY, \\calE')$ are non-vacuous families of lumpable stochastic matrices. Then,\n there exists $L \\in \\bbN$ and a finite sequence of edge sets $\\calE_0, \\calE_1, \\dots, \\calE_L \\subset \\calY^2$, such that\n \\begin{enumerate}\n \\item $\\calE_0 = \\calE$, $\\calE_L = \\calE'$.\n \\item The sequence $\\{\\calE_\\ell \\}_{\\ell = 0, \\dots, L }$ is strictly monotone increasing, that is, for any $\\ell \\in \\set{0, \\dots, L - 1}$, $\\calE_\\ell \\subsetneq \\calE_{\\ell+1}$.\n \\item The sequence $\\{\\calE_\\ell \\}_{\\ell = 0, \\dots, L }$ is consecutive, in the sense where for any $\\ell \\in \\set{0, \\dots, L -1}$, it holds that for any $\\calE'' \\subset \\calY^2$ such that $\\calE_\\ell \\subset \\calE'' \\subset \\calE_{\\ell+1}$ either $\\calE'' = \\calE_\\ell, \\calE'' = \\calE_{\\ell+1}$ or $\\calW_{\\kappa}(\\calY ,\\calE'') = \\emptyset$.\n \\item For any $\\ell \\in \\set{0, \\dots, L}$, $\\calW_\\kappa(\\calY, \\calE_\\ell)$ is non-vacuous.\n \\item For any $\\ell \\in \\set{0, 1\\dots, L - 1}$, $\\calE_\\ell$ and $\\calE_{\\ell+1}$ only differ either on an edge-link or a block-link:\n \\begin{description}\n \\item[Edge-link:] We say that $(y_\\star, y_\\star') \\in \\calE_{\\ell+1}$ is an edge-link between $\\calE_{\\ell}$ and $\\calE_{\\ell+1}$ whenever $$\\kappa_2(\\calE_{\\ell+1}) = \\kappa_2(\\calE_{\\ell}), \\text{ and } \\calE_{\\ell+1} \\setminus \\calE_{\\ell} = \\set{(y_\\star, y_\\star')}.$$\n \\item[Block-link:] We say that $(x_\\star, x'_\\star) \\in \\kappa_2(\\calE_{\\ell+1})$ is a block-link between $\\calE_\\ell$ and $\\calE_{\\ell+1}$ whenever\n $$\\kappa_2(\\calE_{\\ell+1}) \\neq \\kappa_2(\\calE_{\\ell}), \\calE_{\\ell+1} \\setminus \\calE_{\\ell} \\subset \\calS_{x_\\star} \\times \\calS_{x_\\star'} \\text{, and} (x_\\star, x'_\\star) \\text{ is non-merging}.$$\n \\end{description}\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Let us set $\\calE_0 = \\calE$ and let us denote $$\\kappa_2(\\calE') \\setminus \\kappa_2(\\calE_0) = \\set{(x_1,x'_1), \\dots , (x_B, x'_B)},$$\n the indexed collection of $B \\in \\bbN$ non-zero blocks in lumped $\\calE'$ that vanish in lumped $\\calE$.\n For such a block $b \\in [B]$, we denote $s_b = \\abs{\\calS_{x_b}}$, and we let\n \\begin{equation*}\n (y_{b,1}, y'_{b,1}), \\dots, (y_{b, s_b}, y'_{b, s_b}) \\in \\calS_{x_b} \\times \\calS_{x'_b} \\cap \\calE'\n \\end{equation*}\n be a collection of $s_b$ pairs such that for any $i,j \\in [s_b]$, $i \\neq j \\implies y_{b,i} \\neq y_{b,j}$.\n Note that such a collection of $s_b$ pairs must necessarily exist from our assumption that $\\calW_\\kappa(\\calY, \\calE') \\neq \\emptyset$.\n For simplicity, we write\n \\begin{equation*}\n \\Delta \\calE_b \\eqdef \\set{ (y_{b,1}, y'_{b,1}), \\dots, (y_{b, s_b}, y'_{b, s_b}) },\n \\end{equation*}\n and for $b \\in [B]$, we then define inductively\n $$\\calE_{b} = \\calE_{b - 1} \\cup \\Delta \\calE_b.$$\n Observe that by construction, $\\calW_\\kappa(\\calY, \\calE_b) \\neq \\emptyset$ for any $b = 0, \\dots, B$, and that by exhaustion, we obtain $\\kappa_2(\\calE') = \\kappa_2(\\calE_{B})$.\n As a second step, we now denote\n \\begin{equation*}\n \\calE' \\setminus \\calE_B = \\set{ (y_1, y'_1), \\dots, (y_{E}, y'_{E}) }\n \\end{equation*}\n the set of edges in $\\calE'$ which are still missing in $\\calE_B$.\n For $e \\in [E]$, defining inductively\n \\begin{equation*}\n \\calE_{B + e} = \\calE_{B + e - 1} \\cup \\set{ (y_e, y'_e) },\n \\end{equation*}\n we obtain a full chain of non-vacuous lumpable families\n \\begin{equation*}\n \\calE_0 \\subsetneq \\dots \\subsetneq \\calE_{B} \\subsetneq \\calE_{B +1 } \\subsetneq \\dots \\subsetneq \\calE_{B + E},\n \\end{equation*}\n which concludes the claim with $L = B + E$.\n\\end{proof}\n\n\\begin{remark}\n Links offer minimal updates in as much as it is not possible to insert additional (non-vacuous) elements in between links.\n Moreover, while chains are not unique, observe that they all share the same length\n \\begin{equation*}\n L = \\sum_{(x,x') \\in \\kappa_{2}(\\calE)} \\abs{ (\\calE' \\setminus \\calE) \\cap (\\calS_{x} \\times \\calS_{x'}) } + \\sum_{(x,x') \\in \\kappa_2(\\calE') \\setminus \\kappa_2(\\calE)} \\abs{ \\calE' \\cap (\\calS_{x} \\times \\calS_{x'})} - \\abs{\\calS_{x}} + 1.\n \\end{equation*}\n\\end{remark}\n\nWe now state and show a monotonicity property of e-families.\n\n\\begin{theorem}[Monotonicity]\n\\label{theorem:monotonicity}\nLet $\\calE \\subset \\calE'$\nsuch that both $\\calW_\\kappa(\\calY, \\calE)$ and $\\calW_\\kappa(\\calY, \\calE')$ are non-vacuous. If $\\calW_\\kappa(\\calY, \\calE')$ forms an e-family, then $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family.\n\\end{theorem}\n\n\\begin{proof}\n Consider the max-norm on $\\calF(\\calY, \\calE)$ defined by $\\nrm{A} = \\max_{y,y' \\in \\calE} \\abs{A(y,y')}$.\n We prove the contrapositive of the claim.\n Let us suppose that $\\calW_\\kappa(\\calY, \\calE)$ does not form an e-family.\n From \\citet[Corollary~3]{nagaoka2005exponential}, there exist $P_0, P_1 \\in \\calW_\\kappa(\\calY, \\calE)$ and $t_\\star \\in \\bbR_+$ such that $\\gamma^{(e)}_{P_0 P_1}(t_\\star) \\not \\in \\calW_\\kappa(\\calY, \\calE)$. \n Constructing a ball with respect to $\\nrm{\\cdot}$ around any lumpable matrix, it is easy to see that every point is a boundary point. In the boundary, we must also include stochastic matrices where some edges are missing.\n In fact, the boundary set of $\\calW_\\kappa(\\calY, \\calE)$ with respect to the max-norm is given by\n \\begin{equation*}\n \\partial \\calW_\\kappa(\\calY, \\calE) = \\calW_\\kappa(\\calY, \\calE) \\cup \\bigcup_{\\calE'' \\subsetneq \\calE} \\set{ F \\in \\calF_{\\kappa}^{+}(\\calY, \\calE'') \\colon F 1 \\trn = 1 \\trn }.\n \\end{equation*}\n However, since the super-family $\\calW(\\calY, \\calE)$ forms an e-family, it holds that $\\gamma^{(e)}_{P_0, P_1} \\subset \\calW(\\calY, \\calE)$, thus $\\gamma^{(e)}_{P_0, P_1}(t_\\star) \\not \\in \\partial \\calW_\\kappa(\\calY, \\calE)$.\n As a consequence, there must exist $\\delta \\in \\bbR_+$ such that the curve at parameter $t_\\star$ satisfies \n \\begin{equation}\n \\label{eq:parameter-time-outside}\n \\inf_{P \\in \\calW_\\kappa(\\calY, \\calE)} \\nrm{P - \\gamma^{(e)}_{P_0, P_1}(t_\\star)} > \\delta.\n \\end{equation}\n We first treat the two simple cases where $\\calE$ and $\\calE'$ are connected by either an edge-link or a block-link.\n From the chaining Lemma~\\ref{lemma:chaining-edge-sets}, we will then deduce the general claim.\n\n \\paragraph{Edge-link case:}\n Let us first assume that $\\kappa_2(\\calE) = \\kappa_2(\\calE')$ and that $\\calE$ and $\\calE'$ differ on a single edge\n $(y_\\star, y_\\star') \\in \\calE' \\setminus \\calE$, where $(y_\\star, y_\\star') \\in \\calS_{x_\\star} \\times \\calS_{x'_\\star}$. By construction, since $\\calW(\\calY, \\calE)$ is lumpable, there exists $\\bar{y}'_\\star \\in \\calS_{x'_\\star}$ with $\\bar{y}'_\\star \\neq y'_\\star$ such that $(y_\\star, \\bar{y}'_\\star) \\in \\calE$.\n Let $\\eta \\in (0, \\bar{\\eta}/2]$ with $\\bar{\\eta} \\eqdef \\min \\{ P_0(y_\\star, \\bar{y}'_\\star), P_1(y_\\star, \\bar{y}'_\\star) \\}$, and for $k \\in \\{ 0, 1\\}$ and $(y,y') \\in \\calY^2$ set\n \\begin{equation*}\n \\begin{split}\n P'_k(y, y') &= \\begin{cases} 0 &\\text{ when } (y,y') \\not \\in \\calE' \\\\\n \\eta &\\text{ when } (y,y') = (y_\\star, y'_\\star) \\\\\n P_k(y, y') - \\eta &\\text{ when } (y,y') = (y_\\star, \\bar{y}'_\\star) \\\\\n P_k(y, y') &\\text{ otherwise.}\n \\end{cases}\n \\end{split}\n \\end{equation*}\n The resulting $P'_0$ an $P'_1$ are irreducible and $\\kappa$-lumpable stochastic matrices, that is $P'_0, P'_1 \\in \\calW_{\\kappa}(\\calY, \\calE')$.\n We proceed to inspect the curve $\\gamma^{(e)}_{P'_0, P'_1} \\colon \\bbR \\to \\calW(\\calY, \\calE')$ and for simplicity, denote $P'_t \\eqdef \\gamma^{(e)}_{P'_0, P'_1}(t)$ the transition matrix such that for any $(y,y') \\in \\calY^2$,\n \\begin{equation*}\n P'_t(y,y') = P'_0(y,y')^{1-t} P'_1(y,y')^{t} \\frac{v_{t, \\eta}(y')}{\\rho_{t, \\eta} v_{t, \\eta}(y)},\n \\end{equation*}\n where $\\rho_{t, \\eta}$ and $v_{t, \\eta}$ are respectively the Perron--Frobenius root and associated eigenvector of $P_0'^{\\hadamard (1 - t)} \\hadamard P_1'^{\\hadamard t}$.\n Observe that by unicity, for any $t \\in \\bbR$ it holds that $v_{t} = v_{t, 0}, \\rho_{t} = \\rho_{t, 0}$,\n and\n following analyticity of $\\eta \\mapsto v_{t, \\eta}$ and $\\eta \\mapsto \\rho_{t, \\eta}$ on the closed interval $[0, \\bar{\\eta}/2]$ \\citep{kato2013perturbation}, at time parameter $t = t_\\star$ achieving \\eqref{eq:parameter-time-outside}, it will be convenient to introduce\n \\begin{equation*}\n \\begin{split}\n \\underline{\\rho} = \\min_{\\eta \\in [0, \\bar{\\eta} / 2]} \\rho_{t_\\star , \\eta} > 0, \\qquad \\underline{v} = \\min_{y \\in \\calY, \\eta \\in [0, \\bar{\\eta} / 2]} v_{t_\\star , \\eta}(y) > 0, \\qquad \\overline{v} = \\max_{y \\in \\calY, \\eta \\in [0, \\bar{\\eta} / 2]} v_{t_\\star , \\eta}(y).\n \\end{split}\n \\end{equation*}\n Furthermore, \n there exists $\\bar{\\eta}_{\\delta} \\in \\bbR_+$ such that for $\\eta < \\bar{\\eta}_{\\delta}$\n \\begin{equation*}\n \\abs{\\frac{v_{t_\\star, \\eta}(y')}{\\rho_{t_\\star, \\eta} v_{t_\\star, \\eta}(y)} - \\frac{v_{t_\\star}(y')}{\\rho_{t_\\star} v_{t_\\star}(y)}} \\leq \\delta.\n \\end{equation*}\n For $(y,y') \\in \\calE' \\setminus \\set{ (y_\\star, y_\\star'), (y_\\star, \\bar{y}'_\\star) }$ and any $t \\in \\bbR$,\n \\begin{equation*}\n P'_t(y,y') - P_t(y,y') = P_0(y,y')^{1-t} P_1(y,y')^{t} \\left(\\frac{v_{t, \\eta}(y')}{\\rho_{t, \\eta} v_{t, \\eta}(y)} - \\frac{v_{t}(y')}{\\rho_{t} v_{t}(y)}\\right),\n \\end{equation*}\nthus,\n\\begin{equation*}\n \\abs{P'_{t_\\star}(y,y') - P_{t_\\star}(y,y')} \\leq \\abs{\\frac{v_{t_\\star, \\eta}(y')}{\\rho_{t_\\star, \\eta} v_{t_\\star, \\eta}(y)} - \\frac{v_{t_\\star}(y')}{\\rho_{t_\\star} v_{t_\\star}(y)}}.\n \\end{equation*}\n Additionally,\n \\begin{equation*}\n \\begin{split}\n P'_t(y_\\star, y'_\\star) - P_t(y_\\star, y'_\\star) &= P'_t(y_\\star, y'_\\star) = \\eta \\frac{v_{t, \\eta}(y'_\\star)}{\\rho_{t, \\eta} v_{t, \\eta}(y_\\star)}, \\\\\n\\end{split}\n\\end{equation*}\nleading to\n\\begin{equation*}\n\\begin{split}\n \\abs{P'_{t_\\star}(y_\\star, y'_\\star) - P_{t_\\star}(y_\\star, y'_\\star)} &\\leq \\eta \\frac{\\overline{v}}{\\underline{\\rho} \\underline{v}}.\n \\end{split}\n \\end{equation*}\n Finally, observing that\n \\begin{equation*}\n \\begin{split}\n &P'_t(y_\\star, \\bar{y}'_\\star) - P_t(y_\\star, \\bar{y}'_\\star) \\\\\n =& \\left(P_0(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{1 - t}\\left(P_1(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{t} \\frac{v_{t, \\eta}(\\bar{y}'_\\star)}{\\rho_{t, \\eta} v_{t, \\eta}(y_\\star)} - \n \\left(P_0(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{1 - t}\\left(P_1(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{t}\n \\frac{v_{t}(\\bar{y}'_\\star)}{\\rho_{t} v_{t}(y_\\star)} \\\\\n&+ \\left(P_0(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{1 - t}\\left(P_1(y_\\star, \\bar{y}'_\\star) - \\eta\\right)^{t}\n \\frac{v_{t}(\\bar{y}'_\\star)}{\\rho_{t} v_{t}(y_\\star)} - \n P_0(y_\\star, \\bar{y}'_\\star)^{1 - t} P_1(y_\\star, \\bar{y}'_\\star)^{t}\n \\frac{v_{t}(\\bar{y}'_\\star)}{\\rho_{t} v_{t}(y_\\star)} \\\\\n \\end{split}\n \\end{equation*}\nwe obtain that\n \\begin{equation*}\n \\begin{split}\n \\abs{P'_{t_\\star}(y_\\star, \\bar{y}'_\\star) - P_{t_\\star}(y_\\star, \\bar{y}'_\\star)} \\leq& \\abs{\\frac{v_{t_\\star, \\eta}(\\bar{y}'_\\star)}{\\rho_{t_\\star, \\eta} v_{t_\\star, \\eta}(y_\\star)} - \n \\frac{v_{t_\\star}(\\bar{y}'_\\star)}{\\rho_{t_\\star} v_{t_\\star}(y_\\star)}} \\\\\n&+ \\left(1 - \\left(1 - \\frac{\\eta}{P_0(y_\\star, \\bar{y}'_\\star)}\\right)^{1 - t_\\star}\\left(1 - \\frac{\\eta}{P_1(y_\\star, \\bar{y}'_\\star)}\\right)^{t_\\star}\n \\right) \\frac{\\overline{v}}{\\underline{\\rho} \\underline{v}}, \\\\\n \\end{split}\n \\end{equation*}\n and for \n \\begin{equation*}\n \\begin{split}\n \\eta < \\min \\set{ P_0(y_\\star, \\bar{y}'_\\star) \\left(1 - \\left(1 - \\delta \\frac{\\underline{\\rho} \\underline{v}}{4 \\overline{v}} \\right)^{\\frac{1}{2(1 -t_\\star)}}\\right), P_1(y_\\star, \\bar{y}'_\\star) \\left(1 - \\left(1 - \\delta \\frac{\\underline{\\rho} \\underline{v}}{4 \\overline{v}} \\right)^{\\frac{1}{2 t_\\star}}\\right), \\bar{\\eta}_{ \\delta/4} },\n \\end{split}\n \\end{equation*}\n the above is smaller than $\\delta /2$.\n As a result, if we also assume that $\\eta$ such that $\\eta < \\bar{\\eta}_{\\delta/2}$ and $\\eta < \\frac{\\delta \\underline{\\rho} \\underline{v}}{2 \\overline{v}}$, it holds that\n \\begin{equation*}\n \\nrm{P_{t_\\star}' - P_{t_\\star}} \\leq \\delta / 2.\n \\end{equation*}\nFor any $P \\in \\calW_\\kappa(\\calY, \\calE')$, it then holds from the triangle inequality that\n\\begin{equation*}\n \\nrm{P - P_{t_\\star}} \\leq \\nrm{P - P'_{t_\\star}} + \\nrm{P'_{t_\\star} - P_{t_\\star}},\n\\end{equation*}\nthus,\n\\begin{equation*}\n \\nrm{P - P'_{t_\\star}} > \\nrm{P - P_{t_\\star}} - \\delta/2 > \\delta/2,\n\\end{equation*}\nand taking the infimum over $\\calW_\\kappa(\\calY, \\calE')$ finishes proving that $\\gamma^{(e)}_{P'_0, P'_1}(t_\\star) \\not \\in \\calW_\\kappa(\\calY, \\calE')$.\nInvoking \\citet[Corollary~3]{nagaoka2005exponential}, we conclude that $\\calW_\\kappa(\\calY, \\calE')$ does not form an e-family.\n\n\\paragraph{Block-link case:}\n\nLet us now assume that $\\calE'$ and $\\calE$ differ by a block-link, that is $\\kappa_2(\\calE') \\setminus \\kappa_2(\\calE) = \\{ (x_\\star, x'_\\star) \\}$, a single non-merging block, and $\\calE' \\setminus \\calE \\subset \\calS_{x_\\star} \\times \\calS_{x'_\\star}$.\nWe denote $s = \\abs{\\calS_{x_\\star}}$ and we enumerate $(y_1, y_1'), \\dots, (y_s, y_{s}') \\in \\calS_{x_\\star} \\times \\calS_{x'_\\star}$ the elements which are also in $\\calE' \\setminus \\calE$.\nSince elements of $\\calW_\\kappa(\\calY, \\calE)$ are stochastic matrices, there must exist a block $(x_\\star, \\bar{x}_\\star') \\in \\kappa_2(\\calE), $ with $ \\bar{x}_\\star' \\neq x_\\star'$.\nWe let $\\eta \\in (0, \\bar{\\eta}/2]$ where $$\\bar{\\eta} \\eqdef \\min \\{ P_k(y,y') \\colon k \\in \\{ 0, 1 \\}, (y, y') \\in \\calS_{x_\\star} \\times \\calS_{\\bar{x}'_\\star} \\}.$$\nWe further let\n$\\bar{y}'_1, \\dots, \\bar{y}'_s \\in \\calS_{\\bar{x}'_\\star}$ be such that\n$(y_1, \\bar{y}_1'), \\dots, (y_s, \\bar{y}'_{s}) \\in \\calS_{x_\\star} \\times \\calS_{\\bar{x}'_\\star}$.\nFor $k \\in \\set{0,1}$ and $(y,y') \\in \\calY^2$, we construct\n\\begin{equation*}\n P'_{k}(y,y') = \\begin{cases} 0 &\\text{ when } (y,y') \\not \\in \\calE' \\\\\n \\eta &\\text{ when } (y, y' ) \\in \\calS_{x_\\star} \\times \\calS_{x'_\\star}\\\\\n P_{k}(y,y') - \\eta &\\text{ when there exists $i \\in [s]$ such that $(y,y') = (y_i, \\bar{y}_i')$} \\\\ \n P_{k}(y,y') &\\text{ otherwise.}\\\\\n \\end{cases}\n\\end{equation*}\nObserve that $P'_0$ and $P'_1$ are both irreducible and $\\kappa$-lumpable stochastic matrices.\nSimilar to the edge-link case, we inspect the curve $\\gamma^{(e)}_{P_0', P_1'} \\colon \\bbR \\to \\calW(\\calY, \\calE')$, and we denote $P'_t = \\gamma^{(e)}_{P_0', P_1'}(t)$ the point on the curve at time parameter $t$.\nUsing a similar analyticity argument as for the edge-link case, we show that the two curves $\\gamma^{(e)}_{P_0', P_1'}$ and $\\gamma^{(e)}_{P_0, P_1}$ can be made arbitrarily close point-wise, and in particular for $t = t_\\star$. That is, there exists $\\bar{\\eta}_{\\delta}$ such that for $\\eta < \\bar{\\eta}_{ \\delta}$, it holds that $\\nrm{P'_{t_\\star} - P_{t_\\star}} \\leq \\delta / 2$.\nAn application of the triangle inequality and \\citet[Corollary~3]{nagaoka2005exponential} conclude the claim.\n\n\\paragraph{Chaining and conclusion.}\n\nAs a conclusion of the above, if $\\calE$ and $\\calE'$ only differ by an edge-link or a block-link, the claim holds.\nThe general case can be obtained by inductively applying the above result to the chain of consecutive edge sets $\\{\\calE_\\ell\\}_{\\ell = 0 \\dots, L}$ with $\\calE_0 = \\calE, \\calE_L = \\calE', $ obtained from Lemma~\\ref{lemma:chaining-edge-sets}. \n\n\\end{proof}\n\n\\begin{remark}\nAny chain constructed between two e-families can only be linked by non-merging blocks\nas adding a merging block immediately disrupts the property of being an e-family.\nNote that adding either an edge-link or a (non-merging) block-link can still disrupt the quality of being an e-family. The length of a chain between two e-families $\\calW_{\\kappa}(\\calY, \\calE)$ and $\\calE_{\\kappa}(\\calY, \\calE')$ simplifies to\n\\begin{equation*}\n L = \\abs{\\kappa_2(\\calE') \\setminus \\kappa_2(\\calE)} + \\sum_{(x,x') \\in \\kappa_{2}(\\calE)} \\abs{ (\\calE' \\setminus \\calE) \\cap (\\calS_{x} \\times \\calS_{x'}) }.\n \\end{equation*}\n\\end{remark}\n\n\\begin{definition}[Maximal e-families and minimal non e-families]\nWe consider the partial order on non-vacuous lumpable and irreducible families induced from their edge sets ordered by inclusion. We call $\\calW_\\kappa(\\calY, \\calE)$ a maximal e-family, if $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family, and any non-vacuous lumpable family $\\calW_{\\kappa}(\\calY, \\calE')$ such that $\\calE \\subsetneq \\calE'$ does not form an e-family.\nSimilarly, we call $\\calW_\\kappa(\\calY, \\calE')$ a minimal non e-family, if $\\calW_\\kappa(\\calY, \\calE')$ forms an e-family, and any non-vacuous lumpable family $\\calW_{\\kappa}(\\calY, \\calE)$ such that $\\calE \\subsetneq \\calE'$ forms an e-family.\n\\end{definition}\n\n\\begin{remark}[Well-definedness]\n The family $\\calW_\\kappa(\\calY, \\calY^2)$ defined over the complete graph being lumpable for any $\\kappa$, irreducible, and not forming an e-family (in non-trivial settings) it always holds that for any $\\calE \\subsetneq \\calY^2$ such that $\\calW_{\\kappa}(\\calY, \\calE)$ forms an e-family, that there exists\n $\\calE' \\subset \\calY^2$ with $\\calE \\subset \\calE'$ such that $\\calW_{\\kappa}(\\calY, \\calE')$ does not form an e-family.\n Note however that for any $\\calE' \\subset \\calE$ such that $\\calW_\\kappa(\\calY, \\calE')$ is non-vacuous and does not form an e-family, it is not always possible to extract $\\calE \\subset \\calE'$ such that $\\calW_{\\kappa}(\\calY, \\calE)$ forms a non-vacuous e-family.\n The example below demonstrates this fact.\n\\begin{equation*}\n \\calW_{\\kappa}(\\calY, \\calE) \\sim \\left(\\begin{tblr}{cc|[dashed]cc}\n \\zero & \\zero & \\zero & \\plus \\\\\n \\zero & \\zero & \\plus & \\zero \\\\\\hline[dashed]\n \\zero & \\plus & \\zero & \\plus \\\\\n \\plus & \\plus & \\zero & \\plus \\\\\n\\end{tblr}\\right)\n\\end{equation*}\n Observe that if we remove any subset of edges from the family above, it would become either reducible, or non-lumpable. The minimality statement thus only holds vacuously in this case.\n\\end{remark}\n\nIt is instructive to observe that adding or removing diagonal blocks from the diagonal does not alter the e-family property.\n\n\\begin{proposition}[Stability through diagonal modification]\n\\label{proposition:stability-diagonal-modification}\n Let $\\calE, \\calE'$ where $\\calE \\subset \\calE' \\subset \\calY^2$ be such that $\\calW_{\\kappa}(\\calY, \\calE') \\neq \\emptyset$, and suppose additionally that\n $$\\calE' \\setminus \\calE \\subset \\set{(y,y) \\colon y \\in \\calY, \\kappa(y) \\not \\in \\calD = \\kappa_2(\\calE)}.$$\n Then it holds that $\\calW_\\kappa(\\calY, \\calE')$ forms an e-family if and only if $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family.\n\\end{proposition}\n\\begin{proof}\nLet us first assume that $\\calE'$ and $\\calE$ differ on a single diagonal block $(x_0, x_0) \\in \\calD'$ and suppose that $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family.\nWe let $P_0', P_1'$ be an arbitrary pair in $\\calW_\\kappa(\\calY, \\calE')$ and for $t \\in \\bbR_+$, we define $\\widetilde{P'_t} = {P'_0}^{\\hadamard (1 - t)} \\hadamard {P'_1}^{\\hadamard t}$ to be their log-affine combination.\nFor $i \\in \\{0,1\\}$, we additionally define $P_i \\in \\calW_{\\kappa}(\\calY, \\calE)$ be such that\n\\begin{equation*}\n P_i(y,y') = \\begin{cases}\n P'_i(y,y') &\\text{ when } \\kappa(y) \\neq x_0 \\\\\n 0 &\\text{ when } (\\kappa(y), \\kappa(y')) = (x_0, x_0) \\\\\n \\frac{P'_i(y,y')}{\\sum_{x' \\neq x_0} P'^{\\flat}_i(x_0, x') } &\\text{ otherwise}.\n \\end{cases}\n\\end{equation*}\nSince $\\calW_\\kappa(\\calY, \\calE)$ forms an e-family, it must be that $\\stoch(\\widetilde{P}_t) \\in \\calW_\\kappa(\\calY, \\calE)$ where\n$\\widetilde{P}_t= P_0^{\\hadamard (1-t)} \\hadamard P_1^{\\hadamard t}$.\nThus, from Corollary~\\ref{corollary:geometric-intersection-interpretation}, there exists $v \\in \\bbR_+$ such that $\\diag(v) \\widetilde{P}_t \\diag(v)^{-1} \\in \\calF_{\\kappa}(\\calY, \\calE)$.\nObserve now that the same vector $v$ also satisfies $\\diag(v) \\widetilde{P}'_t \\diag(v)^{-1} \\in \\calF_{\\kappa}(\\calY, \\calE')$, thus $\\stoch(\\widetilde{P}'_t) \\in \\calW_\\kappa(\\calY, \\calE')$, which forms an e-family. The case where $\\calE$ and $\\calE'$ differ by more than one diagonal block can be retrieved using chaining (refer to Lemma~\\ref{lemma:chaining-edge-sets}). The converse statement follows more generally by monotonicity (refer to Theorem~\\ref{theorem:monotonicity}).\n\\end{proof}\n\n\n \n\n\\section{Algorithmics}\n\\label{section:algorithmics}\nThe classification problem involves determining whether the family $\\calW_{\\kappa}(\\calY, \\calE)$ forms an exponential family, given a strongly connected graph $(\\calY, \\calE)$ and a lumping map $\\kappa$.\nSpecifically, we will provide a worst-case time complexity analysis of the criteria developed in Section~\\ref{section:sufficient-condition} and Section~\\ref{section:necessary-conditions}. \n\n\\subsection{Deterministic classification}\n\\label{section:deterministic-classification}\nComplexity will be typically expressed as a function of the number of vertices or edges in the graph $(\\calY, \\calE)$ and the lumping map $\\kappa$. \nThe space $\\calY$ is represented by the integers $\\set{1, \\dots, \\abs{\\calY}}$.\nAs is traditional in the literature, we will use the landau notation $\\bigO$.\nWe assume that the graph $(\\calY, \\calE)$ is represented in the machine using adjacency lists leading to a total structure of size $\\bigO(\\abs{\\calY} + \\abs{\\calE})$. Time complexity is measured in terms of elementary field operations (addition, multiplication) and memory accesses.\nStoring the graph already requires $\\bigO(\\abs{\\calY} + \\abs{\\calE})$ operations thus we will take this quantity as our lower bound and be mostly concerned with complexity which exceeds this value.\nRecall that the lumping map can have two representations---either partitional or functional.\nFor the partitional representation, we store an array of size $\\abs{\\calX}$ corresponding to each elements of the lumped space. In each entry $x \\in \\calX$ of this array we store a list of elements of $\\calY$ lumping into $x$.\nFor the functional representation, we store an array of size $\\abs{\\calY}$ where at each entry $y \\in \\calY$ we store $\\kappa(y)$. It is not hard to see that one representation can be constructed from the other in $\\bigO(\\abs{\\calY})$. As a result, both representations can be interchangeably considered, and we will henceforth assume that we can both compute $\\kappa(y)$ in $\\bigO(1)$ and loop over elements of $\\calS_x$ in $\\abs{\\calS_x}$.\n\n\\begin{proposition}[Complexity of basic procedures]\nWe rely on the subroutines below.\n\\begin{enumerate}[$(i)$]\n \\item Determine whether $(\\calY, \\calE)$ is strongly connected: $\\bigO(\\abs{\\calY} + \\abs{\\calE})$.\n \\item Determine whether $P \\in \\calW(\\calY, \\calE)$ is $\\kappa$-lumpable: $\\bigO(\\abs{\\calX}\\abs{\\calY} + \\abs{\\calE})$.\n \\item Determine whether $\\calW_{\\kappa}(\\calY, \\calE)$ is non-vacuous: $\\bigO(\\abs{\\calX}\\abs{\\calY} + \\abs{\\calE})$.\n \\item Construct the lumped graph $(\\calX, \\calD)$ (as adjacency lists) in $\\bigO(\\abs{\\calX}^2 + \\abs{\\calE})$.\n \\item List merging blocks of $(\\calY, \\calE)$ with respect to $\\kappa$: \n$\\bigO(\\abs{\\calX}\\abs{\\calY} + \\abs{\\calE})$.\n\\end{enumerate}\n\\begin{proof}\nSimple algorithms yields $(ii), (iii), (iv), (v)$, while for $(i)$,\n listing strongly connected components can be achieved by running \nTarjan's strongly connected components algorithm \\citep{tarjan1972depth}, \nKosaraju-Sharir's algorithm \\citep{sharir1981strong}, or\nthe path-based strong component algorithm \\citep{dijkstra1976discipline}\nwhich all run in $\\bigO(\\abs{\\calY} + \\abs{\\calE})$.\n\\end{proof}\n\n\\end{proposition}\n\n\\begin{proposition}[Complexity of no multi-row merging block criterion (Corollary~\\ref{corollary:no-multi-row-merging-block-is-sufficient})]\n $\\bigO(\\abs{\\calX} \\abs{\\calY} + \\abs{\\calE})$.\n\\end{proposition}\n\n\\begin{proposition}[Complexity of dimensional criterion (Theorem~\\ref{theorem:dimensional-criterion})]\n $\\bigO(\\abs{\\calE}^{\\omega})$ with $\\omega \\leq 2.371552$.\n\\end{proposition}\n\n\\begin{proof}\n It should be clear that the bottleneck operation is the computation of the rank of the system of matrices.\n Typically, computing the rank of a matrix is done by Gaussian elimination. For a system of $m$ vectors of dimension $n$, this approach can be theoretically implemented in $\\bigO(m n ^{\\omega - 1})$\n\\citep{ibarra1982generalization} where $2 \\leq \\omega \\leq 2.371552$ is the matrix multiplication exponent \\citep{williams2024new}.\nIn our case, we have at most $\\abs{\\calE} + \\abs{\\calY}$ vectors, each of dimension $\\abs{\\calE}$. As a result, we obtain a worst-case time complexity of $\\bigO((\\abs{\\calE} + \\abs{\\calY}) \\abs{\\calE}^{\\omega - 1})$.\n\\end{proof}\n\n\\begin{remark}\nIn practice---that is for most implementations---the complexity is of order $\\bigO(\\abs{\\calE}^3)$.\n Note that with a parallel algorithm, it is possible to deterministically compute this rank in $\\bigO(\\log^2\\abs{\\calY})$ time \\citep{mulmuley1986fast} using a polynomial number of processors.\n Since the rank calculation is the bottleneck, distributing this task would substantially improve the efficiency of our algorithm.\n\\end{remark}\n\n\\begin{proposition}\n There exists a $\\bigO(\\abs{\\calY}^\\omega)$ time verifiable witness that can be used to conclude that a lumpable family is not an e-family.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that $\\calW_{\\kappa}(\\calY , \\calE)$ does not form an e-family. Then there exists a $P_0, P_1 \\in \\calW_{\\kappa}(\\calY , \\calE)$ and $t \\in \\bbR$ such that $\\gamma_{P_0, P_1}^{(e)}(t)$ is not $\\kappa$-lumpable. \nThe triplet $(P_0, P_1, t)$ is a witness and it remains to argue that it can be verified in polynomial time.\nSince constructing a point at parameter $t$ on the e-geodesic amounts to computing the Perron-Frobenius eigenpair of a Hadamard product of two matrices of size $\\abs{\\calY} \\times \\abs{\\calY}$, it follows that\n$\\gamma_{P_0, P_1}^{(e)}(t)$ can be computed in $\\bigO(\\abs{\\calY}^\\omega)$, and it can be verified that it is not $\\kappa$-lumpable in $\\bigO(\\abs{\\calX} \\abs{\\calY} + \\abs{\\calE})$.\n\\end{proof}\n\n\n\\bibliography{bibliography}\n\\bibliographystyle{abbrvnat}\n\n\\end{document}\n\ninput\n\n\n"}}},{"rowIdx":96,"cells":{"id":{"kind":"string","value":"2412.08415v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08415v2"},"title":{"kind":"string","value":"The two-boost problem and Lagrangian Rabinowitz Floer homology"},"full_text":{"kind":"string","value":"\n\\documentclass[a4paper]{amsart}\n\\usepackage{hyperref, enumitem}\n\\usepackage[dvipsnames]{xcolor} \\usepackage{amsfonts,amsmath, amssymb,mathrsfs,mathtools,amsthm} \\usepackage[doi=false, url=false, eprint=false, maxbibnames=10, firstinits=true]{biblatex}\n\\addbibresource{bibliography.bib}\n\\usepackage{todonotes}\n\n\\usepackage[margin=0pt]{caption}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{prop}[thm]{Proposition}\\newtheorem{define}[thm]{Definition}\\newtheorem{lem}[thm]{Lemma}\\newtheorem{cor}[thm]{Corollary}\\newtheorem{rem}[thm]{Remark}\\newtheorem{ex}[thm]{Example}\\numberwithin{equation}{section}\n\\newcommand{\\J}{\\color{MidnightBlue}}\n\\newcommand{\\A}{\\mathcal{A}}\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Nn}{\\mathbb{N^\\ast}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\B}{\\mathcal{B}}\n\\newcommand{\\F}{\\mathscr{F}}\n\\newcommand{\\scinf}{\\mathrm{sc}^\\infty}\n\\newcommand{\\Crit}{\\operatorname{Crit}}\n\\newcommand{\\dom}{\\mathrm{dom}}\n\\newcommand{\\supp}{\\operatorname{supp}}\n\\newcommand{\\Exp}{\\operatorname{Exp}}\n\\newcommand{\\Id}{\\operatorname{Id}}\n\\newcommand{\\reg}{\\operatorname{reg}}\n\\usepackage{verbatim}\n\\parindent=0pt\n\\parskip=4pt\n\\newcommand{\\kai}{\\color{red}}\n\\newcommand{\\ol}{\\overline}\n\\newcommand{\\wt}{\\widetilde}\n\\newcommand{\\wh}{\\widehat}\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\p}{\\partial}\n\\newcommand{\\one}\n{{{\\mathchoice \\mathrm{ 1\\mskip-4mu l} \\mathrm{ 1\\mskip-4mu l}\n\\mathrm{ 1\\mskip-4.5mu l} \\mathrm{ 1\\mskip-5mu l}}}}\n\\newcommand{\\LRFH}{{\\rm LRFH}}\n\n\\begin{document}\n\\title{The two-boost problem and Lagrangian Rabinowitz Floer homology}\n\n\\author{Kai Cieliebak}\n\\address{Department of Mathematics, University of Augsburg}\n\\thanks{Kai Cieliebak is funded by the Deutsche Forschungsgemeinschaft (DFG, German\nResearch Foundation) – 541525489.}\n\\author{Urs Frauenfelder}\n\\address{Department of Mathematics, University of Augsburg}\n\\thanks{ Urs Frauenfelder is funded by the Deutsche Forschungsgemeinschaft (DFG, German\nResearch Foundation) – 541525489.}\n\\author{Eva Miranda} \n\\address{Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Politècnica de Catalunya-IMTech \\& CRM}\n\\thanks{Eva Miranda is funded by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt Foundation via a Friedrich Wilhelm Bessel Research Award. She is also supported by the Spanish State\nResearch Agency, through the Severo Ochoa and Mar\\'{\\i}a de Maeztu Program for Centers and Units\nof Excellence in R\\&D (project CEX2020-001084-M), by the AGAUR project 2021 SGR 00603 and by the project “Computational, dynamical and geometrical complexity in fluid dynamics”, Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021. Eva Miranda and Jagna Wi\\'sniewska are partially supported by the Spanish State Research Agency grants reference PID2019-103849GB-I00 of AEI / 10.13039/501100011033 and PID2023-146936NB-I00 funded by MICIU/AEI/\n10.13039/501100011033 and, by ERDF/EU. }\n\\author{Jagna Wi\\'sniewska}\n\\address{Laboratory of Geometry and Dynamical Systems, Department of Mathematics, Universitat Politècnica de Catalunya}\\thanks{Jagna Wi\\'sniewska's postdoctoral contract is financed under the project “Computational, dynamical and geometrical complexity in fluid dynamics”, Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021.}\n\n\\maketitle\n\n\\begin{abstract}\n\\noindent\nThe two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems related to the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The main technical work goes into dealing with the noncompactness of the corresponding energy hypersurfaces. \n\\end{abstract}\n \n\\section{Introduction}\n\nThe {\\em two-boost problem} goes back to the classical work of W.\\,Hohmann on the attainability of heavenly bodies \\cite{Hohmann}. Although this work was written more than four decades before the first human being stepped on the moon, the Hohmann transfer is still one of the crucial ingredients in space mission design, see e.g.~\\cite{Vallado}. The Hohmann transfer\nis a transfer between two circular orbits in the Kepler problem with the help of a Kepler ellipse which is tangent to the two circles. It requires two tangential boosts, one to transfer from the first circle to the ellipse and a second one to transfer from the ellipse to the second circle.\nGiven two points in the plane different from the origin, there always exists a conic section through the two points with focus in the origin. This means that, for the Kepler problem, two points in phase space can always be connected with the help of two boosts. The motivating question for this paper is whether this continues to hold for more general systems.\n\nThe general setup for the two-boost problem is as follows: Consider the cotangent bundle $T^*Q$ of a manifold $Q$ with its canonical exact symplectic form $\\om=d\\lambda$, $\\lambda=p\\,dq$, and a Hamiltonian $H:T^*Q\\to\\R$. Given two points $q_0,q_1\\in Q$ and an energy value $c$, the two-boost problem asks for the existence of a Hamiltonian orbit of energy $c$ connecting the cotangent fibres $T_{q_0}^*Q$ and $T_{q_1}^*Q$. \nSuch orbits arise as critical points of the {\\em Rabinowitz action functional}\n\\begin{align*}\n\\A^{H-c}_{q_0,q_1} & : \\mathscr{H}_{q_0,q_1}\\times \\R \\to \\R,\\\\\n\\A^{H-c}_{q_0,q_1}(v,\\eta) & := \\int_0^1 \\lambda(\\partial_t v)dt - \\eta \\int_0^1 (H-c)(v(t))dt\n\\end{align*}\nassociated to the path space\n$$\n\\mathscr{H}_{q_0,q_1}:= \\left\\lbrace v \\in W^{1,2}([0,1], T^*Q)\\ \\big|\\ v(i)\\in T^*_{q_i}Q\\quad \\textrm{for}\\quad i=0,1\\right\\rbrace.\n$$\nThus the two-boost problem has a positive answer whenever the corresponding {\\em Lagrangian Rabinowitz Floer homology} $\\LRFH_*(\\A^{H_0-h}_{q_0,q_1})$ is well-defined and nontrivial. \n\nAs observed in~\\cite{CieliebakFrauenfelder2009}, the well-definedness of (Lagrangian) Rabinowitz Floer homology requires some hypothesis on the Hamiltonian $H$. A suitable class is formed by {\\em magnetic Hamiltonians} \n$$\n H(q,p) = \\frac12|p-A(q)|^2 - V(q)\n$$\nfor a magnetic potential $A\\in\\Omega^1(Q)$, a potential\\footnote{\nAs is customary in celestial mechanics, our $V$ is {\\em minus} the physical potential.}\n$V:Q\\to\\R$, and a Riemannian metric on $Q$. Under the assumption that the underlying manifold $Q$ is {\\em compact}, the Rabinowitz Floer homology of such Hamiltonians (with possibly noncompact magnetic field) has been studied in~\\cite{CieliebakFrauenfelderPaternain2010} for the periodic case and in~\\cite{Merry2014} for the Lagrangian case. An important dynamical quantity associated to such a Hamiltonian is its {\\em Ma\\~n\\'e critical value}. By a Theorem of W.\\,Merry~\\cite{Merry2014}, for $c$ above the Ma\\~n\\'e critical value the Lagrangian Rabinowitz Floer homology $\\LRFH_*(\\A^{H-c}_{q_0,q_1})$ is well-defined and nontrivial, hence the two-boost problem is solvable. \n\nReturning to celestial mechanics, let us consider the {\\em planar circular restricted 3-body problem}. Here two heavy bodies (the primaries) move under their mutual gravitational attraction on circles around their center of mass in a plane, and the third body (of negligible mass) moves in the same plane under the gravitational attraction by the primaries. In rotating coordinates, this system is described by the autonomous Hamiltonian\n\\begin{align*}\nH(q_1,q_2,p_1,p_2) &=\\frac{1}{2}(p_1^2+p_2^2)+ p_1q_2-p_2q_1 - V(q_1,q_2), \\end{align*}\nwhere $V$ is the sum of the (negative) Coulomb potentials of the two primaries (see e.g.~\\cite{FrauenfeldervanKoert2018}). Writing it as\n\\begin{align*}\nH(q_1,q_2,p_1,p_2) &= \\frac{1}{2}(|p_1+q_2|^2+|p_2-q_1|^2)-V_{\\rm eff}(q_1,q_2)\n\\end{align*}\nwith the effective potential $V_{\\rm eff}(q_1,q_2)=V(q_1,q_2)+\\frac12(q_1^2+q_2^2)$, we see that this is a magnetic Hamiltonian, with magnetic field generating the Coriolis force. \nIn contrast to the previous paragraph, the configuration space $\\R^2\\setminus\\{q_E,q_M\\}$ is now noncompact due to possible collisions with the primaries at positions $q_E,q_M$ and possible escapes to infinity. \nOur goal is to define and compute Lagrangian Rabinowitz Floer homology in this situation. \nSince the noncompactness due to collisions can be removed by Moser or Levi-Civita regularization (see~\\cite{FrauenfeldervanKoert2018}), we will focus our attention on the noncompactness at infinity in $\\R^2$. For this, we consider the quadratic Hamiltonian\n\\begin{equation}\\label{DefH0}\n\\begin{aligned}\nH_0(q_1,q_2,p_1,p_2) & :=\\frac{1}{2}(p_1^2+p_2^2)+ p_1q_2-p_2q_1 = \\frac{p_r^2}{2}+\\frac{p_\\theta^2}{2r^2}+p_\\theta,\n\\end{aligned}\n\\end{equation} \nwhere $(r,\\theta)$ are polar coordinates on the plane and $(p_r,p_\\theta)$ their conjugate momenta. \nWe introduce the following set of smooth functions constant outside a compact set:\n\\begin{equation}\\label{DefHset}\n\\mathcal{H}:=\\left\\lbrace h\\in C^\\infty(T^*\\R^2)\\ \\Big|\\ dh\\in C_c^\\infty(T^*\\R^2),\\quad h>0,\\quad h-dh(p\\partial_p)> 0\\right\\rbrace.\n\\end{equation}\n\nFor $q_0,q_1\\in \\R^2$ we define the Rabinowitz action functional $\\A^{H_0-h}_{q_0,q_1} : \\mathscr{H}_{q_0,q_1}\\times \\R \\to \\R$ as above, with $Q=\\R^2$ and $H-c$ replaced by $H_0-h$. Our first result is\n\n\\begin{thm}\\label{thm:LRFH}\nFix $q_0,q_1\\in \\R^2$ and let $H_0$ be as in \\eqref{DefH0}. For every $h \\in \\mathcal{H}$ the Lagrangian Rabinowitz Floer homology of $\\A^{H_0-h}_{q_0,q_1}$ is well defined.\nMoreover, for any two $h_0,h_1\\in \\mathcal{H}$ the Lagrangian Rabinowitz Floer homology of $\\A^{H_0-h_0}_{q_0,q_1}$ and $\\A^{H_0-h_1}_{q_0,q_1}$ are isomorphic.\n\\end{thm}\n\nWe denote the Lagrangian Rabinowitz Floer homology of $\\A^{H_0-h}_{q_0,q_1}$ by $\\LRFH_*(\\A^{H_0-h}_{q_0,q_1})$. Here we use coefficients in $\\Z_2=\\Z/2\\Z$ and a half-integer grading by Maslov indices (see~\\S\\ref{ss:grading}). \nMoreover, it comes with a canonical action filtration. Our next result computes the positive action part in the case $q_0\\neq q_1$. \n\n\\begin{thm}\\label{thm:posLRFH}\nFor the Hamiltonian $H_0$ in \\eqref{DefH0}, $h \\in \\mathcal{H}$, and any $q_0,q_1\\in \\R^2$ with $q_0\\neq q_1$, the positive Lagrangian Rabinowitz homology of $\\A^{H_0-h}_{q_0,q_1}$ is well defined\nand equal to\n$$\n\\LRFH_*^+\\left(\\A^{H_0-h}_{q_0,q_1}\\right)=\n\\begin{cases}\n\\mathbb{Z}_2 & \\textrm{for}\\quad *=1/2,\\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n$$ \n\\end{thm}\n\nThis shows that the two-boost problem at energy $0$ is solvable for $H_0-h$ with $h\\in\\mathcal{H}$. Note that $h(q,p)=c+V(q)$ belongs to $\\mathcal{H}$ for each constant $c>0$ and compactly supported potential $V\\geq 0$, so the two-boost problem is solvable for $H_0-V$ with such $V$ at each energy $c>0$.\n\nThe potential in the planar circular restricted 3-body problem (after regularisation) satisfies $V\\geq 0$, but it is not compactly supported. The following result allows us to deal with certain non-compactly supported potentials. \n\n\\begin{prop}\\label{prop:CritH=CritH1}\nLet $H_0$ be the Hamiltonian in \\eqref{DefH0}. Fix $q_0,q_1\\in \\R^2$ with $q_0\\neq q_1$ and a positive constant $c>0$.\nLet $V: \\R^2 \\to \\R$ be a nonnegative potential for which there exist $r_0>0$ and\n$(\\alpha, a) \\in \\{2\\}\\times \\left(0 ,\\frac{c^2}{4}\\right)\\cup (2, +\\infty) \\times \\R_+$,\nsuch that for $r>r_0$ we have \n $$\n V(r,\\theta)\\leq \\frac{a}{r^\\alpha}\\qquad \\textrm{and} \\qquad \\partial_r V (r, \\theta) \\geq -\\frac{a}{r^{\\alpha+1}}.\n $$\nThen there exists a function $V_0\\in \\mathcal{H}$ such that $\\Crit \\A^{H_0-V-c}_{q_0,q_1} = \\Crit \\A^{H_0-V_0}_{q_0,q_1}$.\n\\end{prop}\n\nIn view of Theorem~\\ref{thm:posLRFH}, the two-boost problem is therefore solvable for $H_0-V$ at energy $c>0$ for any $(V,c)$ as in Proposition~\\ref{prop:CritH=CritH1}. In particular, this holds for any $c>0$ and $V\\geq 0$ which decays at infinity as $r^{-\\alpha}$ with $\\alpha>2$. Note that this does {\\em not} cover the planar circular restricted 3-body problem, in which the potential decays as $r^{-1}$. We defer the extension to this case to future work.\n\n\\begin{rem}\\label{rem:contact}\nNote that for $h \\in \\mathcal{H}$ the Liouville vector field $p\\partial_p$ is transverse to $(H_0-h)^{-1}(0)$, so the zero level set of $H_0-h$ is of restricted contact type. Thus the solutions of the two-boost problem resulting from Theorem~\\ref{thm:posLRFH} and Proposition~\\ref{prop:CritH=CritH1} can also be interpreted as solutions of Arnold's chord conjecture (existence of a Reeb chord connecting two Legendrians) for certain noncompact contact manifolds. \n\\end{rem}\n\n\n\\begin{rem}\nWork by R.\\,Nicholls~\\cite{Nicholls2021} provides some evidence that, if defined, the Ma\\~n\\'e critical value for the restricted three-body problem should be zero. It would be interesting to make this rigorous by developing a notion of Ma\\~n\\'e critical value for magnetic Hamiltonians over noncompact base manifolds. \n\\end{rem}\n\n\n\\section{Lagrangian Rabinowitz Floer homology}\n\nLagrangian Rabinowitz Floer homology was first defined by Merry in \\cite{Merry2014} for virtually exact Lagrangian submanifolds and compact hypersurfaces of virtual contact type in symplectically aspherical symplectic manifolds. In his setting, the pull-back of the symplectic form to the universal cover of the symplectic manifold had to be exact and the pre-image of the chosen hypersurface in the universal cover had to be of contact type. The symplectic manifold and the Lagrangian submanifold could be non-compact, but the chosen hypersurface of virtual contact type had to be compact.\n\nOur aim is to use Lagrangian Rabinowitz Floer homology to analyse the Planar Circular Restricted Three Body Problem. Therefore, we are interested in the setting of a cotangent bundle $T^*Q$ with its standard symplectic form and a pair of fibres $T_{q_0}^*Q, T_{q_1}^*Q$ as the Lagrangians. In this section we will explain the definition of the Lagrangian Rabinowitz Floer homology in this setting. It is a far easier setting than the one in \\cite{Merry2014}, as the cotangent bundle is an exact manifold and the fibres are exact Lagrangians. Moreover, the level sets above the 4th and 5th Lagrange point of the Hamiltonian corresponding to the Planar Circular Restricted Three Body Problem are all of exact contact type (see Remark~\\ref{rem:contact}).\nThe only challenge of this setting is that all the energy level sets in the Restricted Three Body Problem are non-compact.\nRabinowitz Floer homology (for periodic orbits) of noncompact energy level sets of ``tentacular Hamiltonians'' on $\\R^{2n}$ has been defined in~\\cite{Pasquotto2017}. However, the Hamiltonian for the Planar Circular Restricted Three Body Problem does not belong to this class, so new arguments are needed for this setting. \n\n \n\\subsection{The Lagrangian Rabinowitz action functional}\n\nThroughout this section, $Q$ denotes a smooth oriented $n$-dimensional manifold, and $T^*Q$ its cotangent bundle equipped with the exact symplectic form $\\omega=d\\lambda$ for the canonical $1$-form $\\lambda:=p\\,dq$. \nFix $q_0,q_1 \\in Q$ and define the space of paths\n$$\n\\mathscr{H}_{q_0,q_1}:= \\left\\lbrace v \\in W^{1,2}([0,1], T^*Q)\\ \\big|\\ v(i)\\in T^*_{q_i}Q\\quad \\textrm{for}\\quad i=0,1\\right\\rbrace.\n$$\nConsider a Hamiltonian $H: T^*Q \\to \\R$ with regular level set $H^{-1}(0)$. \nThe {\\em Lagrangian Rabinowitz action functional} $\\A^H_{q_0,q_1}:\\mathscr{H}_{q_0,q_1}\\times \\R\\to\\R$ associated to $H$ is defined as\n$$\n\\A^H_{q_0,q_1}(v,\\eta):= \\int_0^1 \\lambda(\\partial_t v)dt - \\eta \\int_0^1 H(v(t))dt.\n$$\n\n\\begin{rem}\nWe have a natural bijection $\\mathscr{H}_{q_0,q_1}\\ni v \\mapsto \\ol v\\in \\mathscr{H}_{q_1,q_0}$ with $\\ol v(t):=v(1-t)$. Then\n$$\n\\A^H_{q_0,q_1}(v, \\eta) = -\\A^H_{q_1,q_0}(\\ol v, -\\eta)\\qquad\\forall\\ (v, \\eta) \\in \\mathscr{H}_{q_0,q_1}\\times\\R.\n$$\n\\end{rem}\n\nThe derivative of $\\A^H_{q_0,q_1}$ in direction $(\\xi, \\sigma) \\in T_v \\mathscr{H}_{q_0,q_1}\\times T_\\eta \\R$ equals\n\\begin{equation}\\label{eq:dAH}\nd\\A^H_{q_0,q_1}(v,\\eta)[\\xi,\\sigma] = \\int_0^1 \\omega (\\xi,\\partial_tv -\\eta X_H) - \\sigma\\int_0^1 H(v)dt.\n\\end{equation}\nHere $X_H$ is the Hamiltonian vector field defined by $dH=-i_{X_H}\\om$. \nConsequently, $(v,\\eta) \\in \\Crit (\\A^H_{q_0,q_1})$ if and only if it satisfies\n$$\n\\partial_t v= \\eta X_H(v)\\quad \\textrm{and}\\quad v(t)\\in H^{-1}(0)\\quad \\forall\\ t\\in [0,1].\n$$\nThus we can have three types of critical points: \n\\begin{itemize}\n\\item $\\eta>0$ and $\\wt v(t):=v(t/\\eta)$ is a Hamiltonian chord (i.e., an integral curve of $X_H$) on $H^{-1}(0)$ from $T^*_{q_0}Q$ to $T_{q_1}^*Q$;\n\\item $\\eta<0$ and $\\wt v(t):=v(t/\\eta)$ is a Hamiltonian chord on $H^{-1}(0)$ from $T^*_{q_1}Q$ to $T_{q_0}^*Q$; \n\\item $\\eta=0$ and $v$ is a constant path in $T^*_{q_0}Q\\cap T_{q_1}^*Q\\cap H^{-1}(0)$ (which can only occur if $q_0=q_1$). \n\\end{itemize}\nIn particular, if $T^*_{q_0}Q \\cap H^{-1}(0)=\\emptyset$ or $T_{q_1}^*Q \\cap H^{-1}(0)=\\emptyset$, then $ \\Crit (\\A^H_{q_0,q_1})=\\emptyset$. Therefore, from now on we will assume that \n\\begin{equation}\\label{T*QcapHnonempty}\nT^*_{q_0}Q \\cap H^{-1}(0)\\neq\\emptyset,\\quad T^*_{q_1}Q \\cap H^{-1}(0)\\neq\\emptyset,\\quad\\textrm{and } T^*_{q_0}Q\\pitchfork H^{-1}(0) \\text{ if }q_0=q_1.\n\\end{equation}\nIn order to construct Lagrangian Rabinowitz Floer homology we want the critical set of the action functional to be bounded in $L^\\infty$. It is possible to define Lagrangian Rabinowitz Floer homology without this assumption, but the construction is much more challenging, so we postpone it for future projects. In fact, to construct Lagrangian Rabinowitz Floer homology we will need the boundedness of the critical set of the action functional to persist under compact perturbations of the Hamiltonian. We formalize this in the following definition. \n\n\\begin{define}\nConsider a Hamiltonian $H: T^*Q \\to \\R$, such that $H^{-1}(0)$ is noncompact. Fix $q_0,q_1\\in T^*Q$ satisfying \\eqref{T*QcapHnonempty}. Let $K\\subseteq T^*Q$ be a compact set and let \n$$\n\\mathcal{H}\\subseteq \\{ h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\},\n$$\nbe an open neighbourhood of $0$ in $\\{ h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\}$.\nWe say that the associated Lagrangian Rabinowitz Floer functional $\\A^H_{q_0,q_1}:\\mathscr{H}_{q_0,q_1}\\times \\R\\to \\R$ has critical set \\emph{continuously compact} in $(K,\\mathcal{H})$ if\n$$\n\\forall\\ h\\in\\mathcal{H}\\quad \\textrm{and}\\quad\\forall\\ (v,\\eta) \\in \\Crit \\A^{H+h}_{q_0,q_1} \\quad \\textrm{we have}\\quad v([0,1])\\subseteq K.\n$$\n\\end{define}\n\n\\subsection{Grading}\\label{ss:grading}\n\nIn~\\cite{CieliebakFrauenfelder2009}, Rabinowitz Floer homology (for periodic orbits) is equipped with a shifted integer grading under the hypothesis that the ambient symplectic manifold has vanishing first Chern class. In this subsection we indicate how this carries over to the Lagrangian setting; see~\\cite{Merry2014} for details. \n\nConsider an exact symplectic manifold $(V,\\om=d\\lambda)$ of dimension $2n$ and a Hamiltonian $H:V\\to\\R$ with regular level set $\\Sigma=H^{-1}(0)$ such that $\\xi=\\ker(\\lambda|_\\Sigma)$ is a contact structure. Consider in addition two Lagrangian submanifolds $L_0,L_1\\subset V$ transverse to $\\Sigma$ with $\\lambda|_{L_i}=0$. We assume that $L_0$ and $L_1$ are either disjoint or equal. \nThe condition of vanishing first Chern class in~\\cite{CieliebakFrauenfelder2009} gets replaced by vanishing of the relative first Chern class $c_1(V,L_0\\cup L_1)$. The transversal Conley-Zehnder index in~\\cite{CieliebakFrauenfelder2009} gets replaced by the {\\em transverse Maslov index} defined a follows.\nLet $(v,\\eta)$ be a critical point of the Rabinowitz action functional $\\A^H$ on the space of paths from $L_0$ to $L_1$. Pick a symplectic trivialization $v^*\\xi\\cong [0,1]\\times\\R^{2n-2}$ sending $T_{v(i)}L_i\\cap\\xi_{v(i)}$ to $\\R^{n-1}$ for $i=0,1$. In this trivialization, the image of $T_{v(i)}L_i\\cap\\xi_{v(i)}$ under the linearized flow of $\\eta X_H$ gives a path $\\Lambda:[0,1]\\to\\mathcal{L}_{n-1}$ in the Grassmannian $\\mathcal{L}_{n-1}$ of Lagrangian subspaces of $(\\R^{2n-2},\\om_{\\rm std})$ with $\\Lambda(0)=\\R^{n-1}$. Let $\\mu^{\\rm tr}(v,\\eta)\\in\\frac12\\Z$ be the Maslov index of the path $\\Lambda$ defined by Robbin and Salamon in~\\cite{Robbin1993}. By vanishing of $c_1(V,L_0\\cup L_1)$ this is independent of the choices. In the case $\\eta=0$ (which occurs only if $L_0=L_1$), $v$ is a critical point of an auxiliary Morse function $f$ on $L_0\\cap\\Sigma$. Its {\\em signature index} is ${\\rm ind}^\\sigma_f(v,0) := -\\frac12{\\rm sign}\\,{\\rm Hess}_f(v)$, and we set it to zero if $\\eta\\neq 0$. According to~\\cite[Appendix A]{CieliebakFrauenfelder2009}, the {\\em Maslov index}\n$$\n \\mu(v,\\eta) := \\mu^{\\rm tr}(v,\\eta) + {\\rm ind}^\\sigma_f(v,\\eta) \\in\\frac12\\Z\n$$\ndefines a half-integer grading on the Rabinowitz Floer complex with respect to which the differential has degreee $-1$.\n \nThis discussion applies to the setting in this section where $V=T^*Q$ and $L_i=T_{q_i}^*Q$ for an oriented $n$-dimensional manifold $Q$. \n\n\\begin{ex}\\label{ex:Maslov}\nConsider the free Hamiltonian $H_{\\bullet}(q,p):=\\frac{1}{2}|p|^2$ on $T^*\\R^n$ and fix an energy $c>0$. Assume first that $q_0\\neq q_1\\in\\R^n$. Then $\\A^{H_{\\bullet}-c}_{q_0,q_1}$ has exactly two critical points $(v_\\pm,\\eta_\\pm)$ given by \n$$\n v_\\pm(t)=\\Bigl((1-t)q_0+tq_1,\\frac{q_1-q_0}{\\eta_\\pm}\\Bigr),\\qquad \\eta_\\pm=\\pm\\frac{\\sqrt{2c}}{|q_1-q_0|}.\n$$\nThe linearized flow of $\\eta_\\pm X_{H_\\bullet}$ along $v_\\pm$ equals $\\Phi_t=\\begin{pmatrix}\\one & t\\eta_\\pm\\one \\\\ 0 & \\one\\end{pmatrix}$, so the corresponding path of Lagrangian subspaces $\\Lambda:[0,1]\\to\\mathcal{L}_{n-1}$ is given by $\\Lambda(t)=\\Phi_t(\\R^{n-1})={\\rm graph}\\,A(t)$ with $A(t)=t\\eta_\\pm\\one$. Thus the (Localization) property in~\\cite{Robbin1993} yields \n$$\n \\mu(v_\\pm,\\eta_\\pm) = \\mu^{\\rm tr}(v_\\pm,\\eta_\\pm) = \\frac12{\\rm sign}\\,A(1) - \\frac12{\\rm sign}\\,A(0) = \\pm\\frac{n-1}{2}.\n$$\nIn the case $q_0=q_1$, let $f$ be a Morse function on the $(n-1)$-sphere $T_{q_0}Q\\cap H_\\bullet^{-1}(c)$ with exaxctly two critical points, the maximum $v_+$ and the minimum $v_-$. Then $\\A^{H_{\\bullet}-c}_{q_0,q_0}$ has exactly two critical points $(v_\\pm,0)$ of index\n$$\n \\mu(v_\\pm,0) = {\\rm ind}^\\sigma_f(v_\\pm,0) = \\pm\\frac{n-1}{2}.\n$$\nFor $q_0\\neq q_1$ this immediately implies\n$$\n\\LRFH_*^\\pm\\left(H_\\bullet^{-1}(c), T^*_{q_0}\\R^n,T^*_{q_1}\\R^n\\right)=\n\\begin{cases}\n\\mathbb{Z}_2 & \\textrm{for}\\quad *=\\pm\\frac{n-1}{2},\\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n$$\nMoreover, for $n\\neq 3$ and any $q_0,q_1$ (equal or not) we obtain\n$$\n\\LRFH_*\\left(H_\\bullet^{-1}(c), T^*_{q_0}\\R^n,T^*_{q_1}\\R^n\\right)=\n\\begin{cases}\n\\mathbb{Z}_2 & \\textrm{for}\\quad *=\\frac{n-1}{2}\\textrm{ and }*=-\\frac{n-1}{2},\\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n$$\n(This also holds for $n=2$, but an additional argument is needed to show that the two critical points of index $\\pm1/2$ do not cancel in homology).\n\\end{ex}\n\n\\begin{rem}\\label{rem:Maslov}\nThe previous example generalizes to the free Hamiltonian $H_{\\bullet}(q,p)=\\frac{1}{2}|p|^2$ on the cotangent bundle of an $n$-dimensional Riemannian manifold $(Q,g)$. For $q_0\\neq q_1$ and $(v,\\eta)\\in \\Crit\\A^{H_{\\bullet}-c}_{q_0,q_1}$, the path $v$ projects onto a geodesic $\\bar v$ from $q_0$ to $q_1$ and Proposition 6.38 in~\\cite{RobbinSalamon1995} yields\n\\begin{equation*} \\mu(v,\\eta) = {\\rm ind}(\\bar v) + {\\rm sign}(\\eta)\\frac{n-1}{2},\n\\end{equation*}\nwhere ${\\rm ind}(\\bar v)$ is the Morse index of the geodesic $\\bar v$ (i.e., the number of conjugate points along $\\bar v$). \n\\end{rem}\n\n \n\\subsection{Floer trajectories}\\label{sec:FloerTrajec}\n\nIn this subsection we introduce the fundamental notion for constructing any Floer-type homology - the Floer trajectories. We start by equipping the space $\\mathscr{H}_{q_0,q_1}$ with a metric. An \\emph{almost complex structure} $J$ on a manifold $M$ is a bundle endomorphism $J: TM \\to TM$ satisfying $J^2 = -\\operatorname{Id}$. An almost complex structure $J$ on a symplectic manifold $(M, \\omega)$ is called \\emph{compatible} if $\\omega(\\ \\cdot\\ ,J \\cdot\\ )$ defines a Riemannian metric on $M$. Denote by $\\mathcal{J}(M,\\omega)$ the space of all compatible almost complex structures on $(M,\\omega)$ with the $C^\\infty$-topology.\nA linear algebra argument \\cite[Prop. 13.1]{Silva2001} shows that $\\mathcal{J}(M,\\omega)$ is contractible.\n\nA smooth $2$-parameter family $\\{J_{t,\\eta}\\}_{(t,\\eta)\\in[0,1]\\times \\R}$ of compatible almost complex structures on $(T^*Q, \\omega)$ defines an $L^2$-metric on $\\mathscr{H}_{q_0,q_1}\\times \\R$ by\n$$\n\\langle (\\xi_1, \\sigma_1), (\\xi_2, \\sigma_2)\\rangle := \\int_0^1 \\omega(\\xi_1(t),J_{t,\\eta}(v(t))\\xi_2(t) )+\\sigma_1\\sigma_2.$$\nfor $(\\xi_i,\\sigma_i)\\in T_{(v,\\eta)}(\\mathscr{H}_{q_0,q_1}\\times \\R)$.\nThe gradient of the Lagrangian Rabinowitz action functional $\\A^H_{q_0,q_1}$ with respect to this metric equals\n$$\n\\nabla \\A^H_{q_0,q_1} (v,\\eta)= \\left( \\begin{array}{c}\n-J_{t,\\eta}(v(t)) (\\partial_tv -\\eta X_H) \\\\ - \\int H(v)dt.\n\\end{array}\n\\right)\n$$\nFix an open subset $\\mathcal{V}\\subseteq T^*Q$ and $\\mathbb{J}\\in \\mathcal{J}(T^*Q,\\omega)$.\nWe denote by $\\mathcal{J}(\\mathcal{V}, \\mathbb{J})$ the set of all smooth maps\n$$\n[0,1]\\times \\mathbb{R}\\to \\mathcal{J}(T^*Q,\\omega),\\qquad (t,\\eta)\\mapsto J_{t,\\eta}\n$$\nsatisfying\n\\begin{align*}\nJ_{t,\\eta}(x) = \\mathbb{J}(x)\\text{ for } x\\notin \\mathcal{V}\n\\quad\\textrm{and}\\quad \\sup_{(t,\\eta)\\in [0,1]\\times \\mathbb{R}}\\|J_{t,\\eta}\\|_{C^k}<+\\infty \\text{ for all } k\\in \\mathbb{N}.\n\\end{align*}\nFix Hamiltonians $H_\\pm$ and $J_\\pm\\in\\mathcal{J}(\\mathcal{V},\\mathbb{J})$. A {\\em homotopy} from $(H_-,J_-)$ to $(H_+,J_+)$ is a smooth family $\\Gamma = \\{(H_s, J_s)\\}_{s\\in \\R}$ of Hamiltonians and compatible almost complex structures $J_s\\in \\mathcal{J}(\\mathcal{V},\\mathbb{J})$ which agrees with $(H_-,J_-)$ for $s\\leq s_-$ and with $(H_+,J_+)$ for $s\\geq s_+$, with some $s_\\pm\\in\\R$. \nA solution $u: \\R \\to \\mathscr{H}_{q_0,q_1} \\times \\R$ to the gradient flow equation $\\partial_s u = \\nabla \\A^{H_s}_{q_0,q_1}(u)$ is called a \\emph{Floer trajectory}. In other words, a Floer trajectory $u=(v,\\eta) \\in W^{1,2}(\\R\\times[0,1], T^*Q)\\times W^{1,2}(\\R)$ is a solution to the equations\n\\begin{equation}\\label{FloerEq}\n\\begin{aligned}\n\\partial_s v(s,t) & = - J_{s, t,\\eta}(v(s,t)) (\\partial_sv(s,t)-\\eta(s)X_{H_s}(v(s,t))),\\\\\n\\partial_s \\eta (s) & = - \\int_0^1 H_s\\circ v(s,t)dt,\n\\end{aligned}\n\\end{equation}\n$$\nv(s,0) \\in T^*_{q_0}Q \\quad \\textrm{and} \\quad v(s,1) \\in T^*_{q_1}Q \\quad \\forall\\ s\\in \\R.\n$$\nFor $(x_-, x_+) \\in \\Crit(\\A^{H_-}_{q_0,q_1})\\times \\Crit \\A^{H_+}_{q_0,q_1}$ we denote the space of Floer trajectories from $x_-$ to $x_+$ by\n$$\n\\F_\\Gamma(x_-, x_+):=\n\\left\\lbrace\\begin{array}{c|c}\n& \\partial_s u = \\nabla \\A^{H_s}_{q_0,q_1}(u),\\\\\n\\hspace*{-0.2cm}\\smash{\\raisebox{.5\\normalbaselineskip}{ $u: \\R \\to \\mathscr{H}_{q_0,q_1} \\times \\R$}}& \\lim_{s\\to \\pm \\infty}u(s)=x_\\pm.\n\\end{array}\\right\\rbrace\n$$\nIn case the homotopy $\\Gamma$ is constant in $s$, i.e. $H_s\\equiv H$ and $J_{s,t,\\eta}\\equiv J \\in \\mathcal{J}(\\mathcal{V},\\mathbb{J})$ for some Hamiltonian $H$ and a compatible almost complex structure $J$, we denote $\\F_{H,J}(x_-,x_+):=\\F_\\Gamma(x_-, x_+)$.\n\nMoreover, for every pair $(a,b)\\in \\R^2$ we denote \n\\begin{equation}\\label{DefM(a,b)}\n\\mathcal{M}^\\Gamma(a,b):=\\left\\lbrace\\begin{array}{c|c}\n& (x_-,x_+)\\in \\Crit\\A^{H_-}_{q_0,q_1}\\times \\Crit\\A^{H_+}_{q_0,q_1},\\\\\n\\hspace*{-0.2cm}\\smash{\\raisebox{.5\\normalbaselineskip}{ $u\\in \\F_\\Gamma(x_-,x_+)$}}& \\A^{H_-}_{q_0,q_1}(x_-)\\geq a, \\quad \\A^{H_+}_{q_0,q_1}(x_+)\\leq b.\n\\end{array}\\right\\rbrace\n\\end{equation}\nAnalogously, we denote $\\mathcal{M}^{H,J}(a,b):=\\mathcal{M}^\\Gamma(a,b)$ whenever the homotopy $\\Gamma$ is constant in $s$ and equal to the pair $(H,J)$.\n\nIf s homotopy $\\Gamma$ is constant in $s$, then the action is increasing along Floer trajectories. However, for a nonconstant homotopy $\\Gamma$ this need not be the case. To deal with this phenomenon, we introduce a condition that ensures that the action cannot decrease indefinitely along a Floer trajectory.\nWe say a that homotopy $\\Gamma:= \\{(H_s, J_s)\\}_{s\\in \\R}$ with $H_{\\pm}:=\\lim_{s\\to \\pm \\infty}H_s$ satisfies the \\emph{Novikov finiteness condition} if for all $a,b\\in \\R$ we have\n\\begin{equation}\\label{Novikov}\n\\begin{aligned}\n\\inf\\left\\lbrace\\begin{array}{c|c}\n& y \\in \\Crit\\A^{H_+}_{q_0,q_1}, \\quad \\exists\\ x\\in \\Crit \\A^{H_-}_{q_0,q_1},\\\\\n\\hspace*{-0.2cm}\\smash{\\raisebox{.5\\normalbaselineskip}{ $\\A^{H_+}_{q_0,q_1}(y)$}}& \\A^{H_-}_{q_0,q_1}(x)\\geq a, \\quad \\F_\\Gamma(x,y)\\neq \\emptyset.\n\\end{array}\\right\\rbrace & > -\\infty,\\\\\n\\sup\\left\\lbrace\\begin{array}{c|c}\n& x \\in \\Crit\\A^{H_-}_{q_0,q_1}, \\quad \\exists\\ y\\in \\Crit \\A^{H_+}_{q_0,q_1},\\\\\n\\hspace*{-0.2cm}\\smash{\\raisebox{.5\\normalbaselineskip}{ $\\A^{H_-}_{q_0,q_1}(x)$}}& \\A^{H_+}_{q_0,q_1}(x)\\leq b, \\quad \\F_\\Gamma(x,y)\\neq \\emptyset.\n\\end{array}\\right\\rbrace &< +\\infty.\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Defining Lagrangian Rabinowitz Floer homology}\n\nIn this subsection we will recall the construction of the Lagrangian Rabinowitz Floer homology from \\cite{Merry2014}, using standard Floer techniques introduced in Floer's seminal paper \\cite{Floer1989} and the techniques typical for the setting of the Rabinowitz action functional from the first two author's paper \\cite{CieliebakFrauenfelder2009}. More precisely, we will prove the following theorem.\n\n\\begin{thm}\\label{thm:DefLRFH}\nConsider a cotangent bundle $(T^*Q,\\omega)$ with its standard symplectic form and a Hamiltonian $H:T^*Q\\to \\R$ with regular level set $H^{-1}(0)$.\nFix a pair $q_0,q_1\\in Q$ such that both sets $H^{-1}(0)\\cap T^*_{q_0}Q$ and $H^{-1}(0)\\cap T^*_{q_1}Q$ are compact and nonempty. Fix a compatible almost complex structure $\\mathbb{J}\\in \\mathcal{J}(T^*Q,\\omega)$.\nAssume that there exists\na compact subset $K$, an open subset $\\mathcal{V}$ satisfying $K\\subseteq \\mathcal{V}\\subseteq T^*Q$, and an open neighbourhood $\\mathcal{H}\\subseteq \\{h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\}$ of $0$\nsuch that:\n\\begin{enumerate}\n\\item for all $h\\in \\mathcal{H}$ the Hamiltonian $H+h$ satisfies \\eqref{T*QcapHnonempty};\n\\item the Lagrangian Rabinowitz Floer functional $\\A^H_{q_0,q_1}$ has critical set continuously compact in $(K,\\mathcal{H})$;\n\\item for all $h_0,h_1 \\in \\mathcal{H}$ and $J_0, J_1 \\in \\mathcal{J}(\\mathcal{V}, \\mathbb{J})$ every homotopy $\\Gamma=\\{(H+h_s, J_s)\\}_{s\\in \\R}$ \nfrom $(H+h_0,J_0)$ to $(H+h_1,J_1)$ satisfies the Novikov finiteness condition \\eqref{Novikov}\nand for all $a, b \\in \\R$ the space of Floer trajectories $\\mathcal{M}^\\Gamma(a,b)$ is bounded in the $L^\\infty$-norm.\n\\end{enumerate}\nThen for every $h\\in\\mathcal{H}$ the Lagrangian Rabinowitz Floer homology $\\LRFH_*(\\A^{H+h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*(\\A^{H}_{q_0,q_1})$.\n\\end{thm}\n\n\\begin{proof}\nWe will define Lagrangian Rabinowitz Floer homology in the case $q_0\\neq q_1$, and then briefly explain the case $q_0=q_1$. Therefore, we first assume $q_0\\neq q_1$.\n\nThe first step is to show that for $q_0\\neq q_1$ and generic $h\\in \\mathcal{H}$ the Lagrangian Rabinowitz action functional $\\A^{H+h}_{q_0,q_1}$ is Morse, i.e., its Hessian $\\nabla^2\\A^{H+h}_{q_0,q_1}(x)$ has trivial kernel for all $x \\in \\Crit \\A^{H+h}_{q_0,q_1}$. \nObserve that for $q_0\\neq q_1$ we we have $\\Crit \\A^{H+h}_{q_0,q_1} \\cap \\left( \\mathscr{H}_{q_0,q_1}\\times\\{0\\}\\right)=\\emptyset$. By a standard Sard-Smale argument \\cite[Theorem A.51]{McDuff2012}, there exists a residual subset $\\mathcal{H}^{\\reg}\\subseteq \\mathcal{H}\\subseteq C_0^\\infty(K)$ such that for all $h \\in \\mathcal{H}^{\\reg}$ we have\n\\begin{equation*}D \\phi^\\eta_{H+h} (T^*_{q_0}Q) \\pitchfork T^*_{q_1}Q\\qquad\\forall\\ (v, \\eta)\\in \\Crit\\A^{H+h}_{q_0,q_1},\n\\end{equation*}\nwhere $\\phi^t_{H+h}$ denotes the Hamiltonian flow of $X_{H+h}$.\nThe condition is equivalent to the triviality of the kernel of the Hessian $\\nabla^2 \\A^{H+h}_{q_0,q_1}(x)$ for all $x \\in \\Crit \\A^{H+h}_{q_0,q_1}$.\nA straightforward consequence of the Morse property of the Lagrangian Rabinowitz action functional and continuous compactness of the critical set of $\\A^{H}_{q_0,q_1}$ is that the critical set of $\\A^{H+h}_{q_0,q_1}$ is finite for each $h \\in \\mathcal{H}^{\\reg}$.\n\nFix $h\\in \\mathcal{H}^{\\reg}$. Since $K \\subseteq \\mathcal{V}$, by continuous compactness of the critical set of $\\A^{H}_{q_0,q_1}$ in $(K,\\mathcal{H})$ for every $J\\in \\mathcal{J}(\\mathcal{V}, \\mathbb{J})$, we have\n\\begin{equation}\\label{nonEmpty}\nv(\\R\\times[0,1])\\cap \\mathcal{V}\\neq \\emptyset\\quad \\forall\\ u=(v,\\eta)\\in \\mathscr{F}_{H+h,J}(x,y)\\quad\\forall\\ x,y \\in \\Crit\\A^{H+h}_{q_0,q_1}.\n\\end{equation}\nTherefore, we can apply the Sard-Smale argument in \\cite[Thm. A.51]{McDuff2012} to conclude that there exists a residual set $\\mathcal{J}^{\\reg}_h\\subseteq \\mathcal{J}(T^*Q,\\omega,\\mathcal{V})$ such that for all $x,y\\in \\Crit\\A^{H+h}_{q_0,q_1}$ and all $J \\in \\mathcal{J}^{\\reg}_h$ the space of Floer trajectories $\\mathscr{F}_{H+h,J}(x,y)$ is a transversely cut out smooth manifold. \n\nLet us now fix $h\\in \\mathcal{H}^{\\reg}$ and $J\\in \\mathcal{J}^{\\reg}_h$.\nFor every $x,y\\in \\Crit\\A^{H+h}_{q_0,q_1}$ there is a natural $\\R$-action on $\\F_{H+h,J}(x,y)$ given by\n$$\n\\F_{H+h,J}(x,y) \\times \\R \\ni (u, s) \\longmapsto u(s + \\cdot) \\in \\F_{H+h,J}(x,y).\n$$\nDenote the quotient space $\\overline{\\F}_{H+h,J}(x,y):=\\F_{H+h,J}(x,y)/ \\R$.\nBy \\cite[Thm. 2.23]{Merry2014}, for all $x,y\\in \\Crit\\A^{H+h}_{q_0,q_1}$ with $x\\neq y$ the quotient $\\overline{\\F}_{H+h,J}(x,y)$ is also a smooth manifold and its dimension is given by\n$$\n \\dim \\overline{\\F}_{H+h,J}(x,y)= \\mu (y) - \\mu (x) -1.\n$$\nBy condition 3 in Theorem~\\ref{thm:DefLRFH}, for all $x,y\\in \\Crit\\A^{H+h}_{q_0,q_1}$ the space of Floer trajectories $\\mathscr{F}_{H+h,J}(x,y)$ is bounded in the $L^\\infty$-norm. Now standard compactness arguments (see \\cite[Prop. 3b]{Floer1989} or \\cite[Thm. 9.1.7]{Audin2014})\nimply that $\\overline{\\F}_{H+h,J}(x,y)$ is compact up to breaking in the sense of Floer. \nIf $\\mu(y)-\\mu(x)=1$ this means that $\\overline{\\F}_{H+h,J}(x,y)$ is a finite set of points \\cite[Thm. 9.2.1]{Audin2014}. \n\nNow we are ready to define Lagrangian Rabinowitz Floer homology. For $h\\in \\mathcal{H}^{\\reg}$ and $k\\in\\Z$ let $CF_k(\\A^{H+h}_{q_0,q_1})$ be the $\\mathbb{Z}_2$-vector space of formal sums of the form $\\sum_{x\\in S}x$, where $S\\subseteq \\Crit\\A^{H+h}_{q_0,q_1}$ is a (possibly infinite) set satisfying $\\mu(x)=k$ for all $x\\in S$ and the \\emph{Novikov finiteness condition}\n$$\n\\# \\{ x\\in S\\ |\\ \\A^{H+h}_{q_0,q_1}(x)>a\\}<+\\infty\\qquad\\forall\\ a\\in\\mathbb{R}.\n$$\nNow we fix an almost complex structure $J\\in \\mathcal{J}^{\\reg}_h$ and turn $CF(\\A^{H+h}_{q_0,q_1})$ into a chain complex. For $x,y\\in \\Crit\\A^{H+h}_{q_0,q_1}$ with $\\mu(y)-\\mu(x)=1$ we set\n$$\n n(x,y):=\\# \\overline{\\F}_{H+h,J}(x,y) \\operatorname{mod}2\\in \\mathbb{Z}_2.\n$$\nThe Floer boundary operator $\\partial_k^J: CF_k(\\A^{H+h}_{q_0,q_1})\\to CF_{k-1}(\\A^{H+h}_{q_0,q_1})$ is the linear map defined on generators by \n$$\n \\partial_k^J y := \\sum_x n(x,y)x.\n$$\nCompactness up to breaking implies that $\\partial_k^J\\circ\\partial_{k+1}^J=0$, so we can define the {\\em Lagrangian Rabinowitz Floer homology}\n$$\n \\LRFH_k(\\A^{H+h}_{q_0,q_1};J) := \\ker \\partial_k^J / \\operatorname{im}\\partial_{k+1}^J.\n$$\nA standard continuation argument shows that the Lagrangian Rabinowitz Floer homology does not depend on the choice of $h\\in \\mathcal{H}^{\\reg}$ and $J\\in \\mathcal{J}^{\\reg}_h$. For this, let $h_i\\in \\mathcal{H}^{\\reg}$ and $J_i\\in \\mathcal{J}^{\\reg}_{h_i}$ for $i=0,1$ be given. Pick a homotopy $\\Gamma = \\{(H+h_s, J_s)\\}_{s\\in \\R}$ connecting $(H+h_0,J_0)$ to $(H+h_1,J_1)$ as in condition 3 of Theorem~\\ref{thm:DefLRFH}. \nAs before, it follows that for generic $\\Gamma$ and any pair $(x,y)\\in \\Crit \\A^{H+h_0}_{q_0,q_1} \\times \\Crit \\A^{H+h_1}_{q_0,q_1}$ the associated space of Floer trajectories $\\F_\\Gamma(x,y)$ is a smooth manifold of dimension $\\mu(y)-\\mu(x)$ which is compact up to breaking. In particular, if $\\mu(x)=\\mu(y)$ then $\\F_\\Gamma(x,y)$ is a finite set of points and we set $m(x,y):= \\# \\F_\\Gamma(x,y)\\mod 2$. We define a linear map $\\phi^\\Gamma : (CF_*(\\A^{H+h_1}_{q_0,q_1}),\\p^{J_1})\\to (CF(\\A^{H+h_0}_{q_0,q_1}),\\p^{J_0})$ on generators by\n$$\n \\phi^\\Gamma(y) := \\sum_{x\\in \\Crit \\A^{H+h_0}_{q_0,q_1}}m(x,y)x.\n$$\nHere the Novikov finiteness condition \\eqref{Novikov} on $\\Gamma$ ensures that $\\phi^\\Gamma\\left( CF(\\A^{H+h_1}_{q_0,q_1})\\right)\\subseteq CF(\\A^{H+h_0}_{q_0,q_1})$. Compactness up to breaking implies that $\\phi^\\Gamma$ is a chain map, and composition with a homotopy in the opposite direction shows that $\\phi^\\Gamma$ induces an isomorphism on homology (see \\cite[Chapter 11]{Audin2014} for details on this standard argument).\n\nSince $\\LRFH_k(\\A^{H+h}_{q_0,q_1};J)$ does not depend on $J\\in \\mathcal{J}^{\\reg}_h$, we can drop $J$ from the notation and write is as $\\LRFH_k(\\A^{H+h}_{q_0,q_1})$. Since this homology is the same vector space for all $h\\in \\mathcal{H}^{\\reg}$, we can unambiguously extend it as this vector space to all $h\\in \\mathcal{H}^{\\reg}$. This proves the theorem in the case $q_0\\neq q_1$. \n\nIn the case $q_0=q_1$, by condition 1 for each $h\\in\\mathcal{H}$ we have critical points $(v,0)\\in\\Crit\\A^{H+h}_{q_0,q_0}$ where $v$ is a constant path in the nonempty transverse intersection $S_h:=T^*_{q_0}Q\\cap (H+h)^{-1}(0)$. If $S_h$ has positive dimension, then the functional $\\A^{H+h}_{q_0,q_0}$ will only be Morse-Bott for generic $h\\in\\mathcal{H}$. This phenomenon is well known in Rabinowitz Floer theory and can be dealt with by picking a Morse function on $S_h$ and counting gradient flow lines with cascades, see~\\cite{CieliebakFrauenfelder2009}. With this understood, the rest of the argument works as in the previous case. \n\\end{proof}\n\n\\begin{cor}\nConsider a cotangent bundle $(T^*Q,\\omega)$ with its standard symplectic form and a Hamiltonian $H:T^*Q\\to \\R$ such that $H^{-1}(0)$ is of exact contact type. Assume that there exists an exhausting sequence of compact sets $\\{K_n\\}_{n\\in\\mathbb{N}}$, $K_n\\subseteq K_{n+1}\\subseteq T^*Q$, $\\bigcup_{n\\in\\mathbb{N}}K_n=T^*Q$ and an open neighbourhood $\\mathcal{H}$ of $0$ in $\\{h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_c^\\infty(T^*Q)\\}$ such that for every $n\\in \\mathbb{N}$ the sets $K_n$ and $\\mathcal{H}_n=\\{h\\in\\mathcal{H}\\ |\\ dh\\in C_0^\\infty(K_n)\\}$ satisfy the assumptions of Theorem \\ref{thm:DefLRFH}. Then for any $h\\in\\mathcal{H}$ the Lagrangian Rabinowitz Floer homology $\\LRFH_*(\\A^{H+h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*(\\A^{H}_{q_0,q_1})$.\n\\end{cor}\n\n\\subsection{Positive Lagrangian Rabinowitz Floer homology}\n\nThe action functional $\\A^{H}_{q_0,q_1}$ provides an $\\R$-filtration on $CF_*(\\A^{H}_{q_0,q_1})$ by\n$$CF_*^{\\leq a}\\left(\\A^{H}_{q_0,q_1}\\right)\\coloneqq \\left\\lbrace {\\textstyle \\sum_{x\\in S}x}\\in CF_*\\left(\\A^{H}_{q_0,q_1}\\right) \\left|\\ \\sup_{x\\in S}\\A^{H}_{q_0,q_1}(x) \\leq a\\right.\\right\\rbrace.\n$$Since Floer trajectories are defined by the $L^2$-gradient of the action functional, the boundary operator does not increase the action, i.e.\\\n\\begin{equation}\\label{filtration}\n\\partial\\left( CF^{\\leq a}_{*+1}\\left(\\A^{H}_{q_0,q_1}\\right)\\right)\\subseteq CF^{\\leq a}_*\\left(\\A^{H}_{q_0,q_1}\\right).\n\\end{equation}\nThe {\\em positive Rabinowitz Floer homology} $\\LRFH_*^+(\\A^{H}_{q_0,q_1})$ is the homology of the quotient complex\n$$ CF_*^+\\left(\\A^{H}_{q_0,q_1}\\right) \\coloneqq CF_*\\left(\\A^{H}_{q_0,q_1}\\right)\\Big/CF_*^{\\leq 0}\\left(\\A^{H}_{q_0,q_1}\\right) $$with boundary operator $\\partial^{\\scriptscriptstyle +}$ induced by $\\partial$ on the quotient.\n\nIn order to get the required transversality in the preceding discussion, we need to replace $H$ by $H+h$ with $h\\in\\mathcal{H}^{reg}$ as in the previous subsection. \nFor independence of positive Lagrangian Rabinowitz Floer homology of the choice of $h$ we need a further condition. The regular level set $H^{-1}(0)$ is said to be of {\\em exact contact type} if there exists a Liouville vector field $Y$ on $T^*Q$ (satisfying $L_Y\\om=\\om$) such that $dH(Y)>0$ along $H^{-1}(0)$. \nSetting $(\\xi,\\sigma)=(Y,\\eta)$ in equation~\\eqref{eq:dAH} we get\n\\begin{equation}\\label{dA^H(Y)}\n \\A^H (v,\\eta) - d\\A^H(v,\\eta)[Y, \\eta] = \\eta\\int_0^1 dH(Y)(v(t))dt.\n\\end{equation}\nAt critical points the second term on the left-hand side vanishes and we conclude\n\\begin{equation}\\label{Crit+}\n\\begin{aligned}\n \\Crit^\\pm\\A^{H}_{q_0,q_1} &:= \\left\\lbrace x\\in \\Crit\\A^{H}_{q_0,q_1}\\ |\\ \\pm\\A^{H}_{q_0,q_1}(x) >0\\right\\rbrace \\cr\n &= \\left\\lbrace (v,\\eta)\\in \\Crit\\A^{H}_{q_0,q_1}\\ |\\ \\pm \\eta >0\\right\\rbrace.\n\\end{aligned}\n\\end{equation}\nThe following lemma shows that the exact contact type property provides a set of compactly supported perturbations for which the set of positive action values is bounded away from zero.\n\n\\begin{lem}\\label{lem:posAction}\nConsider a cotangent bundle $(T^*Q,\\omega)$ with its standard symplectic form and a Hamiltonian $H:T^*Q\\to \\R$ such that $H^{-1}(0)$ is of exact contact type. Fix $q_0,q_1\\in Q$ with $q_0\\neq q_1$.\nAssume that there exists a compact subset $K\\subseteq T^*Q$ and an open neighbourhood $\\mathcal{H}$ of $0$ in $\\{h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\}$ such that for all $h\\in \\mathcal{H}$ the following holds:\n\\begin{equation}\n\\forall\\ (v,\\eta) \\in \\Crit \\A^{H+h}_{q_0,q_1} \\quad \\textrm{we have}\\quad v([0,1]) \\subseteq K.\\label{PO_K}\n\\end{equation}\nThen there exists an open neighbourhood $\\mathcal{O}(K)$ of $0$ in $\\mathcal{H}$ such that\n\\begin{equation}\n\\inf\\left\\lbrace\\begin{array}{c|c}\n\\A^{H+h}_{q_0,q_1}(x) & x\\in \\Crit^+ \\A^{H+h}_{q_0,q_1}, \\quad h\\in\\mathcal{O}(K)\\end{array} \\right\\rbrace >0.\\label{infA+}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nBy assumption there exists a Liouville vector field $Y$ such that $dH(Y)>0$ along $H^{-1}(0)$. Since $K$ is compact,\n$$\n\\delta:=\\inf\\{dH(Y)(x)\\ |\\ x\\in K\\cap H^{-1}(0)\\} > 0.\n$$\nFor $h\\in \\mathcal{H}$ small enough in the $C^1$-norm we then have\n\\begin{equation}\\label{nbhd}\n(H+h)^{-1}(0)\\cap K \\subseteq \\left\\lbrace\n\\begin{array}{c|c}\nx\\in K& d(H+h)(Y)(x)\\geq\\delta/2\n\\end{array} \\right\\rbrace.\n\\end{equation}\nDenote by $\\mathcal{O}(K)$ the set of all $h\\in \\mathcal{H}$ satisfying \\eqref{nbhd}. Then $\\mathcal{O}(K)$ is an open neighbourhood of $0$ in $\\{h\\in C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\}$.\nConsider $h\\in\\mathcal{O}(K)$ and $(v,\\eta)\\in \\Crit^+ \\A^{H+h}_{q_0,q_1}$, so $\\eta>0$ in view of~\\eqref{Crit+} applied to $H+h$. \nBy~\\eqref{PO_K} we have $v([0,1])\\subset K\\cap(H+h)^{-1}(0)$, so from~\\eqref{dA^H(Y)} and~\\eqref{nbhd} we obtain\n\\begin{equation*} |\\A^{H+h}_{q_0,q_1}(v,\\eta)| \\geq \\delta\\eta/2.\n\\end{equation*}\nOn the other hand, from $v(i)\\in T_{q_i}^*Q$ for $i=0,1$ and $\\p_sv=\\eta X_{H+h}(v)$ we deduce\n$$\n 0 < \\varepsilon \\leq |v(1)-v(0)| \\leq \\eta\\int_0^1|X_{H+h}(v(t))|dt \\leq C\\eta \n$$\nwith the positive constants\n$$\n \\varepsilon := \\textrm{dist}(K\\cap T_{q_0}^*Q,K\\cap T_{q_1}^*Q),\\qquad C := \\max\\{|X_{H+h}(x)|\\;\\bigl|\\;x\\in K\\}.\n$$\nThe two estimates combine to $|\\A^{H+h}_{q_0,q_1}(v,\\eta)| \\geq \\delta\\varepsilon/2C>0$. \n\\end{proof}\n\nThe next result provides conditions under which positive Lagrangian Rabinowitz Floer homology is not only well defined, but also independent of the auxiliary choices and invariant under compact perturbations.\n\n\\begin{cor}\\label{cor:posLRFH}\nConsider the setting as in Theorem \\ref{thm:DefLRFH} with sets $K\\subseteq\\mathcal{V}\\subseteq T^*Q$ and $\\mathcal{H}\\subseteq \\{C^\\infty(T^*Q)\\ |\\ dh\\in C_0^\\infty(K)\\}$. Assume $q_0\\neq q_1$. Let $\\mathcal{O}\\subseteq\\mathcal{H}$ be an open neighbourhood of $0$ such that for every pair $h_0,h_1 \\in \\mathcal{O}$ there exists a homotopy $\\Gamma:= \\{(H+h_s, J_s)\\}_{s\\in \\R}$ satisfying Condition 3 of Theorem \\ref{thm:DefLRFH}, and such that for every $x\\in \\Crit^+\\A^{H+h_0}_{q_0,q_1}$ and every $y\\in \\Crit\\A^{H+h_1}_{q_0,q_1}$ for which $\\F_\\Gamma(x,y)\\neq \\emptyset$ we have $\\A^{H+h_1}_{q_0,q_1}(y)>0$.\n\nThen for every $h\\in\\mathcal{O}$ its positive Lagrangian Rabinowitz Floer homology is well defined, and for every pair $h_0,h_1 \\in \\mathcal{O}$, $\\LRFH_*^+(\\A^{H+h_0}_{q_0,q_1})$ is isomorphic to $\\LRFH_*^+(\\A^{H+h_1}_{q_0,q_1})$.\n\\end{cor}\n\n\\begin{proof}\nBy assumption the critical set of $\\A^H_{q_0,q_1}$ is continuously compact in $(K,\\mathcal{O})$. Denote by $\\mathcal{O}^{\\reg}$ the subset of $\\mathcal{O}$ consisting of all Hamiltonians $h\\in \\mathcal{O}$, such that $\\A^{H+h}_{q_0,q_1}$ is Morse. By the standard Sard-Smale argument $\\mathcal{O}^{\\reg}$ is dense in $\\mathcal{O}$.\nFix $h_0,h_1 \\in \\mathcal{O}^{\\reg}$. By assumption, there exist $J_i \\in \\mathcal{J}_{h_i}^{\\reg}\\subset \\mathcal{J}(\\mathcal{V}, \\mathbb{J})$ for $i=0,1$ and a homotopy $\\Gamma= \\{(H+h_s, J_s)\\}_{s\\in \\R}$ with $h_s\\in \\mathcal{O}$ and $J_s\\in \\mathcal{J}(\\mathcal{V}, \\mathbb{J})$ from $(H+h_0,J_0)$ to $(H+h_1,J_1)$ with the following properties:\n\\begin{enumerate}[label=\\alph*)]\n\\item the homotopy $\\Gamma$ satisfies the Novikov finiteness condition \\eqref{Novikov};\n\\item for any pair $a,b\\in \\R$ the space of Floer trajectories $\\mathcal{M}^\\Gamma(a,b)$ is bounded in the $L^\\infty$-norm;\n\\item for every $x\\in \\Crit^+\\A^{H+h_0}_{q_0,q_1}$ and every $y\\in \\Crit\\A^{H+h_1}_{q_0,q_1}$ such that $\\F_\\Gamma(x,y)\\neq \\emptyset$ we have $\\A^{H+h_1}_{q_0,q_1}(y)>0$.\n\\end{enumerate}\nFrom properties a) and b) we get a chain map $\\phi^\\Gamma : CF_*(\\A^{H+h_1}_{q_0,q_1})\\to CF_*(\\A^{H+h_0}_{q_0,q_1})$.\nIt satisfies $\\phi^\\Gamma \\left(CF_*^{\\leq 0}(\\A^{H+h_1}_{q_0,q_1})\\right) \\subseteq CF_*^{\\leq 0}(\\A^{H+h_0}_{q_0,q_1})$, \nsince otherwise there would exist $x\\in \\Crit^+\\A^{H+h_0}_{q_0,q_1}$ and $y\\in \\Crit\\A^{H+h_1}_{q_0,q_1}$ such that $\\F_\\Gamma(x,y)\\neq \\emptyset$ and $\\A^{H+h_1}_{q_0,q_1}(y)\\leq 0$, contradicting property c). \nThus $\\phi^\\Gamma$ descends to a chain map $\\phi^\\Gamma_+ : CF_*^+(\\A^{H+h_1}_{q_0,q_1})\\to CF_*^+(\\A^{H+h_0}_{q_0,q_1})$, and the usual argument using a homotopy in the opposite direction (see~\\cite[Proposition 11.2.9]{Audin2014}) shows that the induced map on homology\\linebreak $\\Phi^\\Gamma_+ \\colon \\LRFH_*^+(\\A^{H+h_1}_{q_0,q_1})\\to \\LRFH_*^+(\\A^{H+h_0}_{q_0,q_1})$ is an isomorphism.\n\\end{proof}\n\n\n\n\\section{Bounds on the Floer trajectories}\n\nThe aim of this section is to show that the Copernican Hamiltonian $H_0$ defined in \\eqref{DefH0} together with the set of compactly supported perturbations defined in \\eqref{DefHset} satisfy the assumptions of Theorem \\ref{thm:DefLRFH}, in order to apply it and prove Theorem \\ref{thm:LRFH}.\n\nWe will start by showing that the necessary condition for the existence of Reeb chords on $(H_0-h)^{-1}(0)$ between any two cotangent fibres $T_{q_0}\\R^2$ and $T_{q_1}\\R^2$, assumption 1 of Theorem \\ref{thm:DefLRFH}, holds true in our setting.\nNext, we will prove that the second assumption of Theorem \\ref{thm:DefLRFH} holds true in our setting, i.e., the critical set of the Lagrangian Rabinowitz action functional is bounded in $L^\\infty$. It is possible to define the Lagrangian Rabinowitz Floer homology without this assumption as a limit of homologies defined in an action window, but this is more involved and we postpone it to a future paper.\n\nThe most challenging part, which will occupy most of this section, is to prove that the Floer trajectories are uniformly bounded in the $L^\\infty$-norm. This is essential in first defining the Lagrangian Rabinowitz homology and then constructing the isomorphism from Theorem \\ref{thm:LRFH}.\n\n\\begin{lem}\\label{lem:nonEmpty}\nLet $H_0$ be the Copernican Hamiltonian defined in \\eqref{DefH0} and let $\\mathcal{H}$ be the set of perturbations defined in \\eqref{DefHset}. Then the set $(H_0-h)^{-1}(0)\\cap T_q\\R^2$ is compact and nonempty for every $h\\in \\mathcal{H}$ and every $q\\in \\R^2$.\n\\end{lem}\n\n\\begin{proof}\nFix $q\\in \\R^2$. Since $h$ is constant outside a compact set,\nwe have\n$$\n\\lim_{|p|\\to\\infty} \\left(H_0(q,p)-h(q,p)\\right)=+\\infty.\n$$\nOn the other hand, we have $h>0$ by assumption and $H_0(q,0)=0$, hence\n$$\nH_0(q,0)-h(q,0)=-h(q,0)<0.\n$$\nConsequently, by the intermediate value theorem there exists $p\\in T_q\\R^2$, such that $H_0(q,p)-h(q,p)=0$.\n\\end{proof}\n\nThe following lemma proves that the Copernican Hamiltonian $H_0$ together with the set of compactly supported perturbations defined in \\eqref{DefHset} satisfies the second assumption of Theorem \\ref{thm:DefLRFH}.\n\n\\begin{lem}\\label{lem:H0Chord}\nLet $H_0:T^*\\R^2 \\to \\R$ be the Copernican Hamiltonian defined in \\eqref{DefH0}. Fix two points $q_0,q_1\\in \\R^2$. For $n\\in\\mathbb{N}$ and $m>0$ denote $r_n := \\max\\{|q_0|,|q_1|\\}+n$ and define \nthe compact set $K_{n,m} \\subseteq T^*\\R^2$ by\n$$\nK_{n,m}:=\n\\left\\lbrace\\begin{array}{c|c}\n& r\\leq r_n, \\quad p_r \\leq \\sqrt{r_n^2+2m},\\\\\n\\hspace*{-0.2cm}\\smash{\\raisebox{.5\\normalbaselineskip}{ $(r, p_r,\\theta,p_\\theta)\\in T^*\\R_+\\times T^*S^1$}}& p_\\theta \\leq r_n\\sqrt{ 2m}.\n\\end{array}\\right\\rbrace\\,.\n$$\nThen for every $h \\in \\mathcal{H}$ such that $dh \\in C_0^\\infty( K_{n,m})$ and $\\|h\\|_{L^\\infty}<m$ and any \n$(v,\\eta)\\in \\Crit \\A^{H_0-h}_{q_0,q_1}$ we have $v([0,1])\\subseteq K_{n,m}$.\n\\end{lem}\n\n\\begin{proof}\nWe will use the Poisson bracket defined by $\\{f,g\\}=\\om(X_f,X_g)=dg(X_f)$.\nUsing \\eqref{DefH0} we calculate\n$$\n\\{H_0,r\\} = p_r, \\quad \\textrm{and}\\quad \\{H_0,\\{H_0, r\\}\\}=\\frac{p_\\theta^2}{r^3}.\n$$\nThus at a point where $\\{\\{H_0, r\\}=0\\}$ and $\\{\\{H_0,\\{H_0, r\\}\\}\\leq 0$ we have $p_r=p_\\theta=0$. Since $H_0(r,\\theta, 0, 0) = 0$, this implies that for each constant $c>0$ we have\n\\begin{equation}\\label{supp_h}\nH_0^{-1}(c)\\cap \\{\\{H_0, r\\}=0\\}\\cap \\{\\{H_0,\\{H_0, r\\}\\}\\leq 0\\} = \\emptyset.\n\\end{equation}\nLet now $(v, \\eta) \\in \\Crit \\A^{H_0-h}_{q_0,q_1}$. We will first show that\n$$\n \\max r \\circ v \\leq r_n.\n$$\nArguing by contradiction, suppose that $r\\circ v(t_0)=\\max r \\circ v > r_n$ for some $t_0\\in [0,1]$. \nSince $r\\circ v(i)=|q_i|<r_n$ for $i=0,1$, we have $t_0\\in (0,1)$ and the condition that $r\\circ v$ attains its maximum at $t_0$ gives \n\\begin{gather*}\n\\frac{d}{dt}r\\circ v (t_{0}) = \\eta\\{H_0-h, r\\}\\circ v (t_{0})= 0, \\\\\n \\textrm{and} \\quad \\frac{d^{2}}{dt^{2}}r\\circ v (t_{0})=\\eta^2\\{H_0-h, \\{H_0-h, r\\}\\} \\circ v (t_{0})\\leq 0.\n\\end{gather*}\nThe definition of $K_{n,m}$ and $dh \\in C_0^\\infty( K_{n,m})$ imply that $h$ is equal to a constant $c>0$ near $v(t_0)$ and can therefore be ignored in the Poisson brackets, and we obtain a contradiction to~\\eqref{supp_h}. \n\nFor the bounds on $p_r\\circ v$ and $p_\\theta\\circ v$, recall that $v(t)\\in (H_0-h)^{-1}(0)$ for all $t\\in [0, 1]$. Consequently, we have\n\\begin{align*}\n\\frac{p_r^2}{2}+\\frac{p_\\theta^2}{2r^2} +p_\\theta & =h(r,\\theta, p_r, p_\\theta),\\\\\np_r^2+\\left(\\frac{p_\\theta}{r}+r\\right)^2 & =r^2+2h \\leq r_n^2+2m,\\\\\n|p_r \\circ v| & \\leq \\sqrt{r_n^2+2m},\\\\\n|p_\\theta \\circ v| & \\leq r_n\\left(\\sqrt{r_n^2+2m}-r_n\\right)\\leq r_n\\sqrt{2m}.\n\\end{align*}\n\\end{proof}\n\n\nBefore formulating the main theorem of this section we will first introduce the following notation. \nFirst, to avoid unnecessary clutter we will abbreviate $\\A^{H_0-h}:=\\A^{H_0-h}_{q_0,q_1}$ as in this section dependence on $q_0,q_1\\in \\R^2$ is not very relevant.\nNext, recall the set $\\mathcal{H}$ from \\eqref{DefHset}. For a compact subset of $K\\subseteq T^*\\R^2$ we define\n$$\\mathcal{H}(K):= \\{h\\in \\mathcal{H}\\ |\\ dh\\in C_0^\\infty(K)\\}.$$\nMoreover, for an open precompact subset $\\mathcal{V}\\subseteq T^*\\R^2$ we denote by $\\mathcal{J}(\\mathcal{V},\\mathbb{J})$ \nthe set of all the $2$-parameter families of $\\omega_0$-compatible almost complex structures on $(T^*\\R^2, \\omega_0)$ which are equal to the standard complex structure $\\mathbb{J}$ outside of $\\mathcal{V}$ (see the definition in Section \\ref{sec:FloerTrajec}).\n\n\\begin{thm}\\label{thm:FloerBounds}\nLet $H_0$ be the Hamiltonian defined in \\eqref{DefH0} and let $h_0,h_1 \\in \\mathcal{H}$. Fix two points $q_0,q_1 \\in \\R^2$. Let $K$ be a compact subset of $T^*\\R^2$, such that for $i=0,1$ it satisfies\n$$\n\\supp h_i \\subseteq K\\qquad \\textrm{and}\\qquad v([0,1])\\subseteq K\\qquad \\textrm{for all}\\quad (v,\\eta)\\in \\Crit \\A^{H_0-h_i}.\n$$\nLet $\\mathcal{V}\\subseteq T^*\\R^2$ be an open, but precompact subset, such that $K\\subseteq \\mathcal{V}$. Let $\\Gamma:=\\{(h_s, J_s)\\}_{s\\in \\R}$ be a smooth homotopy of Hamiltonians $h_s\\in \\mathcal{H}(K)$ and $2$-parameter families of almost complex structures $J_s \\in C^\\infty([0,1]\\times \\R,\\mathcal{J}(\\mathcal{V},\\mathbb{J}))$ constant in $s$ outside $[0,1]$, such that\n\\begin{gather}\n\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\left(\\frac{1}{c}+\\|J\\|_{L^{\\infty}}^2\\right) \\leq \\frac{1}{3},\\label{inqGamma}\\\\\n\\textrm{where}\\qquad c:=\\inf_{s\\in[0,1]}\\{h_s-dh_s(p\\partial_p)\\}.\\label{Defc}\n\\end{gather}\nThen for every pair $(a,b)\\in \\R^2$ the corresponding space $\\mathcal{M}^\\Gamma(a,b)$ of Floer trajectories defined in \\eqref{DefM(a,b)} is bounded in the $L^\\infty$-norm.\n\\end{thm}\n\n\\subsection{The bounds on the action} \nIf the family of Hamiltonians $\\{H_s\\}_{s\\in \\R}$ is constant in $s$, i.e. $H_s = H$ for all $s\\in \\R$ then $\\mathcal{M}^\\Gamma(a,b)\\neq \\emptyset$ implies $b>a$. However, since in the parametric case the action along the Floer trajectories may not be monotonically increasing, we need to ensure that we have a bound on how much the action can decrease along a Floer trajectory.\nProving the bounds on the action conditions will be the main aim of this section. \n\nWe will start by proving the following lemma:\n\n\\begin{lem}\\label{lem:inqEtaAct}\nFor every $h\\in\\mathcal{H}$\n\\begin{gather*}\n\\textrm{if}\\qquad \\|\\nabla \\A^{H_0-h}(v,\\eta)\\|_{L^2\\times\\R} <1,\\\\\n\\textrm{then}\\qquad |\\eta | \\leq \\frac{1}{c}|\\A^{H_0-h}(v,\\eta)|+\\frac{1}{\\sqrt{2c}},\n\\end{gather*}\nwhere $c:=\\inf\\{h-dh(p\\partial_p\\}>0$.\n\\end{lem}\n\\begin{proof}\nUsing \\eqref{DefH0} we can compute the Hamiltonian $H_0$ satisfies the following equality:\n\\begin{equation}\\label{H0pdp}\ndH_0(p\\partial_p) = p_1^2+p_2^2+q_2p_1-q_1p_2 =\\frac{1}{2}|p|^2+H_0(p,q).\n\\end{equation}\nCombining that with equality \\eqref{dA^H(Y)} for all $(q,p,\\eta)=(v,\\eta)\\in \\mathscr{H}_{q_0,q_1}$ we obtain\n\\begin{align}\n\\A^{H_0-h}(v,\\eta) -d\\A^{H_0-h}[p\\partial_p,0] & =\\eta\\int_0^1\\left(d(H_0-h)(p\\partial_p)-H_0(v)+h(v)\\right)\\nonumber \\\\\n& = \\eta \\left(\\frac{1}{2}\\|p\\|_{L^2} +\\int_0^1 \\left(h(v)-dh(p\\partial p)\\right)\\right)\\\\\n& \\geq \\eta \\left(\\frac{1}{2}\\|p\\|_{L^2}+c\\right), \\label{IntdH(Y)}\n\\end{align}\nwhere the last inequality comes from the fact that\n$c:=\\inf\\{h-dh(p\\partial_p\\}>0$.\n\nConsequently, for all $(q,p,\\eta)=(v,\\eta)\\in \\mathscr{H}_{q_0,q_1}$, such that $\\|\\nabla \\A^{H_0-h}(v,\\eta)\\|_{L^2\\times\\R} <1$, we obtain\n$$\n|\\eta| < \\frac{|\\A^{H_0-h}(v,\\eta)|+\\|p\\|_{L^2}}{\\frac{1}{2}\\|p\\|_{L^2}^2+c}.\n$$\nNow let us analyze the right-hand side as functions of $\\|p\\|_{L^2}$:\n\\begin{align*}\n\\max_{x\\geq 0}\\left(\\frac{1}{\\frac{1}{2} x^2+c}\\right) & =\\frac{1}{c},\\\\\n\\frac{d}{dx}\\left( \\frac{x}{\\frac{1}{2}x^2+c}\\right) & = \\frac{c-\\frac{1}{2}x^2}{\\left(\\frac{1}{2}x^2+c\\right)^2},\\\\\n\\max_{x\\geq 0} \\left( \\frac{x}{\\frac{1}{2}x^2+c}\\right) & = \\frac{1}{\\sqrt{2c}}.\n\\end{align*}\nThese estimates give the desired inequality.\n\\end{proof}\n\\begin{lem}\\label{lem:ActBound}\nConsider the setting as in Theorem \\ref{thm:FloerBounds}. Fix $a, b\\in \\mathbb{R}$. Then $\\|\\eta\\|_{L^\\infty}$, $\\|\\A^{H_0-h_s}\\circ u(s)\\|_{L^\\infty}$ and $\\|\\nabla^{J_s}\\A^{H_0-h_s}\\circ u(s)\\|_{L^2([0,1])\\times\\R}$ are uniformly bounded for all $u \\in \\mathcal{M}^\\Gamma(a,b)$.\n\\end{lem}\nThe following proof follows closely the arguments presented in \\cite[Prop. 3.3]{Pasquotto2017}. We will not present all of them here, but we encourage a curious reader to see the details in \\cite{Pasquotto2017}.\n\n\\begin{proof}\nUsing the fact that $u\\in \\mathcal{M}^\\Gamma(a,b)$ is a Floer trajectory one can calculate the derivative of the action functional over $s$ and obtain the following inequalities (see \\cite[Prop. 3.3]{Pasquotto2017}):\n\\begin{align}\n\\|\\mathcal{A}^{H_0-h_{s}}( u)\\|_{L^{\\infty}} & \\leq \\max\\{|a|,|b|\\} + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}},\\label{inqAct}\\\\\n\\|\\nabla^{J_{s}} \\mathcal{A}^{H_0-h_{s}}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2}\n& \\leq \\|J\\|_{L^{\\infty}} (b-a + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}). \\label{inqEnergy}\n\\end{align}\n\nIn particular the convergence of the integral\n$$\n\\|\\nabla \\mathcal{A}^{H_0-h_{s}}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])} \\leq \\|J\\|_{L^{\\infty}} \\|\\nabla^{J_{s}} \\mathcal{A}^{H_0-h_{s}}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])},\n$$\nimplies that for small enough $s$ we have $\\|\\nabla \\mathcal{A}^{H_0-h_{s}}(u(s))\\|_{L^{2}\\times\\mathbb{R}}<1$.\nThis ensures that for all $s \\in \\mathbb{R}$ the following value $\\tau_{0}(s)$ is well defined and finite\n$$\n\\tau_{0}(s): = \\sup\\{ \\tau \\leq s\\ |\\ \\|\\nabla \\mathcal{A}^{H_0-h_{\\tau}}(u(\\tau))\\|_{L^{2}\\times\\mathbb{R}}<1\\}.\n$$\nFor $\\tau \\in [\\tau_{0}(s),s]$ we have $\\|\\nabla \\mathcal{A}^{H_0-h_{\\tau}}(u(\\tau))\\|_{L^{2}\\times\\mathbb{R}}\\geq 1$, which allows us to estimate (see \\cite[Prop. 3.3]{Pasquotto2017}):\n\\begin{align}\n|s -\\tau_0(s)| & \\leq \\|J\\|_{L^{\\infty}}^3 (b-a + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}),\\\\\n|\\eta(s) -\\eta(\\tau_0(s))| & \\leq \\|J\\|_{L^{\\infty}}^2 (b-a + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}). \\label{inqeta}\n\\end{align}\n\nNow using \nthe constant $c>0$ as in \\eqref{Defc} together with the\nthe estimates from Lemma \\ref{lem:inqEtaAct} and \\eqref{inqAct} we can calculate:\n\\begin{align*}\n|\\eta(s)|& \\leq |\\eta(\\tau_0(s))|+|\\eta(s) -\\eta(\\tau_0(s))|\\\\\n& \\leq \\frac{1}{c}|\\A^{H_0-h_s}(u(s))|+\\frac{1}{\\sqrt{2c}}+\\|J\\|_{L^{\\infty}}^2 (b-a + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}})\\\\\n& \\leq \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\left(\\frac{1}{c}+\\|J\\|_{L^{\\infty}}^2\\right)+\\frac{1}{c}\\max\\{|a|,|b|\\}+\\frac{1}{\\sqrt{2c}}+\\|J\\|_{L^{\\infty}}^2 (b-a).\n\\end{align*}\nSince this inequality has to hold for all $s\\in \\R$ it also has to hold for $\\|\\eta\\|_{L^\\infty}$. Using \\eqref{inqGamma} we obtain the following estimate:\n\\begin{align}\n\\|\\eta\\|_{L^{\\infty}} & \\leq \\frac{\\frac{1}{c}\\max\\{|a|,|b|\\}+\\frac{1}{\\sqrt{2c}}+\\|J\\|_{L^{\\infty}}^2 (b-a)}{1-\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\left(\\frac{1}{c}+\\|J\\|_{L^{\\infty}}^2\\right)}\\nonumber\\\\\n&\\leq \\frac{3}{2} \\left( \\frac{1}{c}\\max\\{|a|,|b|\\}+\\frac{1}{\\sqrt{2c}}+\\|J\\|_{L^{\\infty}}^2 (b-a)\\right)=:\\mathfrak{y},\\label{eqEta}\n\\end{align}\nNow using \\eqref{inqGamma}, \\eqref{inqAct}, \\eqref{inqEnergy} and \\eqref{eqEta} we obtain the desired uniform bounds:\n\\begin{align}\n\\|\\mathcal{A}^{H_0-h_{s}-c}( u)\\|_{L^{\\infty}} &\\leq \\max\\{|a|,|b|\\} + \\frac{c}{3}\\mathfrak{y}=:\\mathfrak{a},\\label{DefA}\\\\ \n\\|\\nabla^{J_{s}} \\mathcal{A}^{H_0-h_{s}-c}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2} & \\leq \\|J\\|_{L^{\\infty}} \\left(b-a + \\frac{c}{3}\\mathfrak{y}\\right)=:\\mathfrak{e}.\\label{DefE}\n\\end{align}\n\\end{proof}\nHaving obtained the bounds on the action we are ready to prove the Novikov finiteness condition:\n\\begin{lem}\\label{lem:Novikov}\nConsider the setting as in Theorem \\ref{thm:FloerBounds}. Then $\\mathcal{M}_\\Gamma(a,b)\\neq \\emptyset$ implies\n$$\na \\leq \\max\\left\\lbrace 3 b, 3\\sqrt{\\frac{c}{2}}\\right\\rbrace \\qquad \\textrm{and} \\qquad b \\geq \\min\\left\\lbrace 3a, -3\\sqrt{\\frac{c}{2}}\\right\\rbrace.\n$$\n\\end{lem}\n\\begin{proof}\nThis proof follows arguments presented in \\cite[Cor. 3.8]{CieliebakFrauenfelder2009}. We will prove the first inequality since the second is completely analogous.\n\\begin{align}\n\\textrm{First assume}\\ \\qquad 3\\sqrt{\\frac{c}{2}} & \\leq a \\quad \\textrm{and} \\quad |b|\\leq a.\\label{aBound}\\\\\n\\textrm{Then} \\qquad\\max\\{|a|,|b|\\} & = a \\quad \\textrm{and} \\quad b-a\\leq 0.\\nonumber\n\\end{align}\nBy \\eqref{eqEta} and \\eqref{aBound} we get:\n$$\n\\|\\eta\\|_{L^\\infty}\\leq \\frac{3}{2}\\left(\\frac{a}{c}+\\frac{1}{\\sqrt{2c}}\\right)=2\\frac{a}{c}.\n$$\nOn the other hand, by \\eqref{inqGamma} we have $\\|\\partial_sh_s\\|_{L^\\infty}\\leq \\frac{c}{3}$. This, together with \\eqref{inqEnergy}, implies\n$$\nb \\geq a- \\|\\eta\\|_{L^\\infty}\\|\\partial_sh_s\\|_{L^\\infty} \\geq a -2 \\frac{a}{c}\\cdot \\frac{c}{3}=\\frac{1}{3}a.\n$$\n\nNow assume that $ 3\\sqrt{\\frac{c}{2}} \\leq a < |b|$. To finish the proof it suffices to exclude the case $a< - b$.\n$$\n\\textrm{Then} \\qquad\\max\\{|a|,|b|\\} = -b \\quad \\textrm{and} \\quad b-a\\leq 0.\n$$\nBy \\eqref{eqEta} and \\eqref{aBound} we get:\n$$\n\\|\\eta\\|_{L^\\infty}\\leq \\frac{3}{2}\\left(\\frac{-b}{c}+\\frac{1}{\\sqrt{2c}}\\right)\\leq \\frac{3}{2}\\left(\\frac{-b}{c}+\\frac{a}{3c}\\right)\\leq -2 \\frac{b}{c}.\n$$\nThis, together with \\eqref{inqGamma} and \\eqref{inqEnergy}, gives us a contradiction:\n\\begin{gather*}\nb \\geq a- \\|\\eta\\|_{L^\\infty}\\|\\partial_sh_s\\|_{L^\\infty}\\geq a + 2 \\frac{b}{c} \\cdot \\frac{c}{3}=a+\\frac{2}{3}b,\\\\\n0 \\geq \\frac{1}{3} b \\geq a \\geq 3\\sqrt{\\frac{c}{2}}>0.\n\\end{gather*}\n\\end{proof}\n\n\\subsection{The set of infinitesimally small action derivation}\n\nLet $H_0$ be the Hamiltonian defined in \\eqref{DefH0} and let $\\mathcal{H}$ be the set of Hamiltonians as in \\eqref{DefHset}. For a Hamiltonian $h \\in \\mathcal{H}$ and fixed constants $\\mathfrak{a}, \\mathfrak{y}, \\varepsilon>0$ we define the following set:\n\\begin{equation}\n\\mathcal{B}_h(\\mathfrak{a}, \\mathfrak{y}, \\varepsilon):= \\left\\lbrace\\begin{array}{c|c}\n & |\\nabla\\A^{H_0-h}(v,\\eta)|_{L^2\\times\\R}<\\varepsilon,\\\\\n{\\smash{\\raisebox{.5\\normalbaselineskip}{ $(v, \\eta) \\in\\mathscr{H}_{q_0,q_1}\\times\\R$}}}& |\\A^{H_0-h}(v,\\eta)|\\leq \\mathfrak{a},\\ |\\eta|\\leq \\mathfrak{y}.\n\\end{array}\\right\\rbrace\n\\end{equation}\nWe will call this set the set of infinitesimally small action derivation.\n\nThe main aim of this subsection will be to prove the following lemma:\n\n\\begin{prop}\\label{prop:smallDerivSetBound}\nFor fixed constants $\\mathfrak{a}, \\mathfrak{y}>0$ and $\\varepsilon>0$ small enough, the corresponding set $\\mathcal{B}_h(\\mathfrak{a}, \\mathfrak{y}, \\varepsilon)$ is bounded both in the $L^{\\infty}\\times\\R$-norm and in the $L^2\\times \\R$-norm.\n\\end{prop}\n\nWe will prove Proposition \\ref{prop:smallDerivSetBound} in a series of smaller lemmas:\n\n\\begin{lem}\\label{lem:partial_p}\nLet $h\\in \\mathcal{H}$ and let $c:=\\inf\\{h-dh(p\\partial_p)\\}>0$.\nIf we fix $\\mathfrak{a}, \\mathfrak{y}, \\varepsilon>0$ then for every $(q,p,\\eta) \\in \\mathcal{B}_h (\\mathfrak{a}, \\mathfrak{y}, \\varepsilon)$ we have:\n$$\n|\\eta|\\|p\\|_{L^2}\\leq 2\\varepsilon +\\frac{\\mathfrak{a}}{\\sqrt{2c}}\\qquad\\textrm{and}\\qquad \\|\\partial_t p\\|_{L^2}\\leq 3\\varepsilon +\\frac{\\mathfrak{a}}{\\sqrt{2c}} + \\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\n$$\n\\end{lem}\n\n\\begin{proof}\nUsing \\eqref{IntdH(Y)} for $(q,p,\\eta) \\in \\mathcal{B}_h (\\mathfrak{a}, \\mathfrak{y}, \\varepsilon)$ we obtain\n\\begin{gather*}\n\\eta \\left(\\frac{1}{2}\\|p\\|_{L^2}^2+c\\right) \\leq \\A^{H_0-h}(v,\\eta)-d\\A^{H_0-h}[p\\partial_p,0] \\leq \\mathfrak{a}+\\varepsilon\\|p\\|_{L^2},\\\\\n|\\eta| \\|p\\|_{L^2}\\leq \\frac{\\mathfrak{a}\\|p\\|_{L^2}+\\varepsilon\\|p\\|_{L^2}^2}{\\frac{1}{2}\\|p\\|_{L^2}^2+c}.\n\\end{gather*}\nNow we would like to estimate the maximum of the function on the right-hand side of the inequality:\n\\begin{gather*}\n\\frac{d}{dx}\\left(\\frac{\\mathfrak{a}x+\\varepsilon x^2}{\\frac{1}{2}x^2+c}\\right) = -\\frac{\\frac{1}{2}\\mathfrak{a}x^2-2\\varepsilon c x -\\mathfrak{a} c}{\\left(\\frac{1}{2}x^2+c\\right)^2},\\\\\n\\frac{d}{dx}\\left(\\frac{\\mathfrak{a}x+\\varepsilon x^2}{\\frac{1}{2}x^2+c}\\right) = 0 \\quad \\iff \\quad x=\\frac{1}{\\mathfrak{a}}\\left(2\\varepsilon c \\pm \\sqrt{4 \\varepsilon^2 c^2 + 2 \\mathfrak{a}^2 c}\\right),\\\\\n\\max_{x\\geq 0} \\left(\\frac{\\mathfrak{a}x+\\varepsilon x^2}{\\frac{1}{2}x^2+c}\\right)= \\varepsilon+ \\sqrt{\\frac{\\mathfrak{a}^2}{2c}+\\varepsilon^2} \\leq 2\\varepsilon +\\frac{\\mathfrak{a}}{\\sqrt{2c}}.\n\\end{gather*}\nThis gives us the bound on $|\\eta| \\|p\\|_{L^2}$, i.e. the first inequality.\n\nLet us denote by \n$$\n(X_{H_0-h})_p:= \\left(p_2 +\\frac{\\partial h}{\\partial_{q_1}}\\right)\\partial_{p_1}-\\left(p_1-\\frac{\\partial h}{\\partial_{q_2}}\\right)\\partial_{p_2}.\n$$\nThen we can estimate\n\\begin{align*}\n\\|\\partial_tp\\|_{L^2} & \\leq \\| \\partial_t p-\\eta (X_{H_0-h})_p\\|_{L^2}+|\\eta| \\|(X_{H_0-h})_p\\|_{L^2}\\nonumber\\\\\n& \\leq \\|\\nabla\\A^H(q,p,\\eta)\\|_{L^2\\times \\R}+|\\eta|(\\|p\\|_{L^2}+\\|\\nabla h \\|_{L^\\infty}),\\\\\n& \\leq 3\\varepsilon +\\frac{\\mathfrak{a}}{\\sqrt{2c}} + \\mathfrak{y}\\|\\nabla h \\|_{L^\\infty}.\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\\label{lem:qBound}\nIf we fix $\\mathfrak{a}, \\mathfrak{y}, \\varepsilon>0$ then for every $(q,p,\\eta) \\in \\mathcal{B}_h (\\mathfrak{a}, \\mathfrak{y}, \\varepsilon)$ we have:\n$$\n\\|q\\|_{L^2}, \\| q\\|_{L^\\infty} \\leq \\min\\{|q_0|,|q_1|\\}+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right).\n$$\n\\end{lem}\n\n\\begin{proof}\nBegin first observe that\n$$\nqdq(X_{H_0})=q_1(p_1+q_2)+q_2(p_2-q_1)=q_1 p_1 + q_2 p_2.\n$$\n\nTherefore, for all $t\\in [0,1]$ we have\n\\begin{align}\n|q(t)|^2 - |q_0|^2& =2\\int_0^t qdq\\left(\\partial_tv - \\eta X_{H_0-h}\\right)+2\\eta \\int_0^t \\left(qdq(X_{H_0})-qdq(X_h)\\right)\\nonumber\\\\\n& \\leq 2\\|q\\|_{L^2}\\|\\nabla \\A^H(q,p,\\eta)\\|_{L^2\\times \\R}+2|\\eta|\\|q\\|_{L^2}\\left(\\|p\\|_{L^2} + \\|\\nabla h\\|_{L^\\infty}\\right),\\nonumber\\\\\n& \\leq 2 \\|q\\|_{L^2}\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right), \\label{ineq1}\n\\end{align}\nwhere the last inequality we obtained using the result form Lemma \\ref{lem:partial_p}. Integrating both sides, we obtain\n$$\n\\|q\\|_{L^2}^2\\leq |q_0|^2+2 \\|q\\|_{L^2}\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right).\n$$\nBy solving this quadratic inequality we obtain the following bound:\n\\begin{align*}\n\\|q\\|_{L^2} & \\leq 3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty} + \\sqrt{\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)^2+|q_0|^2}\\\\\n& \\leq |q_0|+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)\n\\end{align*}\nBy repeating this procedure with the equation $|q(t)|^2=|q_1|^2-2\\int_t^1 q\\partial_tq$, we obtain the bound on $\\|q\\|_{L^2}$ we were looking for.\n\nTo obtain the bound for $\\|q\\|_{L^\\infty}$ we will use \\eqref{ineq1} again:\n\\begin{align*}\n\\|q\\|_{L^\\infty} & \\leq \\sqrt{|q_0|^2+ 2 \\|q\\|_{L^2}\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)}\\\\\n& \\leq \\sqrt{|q_0|^2+ 2 \\left( |q_0|+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)\\right)\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)}\\\\\n& \\leq |q_0|+ 2\\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)\n\\end{align*}\nAnalogously as before, we repeat this procedure with the equation\\linebreak $|q(t)|^2=|q_1|^2-2\\int_t^1 q\\partial_tq$ to obtain the bound on $\\|q\\|_{L^\\infty}$ we were looking for.\n\\end{proof}\n\\begin{lem}\\label{lem:pBound}\nIf we fix $\\mathfrak{a}, \\mathfrak{y}, \\varepsilon>0$ then for every $(q,p,\\eta) \\in \\mathcal{B}_h (\\mathfrak{a}, \\mathfrak{y}, \\varepsilon)$ we have:\n\\begin{align*}\n\\|p\\|_{L^2} & \\leq 2\\mathfrak{q}+\\sqrt{2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)}=:\\mathfrak{p},\\\\\n\\|p\\|_{L^\\infty} & \\leq \\mathfrak{p}+\\left( \\varepsilon+\\|\\nabla h\\|_{L^\\infty}\\right),\\\\\n\\textrm{where}\\qquad \\mathfrak{q} & :=\\min\\{|q_0|,|q_1|\\}+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nBy \\eqref{H0pdp} we have\n\\begin{align*}\n\\frac{1}{2}\\|p\\|_{L^2}^2 & = \\int_0^1 (H_0-h)(q,p)dt -\\int_0^1\\left(p_1q_2-p_2q_1\\right)dt +\\int_0^1 h(q,p)dt\\\\\n& \\leq \\|\\nabla \\A^{H_0-h-c}\\|_{L^2\\times \\R}+\\|p\\|_{L^2}\\|q\\|_{L^2}+\\|h\\|_{L^\\infty}+c\\\\\n& \\leq \\varepsilon + \\mathfrak{q}\\|p\\|_{L^2} +\\|h\\|_{L^\\infty}+c\n\\end{align*}\nwhere $\\mathfrak{q}:=\\min\\{|q_0|,|q_1|\\}+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)$ as in Lemma \\ref{lem:qBound}. By solving this quadratic inequality we obtain the following bound:\n\\begin{align*}\n\\|p\\|_{L^2} & \\leq \\mathfrak{q} + \\sqrt{\\mathfrak{q}^2 + 2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)} \\leq 2 \\mathfrak{q}+\\sqrt{2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)}\\\\\n& = 2\\min\\{|q_0|,|q_1|\\}+4 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y}\\|\\nabla h\\|_{L^\\infty}\\right)+\\sqrt{2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)}.\n\\end{align*}\n\nTo prove the second bound first observe that there exists $t_0\\in [0,1]$, such that $|p(t_0)|\\leq 2 \\mathfrak{q}+\\sqrt{2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)}$. Thus can estimate:\n\\begin{align*}\n|p(t)| & \\leq |p(t_0)| + \\int_{t_0}^t |\\partial_tv|= |p(t_0)| + \\int_{t_0}^t |\\partial_tv- X_{H_0-h}|+\\int_{t_0}^t |X_h|\\\\\n& \\leq |p(t_0)| +\\|\\nabla\\A^{H_0-h-c}\\|_{L^2\\times \\R} + \\|\\nabla h\\|_{L^\\infty} \\leq |p(t_0)| +\\varepsilon+\\|\\nabla h\\|_{L^\\infty}\\\\\n& \\leq 2\\min\\{|q_0|,|q_1|\\}+13\\varepsilon + 4\\frac{\\mathfrak{a}}{\\sqrt{2c}}+(4\\mathfrak{y}+1)\\|\\nabla h\\|_{L^\\infty} +\\sqrt{2 \\left( \\|h\\|_{L^\\infty}+c+\\varepsilon\\right)}.\n\\end{align*}\n\\end{proof}\n\n\\begin{rem}\nNote that in the proof above we did not use the assumption that $dh \\in C_c^\\infty(T^*\\R^2)$. Thus we can conclude that for any $\\mathfrak{a}$, $\\mathfrak{y}$, $\\varepsilon>0$ and any Hamiltonian $h\\in C^\\infty(T^*\\R^2)\\cap W^{1,\\infty}(T^*\\R^2)$, such that $h>0$ and $h-dh(p\\partial_p)>0$ the corresponding set $\\mathcal{B}_h(\\mathfrak{a},\\mathfrak{y},\\varepsilon)$ is bounded in the $L^{\\infty}\\times \\R$-norm and the $L^2\\times \\R$-norm.\n\\end{rem}\n\n\\begin{rem}\\label{rem:smallDeriv}\nNote that for $h \\in \\mathcal{H}$ the bounds on $\\mathcal{B}_h(\\mathfrak{a},\\mathfrak{y}, \\varepsilon)$ depend smoothly on the constants $\\mathfrak{a}$, $\\mathfrak{y}$, $\\varepsilon$, $c:=\\inf\\{h-dh(p\\partial_p)\\}$ and $\\|h\\|_{W^{1,\\infty}}$. Thus if we take a subset $\\mathcal{H}'\\subseteq \\mathcal{H}$ bounded in the $W^{1,\\infty}$-norm such that $\\inf_{h\\in\\mathcal{H}'}\\inf\\{h-dh)p\\partial_p)\\}>0$, then the corresponding set\n$$\n\\bigcup_{h\\in\\mathcal{H}'}\\mathcal{B}_h(\\mathfrak{a},\\mathfrak{y},\\varepsilon)\\subseteq \\mathscr{H}_{q_0,q_1}\\times \\R,\n$$\nwill be also bounded in the $L^{\\infty}\\times \\R$-norm and the $L^2\\times \\R$-norm.\n\\end{rem}\n\n\\subsection{The $L^2$ bounds}\nThe aim of this subsection will be to prove the following Proposition:\n\\begin{prop}\\label{prop:L2bounds}\nConsider a setting as in Theorem \\ref{thm:FloerBounds}. Then for every pair $a,b\\in \\R$ the corresponding space $\\mathcal{M}^\\Gamma(a,b)$ is bounded in the $L^2\\times\\R$-norm.\n\\end{prop}\n\\begin{proof}\nBy Lemma \\ref{lem:ActBound} there exist $\\mathfrak{a,e, y}>0$, such that for every $u:=(v,\\eta)\\in \\mathcal{M}^\\Gamma(a,b)$ we have\n\\begin{align*}\n\\|\\eta\\|_{L^\\infty} \\leq \\mathfrak{y}, \\qquad\\|\\mathcal{A}^{H_0-h_s}( u)\\|_{L^{\\infty}} &\\leq\\mathfrak{a},\\\\ \\textrm{and}\\qquad \\|\\nabla^{J_s} \\mathcal{A}^{H_0-h_s}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2}& \\leq \\mathfrak{e}.\n\\end{align*}\nMoreover, by \\eqref{DefE} the convergence of the integral\n$$\n\\|\\nabla \\mathcal{A}^{H_0-h_s}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2} \\leq \\mathfrak{e}\\|J\\|_{\\infty}^2,\n$$\nimplies that for every $\\varepsilon>0$ there exists $s\\in \\R$ small enough such that\\linebreak $\\|\\nabla \\mathcal{A}^{H_0-h_s}(u(s))\\|_{L^{2}\\times\\mathbb{R}}<\\varepsilon$. This ensures that for all $s \\in \\mathbb{R}$ the following value $\\tau_\\varepsilon(s)$ is well defined and finite\n$$\n\\tau_\\varepsilon(s): = \\sup\\left\\lbrace \\tau \\leq s\\ \\Big|\\ u(\\tau) \\in \\mathcal{B}_{h_\\tau}(\\mathfrak{a},\\mathfrak{y}, \\varepsilon)\\right\\rbrace.\n$$\nNote that for all $\\tau \\in [\\tau_\\varepsilon, s]$ we have $\\|\\nabla \\mathcal{A}^{H_0-h_\\tau-c}(u(\\tau))\\|_{L^{2}\\times\\mathbb{R}}\\geq \\varepsilon$, thus we can estimate\n\\begin{gather} \n\\varepsilon^2 |s-\\tau_\\varepsilon(s)| \\leq\\|\\nabla \\mathcal{A}^{H_0-h_s}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2}\\leq\\mathfrak{e}\\|J\\|_{\\infty}^2,\\nonumber\\\\\n|s-\\tau_\\varepsilon(s)|\\leq\\frac{ \\mathfrak{e}\\|J\\|_{\\infty}^2}{\\varepsilon^2}.\\label{sBoundEpsi}\n\\end{gather}\n\nOn the other hand, we know that\n\\begin{gather*}\nc:=\\inf_{s\\in[0,1]}\\{h_s-dh_s(p\\partial_p)\\}>0,\\\\\n\\sup_{s\\in[0,1]}\\|h_s\\|_{L^\\infty} < +\\infty , \\qquad\\textrm{and} \\qquad \\sup_{s\\in[0,1]}\\|\\nabla h_s\\|_{L^2} < +\\infty,\n\\end{gather*}\ntherefore by Remark \\ref{rem:smallDeriv} we know that the set\n\\begin{equation}\\label{DefBGamma}\n\\mathcal{B}^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon):=\\bigcup_{s\\in [0,1]}\\mathcal{B}_{h_s}(\\mathfrak{a},\\mathfrak{y},\\varepsilon)\\subseteq \\mathscr{H}_{q_0,q_1}\\times \\R,\n\\end{equation}\nis bounded in $L^\\infty\\times \\R$- and $L^2\\times \\R$-norm. In fact, we can use the same bounds as in Lemmas \\ref{lem:qBound} and \\ref{lem:pBound} denoting \n\\begin{align*}\n\\mathfrak{h} & :=\\sup_{s\\in[0,1]}\\|h_s\\|_{W^{1,\\infty}},\\\\\n\\mathfrak{q} & := \\min\\{|q_0|,|q_1|\\}+2 \\left(3\\varepsilon + \\frac{\\mathfrak{a}}{\\sqrt{2c}}+\\mathfrak{y h}\\right),\\\\\n\\mathfrak{p} & := 2\\mathfrak{q}+\\sqrt{2 \\left( \\mathfrak{h}+c+\\varepsilon\\right)}.\n\\end{align*}\n\nNow using \\eqref{sBoundEpsi} together with Lemma \\ref{lem:qBound} we obtain\n\\begin{align}\n|q(s,t)| & \\leq |q(\\tau_\\varepsilon(s),t)|+\\int_{\\tau_\\varepsilon(s)}^s |\\partial_s q(\\tau,t)|d\\tau,\\nonumber\\\\\n& \\leq |q(\\tau_\\varepsilon(s),t)|+\\int_{\\tau_\\varepsilon(s)}^s |\\partial_s v(\\tau,t)|d\\tau,\\nonumber\\\\\n\\|q(s)\\|_{L^2}& \\leq \\|q \\circ \\tau_0(s)\\|_{L^2}+\\left( \\int_0^1\\left| \\int_{\\tau_0(s)}^s\\partial_sv d\\tau\\right|^2dt\\right)^{\\frac{1}{2}}\\nonumber\\\\\n& \\leq \\|q \\circ \\tau_0(s)\\|_{L^2}+\\sqrt{|s-\\tau_0(s)|}\\left( \\int_0^1\\int_{\\tau_0(s)}^s\\left|\\partial_sv\\right|^2d\\tau dt\\right)^{\\frac{1}{2}}\\nonumber\\\\\n& \\leq \\|v \\circ \\tau_0(s)\\|_{L^2}+\\frac{\\sqrt{\\mathfrak{e}}\\|J\\|_\\infty}{\\varepsilon}\\ \\|\\nabla^{J_s}\\A^{H_0-h_s}\\|_{L^2(\\R\\times[0,1])}\\nonumber\\\\\n& \\leq \\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}. \\label{qL2Bound}\n\\end{align}\nAnalogously, using Lemma \\ref{lem:pBound} we obtain\n\\begin{align}\n\\|p(s)\\|_{L^2}& \\leq \\mathfrak{p} +\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon},\\label{pL2Bound}\\\\\n\\|v(s)\\|_{L^2}& \\leq \\mathfrak{q}+\\mathfrak{p} +\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\label{vL2Bound}.\n\\end{align}\n\\end{proof}\n\n\\subsection{The maximum principle}\nIn this section we will explain how to use Aleksandrov’s maximum principle to find $L^\\infty$-bounds on the Floer trajectories outside $\\B(a, \\varepsilon)$, following the argument of Abbondandolo and Schwartz in \\cite{Abbon2009}. The contents of this section are word to word from the last author's paper with Pasquotto \\cite{Pasquotto2017}, but we include it here for the completeness of the argument.\n\n\\begin{thm}(\\textbf{Aleksandrov's maximum principle})\\\\\nLet $\\Omega$ be a domain in $\\mathbb{R}^{2}$ and let $\\rho:\\Omega \\to \\mathbb{R}$ be a function in $C^{2}(\\Omega)\\cap C^(\\overline{\\Omega})$ function satisfying the elliptic differential inequality\n$$\n\\triangle \\rho + \\langle h, \\nabla \\rho \\rangle \\geq f,\n$$\nwhere $g\\in L^2(\\Omega,\\R^2)$ and $f\\in L_{\\operatorname{loc}}^1(\\Omega)$. Then there exists $C>0$ which depends only on $\\operatorname{diam}\\Omega$ and $\\|g\\|_{L^2(\\Omega)}$, such that\n$$\n\\sup_{\\Omega} \\rho \\leq \\sup_{\\partial \\Omega}\\rho +C\\left(\\|g\\|_{L^{2}(\\Omega)}\\right)\\|f^{-}\\|_{L^{2}(\\Omega)},\n$$\nprovided $g$ and the negative part of $f$ are in $L^{2}(\\Omega)$.\n\\end{thm}\n\nIn order to apply Aleksandrov's maximum principle and find $L^{\\infty}$ bounds on the Floer trajectories, one first has to construct a function $F$ with compact level sets, whose composition with a Floer trajectory $u= (v,\\eta)\\in \\mathscr{H}_{q_0,q_1}\\times \\R$ satisfies the elliptic differential inequality\n$$\n\\triangle (F\\circ v) + \\langle g, \\nabla (F\\circ v) \\rangle \\geq f $$\noutside of the set of infinitesimal action derivation $\\mathcal{B}^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon)$, i.e. on every connected component $\\Omega$ of\n$$\n\\Omega\\subseteq \\left(\\R \\setminus u^{-1}\\left(\\mathcal{B}^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon)\\right)\\right)\\times[0,1].\n$$\nHaving such an inequality, one can apply the Aleksandrov maximum principle, which gives us\n$$\n\\sup_{\\Omega} (F\\circ v) \\leq \\sup_{\\partial \\Omega}(F\\circ v)+ C(\\|g\\|_{L^{2}(\\Omega)})\\|f^{-}\\|_{L^{2}(\\Omega)},\n$$\nprovided $g$ and the negative part of $f$ are in $L^{2}(\\Omega)$.\n\nThe core of this method is to find a function satisfying all the required properties. The classical approach is to use plurisubharmonic functions.\n\\begin{define}\nLet $(M,\\omega)$ be a symplectic manifold and $J$ a compatible almost complex structure. Then a $C^{2}$ function $F:M\\to\\mathbb{R}$ is called plurisubharmonic if\n$$\n-dd^{\\mathbb{C}}F=\\omega,\n$$\nwhere $d^{\\mathbb{C}}F=dF\\circ J$.\n\\end{define}\n\n\\begin{rem}\\label{rem:pluri}\nLet $(M, \\omega)$ be a symplectic manifold and let $J$ be a compatible almost complex structure on $M$. Then a function $F:M\\to\\mathbb{R}$ is plurisubharmonic if and only if its gradient with respect to $g$ is a Liouville vector field.\n\\end{rem}\n\nThe reason one uses plurisubharmonic functions is because their composition with a $2$-dimensional curve $v:\\Omega \\to M,\\Omega \\subseteq \\R^2$ satisfies\n$$\n-dd^\\C (F \\circ v)=\\triangle (F \\circ v) ds\\wedge dt.\n$$\nIn particular, if $v$ is a $J$-holomorphic curve then the elliptic inequality is trivially satisfied. Unfortunately, in the case of Floer trajectories proving the elliptic inequality is a little more complicated. One has to investigate how the plurisubharmonic function interacts with the Hamiltonian vector field, in particular one needs to understand the functions\n$dF(X_H)$ and $d^{\\mathbb{C}}F(X_H)$, which appear if we calculate $d^{\\mathbb{C}}(F\\circ u)$.\n\nThe proof of the following Lemma you can find in \\cite{Pasquotto2017} as a part of the proof of Proposition 7.1:\n\\begin{lem}\nLet $H$ be a Hamiltonian function on an exact symplectic manifold $(M, \\omega=d\\lambda)$ and let $J$ be a compatible almost complex structure on $M$. Let $u: \\R \\times [0,1]\\to M \\times \\R$ be a Floer trajectory satisfying $\\partial_s u =\\nabla \\A^H(u)$ (with constant $H$ and $J$). Then for a plurisubharmonic function $F: M \\to \\R$ we have\n\\begin{equation}\\label{Laplace}\n\\begin{aligned}\n\\triangle (F \\circ v) & = \\|\\partial_{s}v\\|^{2}+\\eta \\left(dH +d(d^{\\mathbb{C}}F(X_H))+d^\\mathbb{C}(dF(X_H))\\right)(\\partial_{s}v)\\\\\n& +\\eta^2 d(dF(X_H))(X_H)+\\partial_{s}\\eta\\ d^{\\mathbb{C}}F(X_H).\n\\end{aligned}\n\\end{equation}\n\n\\end{lem}\n\n\\subsection{The $L^\\infty$ bounds}\nIn this subsection we will apply the Maximum Principle explained in the previous subsection to the setting of the Floer trajectories from $\\mathcal{M}^\\Gamma(a,b)$ to establish the $L^\\infty$-bounds.\n\n\\vspace*{.25cm}\n\\noindent \\textit{Proof of Theorem \\ref{thm:FloerBounds}:} \n\nLet $\\Gamma:={(h_s,J_s)}_{s\\in \\R}$ be a smooth homotopy of Hamiltonians $h_s\\in\\mathcal{H}$ and almost complex structures $J_s\\in C^\\infty([0,1]\\times \\R, \\mathcal{J}(\\mathcal{V},\\mathbb{J}))$ satisfying \\eqref{inqGamma}. Here $\\mathcal{V}\\subseteq T^*\\R^2$ is the open, but pre-compact subset defined in Theorem \\ref{thm:FloerBounds} with the property that for all $s\\in [0,1]$ $\\supp dh_s\\subseteq \\mathcal{V}$ and $J_s\\big|_{T^*\\R^2\\setminus \\mathcal{J}}\\equiv \\mathbb{J}$.\n\nLet $c>0$ be as in \\eqref{Defc}.\nIf we fix $a,b \\in \\R$ then by Lemma \\ref{lem:ActBound} we know that there exists $\\mathfrak{y,a,e}>0$, which depend only on $a,b,c$ and $\\|J\\|_\\infty$ such that\n\\begin{equation}\\label{EAEtaBounds}\n\\begin{aligned}\n\\sup\\left\\lbrace|\\eta(s)|\\ |\\ (v,\\eta)\\in \\mathcal{M}^\\Gamma(a,b), \\ s\\in \\R\\right\\rbrace & \\leq \\mathfrak{y},\\\\\n\\sup\\left\\lbrace|\\A^{H_0-h_s-c}(u)|\\ |\\ u\\in \\mathcal{M}^\\Gamma(a,b)\\right\\rbrace & \\leq \\mathfrak{a},\\\\\n\\sup\\left\\lbrace\\|\\nabla^{J_s}\\A^{H_0-h_s-c}(u)\\|_{L^2(\\R\\times[0,1])\\times\\R}\\ |\\ u\\in \\mathcal{M}^\\Gamma(a,b)\\right\\rbrace & \\leq \\mathfrak{e}.\n\\end{aligned}\n\\end{equation}\n\nOn the other hand, if we fix $\\varepsilon>0$ then by Proposition \\ref{prop:smallDerivSetBound} and Remark \\ref{rem:smallDeriv} we know that the set $\\mathcal{B}^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon)$, defined in \\eqref{DefBGamma}, is bounded in the $L^\\infty\\times\\R$ norm and the bounds depend only on $a,b,\\varepsilon$ and $\\Gamma$. Consequently there exists a compact subset $K_\\varepsilon\\subseteq T^*\\R^2$, such that\n\\begin{equation}\\label{DefKepsi}\n v([0,1])\\subseteq K_\\varepsilon\\qquad\\textrm{for all}\\qquad (v,\\eta)\\in \\B^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon).\n\\end{equation}\nWithout loss of generality, we assume that $\\overline{\\mathcal{V}} \\subseteq K_\\varepsilon$.\n\nLet us fix a Floer trajectory $u=(v,\\eta)\\in \\mathcal{M}^\\Gamma(a,b)$ and denote a connected component \n\\begin{equation}\\label{Omega}\n\\Omega \\subseteq \\left(\\mathbb{R} \\times [0,1]\\right)\\setminus v^{-1}(K_\\varepsilon).\n\\end{equation}\nBy assumption $\\supp dh_s \\subseteq \\overline{V} \\subseteq K_\\varepsilon$, hence $dh_s \\circ v |_{\\Omega}\\equiv 0$ for all $s\\in [0,1]$. This means that instead of $H-h_s$ we can use the Hamiltonian $H_0-\\mathbf{c}_s$ for some positive function $\\mathbf{c}_s>0$ on $v(\\Omega)$, which will simplify our computations. More precisely, for $(s,t)\\in \\Omega$ the Floer trajectory $(v,\\eta)$ satisfies\n\\begin{equation}\\label{FloerH0}\n\\partial_sv = \\mathbb{J}(\\partial_t v - \\eta X_{H_0}).\n\\end{equation}\nNaturally, $\\mathfrak{c}_s\\leq \\sup_{s\\in[0,1]}\\|h_s\\|_{L^\\infty}$.\n\nBy the convergence of the integral\n$$\\|\\nabla\\A^{H_0-h_s}(u)\\|_{L^2(\\R\\times[0,1])\\times\\R} \\leq \\|J\\|_{L^\\infty}\\sqrt{\\mathfrak{e}},$$\nwe know that $\\lim_{s\\to\\pm\\infty}u(s)\\in \\B^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon)$. Therefore, \n$$v(\\partial\\Omega)\\subseteq K_\\varepsilon\\cup T^*_{q_0}\\R^2\\cup T^*_{q_1}\\R^2$$\nConsequently, we have $\\sup_{\\partial \\Omega}|q(s,t)|\\leq \\sup_{K_\\varepsilon}|q|$. Unfortunately, we do not have a uniform bound on $\\sup_{\\partial\\Omega}|p|$. Therefore, we will need to treat the cases of the position and momenta coordinates separately. We introduce the following functions on $T^*\\R^2$:\n\\begin{equation}\\label{DefF}\nF_1:= \\frac{1}{2}|q|^2\\qquad \\textrm{and}\\qquad F_2:=\\frac{1}{2}|p|^2,\n\\end{equation}\nBoth of the functions $F_1$ and $F_2$ are plurisubharmonic, since their corresponding gradients $q\\partial_q$ and $p\\partial_p$ are Liouville vector fields (see Remark \\ref{rem:pluri}).\n\nIn Lemma \\ref{lem:LaplaceF} we will show that on $\\Omega$ the functions $F_1\\circ v$ and $F_2\\circ v$ satisfy\n\\begin{align*}\n\\triangle (F_1 \\circ v) &\\geq -\\frac{1}{2}\\eta^2 F_1\\circ v+ \\partial_{s}\\eta \\left(F_2\\circ v -H_0\\circ v\\right)=:f_1(s,t),\\\\\n\\triangle (F_2 \\circ v)&\\geq -\\frac{1}{2}\\eta^2F_1\\circ v-\\partial_s\\eta\\left(F_2\\circ v+H_0\\circ v\\right)=:f_2(s,t).\n\\end{align*}\nIn Lemma \\ref{lem:f1W11Bound} and Lemma \\ref{lem:f2W11Bound} we will prove that $f_1$ and $f_2$, respectively, are bounded in the $W^{1,1}$-norm on $\\Omega$\nand the bounds depend only on $a,b,c,\\varepsilon$ and $\\Gamma$, but not on our choice of the Floer trajectory $u\\in \\mathcal{M}^\\Gamma(a,b)$ or our choice of the connected component $\\Omega\\subseteq \\left(\\R\\times[0,1] \\right)\\setminus v^{-1}(K_\\varepsilon)$. By the continuity of the Sobolev embedding $W^{1,1}\\hookrightarrow L^2$ we obtain $L^2$-bounds on $f_1$ and $f_2$.\n\nSince we know that $\\sup_{\\partial \\Omega}|q(s,t)|\\leq \\sup_{K_\\varepsilon}|q|$ we can now apply the Alexandrov maximum principle to obtain\n$$\n\\sup_{(s,t)\\in\\R\\times[0,1]}F_1\\circ v \\leq \\sup_{K_\\varepsilon}f_1\\circ v+C\\|f_1\\|_{L^2(\\Omega)},\n$$\nand the right-hand side of the inequality does not depend on our choice of the Floer trajectory $u\\in \\mathcal{M}^\\Gamma(a,b)$ or our choice of the connected component $\\Omega\\subseteq \\left(\\R\\times[0,1] \\right)\\setminus v^{-1}(K_\\varepsilon)$. In other words, we obtain that the set\n$$\n\\{\\pi \\circ v \\ |\\ u=(v,\\eta)\\in\\mathcal{M}^\\Gamma(a,b)\\},\n$$\nis bounded in $\\R^2$.\n\nNow we would like to establish the $L^\\infty$-bounds on the $p$-variable of the Floer trajectories. To establish uniform bounds on $F_2 \\circ v$ for the Floer trajectory $u=(v,\\eta)$ we extend the domain of the Floer trajectory $v:\\R\\times[0,1]\\to T^*\\R^2$ to a cylinder $\\R\\times \\left([-1,1]/-1\\sim 1\\right)$ in the following way\n\\begin{equation}\\label{barv}\n\\bar{v}(s,t):=\\begin{cases}\nv(s,-t) & \\textrm{for}\\quad t\\leq 0,\\\\\nv(s,t) & \\textrm{for}\\quad t\\geq 0.\n\\end{cases}\n\\end{equation}\nIf we extend the domain $\\Omega$ to the cylinder $\\R\\times \\left([-1,1]/-1\\sim 1\\right)$ in the following way\n\\begin{align}\n&\\Theta:=\\Omega\\cup\\{(s,-t)\\ |\\ (s,t)\\in \\Omega\\},\\label{Theta}\\\\\nthen \\qquad &\\bar{v}(\\partial\\Theta)\\subseteq K_\\varepsilon\\qquad\\textrm{and}\\qquad v(\\Omega) =\\bar{v}(\\Theta). \\nonumber\n\\end{align}\nIn particular, we have that $\\sup_{\\Omega}f\\circ v=\\sup_\\Theta f\\circ \\bar{v}$ and $\\|f\\circ \\bar{v}\\|_{L^2(\\Theta)}=2\\|f\\circ v\\|_{L^2(\\Omega)}$ for any smooth function $f\\in T^*\\R^2$. \n\nAfter proving in Lemma \\ref{lem:F2smooth} that $F_2\\circ \\bar{v}$ is in $C^2(\\Theta)$ we can apply the Aleksandrov maximum principle and obtain\n\\begin{align*}\n\\sup_{(s,t)\\in\\R\\times[0,1]}F_2\\circ v & \\leq \\sup_{\\partial\\Theta}F_2\\circ\\bar{v}+C\\|f_2\\circ \\bar{v}\\|_{L^2(\\Theta)}\\\\\n& \\leq \\sup_{K_\\varepsilon}f_2\\circ v+2C\\|f_2\\circ v\\|_{L^2(\\Omega)}.\n\\end{align*}\nSince the established bounds do not depend on the choice of $u\\in\\mathcal{M}^\\Gamma(a,b)$ or the choice of the connected component $\\Omega\\subseteq \\left(\\R\\times[0,1] \\right)\\setminus v^{-1}(K_\\varepsilon)$, this concludes the proof that $\\mathcal{M}^\\Gamma(a,b)$ is bounded in the $L^\\infty\\times\\R$-norm.\n\\hfill $\\square$\n\n\\vspace*{.25cm}\nTo complete the proof of Theorem \\ref{thm:FloerBounds} we need to prove the following technical lemmas:\n\\begin{lem}\\label{lem:LaplaceF}\nConsider the standard symplectic space $(T^*\\R^2, \\omega_0)$ with the Hamiltonian $H_0$ defined as in \\eqref{DefH0} and functions $F_1$ and $F_2$ defined in \\eqref{DefF}.\n Let $u:\\R \\to \\mathscr{H}_{q_0,q_1}\\times \\R$, $u=(v,\\eta)$ be a Floer trajectory corresponding to the action functional $\\A^{H_0-c}$, i.e. satisfying the relation\n\\eqref{FloerH0}.\nThen\n\\begin{align*}\n\\triangle (F_1 \\circ v) & \\geq -\\frac{1}{2}\\eta^2 F_1\\circ v+ \\partial_{s}\\eta \\left(F_2\\circ v -H_0\\circ v \\right),\\\\\n\\triangle (F_2 \\circ v) & \\geq -\\frac{1}{2}\\eta^2 F_1\\circ v-\\partial_s\\eta\\left(F_2\\circ v+H_0\\circ v\\right).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nUsing \\eqref{DefH0} we can calculate that\n\\begin{align}\nX_{H_0} & = (p_1+q_2)\\partial_{q_1}+(p_2-q_1)\\partial_{q_2}+p_2\\partial_{p_1}-p_1\\partial_{p_2},\\nonumber\\\\\ndF_1 (X_{H_0}) & = q_1p_1+q_2p_2,\\nonumber\\\\d^\\mathbb{C}(dF_1(X_{H_0})) & = qd^\\mathbb{C}p+pd^\\mathbb{C}q=pdp-qdq=dF_2-dF_1,\\nonumber\\\\\ndF_2 (X_{H_0}) & = 0,\\label{dF2XH}\\\\\nd^\\C F_1 (X_{H_0}) & = qdp(X_{H_0})=q_1p_2-q_2p_1 = F_2-H_0 ,\\label{dCF1XH}\\\\\nd^\\C F_2 (X_{H_0}) & =-\\left(|p|^2 +q_1p_2-q_2p_1\\right) = -F_2-H_0,\\label{dCF2XH}\\\\\nd(dF_1 (X_{H_0}))(X_{H_0}) & = |p|^2=2F_2,\\nonumber\\\\\n\\big(dH_0+d(d^\\C F_1 (X_{H_0}))&+d^\\C \\left( dF_1(X_{H_0})\\right)\\big) = 2dF_2-dF_1,\\nonumber\\\\\n\\big(dH_0+d(d^\\C F_2 (X_{H_0}))&+d^\\C \\left( dF_2(X_{H_0})\\right)\\big) = -dF_1.\\nonumber\n\\end{align}\nPlugging these qualities into \\eqref{Laplace} for a Floer trajectory $u=(v,\\eta)$ satisfying the relation \\eqref{FloerH0} we can calculate\n\\begin{align*}\n\\triangle (F_1 \\circ v) & = \\|\\partial_{s}v\\|^{2}+\\eta \\left(dH_0 +d(d^{\\mathbb{C}}F_1(X_{H_0}))+d^\\mathbb{C}(dF_1(X_{H_0}))\\right)(\\partial_{s}v)\\\\\n& +\\eta^2 d(dF_1(X_{H_0}))(X_{H_0})+\\partial_{s}\\eta\\ d^{\\mathbb{C}}F_1(X_{H_0})\\\\\n& = \\|\\partial_{s}v\\|^{2}+\\eta \\left(2pdp-qdq\\right)(\\partial_{s}v)+\\eta^2|p|^2+ \\partial_{s}\\eta \\left(\\frac{1}{2}|p|^2-H_0\\circ v \\right)\\\\\n& \\geq -\\eta^2 \\frac{1}{4}|q|^2+ \\partial_{s}\\eta \\left(\\frac{1}{2}|p|^2-H_0 \\right)\\\\\n& = -\\frac{1}{2}F_1\\circ v+ \\partial_{s}\\eta \\left(F_2\\circ v-H_0\\circ v \\right),\\\\\n\\triangle (F_2 \\circ v) & = \\|\\partial_{s}v\\|^{2}+\\eta \\left(dH_0 +d(d^{\\mathbb{C}}F_2(X_{H_0}))+d^\\mathbb{C}(dF_2(X_{H_0}))\\right)(\\partial_{s}v)\\\\\n& +\\eta^2 d(dF_2(X_{H_0}))(X_{H_0})+\\partial_{s}\\eta\\ d^{\\mathbb{C}}F_2(X_{H_0})\\\\\n& = \\|\\partial_{s}v\\|^{2}-\\eta qdq(\\partial_sv)-\\partial_s\\eta\\left( \\frac{1}{2}|p|^2+H_0\\right)\\\\\n& \\geq -\\frac{1}{4}\\eta^2|q|^2-\\partial_s\\eta\\left(\\frac{1}{2}|p|^2+H_0\\right)\\\\\n& = -\\frac{1}{2}\\eta^2F_1\\circ v-\\partial_s\\eta\\left(F_2\\circ v+H_0\\circ v\\right)\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\\label{lem:F2smooth}\nConsider the standard symplectic space $(T^*\\R^2, \\omega_0)$ with the Hamiltonian $H_0$ defined as in \\eqref{DefH0}. Let $u:\\R \\to \\mathscr{H}_{q_0,q_1}\\times \\R$, $u=(v,\\eta)$ be a Floer trajectory associated to $\\A^{H_0-\\mathbf{c}_s}$, i.e. satisfying the relation \\eqref{FloerH0} and let $\\bar{v}:\\R\\times([-1,1]/-1\\sim 1) \\to T^*\\R^2$ be the extension of $v$ defined in \\eqref{barv}. Then the composition $F_2\\circ\\bar{v}$ with the function $F_2:T^*\\R^2\\to\\R$, $F(q,p)=\\frac{1}{2}|p|^2$ is $C^2$.\n\\end{lem}\n\\begin{proof}\nFirst, observe that $\\bar{v}$ is continuous. Moreover, $\\bar{v}$ is smooth on\\linebreak $\\R\\times\\left((-1,0)\\cup(0,1)\\right)$. Therefore, $F_2\\circ \\bar{v}$ is also everywhere continuous and smooth on $\\R\\times\\left((-1,0)\\cup(0,1)\\right)$. What is left to check is that $F_2\\circ \\bar{v}$ is $C^2$ on $\\R\\times \\{-1,0,1\\}$.\nNote that\n$$\n\\frac{d}{dt} (F_2\\circ \\bar{v})(s,t)=\\begin{cases}\ndF_2( \\partial_tv)(s,t) & \\textrm{for}\\quad t\\geq 0,\\\\\n-dF_2(\\partial_tv)(s,-t) & \\textrm{for}\\quad t\\leq 0.\n\\end{cases}\n$$\nHence, the function $F_2\\circ\\bar{v}$ is $C^1$ if and only if \n\\begin{equation}\ndF_2( \\partial_tv)(s,0)=dF_2( \\partial_tv)(s,1)=0\\qquad \\forall\\ s\\in \\R.\n\\label{pdpPartialtv}\n\\end{equation}\nBy assumption $v: \\R \\to \\mathscr{H}_{q_0,q_1}$, so\n$v(s,0)\\equiv q_0$ and $v(s,1)\\equiv q_1$ for all $s\\in\\R$. Consequently,\n\\begin{equation}\ndq_i(\\partial_sv)(s,0)=dq_i(\\partial_sv)(s,1)=0\\qquad \\textrm{for}\\quad i=0,1 \\quad \\textrm{and}\\quad \\forall\\ s\\in \\R.\n\\label{dqPartialsv}\n\\end{equation}\nNow using the Floer equation \\eqref{FloerH0} for the Hamiltonian $H_0$ and the relation \\eqref{dF2XH} we can calculate for $t\\geq 0$\n\\begin{align*}\ndF_2( \\partial_tv) & = d^\\mathbb{C}F_2(\\partial_sv)+\\eta dF_2(X_{H_0})=-pdq(\\partial_sv).\n\\end{align*}\nPlugging \\eqref{dqPartialsv} into the equation above we obtain \\eqref{pdpPartialtv}, which proves that $F_2\\circ \\bar{v}$ is $C^1$ everywhere.\n\nTo prove that $F_2\\circ \\bar{v}$ is $C^2$ we first calculate:\n\\begin{align*}\n\\frac{d^2}{dt^2}F_2\\circ v & = D^2F_2(\\partial_t v,\\partial_t v)+dF_2(\\partial_{tt}v)= \\eta^2 D^2F_2(X_{H_0},X_{H_0})+dF_2(\\partial_{tt}v)\\\\ &=\\eta^2|p|^2+dF_2(\\partial_{tt}v).\n\\end{align*}\nObviously $|p|^2$ does not depend on the sign of $\\partial_tv$. On the other hand, the function $\\partial_{tt}\\bar{v}= \\partial_{tt}v$ is continuous, since the two negative signs cancel each other. Therefore, the sum of these two functions is also continuous on the whole cylinder $\\R\\times([-1,1]/-1\\sim 1)$.\n\nNow $\\frac{d^2}{dsdt}F_2\\circ \\bar{v}$ is well defined and continuous on the whole $\\R\\times((-1,0)\\cup(0,1))$. On the other hand by \\eqref{pdpPartialtv} we obtain that $\\frac{d^2}{dsdt}F_2\\circ v (s,0)=\\linebreak\\frac{d^2}{dsdt}F_2\\circ v (s,1)=0$, thus $\\frac{d^2}{dsdt}F_2\\circ \\bar{v}$ is a continuous function on the whole cylinder $\\R\\times([-1,1]/-1\\sim 1)$.\n\nFinally, we observe that $\\frac{d^2}{ds^2}(F_2\\circ v (s,t))=\\frac{d^2}{ds^2}(F_2\\circ v (s,-t))$ as it does not depend on the sign of the $t$-variable.\n\\end{proof}\n\n\\begin{lem}\nLet \n$\\Omega \\subseteq \\left(\\mathbb{R} \\times [0,1]\\right)\\setminus v^{-1}(K_\\varepsilon)$\nbe a connected component as defined in \\eqref{Omega}. Then the following values are well-defined and finite:\n\\begin{equation}\\label{Defs0s1}\ns_0 :=\\inf\\{s\\ |\\ (s,t)\\in \\Omega\\}\\qquad \\textrm{and}\\qquad\ns_1 :=\\sup\\{s\\ |\\ (s,t)\\in \\Omega\\}.\n\\end{equation}\nMoreover, $s_0$ and $s_1$ satisfy\n\\begin{equation}\\label{s0s1Bound}\n|s_1-s_0|\\leq \\frac{\\mathfrak{e}\\|J\\|^2_\\infty}{\\varepsilon^2},\n\\end{equation}\n\n\\end{lem}\n\\begin{proof}\nBy Lemma \\ref{lem:ActBound} we obtain uniform bounds on the action, energy and the $\\eta$-parameter for all $u\\in\\mathcal{M}^\\Gamma(a,b)$. By the convergence of the integral\n$$\\|\\nabla\\A^{H_0-h_s}(u)\\|_{L^2(\\R\\times[0,1])\\times\\R} \\leq \\|J\\|_{L^\\infty}\\sqrt{\\mathfrak{e}},$$\nwe know that $\\lim_{s\\to\\pm\\infty}u(s)\\in \\B^\\Gamma(\\mathfrak{a},\\mathfrak{y},\\varepsilon)$. For the compact subset $K_\\varepsilon\\subseteq T^*\\R^2$ defined in \\eqref{DefKepsi} we have $v([0,1])\\subseteq K_\\varepsilon$ for all $(v,\\eta)\\in\\mathcal{M}^\\Gamma(a,b)$. Since\\linebreak $\\Omega \\subseteq \\left(\\mathbb{R} \\times [0,1]\\right)\\setminus v^{-1}(K_\\varepsilon)$, thus the values $s_0$ and $s_1$ as in \\eqref{Defs0s1} are well defined and finite. Moreover, for all $s\\in (s_0,s_1)$, $u(s)\\notin \\mathcal{B}^\\Gamma(\\mathfrak{a},\\mathfrak{y}, \\varepsilon)$, therefore by \\eqref{sBoundEpsi} we can estimate\n$$\n|s_1-s_0|\\leq \\frac{\\mathfrak{e}\\|J\\|^2_\\infty}{\\varepsilon^2},\n$$\nwhere $\\mathfrak{e}=\\|J\\|_\\infty(b-a+\\frac{c}{3}\\mathfrak{y})$ as in Lemma \\ref{lem:ActBound}. \n\\end{proof}\n\n\\begin{lem}\\label{lem:f1L1Bound}\nConsider the setting as in Theorem \\ref{thm:FloerBounds}. Fix $a,b\\in \\R$ and let $u\\in \\mathcal{M}^\\Gamma(a,b)$ be a Floer trajectory. Let \n$\\Omega \\subseteq \\left(\\mathbb{R} \\times [0,1]\\right)\\setminus v^{-1}(K_\\varepsilon)$\nbe a connected component as defined in \\eqref{Omega}. Then the function\n$$\nf_1(s,t):= -\\frac{1}{2}\\eta^2(s) F_1 \\circ v(s,t) + \\partial_{s}\\eta(s) \\left(F_2-H_0\\right)\\circ v(s,t).\n$$\nis bounded in the $L^1(\\Omega)$ norm and the bounds do not depend on $u$ or $\\Omega$.\n\\end{lem}\n\\begin{proof}\nLet $s_0$ and $s_1$ be as in \\eqref{Defs0s1}.\nDenote $\\overline{\\Omega}:= [s_0,s_1]\\times[0,1]$. Naturally,\\linebreak $\\Omega\\subseteq \\overline{\\Omega}$. Before we estimate $\\|f_1\\|_{L^1(\\Omega)}$ we first use the Cauchy-Schwartz inequality together with \\eqref{DefE}, \\eqref{qL2Bound}, \\eqref{pL2Bound} and \\eqref{s0s1Bound} to do the following estimates:\n\\begin{align}\n\\|\\partial_s\\eta\\|_{L^1(\\Omega)}& \\leq \\|\\partial_s\\eta\\|_{L^1(\\overline{\\Omega})}\\leq \\sqrt{|s_1-s_0|}\\|\\nabla^{J_s} \\A^{H_0-h_s}(u(s))\\|_{L^2\\times\\R}\\nonumber\\\\\n&\\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\label{partialEtaBound}\\\\\n\\|\\partial_s\\eta(H_0-\\mathbf{c}_s)\\|_{L^1(\\Omega)}&\\leq\n\\|\\partial_s\\eta (H_0-h_s)\\circ v\\|_{L^1(\\overline{\\Omega})}=\\|\\partial_s\\eta\\|_{L^2(\\overline{\\Omega})}^2\\nonumber\\\\\n& \\leq \\|\\nabla^{J_s} \\A^{H_0-h_s}(u(s))\\|_{L^2\\times\\R}^2\\leq \\mathfrak{e}\\label{H0Bound}\\\\\n\\|q\\|_{L^2(\\Omega)} & \\leq \\|q \\|_{L^2(\\overline{\\Omega})}\\leq \\sqrt{|s_1-s_0|} \\sup_{s\\in\\R}\\|q(s)\\|_{L^2([0,1])}\\nonumber\\\\\n& \\leq \\frac{\\sqrt{\\mathfrak{e}}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\label{qOmegaBound}\\\\\n\\|p\\|_{L^2(\\Omega)} & \\leq \\frac{\\sqrt{\\mathfrak{e}}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\label{pOmegaBound}\\\\\n\\|\\partial_s\\eta\\ p^2\\|_{L^1(\\Omega)} & \\leq \\|\\partial_s\\eta\\|_{L^1(\\overline{\\Omega})}\\sup_{s\\in\\R}\\|p(s)\\|_{L^2([0,1])}^2 \\leq\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2 \\label{partialEtaPBound}\n\\end{align}\nUsing the bounds obtained above and the inequality \\eqref{eqEta} we can calculate:\n\\begin{align*}\n\\|f_1\\|_{L^1(\\Omega)}& \\leq \\frac{1}{4}\\mathfrak{y}^2 \\|q\\|_{L^2(\\Omega)}^2+\\frac{1}{2}\\|\\partial_s\\eta\\ p^2\\|_{L^1(\\Omega)}+\\|\\partial_s\\eta(H_0-\\mathbf{c}_s)\\|_{L^1(\\Omega)}+\\|\\mathbf{c}_s\\partial_s\\eta\\|_{L^1(\\Omega)}\\\\\n&\\leq \\frac{\\mathfrak{e}\\mathfrak{y}^2\\|J\\|_\\infty^2}{4\\varepsilon^2}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+\\frac{\\mathfrak{e}\\|J\\|_\\infty^2}{2\\varepsilon^2}\\left( \\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+\\mathfrak{e}\\left(\\frac{\\mathfrak{h}\\|J\\|_\\infty}{\\varepsilon}+1\\right).\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\\label{lem:f1W11Bound}\nConsider the setting as in Lemma \\ref{lem:f1L1Bound}. Then the function $f_1$ is bounded in the $W^{1,1}$-norm and the bounds do not depend on $u$ or $\\Omega$.\n\\end{lem}\n\n\\begin{proof}\nTo prove that $f_1\\in W^{1,1}(\\Omega)$ we first need to calculate its derivatives:\n\\begin{align*}\n\\partial_sf_1 & = - \\eta\\partial_s\\eta\\ F_1+\\partial_{ss}\\eta\\ d^\\mathbb{C}F_1(X_{H_0})+\\left(\\partial_s\\eta\\ d(F_2-H_0)-\\frac{1}{2}\\eta^2 dF_1\\right)(\\partial_sv)\\\\\n\\partial_tf_1 & = \\left(\\partial_s\\eta\\ d(F_2-H_0)-\\frac{1}{2}\\eta^2 dF_1\\right)(\\mathbb{J}\\partial_sv+\\eta X_{H_0})\\\\\n& = \\left(\\partial_s\\eta\\ d^\\mathbb{C}(F_2-H_0)-\\frac{1}{2}\\eta^2 d^\\mathbb{C} F_1\\right)(\\partial_sv)-\\frac{1}{2}\\eta^3dF_1(X_{H_0}),\n\\end{align*}\nwhere the last equality comes from the fact that $dF_2(X_{H_0})(q,p)=0$.\nUsing \\eqref{qL2Bound}, \\eqref{vL2Bound} and \\eqref{s0s1Bound} we can estimate:\n\\begin{align}\n\\|dF_1(X_{H_0})\\|_{L^1(\\Omega)}& = \\|q_1p_1+q_2p_2\\|_{L^1(\\Omega)}\\leq\\frac{1}{2}\\| v\\|_{L^2(\\Omega)}^2\\leq \\frac{1}{2}|s_1-s_0|\\sup_{s\\in\\R}\\|v(s)\\|_{L^2([0,1])}^2\\nonumber\\\\\n& \\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty^2}{2\\varepsilon^2}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\label{eq0}\\\\\n\\|dF_1(\\partial_sv)\\|_{L^1(\\Omega)}& \\leq \\|\\nabla F_1\\|_{L^2(\\Omega)}\\|\\partial_sv\\|_{L^1(\\Omega)}\\leq \\|q\\|_{L^2(\\overline{\\Omega})}\\|\\nabla^{J_s}\\A^{H_0-h_s}(u)\\|_{L^2\\times\\R}\\nonumber\\\\\n& \\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\label{eq1}\n\\end{align}\n\\begin{align}\n\\|&\\partial_s\\eta\\ d(F_2 -H_0)(\\partial_sv)\\|_{L^1(\\Omega)} \\leq \\int_{s_0}^{s_1}|\\partial_s\\eta|\\int_0^1|\\nabla (F_2-H_0)||\\partial_sv|dtds\\nonumber\\\\\n &= \\int_{s_0}^{s_1}|\\partial_s\\eta|\\int_0^1|v||\\partial_sv|dtds \\leq \\sup_{s\\in\\R}\\|v(s)\\|_{L^2([0,1])}\\int_{s_0}^{s_1}|\\partial_s\\eta|\\|\\partial_sv(s)\\|_{L^2([0,1])}ds\\nonumber\\\\\n & \\leq\\sup_{s\\in\\R}\\|v(s)\\|_{L^2([0,1])}\\|\\nabla^{J_s}\\A^{H_0-h_s}(u)\\|_{L^2\\times\\R}^2 \\leq \\mathfrak{e}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\label{eq2}\n\\end{align}\nSimilarly, we have\n\\begin{align}\n\\|d^\\C F_1(\\partial_sv)\\|_{L^1(\\Omega)} & \\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right),\\label{eq3}\\\\\n\\|\\partial_s\\eta\\ d^\\C(F_2-H_0)(\\partial_sv)\\|_{L^1(\\Omega)} &\\leq \\mathfrak{e}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right).\n\\end{align}\nUsing the estimates above and \\eqref{eqEta} we can combine them to obtain a uniform $L^1$-bound on $\\partial_tf_1$:\n\\begin{align*}\n\\|\\partial_tf_1\\|_{L^1(\\Omega)} &\\leq \\|\\partial_s\\eta d^\\C( F_2-H_0)(\\partial_sv)\\|_{L^1(\\Omega)}+\\frac{1}{2}\\|\\eta^2 d^\\C F_1(\\partial_sv)\\|_{L^1(\\Omega)}+\\frac{1}{2}\\|\\eta^3 dF_1(X_{H_0})\\|_{L^1(\\Omega)}\\\\\n& \\leq \\mathfrak{e}\\mathfrak{p}+ \\mathfrak{e}\\left(1+\\frac{\\mathfrak{ey}^2\\|J\\|_\\infty}{2\\varepsilon}\\right)\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{\\mathfrak{ey}^3\\|J\\|_\\infty^2}{4\\varepsilon^2}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\n\\end{align*}\nThis gives us a bound on $\\|\\partial_tf_1\\|_{L^1(\\Omega)}$, which does not depend on the choice of the Floer trajectory $u$ or the connected component $\\Omega$.\n\nTo estimate $\\|\\partial_sf_1\\|_{L^1(\\Omega)}$ we first use \\eqref{qL2Bound} and \\eqref{partialEtaBound} to estimate\n\\begin{align}\n\\|\\partial_s\\eta\\ F_1\\circ v\\|_{L^1(\\Omega)} & \\leq \\|\\partial_s\\eta\\ F_1\\circ v\\|_{L^1(\\overline{\\Omega})} = \\frac{1}{2}\\int_{s_0}^{s_1}|\\partial_s\\eta|\\|q(s)\\|_{L^2([0,1])}^2ds\\nonumber\\\\\n& \\leq \\frac{1}{2}\\|\\partial_s\\eta\\|_{L^1(\\overline{\\Omega})}\\sup_{s\\in\\R}\\|q(s)\\|_{L^2([0,1])}^2\\nonumber\\\\\n&\\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty}{2\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\label{partialsEtaF1}\n\\end{align}\nSecondly, we analyze $\\partial_{ss}\\eta$:\n$$\n\\partial_{ss}\\eta = -\\int_0^1d(H_0-h_s)(\\partial_sv)dt-\\int_0^1 \\partial_sh_s (v)dt.\n$$\nUsing \\eqref{inqGamma}, \\eqref{pL2Bound} and \\eqref{vL2Bound} we can estimate:\n\\begin{align}\n\\|\\partial_{ss}\\eta\\|_{L^1(\\Omega)} & = \\int_{\\Omega}\\bigg|\\int_0^1\\left(d(H_0-h_s)(\\partial_sv)+\\partial_sh_s(v)\\right)dt\\bigg|dt ds\\nonumber\\\\\n& \\leq \\|d(H_0-h_s)(\\partial_sv)\\|_{L^1(\\overline{\\Omega})}+\\|\\partial_sh_s\\|_{L^\\infty}\\nonumber\\\\\n&\\leq \\left(\\|\\nabla H_0\\|_{L^2(\\overline{\\Omega})}+\\|\\nabla h_s\\|_{L^2(\\overline{\\Omega})}\\right)\\|\\partial_sv\\|_{L^2(\\overline{\\Omega})}\\|\\partial_sh_s\\|_{L^\\infty}\\nonumber\\\\\n& \\leq \\left(\\|v+p\\|_{L^2(\\overline{\\Omega})}+\\|\\nabla h_s\\|_{L^2(\\overline{\\Omega})}\\right)\\|\\nabla^{J_s}\\A^{H_0-h_s}(u)\\|_{L^2\\times\\R}+\\|\\partial_sh_s\\|_{L^\\infty}\\nonumber\\\\\n&\\leq \\sqrt{\\mathfrak{e}|s_1-s_0|}\\left(\\sup_{s\\in\\R}\\|v(s)\\|_{L^2([0,1])}+\\sup_{s\\in\\R}\\|p(s)\\|_{L^2([0,1])}+\\sup_{s\\in\\R}\\|\\nabla h_s\\|_{L^\\infty}\\right)+\\|\\partial_sh_s\\|_{L^\\infty}\\nonumber\\\\\n&\\leq \\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left(\\mathfrak{h}+ \\mathfrak{q}+2\\mathfrak{p}+\\frac{2\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\label{partialSS}\n\\end{align}\nNow we will use the Floer equations, the Cauchy-Schwartz inequality together with \\eqref{DefE}, \\eqref{sBoundEpsi}, \\eqref{qL2Bound}, \\eqref{pL2Bound} and \\eqref{dCF1XH} to calculate the following:\n\\begin{align}\n\\|\\partial_{ss}\\eta\\ d^\\mathbb{C}F_1(X_{H_0})\\|_{L^1(\\Omega)} & \\leq \\|\\partial_{ss}\\eta\\ d^\\mathbb{C}F_1(X_{H_0})\\|_{L^1(\\overline{\\Omega})} =\\int_{s_0}^{s_1}|\\partial_{ss}\\eta|\\int_0^1|q_1p_2-q_2p_1|dtds\\nonumber\\\\\n& \\leq \\frac{1}{2}\\int_{s_0}^{s_1}|\\partial_{ss}\\eta|(\\|q(s)\\|_{L^2([0,1])}^2+\\|p(s)\\|_{L^2([0,1])}^2)ds\\nonumber\\\\\n&\\leq \\frac{1}{2}\\|\\partial_{ss}\\eta\\|_{L^1(\\overline{\\Omega})}\\sup_{s\\in\\R}\\|v(s)\\|_{L^2([0,1])}^2\\nonumber\\\\\n& \\leq \\left(\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left( \\frac{1}{2}(\\mathfrak{h}+\\mathfrak{q})+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\right)\\left(\\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\label{EtaSSdCF1XH0}\n\\end{align}\nUsing the estimates above together with \\eqref{eqEta}, \\eqref{eq1}, \\eqref{eq2} and \\eqref{partialsEtaF1} we can estimate:\n\\begin{align*}\n\\|\\partial_sf_1\\|_{L^1(\\Omega)} & \\leq \\mathfrak{y}\\|\\partial_s\\eta\\ F_1\\circ v\\|_{L^1(\\Omega)}+\\|\\partial_{ss}\\eta d^\\mathbb{C}F_1(X_{H_0})\\|_{L^1(\\Omega)}+\\frac{1}{2}\\mathfrak{y}^2\\|d F_1(\\partial_sv)\\|_{L^1(\\Omega)}\\\\\n& +\\|\\partial_s\\eta\\ d(F_2-H)(\\partial_sv)\\|_{L^1(\\Omega)} \\\\\n& \\leq \\frac{\\mathfrak{e y}\\|J\\|_\\infty}{2\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2 + \\frac{\\mathfrak{e y}^2\\|J\\|_\\infty}{2\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\\\\n& + \\left(\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left( \\frac{1}{2}(\\mathfrak{h}+\\mathfrak{q})+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\right)\\left(\\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\\\\n& +\\mathfrak{e}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\n\\end{align*}\nThis gives us a bound on $\\|\\partial_sf_1\\|_{L^1(\\Omega)}$, which does not depend on the choice of the Floer trajectory $u$ or the connected component $\\Omega$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:f2L1Bound}\nConsider the setting as in Theorem \\ref{thm:FloerBounds}. Fix $a,b\\in \\R$ and let $u\\in \\mathcal{M}^\\Gamma(a,b)$ be a Floer trajectory. Let \n$\\Omega \\subseteq \\left(\\mathbb{R} \\times [0,1]\\right)\\setminus v^{-1}(K_\\varepsilon)$\nbe a connected component as defined in \\eqref{Omega}. Then the function\n$$\nf_2(s,t):= -\\frac{1}{2}\\eta^2(s)F_1\\circ v(s,t)-\\partial_s\\eta(s)\\left(F_2+H_0\\right)\\circ v(s,t).\n$$\nis bounded in the $L^1(\\Omega)$ norm and the bounds do not depend on $u$ or $\\Omega$.\n\\end{lem}\n\\begin{proof}\nTo estimate $\\|f_2\\|_{L^1(\\Omega)}$ we use the bounds obtained in \\eqref{eqEta}, \\eqref{partialEtaBound}, \\eqref{H0Bound}, \\eqref{qOmegaBound} and \\eqref{partialEtaPBound} \nto calculate:\n\\begin{align*}\n\\|f_2\\|_{L^1(\\Omega)} & \\leq \\frac{\\mathfrak{y}^2}{4}\\|q\\|^2_{L^2(\\Omega)} +\n\\frac{1}{2}\\|\\partial_s\\eta\\ p^2\\|_{L^2(\\Omega)}+\\|\\partial_s\\eta (H_0-\\mathfrak{c}_s)\\|_{L^2(\\Omega)}+\\|\\mathfrak{c}_s\\partial_s\\eta\\|_{L^1(\\Omega)}\\\\\n&\\leq \\frac{\\mathfrak{ey}^2\\|J\\|_\\infty^2}{4\\varepsilon^2}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+ \\frac{\\mathfrak{e}}{2\\varepsilon}\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+\\mathfrak{e}\\left(\\frac{c}{\\varepsilon}+\\|J\\|_\\infty^2\\right).\n\\end{align*}\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:f2W11Bound}\nConsider the setting as in Lemma \\ref{lem:f2L1Bound}. Then the function $f_2$ is bounded in the $W^{1,1}$-norm and the bounds do not depend on $u$ or $\\Omega$.\n\\end{lem}\n\n\\begin{proof}\nTo prove that $f_2\\in W^{1,1}(\\Omega)$ we first need to calculate its derivatives:\n\\begin{align*}\n\\partial_sf_2 & = - \\eta\\partial_s\\eta F_1 + \\partial_{ss}\\eta\\ d^\\mathbb{C}F_2(X_{H_0})-\\left(\\partial_s\\eta\\ d(F_2+H_0)+\\frac{1}{2}\\eta^2 dF_1\\right)(\\partial_sv)\\\\\n\\partial_tf_2 & = -\\left(\\partial_s\\eta\\ d(F_2+H_0)+\\frac{1}{2}\\eta^2 dF_1\\right)(\\mathbb{J}\\partial_sv+\\eta X_{H_0})\\\\\n& = -\\left(\\partial_s\\eta\\ d^\\mathbb{C}(F_2+H_0)+\\frac{1}{2}\\eta^2 d^\\mathbb{C} F_1\\right)(\\partial_sv)-\\frac{1}{2}\\eta^3 dF_1(X_{H_0}),\n\\end{align*}\nwhere the last equality comes from \\eqref{dF2XH}.\n\nLet $s_0$ and $s_1$ be as in \\eqref{Defs0s1}.\nDenote $\\overline{\\Omega}:= [s_0,s_1]\\times[0,1]$. Naturally, $\\Omega\\subseteq \\overline{\\Omega}$.\nBefore we estimate $\\|\\partial_t f_2\\|_{L^1(\\Omega)}$ we will first\nrecall \\eqref{dCF1XH} and \\eqref{dCF2XH} which give us the following relation\n\\begin{equation}\nd^\\mathbb{C}F_1(X_{H_0})=F_2-H_0=2F_2+d^\\mathbb{C}F_2(X_{H_0}).\n\\label{F2-H0}\n\\end{equation}\nFurther on, we use \\eqref{DefE}, \\eqref{pL2Bound}, \\eqref{pOmegaBound} and \\eqref{eq2} to calculate the following bounds:\n\\begin{align*}\n\\|\\partial_s\\eta\\ dF_2(\\partial_sv)\\|_{L^1(\\Omega)}& \\leq \\|\\partial_s\\eta\\ dF_2(\\partial_sv)\\|_{L^1(\\overline{\\Omega})}\\\\\n& \\leq \\int_{s_0}^{s_1}|\\partial_s\\eta|\\int_0^1|p(s,t)||\\partial_sv(p,s)|dtds\\\\\n&\\leq \\int_{s_0}^{s_1}|\\partial_s\\eta|\\|p(s)\\|_{L^2([0,1])}\\|\\partial_s v(s)\\||_{L^2([0,1])}ds\\\\\n&\\leq \\sup_{s\\in \\R}\\|p(s)\\|_{L^2([0,1])}\\|\\partial_s\\eta\\|_{L^2(\\R)}\\|\\partial_sv\\|_{L^2(\\R\\times[0,1])}\\\\\n&\\leq \\mathfrak{e}\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right).\n\\end{align*}\nThis, together with \\eqref{F2-H0} gives us:\n\\begin{align}\n\\|\\partial_s\\eta\\ d(F_2+H_0)(\\partial_sv)\\|_{L^1(\\Omega)} & \\leq 2\\|\\partial_s\\eta\\ dF_2(\\partial_sv)\\|_{L^1(\\Omega)}+\\|\\partial_s\\eta\\ d(F_2-H_0)(\\partial_sv)\\|_{L^1(\\Omega)}\\nonumber\\\\\n&\\leq \\mathfrak{e}\\left( \\mathfrak{q}+3\\mathfrak{p}+2\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\label{eq4}\n\\end{align}\nSimilarly, we have\n$$\n\\|\\partial_s\\eta\\ d^\\mathbb{C}(F_2+H_0)(\\partial_sv)\\|_{L^1(\\Omega)}\\leq \\mathfrak{e}\\left( \\mathfrak{q}+3\\mathfrak{p}+2\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right).\n$$\nUsing the bounds obtained above together with \\eqref{eqEta}, \\eqref{eq0} and \\eqref{eq3} we can estimate:\n\\begin{align*}\n\\|\\partial_tf_2\\|_{L^1(\\Omega)} & \\leq \\|\\partial_s\\eta\\ d^\\mathbb{C}(F_2+H_0)(\\partial_s v)\\|_{L^1(\\Omega)}+\\frac{1}{2}\\mathfrak{y}^2 \\left(\\|d^\\mathbb{C} F_1(\\partial_sv)\\|_{L^1(\\Omega)}+\\mathfrak{y} \\|dF_1(X_{H_0})\\|_{L^1(\\Omega)}\\right)\\\\\n&\\leq \\mathfrak{e}\\left( \\mathfrak{q}+3\\mathfrak{p}+2\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{\\mathfrak{ey}^2\\|J\\|_\\infty}{2\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\\\\n&+\\frac{\\mathfrak{ey}^3\\|J\\|_\\infty^2}{4\\varepsilon^2}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2.\n\\end{align*}\n\nBefore we estimate $\\|\\partial_sf_2\\|_{L^2}$ we first use \\eqref{pL2Bound}, \\eqref{partialSS} and \\eqref{EtaSSdCF1XH0}\nto calculate the following bounds:\n\\begin{align*}\n\\|\\partial_{ss}\\eta\\ F_2\\circ v \\|_{L^1(\\Omega)} & \\leq \\|\\partial_{ss}\\eta\\ F_2\\circ v \\|_{L^1(\\overline{\\Omega)}}=\\frac{1}{2}\\int_{s_0}^{s_1}|\\partial_{ss}\\eta|\\int_0^1|p(s,t)|^2dtds\\\\\n&\\leq \\frac{1}{2}\\|\\partial_{ss}\\eta\\|_{L^1(\\overline{\\Omega})}\\sup_{s\\in\\R}\\|p(s)\\|_{L^2([0,1])}^2\\\\\n& \\leq \\left(\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left(\\frac{1}{2}(\\mathfrak{h}+ \\mathfrak{q})+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\right)\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\\\\n\\|\\partial_{ss}\\eta\\ d^\\mathbb{C}F_2(X_{H_0})\\|_{L^1(\\Omega)} & = \\|\\partial_{ss}\\eta\\left(d^\\mathbb{C}F_1(X_{H_0})-2F_2\\right)\\|_{L^1(\\Omega)}\\\\\n& \\leq \\|\\partial_{ss}\\eta\\ d^\\mathbb{C}F_1(X_{H_0})\\|_{L^1(\\Omega)}+2\\|\\partial_{ss}\\eta\\ F_2\\circ v\\|_{L^1(\\Omega)}\\\\\n& \\leq \\left(\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left( \\frac{1}{2}(\\mathfrak{h}+\\mathfrak{q})+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\right)\\\\\n&\\cdot \\left(\\left(\\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+2\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\right)\n\\end{align*}\nFinally, using the estimates above together with \\eqref{eqEta}, \\eqref{partialEtaPBound}, \\eqref{eq2}, \\eqref{partialsEtaF1} and \\eqref{eq4} we can calculate.\n\\begin{align*}\n\\|\\partial_sf_2\\|_{L^1(\\Omega)} & \\leq \\mathfrak{y}\\|\\partial_s\\eta\\ F_1\\|_{L^1(\\Omega)}+\\|\\partial_s\\eta\\ d(F_2+H_0)(\\partial_sv)\\|_{L^1(\\Omega)}\\\\\n& +\\frac{\\mathfrak{y}^2}{2} \\|dF_1(\\partial_sv)\\|_{L^1(\\Omega)}+\\|\\partial_{ss}\\eta\\left(F_2+H_0\\right)\\|_{L^1(\\Omega)}\\\\\n & \\leq \\frac{\\mathfrak{ey}\\|J\\|_\\infty}{2\\varepsilon}\\left(\\mathfrak{q}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+\\mathfrak{e}\\left( \\mathfrak{q}+3\\mathfrak{p}+2\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\\\\n &+\\frac{\\mathfrak{ey}^2}{2}\\left( \\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)\\\\\n& +\\left(\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\left( \\frac{1}{2}(\\mathfrak{h}+\\mathfrak{q})+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)+\\frac{c}{3}\\right)\\\\\n&\\cdot \\left(\\left(\\mathfrak{q}+\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2+2\\left(\\mathfrak{p}+\\frac{\\mathfrak{e}\\|J\\|_\\infty}{\\varepsilon}\\right)^2\\right)\n\\end{align*}\nThis concludes the proof that $f_2$ is bounded in the $W^{1,1}$-norm and the bounds do not depend on the choice of the Floer trajectory $u\\in \\mathcal{M}^\\Gamma(a,b)$ or the connected component $\\Omega\\subseteq \\R\\times[0,1]\\setminus v^{-1}(K_\\varepsilon)$, but only on the constants $a,b, \\varepsilon$ and the smooth homotopy $\\Gamma$.\n\\end{proof}\n\n\\subsection{The Lagrangian Rabinowitz Floer homology of the Copernican Hamiltonian}\n\nHaving proven the $L^\\infty$-bounds on the Floer trajectories we are finally ready to prove the main theorem of this paper:\n\n\\vspace*{.25cm}\n\\textit{Proof of Theorem \\ref{thm:LRFH}:}\nLet $H_0:T^*\\R^2\\to\\R$ be the Copernican Hamiltonian defined in \\eqref{DefH0} and let $\\mathcal{H}$ be the set of perturbations as defined in \\eqref{DefHset}.\nFix a pair $q_0,q_1\\in \\R^2$. Let $\\{K_{n,m}\\}_{n,m\\in\\mathbb{N}}$ be the sequence of compact sets\nas in Lemma \\ref{lem:H0Chord}, such that \n\\begin{equation}\\label{Knm}\n\\begin{aligned}\nK_{n,m}\\subseteq K_{n+1,m}\\quad &\\textrm{and} \\quad \\bigcup_{n\\in\\mathbb{N}}K_{n,m}=T^*\\R^2\\quad &&\\forall\\ m\\in \\mathbb{N},\\\\\nK_{n,m}\\subseteq K_{n,m+1} \\quad &\\textrm{and} \\quad T_{q_0}^*\\R^2, T_{q_1}^*\\R^2\\subseteq \\bigcup_{m\\in\\mathbb{N}}K_{n,m}\\quad &&\\forall\\ n\\in\\mathbb{N}.\n\\end{aligned}\n\\end{equation}\n\n\n\nFix a Hamiltonian $h_0\\in \\mathcal{H}$ and denote $c(h_0):=\\inf (h-dh(p\\partial_p))$. Naturally, $c(h_0)>0$.\nLet $K_{n,m}$ be any of the compact sets in the sequence, such that $\\supp dh_0\\subseteq K_{n,m}$ and $\\|h_0\\|_{L^\\infty}<m$. Denote\n$$\n\\mathcal{H}_{h_0}(K_{n,m}):=\\left\\lbrace\\begin{array}{c|c}\n& dh \\in C_0^\\infty(K_{n,m}), \\quad \\|h\\|_{L^\\infty}<\\frac{c(h_0)}{50},\\\\\nh \\in C^\\infty(T^*\\R^2) & h_0+h \\in \\mathcal{H}, \\quad \\|h_0+h\\|_{L^\\infty}<m,\\\\\n& \\inf((h+h_0)-d(h+h_0)(p\\partial_p))>\\frac{1}{2} c(h_0).\n\\end{array}\\right\\rbrace.\n$$\nThis way $\\mathcal{H}_{h_0}(K_{n,m})$ is an open neighbourhood of $0$ in $\\{h\\in C^\\infty(T^*\\R^2)\\ |\\ dh\\in C_0^\\infty(K_{n,m})\\}$. By Lemma \\ref{lem:H0Chord} the critical set of the action functional $\\A^{H-h_0}_{q_0,q_1}$ is continuously compact in $(K_{n,m}, \\mathcal{H}_{h_0}(K_{n,m}))$. On the other hand, by construction if $h\\in \\mathcal{H}_{h_0}(K_{n,m})$, then $h_0+h\\in \\mathcal{H}$, hence by Lemma \\ref{lem:nonEmpty} every Hamiltonian $H_0-h_0-h$ satisfies the condition \\eqref{nonEmpty}. This proves that the compact set $K_{n,m}$ and the set of perturbations $\\mathcal{H}_{h_0}(K_{n,m})$ satisfy assumptions 1. and 2. of Theorem \\ref{thm:DefLRFH} for the Hamiltonian $H_0-h_0$.\n\nNow, if we take any open, pre-compact set $\\mathcal{V}\\subseteq T^*\\R^2$, such that $K_{n,m}\\subseteq \\mathcal{V}$, any two almost complex structures $J_0,J_1\\in \\mathcal{J}(\\mathcal{V},\\mathbb{J})$ and any $h\\in\\mathcal{H}_{h_0}(K_{n,m})$, then every smooth homotopy $\\Gamma:=\\{(h_0+h,J_s)\\}_{s\\in\\R}$, with $J_s\\in C^\\infty([0,1]\\times\\R,\\mathcal{J}(\\mathcal{V},\\mathbb{J}))$ such that\n\\begin{equation}\\label{AC}\nJ_s=\\begin{cases}\n J_0 & \\textrm{for}\\quad s\\leq 0,\\\\\n J_1 & \\textrm{for}\\quad s\\geq 1.\n\\end{cases}\n\\end{equation}\nwill automatically satisfy condition \\eqref{inqGamma}. Consequently, by Theorem \\ref{thm:FloerBounds} for every pair $(a,b)\\in \\R^2$ the corresponding space $\\mathcal{M}^\\Gamma(a,b)$ of Floer trajectories is bounded in $L^\\infty$-norm. Therefore, we can apply Theorem \\ref{thm:DefLRFH} and conclude that there exists a residual set {$\\mathcal{H}^{\\reg}_{h_0}(K_{n,m})\\subseteq \\mathcal{H}_{h_0}(K_{n,m})$}, such that\\linebreak for every $h\\in \\mathcal{H}^{\\reg}_{h_0}(K_{n,m})$ the Lagrangian Rabinowitz Floer homology\\linebreak $\\LRFH_*(\\A^{H_0-h_0-h}_{q_0,q_1})$ is well defined and independent of the choice of the almost complex structure.\n\nNow if we fix $h_1,h_2\\in \\mathcal{H}^{\\reg}_{h_0}(K_{n,m})$ then we can construct a homotopy $\\Gamma$ satisfying \\eqref{inqGamma}. Indeed, let $J_0\\in \\mathcal{J}^{\\reg}_{h_0}$ and $J_1\\in \\mathcal{J}^{\\reg}_{h_2}$ will be two almost complex structures close enough, so that there exists a smooth homotopy $\\{J_s\\}_{s\\in\\R}$ with $J_s\\in C^\\infty([0,1]\\times\\R,\\mathcal{J}(\\mathcal{V},\\mathbb{J}))$ satisfying \\eqref{AC} and $\\sup_{s\\in\\R}\\|J_s\\|_{L^\\infty}<\\frac{1}{\\sqrt{6c(h_0)}}$. Moreover, let $\\chi\\in C^\\infty(\\R)$ be a smooth function satisfying $\\|\\chi'\\|_{L^\\infty} \\leq 2$ and\n\\begin{equation}\\label{chi}\n\\chi(s)=\\begin{cases}\n 0 & \\textrm{for}\\quad s\\leq 0,\\\\\n 1 & \\textrm{for}\\quad s\\geq 1.\n\\end{cases}\n\\end{equation}\nDefine $h_s:=h_0+h_1(1-\\chi(s))+\\chi(s)h_2$. Then the smooth homotopy $\\Gamma=\\{(h_s,J_s)\\}_{s\\in\\R}$ satisfies\n\\begin{align*}\n\\|\\partial_sh_s\\|\\left(\\frac{4}{c(h_0)}+\\|J\\|_{L^\\infty}^2\\right) &\\leq \\|\\chi'\\|_{L^\\infty}\\|h_1-h_2\\|_{L^\\infty}\\left(\\frac{4}{c(h_0)}+\\frac{1}{6c(h_0)}\\right)\\\\\n&< 2\\cdot \\frac{c(h_0)}{25}\\cdot \\frac{25}{6c(h_0)}=\\frac{1}{3}.\n\\end{align*}\nIn other words, $\\Gamma$ satisfies \\eqref{inqGamma} and thus by Theorem \\eqref{thm:FloerBounds} for every pair $(a,b)\\in \\R^2$ the corresponding space $\\mathcal{M}^\\Gamma(a,b)$ of Floer trajectories is bounded in $L^\\infty$-norm. Therefore, we can apply Theorem \\ref{thm:DefLRFH} and conclude that for every $h\\in \\mathcal{H}_{h_0}(K_{n,m})$ the Lagrangian Rabinowitz Floer homology is well-defined and isomorphic to $\\LRFH_*(\\A^{H_0-h_0}_{q_0,q_1})$.\n\nTaking all the sets of the form $\\mathcal{H}_{h_0}(K_{n,m})$\nwith $h_0\\in \\mathcal{H}$ and $n,m\\in\\mathbb{N}$ gives us an open cover of $\\mathcal{H}$. Since $\\mathcal{H}$ is path connected and the Lagrangian Rabinowitz Floer homology is constant on every open set $\\mathcal{H}_{h_0}(K_{n,m})$ a basic topological argument gives us that for any pair $h_0,h_1\\in \\mathcal{H}$ the respective homologies $\\LRFH_*(\\A^{H_0-h_0}_{q_0,q_1})$ and $\\LRFH_*(\\A^{H_0-h_1}_{q_0,q_1})$ are isomorphic.\n\\hfill $\\square$\n\n\n\\section{Positive Lagrangian Rabinowitz Floer homology}\n\nThe aim of this section is to prove Theorem \\ref{thm:posLRFH}. We will start by showing that the positive Lagrangian Rabinowitz homology of the Copernican Hamiltonian is well-defined and invariant of perturbations. Further on we will prove that for fixed endpoints of the chords and high enough energy, the Copernican Hamiltonian has exactly one Reeb chord with positive action. Consequently, the corresponding positive Lagrangian Rabinowitz Floer homology has only one generator. Finally, we will show that the generator's Maslov index is in fact $0$.\n\n\\subsection{Definition and invariance of perturbations}\nFirst, we will prove that if the perturbation is small enough then there exists a homotopy, such that any associated Floer trajectory starting at a critical point with a positive action has to end at a critical point with positive action.\n\n\\begin{lem}\\label{lem:posGamma}\nLet $H_0$ be the Hamiltonian defined in \\eqref{DefH0} and let $\\mathcal{H}$ be the set of perturbations defined in \\eqref{DefHset}. Fix two points $q_0,q_1 \\in \\R^2$ and a perturbation\\linebreak $h_0\\in \\mathcal{H}$. Denote $c(h_0):=\\inf(h_0-dh_0(p\\partial_p))$. Let $K\\subseteq T^*\\R^2$ be a compact subset, such that $\\supp dh_0 \\subseteq K$ and for which there exists $\\mathcal{O}_{h_0}(K)$ an open neighbourhood of $0$ in $\\{h\\in C^\\infty(T^*\\R^2)\\ |\\ dh \\in C_0^\\infty(K)\\}$, satisfying:\n\\begin{enumerate}\n\\item $\\{h_0+h\\ |\\ h\\in \\mathcal{O}_{h_0}(K)\\}\\subseteq \\mathcal{H}$;\n\\item The critical set of $ \\A^{H_0-h_0}$ is continuously compact in $(K, \\mathcal{O}_{h_0}(K))$;\n\\item $\\inf\\left\\lbrace\\begin{array}{c|c}\n\\inf((h_0+h)-d(h_0+h)(p\\partial_p)) & h\\in\\mathcal{O}_{h_0}(K)\\end{array} \\right\\rbrace >\\frac{1}{2}c(h_0)$;\n\\item $\\delta(h_0):=\\inf\\left\\lbrace\\begin{array}{c|c}\n\\A^{H_0-h_0-h}_{q_0,q_1}(x) & x\\in \\Crit^+ \\A^{H_0-h_0-h}_{q_0,q_1}, \\quad h\\in\\mathcal{O}_{h_0}(K)\\end{array} \\right\\rbrace >0$.\n\\end{enumerate}\n \nFix $h \\in \\mathcal{O}_{h_0}(K)$ and denote $h_1:=h_0+h$. Let $\\mathcal{V}\\subseteq T^*\\R^2$ be an open, but precompact subset, such that $K\\subseteq \\mathcal{V}$.\n Let $\\Gamma:=\\{(h_s, J_s)\\}_{s\\in \\R}$ be a smooth homotopy of Hamiltonians $h_s\\in h_0+\\mathcal{O}_{h_0}(K)$ and $2$-parameter families of almost complex structures $J_s \\in C^\\infty([0,1]\\times \\R,\\mathcal{J}(\\mathcal{V},\\mathcal{J}))$ constant in $s$ outside $[0,1]$ satisfying \\eqref{AC} and such that\n\\begin{equation}\\label{inq2Gamma}\n\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\leq \\frac{1}{3}\\min\\left\\lbrace\\left(\\frac{4}{c(h_0)}+\\|J\\|_{L^{\\infty}}^2\\right)^{-1},\\ \\frac{c(h_0)}{2}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right)^{-1}\\right\\rbrace.\n\\end{equation}\nThen for every $x\\in \\Crit^+\\A^{H_0-h_0}_{q_0,q_1}$ and every $y\\in \\Crit\\A^{H_0-h_1}_{q_0,q_1}$, such that $\\F_\\Gamma(x,y)\\neq \\emptyset$ we have $\\A^{H-h_1}_{q_0,q_1}(y)>0$.\n\n\\end{lem}\n\n\\begin{proof}\nThe following proof is an adjustment of the proof of \\cite[Cor. 3.8]{CieliebakFrauenfelder2009} to our setting and it\nis similar to the proof of Lemma \\ref{lem:Novikov} where we have proven the Novikov finiteness condition.\n\nFix $x\\in \\Crit^+\\A^{H_0-h_0}_{q_0,q_1}$ and $y\\in \\Crit\\A^{H_0-h_1}_{q_0,q_1}$ and abbreviate:\n$$\na:= \\A^{H_0-h_0}_{q_0,q_1}(x) \\qquad\\textrm{and}\\qquad \\A^{H_0-h_1}_{q_0,q_1}(y).\n$$\nBy assumption $a\\geq \\delta(h_0)$. Let $u=(v,\\eta)\\in \\F_\\Gamma(x,y)$. Since $h_0,h_1\\in \\mathcal{H}_c$ and $\\Gamma$ satisfies \\eqref{inqGamma}, thus our setting satisfies the assumptions of Lemma \\ref{lem:ActBound}. Suppose $|b|\\leq a$. Then $b-a\\leq 0$ and \\eqref{eqEta} gives\n\\begin{align}\n\\|\\eta\\|_{L^\\infty} &\\leq \\frac{3}{2} \\left(\\frac{2}{c(h_0)}\\max\\{|a|,|b|\\}+\\frac{1}{\\sqrt{c(h_0)}}+\\|J\\|_{L^{\\infty}}^2 (b-a)\\right)\\nonumber\\\\\n&\\leq\\frac{3}{c(h_0)}\\left(a+\\frac{\\sqrt{c(h_0)}}{2}\\right)\\leq \\frac{3a}{c(h_0)}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right).\\label{etaBDdelta}\n\\end{align}\nOn the other hand, by equation \\eqref{inqEnergy} we have\n$$\n\\|\\nabla^{J_{s}} \\mathcal{A}^{H_0-h_{s}}(u)\\|_{L^{2}(\\mathbb{R}\\times [0,1])}^{2} \\leq \\|J\\|_{L^{\\infty}} (b-a + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}).\n$$\nCombined with \\eqref{inq2Gamma} and \\eqref{etaBDdelta}, we obtain the following estimate:\n$$\nb \\geq a- \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}} \\geq a \\left( 1-\\frac{3}{c(h_0)}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right)\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\right)\\geq \\frac{1}{2}a.\n$$\nIn particular, $\\A^{H_0-h_1}_{q_0,q_1}(y)=b\\geq \\frac{1}{2}\\delta(h_0)$. By assumption $h_1 \\in \\mathcal{O}_{h_0}(K)$, hence by Lemma \\ref{lem:posAction} we infer that $\\A^{H_0-h_1}_{q_0,q_1}(y)\\geq\\delta(h_0)$. This proves the claim under the assumption $|b|\\leq a$.\n\nNow suppose $b< -a\\leq -\\delta(h_0)$. Then $b-a\\leq 0$ and by \\eqref{eqEta} we have\n\\begin{align*}\n\\|\\eta\\|_{L^\\infty} &\\leq \\frac{3}{2} \\left( \\frac{2}{c(h_0)}\\max\\{|a|,|b|\\}+\\frac{1}{\\sqrt{c(h_0)}}+\\|J\\|_{L^{\\infty}}^2 (b-a)\\right)\\nonumber\\\\\n&\\leq \\frac{3}{c(h_0)}\\left(\\frac{\\sqrt{c(h_0)}}{2}-b\\right)<\\frac{-3b}{c(h_0)}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right)\n\\end{align*}\nCombining it with \\eqref{inqEnergy} and \\eqref{inq2Gamma}, we obtain the following inequality:\n$$\na \\leq b + \\|\\eta\\|_{L^{\\infty}}\\|\\partial_{s}h_{s}\\|_{L^{\\infty}} \\leq b\\left(1-\\frac{3b}{c(h_0)}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right)\\|\\partial_{s}h_{s}\\|_{L^{\\infty}}\\right)\\leq \\frac{1}{2}b<-\\frac{1}{2}a,\n$$\nwhich contradicts the assumption $a\\geq \\delta(h_0)>0$. That excludes the case $b<-a$ and proves the lemma.\n\\end{proof}\n\nWe are now ready to prove the first part of Theorem \\ref{thm:posLRFH}, i.e.\nto prove that the positive Lagrangian Rabinowitz homology of the Copernican Hamiltonian is well-defined and invariant under compact perturbations.\n We will do that by showing that all conditions of Corollary \\ref{cor:posLRFH} are satisfied in our setting.\n\n\\begin{prop}\\label{prop:posLRFH}\nLet $H_0$ be the Copernican Hamiltonian defined in \\eqref{DefH0} and let $\\mathcal{H}$ be the corresponding set of compact perturbations as in \\eqref{DefHset}. Then for all $q_0,q_1\\in \\R^2$ and all $h\\in \\mathcal{H}$ the positive Lagrangian Rabinowitz Floer homology $\\LRFH_*^+(\\A^{H_0-h}_{q_0,q_1})$ is well defined and isomorphic to\\linebreak $\\LRFH_*^+(\\A^{H_0-c}_{q_0,q_1})$ for any $c>0$.\n\\end{prop}\n\n\\begin{proof}\nFix two points $q_0,q_1 \\in \\R^2$ \nand let $\\{K_{n,m}\\}_{n,m\\in\\mathbb{N}}$ be the sequence\\linebreak of compact sets\nas in Lemma \\ref{lem:H0Chord} satisfying \\eqref{Knm}.\nFix $h_0\\in \\mathcal{H}^{\\reg}$ and denote $c(h_0):=\\inf(h_0-dh_0(p\\partial_p))>0$.\nLet $K_{n,m}\\subseteq T^*\\R^2$ be any compact set from the sequence, such that $\\supp dh_0 \\subseteq K_{n,m}$ and $\\|h_0\\|_{L^\\infty}<m$. By Lemma \\ref{lem:posAction} and \\ref{lem:H0Chord} there exists an open neighbourhood $\\mathcal{O}_{h_0}(K_{n,m})$ of $0$ in $\\{h\\in C^\\infty(T^*\\R^2)\\ |\\ dh\\in C_0^\\infty(K_{n,m})\\}$, such that\n\\begin{enumerate}\n\\item $\\{h_0+h\\ |\\ h\\in \\mathcal{O}_{h_0}(K_{n,m})\\}\\subseteq \\mathcal{H}$;\n\\item The critical set of $ \\A^{H_0-h_0}$ is continuously compact in $(K_{n,m}, \\mathcal{O}_{h_0}(K_{n,m}))$;\n\\item For all $h\\in\\mathcal{O}_{h_0}(K_{n,m})$ we have\n$\\inf((h_0+h)-d(h_0+h)(p\\partial_p)) >\\frac{1}{2}c(h_0)$;\n\\item There exists $\\delta(h_0)>0$, such that for all $h\\in\\mathcal{O}_{h_0}(K_{n,m})$ and all\\linebreak $x\\in \\Crit^+ \\A^{H_0-h_0-h}_{q_0,q_1}$ we have\n$\\A^{H_0-h_0-h}_{q_0,q_1}(x)\\geq \\delta(h_0)$;\n\\item For all $h\\in \\mathcal{O}_{h_0}(K_{n,m})$ we have\n$$\n\\|h_0-h\\|_{L^\\infty}<\\frac{1}{2}c(h_0)\\inf \\left\\lbrace\\frac{1}{13}, \\frac{1}{6}\\left(1+\\frac{\\sqrt{c(h_0)}}{2\\delta(h_0)}\\right)^{-1}\\right\\rbrace.\n$$\n\\end{enumerate}\n\nWe will show that for every $h\\in \\mathcal{O}_{h_0}(K_{n,m})$ the positive Lagrangian Rabinowitz Floer homology $\\LRFH_*^+(\\A^{H_0-h_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*^+(\\A^{H_0-h_0}_{q_0,q_1})$.\n\nFix $h_1 \\in\\{h_0+h\\ |\\ h\\in\\mathcal{O}_{h_0}(K_{n,m})\\}\\cap \\mathcal{H}^{\\reg}$. Let $J_0\\in \\mathcal{J}^{\\reg}_{h_0}$ and $J_1\\in \\mathcal{J}^{\\reg}_{h_1}$ will be two almost complex structures close enough, so that there exists a smooth homotopy $\\{J_s\\}_{s\\in\\R}$ with $J_s\\in C^\\infty([0,1]\\times\\R,\\mathcal{J,\\mathbb{J}}))$ satisfying \\eqref{AC} and $\\sup_{s\\in\\R}\\|J_s\\|_{L^\\infty}<\\frac{1}{\\sqrt{3c(h_0)}}$. Moreover, let $\\chi\\in C^\\infty(\\R)$ be a smooth function satisfying $\\|\\chi'\\|_{L^\\infty} \\leq 2$ and \\eqref{chi}. Define $h_s:=h_0(1-\\chi(s))+\\chi(s)h_1$. Then the smooth homotopy $\\Gamma=\\{(h_s,J_s)\\}_{s\\in\\R}$ satisfies \\eqref{inq2Gamma}.\n\nIn particular, the homotopy $\\Gamma$ satisfies \\eqref{inqGamma}, thus by Lemma \\ref{lem:Novikov} it satisfies the Novikov finiteness condition. Moreover, by Theorem \\ref{thm:FloerBounds} for any pair\\linebreak $a,b\\in \\R$ the space of Floer trajectories $\\mathcal{M}^\\Gamma(a,b)$ is bounded in $L^\\infty$-norm. Finally, by Lemma \\ref{lem:posGamma} we know that for every $x\\in \\Crit^+\\A^{H_0-h_0}_{q_0,q_1}$ and every $y\\in \\Crit\\A^{H_0-h_1}_{q_0,q_1}$, such that $\\F_\\Gamma(x,y)\\neq \\emptyset$ we have $\\A^{H-h_1}_{q_0,q_1}(y)>0$. This means that for every $h_1 \\in\\{h_0+h\\ |\\ h\\in\\mathcal{O}_{h_0}(K_{n,m})\\}\\cap \\mathcal{H}^{\\reg}$ there exists a homotopy $\\Gamma$ satisfying all three conditions of Corollary \\ref{cor:posLRFH}. Hence for every $h \\in\\mathcal{O}_{h_0}(K_{n,m})$ the positive Lagrangian Rabinowitz Floer homology $\\LRFH_*^+(\\A^{H_0-h_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*^+(\\A^{H_0-h_0}_{q_0,q_1})$.\n\nTaking all the sets of the form $\\mathcal{O}_{h_r}(K_{n,m})$\nwith $h_r\\in \\mathcal{H}^{\\reg}$ and $n,m\\in\\mathbb{N}$ gives us an open cover of $\\mathcal{H}$.\nSince $\\mathcal{H}$ is path connected and the positive Lagrangian Rabinowitz Floer homology is constant on every open set $\\mathcal{O}_{h_r}(K_{n,m})$ a basic topological argument gives us that for any pair $h_0,h_1\\in \\mathcal{H}$ the respective homologies $\\LRFH_*^+(\\A^{H_0-h_0}_{q_0,q_1})$ and $\\LRFH_*^+(\\A^{H_0-h_1}_{q_0,q_1})$ are isomorphic.\n\\end{proof}\n\n\\subsection{The critical set of the Copernican Hamiltonian}\nIn this subsection we will analyse the Reeb chords corresponding to the Copernican Hamiltonian $H_0$ defined in \\eqref{DefH0}.\nMore precisely, we will show that the positive Lagrangian Rabinowitz action functional $\\A^{H_0-c}_{q_0,q_1}$\nhas an odd number of Reeb chords with positive $\\eta>0$ under specific conditions relating the energy level set of the Hamiltonian $H_0^{-1}(c)>0$ and the endpoints of Reeb chords $q_0,q_1\\in \\R^2$, $q_0\\neq q_1$.\n\n\\begin{prop}\\label{prop:CritPM}\nLet $H_0$ be the Hamiltonian defined in \\eqref{DefH0} and let $\\A^{H_0-c}_{q_0,q_1}:\\mathscr{H}_{q_0,q_1}\\times\\R \\to \\R$ be the Lagranian Rabinowitz action functional corresponding to the energy $c>0$ and a pair $(q_0,q_1)\\in \\R^4, q_0\\neq q_1$.\nThen $\\Crit\\A^{H_0}_{q_0,q_1}$ has the following properties:\n\\begin{enumerate}\n\\item If $q_0\\neq q_1$ and $|q_0||q_1|\\leq 2c$ then $\\#\\Crit^+\\A^{H_0-c}_{q_0,q_1}=\\#\\Crit^-\\A^{H_0-c}_{q_0,q_1}=1$.\n\\item For a fixed $c>0$ there exists a residual set $\\mathcal{Q}^c\\subseteq \\R^4\\setminus \\Delta$, where\\linebreak $\\Delta:=\\left\\lbrace (q,q)\\ |\\ q\\in \\R^2\\right\\rbrace$, such that for all pairs $(q_0,q_1)\\in \\mathcal{Q}^c$ both\\linebreak $\\#\\Crit^+\\A^{H_0-c}_{q_0,q_1}$ and $\\#\\Crit^-\\A^{H_0-c}_{q_0,q_1}$ are odd numbers.\n\\item For a fixed pair $(q_0,q_1)\\in \\R^4,\\ q_0\\neq q_1$ there exists a residual set\\linebreak $\\mathcal{I}_{q_0,q_1}\\subseteq \\R_+$, such that for all $c\\in \\mathcal{I}_{q_0,q_1}$ both $\\#\\Crit^+\\A^{H_0-c}_{q_0,q_1}$ and\\linebreak $\\#\\Crit^-\\A^{H_0-c}_{q_0,q_1}$ are odd numbers.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nFor clarity of the argument, we will divide the proof of this proposition in a sequence of lemmas. First recall that by equation~\\eqref{Crit+} we have\n$$\n\\Crit^\\pm\\A^{H}_{q_0,q_1}=\\left\\lbrace (v,\\eta)\\in \\Crit\\A^{H}_{q_0,q_1}\\ |\\ \\pm \\eta >0\\right\\rbrace.\n$$\nIn Lemma \\ref{lem:bijection} we will show that for each $c>0$ and each pair $(q_0,q_1)\\in \\R^4$ there exists a smooth function $f\\in C^\\infty(\\R)$, such that $(v,\\eta)\\in \\Crit\\A^{H_0-c}_{q_0,q_1}$ if and only if $f(\\eta)=0$. We also show that for $q_0\\neq q_1$ the critical set of $\\A^{H_0-c}_{q_0,q_1}$ is discrete. Denote by $Z(f)$ the set of roots of the function $f$, $Z(f):=\\{\\eta\\in\\R |\\ f(\\eta)=0\\}$.\nIn Lemma \\ref{lem:2c>q0q1} we will consider pairs $(q_0,q_1)\\in \\R^4$, such that $|q_0||q_1|\\leq 2c$ and show that for $q_0\\neq q_1$ the corresponding function $f$ satisfies $\\#\\{\\eta>0\\ |\\ f(\\eta)=0\\}=\\#\\{\\eta<0\\ |\\ f(\\eta)=0\\}=1$, whereas for $q_0=q_1$ we have $Z(f)=\\{0\\}$.\n\nDenote $\\Delta:=\\{(q,q)\\ |\\ q\\in \\R^2\\}\\subseteq \\R^4$. In Lemma \\ref{lem:Res_q0q1} we prove that for a fixed $c>0$ there exists a residual set $\\mathcal{Q}^c\\subseteq \\R^4\\setminus \\Delta$, such that for all pairs $(q_0,q_1)\\in \\mathcal{Q}^c$ the corresponding function $f$ satisfies:\n \\begin{equation}\\label{f'neq0}\n f'(\\eta)\\neq 0\\qquad \\forall\\ \\eta \\in Z(f), \\qquad \\textrm{where}\\quad Z(f):=\\{\\eta \\in \\R\\ |\\ f(\\eta)=0\\}. \n \\end{equation}\n \n In Lemma \\ref{lem:Res_c} we show that for a fixed pair $(q_0,q_1)\\in \\R^4, q_0\\neq q_1$ there exists a residual set $\\mathcal{I}_{q_0,q_1}\\subseteq \\R_+$, such that for all $c\\in \\mathcal{I}_{q_0,q_1}$ the corresponding function $f$ satisfies property \\eqref{f'neq0}.\n \n In Lemma \\ref{lem:Odd} we prove that if $c>0$ and $(q_0, q_1)\\in \\R^4, q_0\\neq q_1$ are such that the corresponding function $f$ satisfies \\eqref{f'neq0}, then $f$ has an odd number of roots on each positive and negative half line. This together with the bijection established in Lemma \\ref{lem:bijection} concludes the proof.\n\\end{proof}\nBefore we prove the first lemma we will start by recalling the properties of the group of matrices corresponding to rotations: if we define\n\\begin{equation}\nR(t) :=\\left(\\begin{array}{c c}\n\\cos(t) & \\sin (t) \\\\\n-\\sin(t)& \\cos(t) \\\\\n\\end{array}\\right).\\label{DefR}\n\\end{equation}\nthen $\\{R(t)\\}_{t\\in\\R}$ form a group with the following properties:\n\\begin{align}\n R(0)=\\Id\\quad \\textrm{and}\\quad R\\left(\\frac{\\pi}{2}\\right) = & \\left(\\begin{array}{c c}\n0 & 1 \\\\\n-1 & 0 \\\\\n\\end{array}\\right),\\nonumber\\\\\nR(t)R(\\tau)=R(t+\\tau)\\qquad & \\forall\\ t,\\tau\\in\\R,\\label{Radd}\\\\\nR^{-1}(t)=R^T(t)=R(-t)\\qquad & \\forall\\ t\\in\\R.\\label{Rinv}\n\\end{align}\nMoreover, we have\n\\begin{align}\nR'(t) &= \\left(\\begin{array}{c c}\n-\\sin(t) & \\cos (t) \\\\\n-\\cos(t)& -\\sin(t) \\\\\n\\end{array}\\right)=\\left(\\begin{array}{c c}\n0 & 1 \\\\\n-1 & 0 \\\\\n\\end{array}\\right)\n\\left(\\begin{array}{c c}\n\\cos(t) & \\sin (t) \\\\\n-\\sin(t)& \\cos(t) \\\\\n\\end{array}\\right)\\nonumber\\\\ \n&= R\\left(\\frac{\\pi}{2}\\right)R(t)= R\\left( t +\\frac{\\pi}{2}\\right).\\label{DerR}\n\\end{align}\n\nIn the following lemma we will show that the critical set of $\\A^{H_0-c}_{q_0,q_1}$ consists of isolated points and, in case $q_0=q_1$ it is a circle. \n\n\\begin{lem}\\label{lem:bijection}\nLet $H_0$ be the Hamiltonian defined in \\eqref{DefH0}. For every $c>0$ and every pair $q_0,q_1\\in \\R^2$ we define\n\\begin{equation}\\label{Deff}\nf(\\eta):=-c\\eta^2+\\eta q_1^TR\\left(\\eta+\\frac{\\pi}{2}\\right)q_0+\\frac{1}{2}\\left(|q_0|^2+|q_1|^2\\right)-q_1^TR(\\eta)q_0.\n\\end{equation}\nIf we denote \n\\begin{equation}\\label{DefZ(f)}\nZ(f):=\\{\\eta\\in \\R \\ |\\ f(\\eta)=0\\},\n\\end{equation}\nthen the map $Z(f)\\setminus\\{0\\}\\ni \\eta \\longmapsto ( v_\\eta, \\eta),$\n$$\n\\textrm{with}\\qquad v_\\eta(t) := \\left(\\begin{array}{c c}\n(1-t) R(t\\eta) & t R(\\eta(t-1))\\\\\n -\\frac{1}{\\eta}R(t\\eta) & \\frac{1}{\\eta} R(\\eta(t-1))\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right).\n$$\ndefines a bijection between the set $Z(f)\\setminus\\{0\\}$ and $\\Crit\\A^{H_0-c}_{q_0,q_1}\\setminus (\\{0\\}\\times \\mathscr{H}_{q_0,q_1})$.\n\\end{lem}\n\\begin{proof}\nBy definition a pair $(v,\\eta)\\in \\Crit\\A^{H_0-c}_{q_0,q_1}$ if and only if it satisfies the following three conditions:\n\\begin{enumerate}\n\\item The curve $v$ starts in the Lagrangian $T^*_{q_0}\\R^2$ and ends in $T^*_{q_1}\\R^2$;\n\\item The curve $v$ is a Reeb chord and $\\eta$ is its period;\n\\item The curve $v$ lies on the level set $H_0^{-1}(c)$.\n\\end{enumerate}\nNote that if $q_0\\neq q_1$ we have $\\Crit\\A^{H_0-c}_{q_0,q_1}\\cap\\left(\\mathscr{H}_{q_0,q_1}\\times\\{0\\}\\right)=\\emptyset$. In case $q_0= q_1$ we always have a submanifold of constant solutions \n$$\n\\Crit\\A^{H_0-c}_{q_0,q_1}\\cap\\left(\\mathscr{H}_{q_0,q_0}\\times\\{0\\}\\right)=\\{q_0\\}\\times\\{p_0\\in T^*_{q_0}\\R^2\\ |\\ H_0(q_0,p_0)=c\\}.\n$$\nOne can easily calculate that for a fixed $q_0\\in \\R^2$ the set $\\{ p \\in T^*\\R^2\\ |\\ H_0(q_0,p)=c\\}$ is a circle with origin at $-\\mathbb{J}q_0$ and radius $\\sqrt{|q_0|^2+2c}$:\n$$\nH_0(q,p)=c \\quad \\iff \\quad |p+ \\mathbb{J}q_0|^2=2c+|q_0|^2.\n$$\n\nFrom now on we will assume $\\eta\\neq 0$.\n\nUsing \\eqref{DefH0} and the notation from \\eqref{DefR} we can express $H_0$ and $X_{H_0}$ in the following way:\n\\begin{align}\nH_0(q,p) & = \\frac{1}{2}(q,p) A (q,p)^T, \\nonumber\\\\\nX_{H_0}(q,p) &= \\mathbb{J}A (q,p)^T\\nonumber\\\\\n\\textrm{where}\\qquad A & =\\left(\\begin{array}{c c}\n0 & -R\\left(\\frac{\\pi}{2}\\right)\\\\\nR\\left(\\frac{\\pi}{2}\\right) & \\Id\n\\end{array}\\right).\\label{DefMatrixA}\n\\end{align}\nIn particular, the flow $\\varphi^t$ of the Hamiltonian vector field $X_{H_0}$ can be easily calculated to be\n\\begin{equation}\n\\varphi^t(q,p)=\\Exp(t\\mathbb{J}A)\\left(\\begin{array}{c}\nq \\\\ p\n\\end{array}\\right) =\n\\left(\\begin{array}{c c}\n R(t) & t R(t)\\\\\n 0 & R(t)\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq \\\\ p\n\\end{array}\\right)\\label{flow}\n\\end{equation}\n\nBy definition, a critical point $(v,\\eta)\\in \\Crit\\A^{H_0-c}_{q_0,q_1}$ satisfies $\\varphi^\\eta \\circ v (0)=v(1)$. Moreover, we have $\\pi \\circ v(0)=q_0$ and $\\pi\\circ v(1)=q_1$. \nIf we denote $v(0)=(q_0,p_0)$ and $v(1)=(q_1,p_1)$ and\ncombine it with the matrix formula for the Hamiltonian flow, we obtain\n$$\n\\left(\\begin{array}{c}\nq_1 \\\\ p_1\n\\end{array}\\right)=\n\\left(\\begin{array}{c c}\nR(\\eta) & \\eta R(\\eta)\\\\\n0 & R(\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ p_0\n\\end{array}\\right).\n$$\nAssuming, that $\\eta\\neq 0$ we can use \\eqref{Rinv} to solve for $p_0$ and $p_1$:\n$$\n\\left(\\begin{array}{c}\np_0 \\\\ p_1\n\\end{array}\\right)\n =\\frac{1}{\\eta}\n\\left(\\begin{array}{c c}\n- \\Id & R(-\\eta)\\\\\n-R(\\eta) & \\Id\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right).\n$$\nUsing the equation above we obtain the following formula for $v(0)$:\n\\begin{equation}\\label{q0p0}\n\\left(\\begin{array}{c}\nq_0 \\\\ p_0\n\\end{array}\\right) =\n\\left(\\begin{array}{c c}\n \\Id &0\\\\\n-\\frac{1}{\\eta} & \\frac{1}{\\eta}R(-\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right)\n\\end{equation}\nNow, by combining \\eqref{flow} and \\eqref{q0p0} we can express a Reeb chord $v$ in terms of $q_0$ and $q_1$ in the following way:\n\\begin{align*}\nv_\\eta(t) & := \\left(\\begin{array}{c c}\n R(t\\eta) & t \\eta R(t\\eta)\\\\\n 0 & R(t\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c c}\n\\Id & 0\\\\\n-\\frac{1}{\\eta} & \\frac{1}{\\eta}R(-\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right)\\\\\n& = \\left(\\begin{array}{c c}\n(1-t) R(t\\eta) & t R(\\eta(t-1))\\\\\n -\\frac{1}{\\eta}R(t\\eta) & \\frac{1}{\\eta} R(\\eta(t-1))\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right).\n\\end{align*}\nSuch defined $v_\\eta$ is a solution of the Hamiltonian flow and it belongs to $\\mathscr{H}_{q_0,q_1}$.\nTo assure that $v_\\eta(t)\\in H_0^{-1}(c)$ for all $t\\in [0,1]$ we use \\eqref{Radd}, \\eqref{Rinv}, \\eqref{DefMatrixA} and \\eqref{q0p0} to express the condition $(q_0,p_0)\\in H_0^{-1}(c)$ in the following way:\n\\begin{align*}\nc & =\\frac{1}{2}\n\\left(q_0,\\ p_0\\right)\n\\left(\\begin{array}{c c}\n \\Id & -\\frac{1}{\\eta}\\\\\n0 & \\frac{1}{\\eta}R(\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c c}\n0 & -R\\left(\\frac{\\pi}{2}\\right)\\\\\nR\\left(\\frac{\\pi}{2}\\right) & \\Id\n\\end{array}\\right)\n\\left(\\begin{array}{c c}\n \\Id &0\\\\\n-\\frac{1}{\\eta} & \\frac{1}{\\eta}R(-\\eta)\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right)\\\\\n& =\\frac{1}{2} \\left(q_0,\\ p_0\\right)\n\\left(\n\\frac{1}{\\eta^2}\n\\left(\\begin{array}{c c}\n \\Id &-R(-\\eta)\\\\\n-R(\\eta) & \\Id\n\\end{array}\\right)\n+\\frac{1}{\\eta}\n\\left(\\begin{array}{c c}\n 0 & -R\\left(\\frac{\\pi}{2}-\\eta\\right)\\\\\nR\\left(\\frac{\\pi}{2}+\\eta\\right) & 0\n\\end{array}\\right)\n\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right)\\\\\n& = \\frac{1}{\\eta^2}\\left(\\frac{1}{2}\\left( |q_0|^2+|q_1|^2\\right)-q_1R(\\eta)q_0^T\\right)\n+\\frac{1}{\\eta}q_1R\\left(\\eta+\\frac{\\pi}{2}\\right)q_0^T.\n\\end{align*}\nNow if we define $f\\in C^\\infty(R)$ as in \\eqref{Deff}\nthen $(v_\\eta,\\eta)\\in\\Crit\\A^{H_0-c}_{q_0,q_1}$ if and only if $f(\\eta)=0$. \n\nObserve that\n$f(0)=\\frac{1}{2}|q_1-q_0|^2$, thus $0\\in Z(f)$ if and only if $q_0=q_1$. In that case, $\\left(\\A^{H_0-c}_{q_0,q_1}\\right)^{-1}(0)$ is a circle.\n\\end{proof}\n\\begin{lem}\\label{lem:2c>q0q1}\nIn case $|q_0||q_1|\\leq 2c$ the critical set of $\\A^{H_0-c}_{q_0,q_1}$ satisfies one of the following properties:\n\\begin{itemize}[leftmargin=2cm]\n\\item[either] $q_0 \\neq q_1$ and then $\\#\\Crit^+\\A^{H_0-c}_{q_0,q_1}=\\#\\Crit^-\\A^{H_0-c}_{q_0,q_1}=1$,\n\\item[or] $q_0=q_1$ and then $\\#\\Crit^+\\A^{H_0-c}_{q_0,q_1}=\\#\\Crit^-\\A^{H_0-c}_{q_0,q_1}=0$.\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\nBy Lemma \\ref{lem:bijection} we know that we can restrict ourselves to the analysis of function $f$ defined in \\eqref{Deff}.\nNote that $f(0)=\\frac{1}{2}|q_1-q_0|^2$, thus \n\\begin{align*}\n\\textrm{for}\\quad q_0 & \\neq q_1\\quad \\textrm{we have}\\quad f(0)>0,\\\\\n\\textrm{and for} \\quad q_0 & =q_1 \\quad \\textrm{we have}\\quad f(0)=0.\n\\end{align*}\nIn both cases, we know that $\\lim_{\\eta \\to \\pm \\infty}f(\\eta)=-\\infty$ due to the dominance of the summand $-c\\eta^2$ for $|\\eta|$ large enough. In case $q_0\\neq q_1$ we can use the intermediate value theorem to prove that $f$ has a root both on the positive and the negative half line. What is left to show is that $f$ has only one root on each half-line.\n\nUsing \\eqref{DerR} we can now calculate the derivative of $f$:\n\\begin{align*}\nf'(\\eta) & = -2c\\eta +q_1^T\\left(R\\left(\\eta+\\frac{\\pi}{2}\\right)-R'\\left(\\eta\\right)\\right)q_0+\\eta q_1^TR'\\left(\\eta+\\frac{\\pi}{2}\\right)q_0\\\\\n& = -\\eta (2 c +q_1^TR(\\eta)q_0).\n\\end{align*}\nThus in case $|q_0||q_1|\\leq 2c$ we have\n$$\n2 c +q_1^TR(\\eta)q_0) \\geq 2c -|q_0||q_1|\\geq 0,\n$$\nand consequently $f'(\\eta)\\leq 0$ for $\\eta>0$ and $f'(\\eta)\\geq 0$ for $\\eta<0$. Moreover, in case $|q_0||q_1|=2c$ we have $f'(\\eta)=0$ if and only if $\\eta=0$ or $q_1=-\\frac{2c}{|q_1|^2}R(\\eta)q_0$. In particular, in case $|q_0||q_1|=2c$ the critical points of $f$ are isolated. Consequently, $f$ obtains its maximum at $0$ and is non-decreasing on the negative half line and non-increasing on the positive half line. This proves that in case $q_0\\neq q_1$ there exists only one root on each positive and negative half line. On the other hand, since $f$ obtains its maximum at $0$, thus in case $q_0=q_1$ we know that $f(\\eta)<0$ for all $\\eta\\neq 0$, which concludes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lem:Res_q0q1}\n For a fixed $c>0$ there exists a residual set $\\mathcal{Q}^c\\subseteq \\R^4\\setminus \\Delta$, such that for all pairs $(q_0,q_1)\\in \\mathcal{Q}^c$ the corresponding function $f$ as defined in \\eqref{Deff} satisfies property \\eqref{f'neq0}.\n \\end{lem}\n\n\\begin{proof}\nTo prove the statement of the lemma, we will first extend the function $f\\in C^\\infty(\\R)$ to $\\bar{f}\\in C^\\infty(\\R^5)$ by taking $\\bar{f}(q_0,q_1,\\eta):=f(\\eta)$ where $f$ depends on $q_0$ and $q_1$ in the way described in \\eqref{Deff}. \n\nRecall, that $f(0)=\\frac{1}{2}|q_1-q_0|^2$, thus \n$$\n\\bar{f}^{-1}(0)\\cap \\left(\\R^4\\times\\{0\\}\\right)=\\Delta\\times\\{0\\}.\n$$\nWe will show now that $\\bar{f}^{-1}(0)\\setminus (\\Delta\\times\\{0\\})$ is a smooth manifold. Using \\eqref{Deff} we can calculate $D\\bar{f}$:\n\\begin{align*}\n\\partial_\\eta \\bar{f} & = -\\eta \\left(2 c +q_1^TR(\\eta)q_0\\right),\\\\\n\\partial_{q_0}\\bar{f} & = \\eta q_1^T R\\left(\\eta+\\frac{\\pi}{2}\\right)- q_1^T R(\\eta)+q_0,\\\\\n\\partial_{q_1}\\bar{f} & = \\eta q_0^T R\\left(-\\eta-\\frac{\\pi}{2}\\right)- q_0^T R(-\\eta)+q_1.\n\\end{align*}\nA straightforward computation shows that $\\bar{f}$ satisfies the following relation:\n$$\n\\bar{f}(q_0,q_1,\\eta)=-c\\eta^2+\\frac{1}{2}\\left( q_0\\partial_{q_0}\\bar{f}+ q_1\\partial_{q_1}\\bar{f}\\right).\n$$\nConsequently, for all $(q_0,q_1,\\eta)\\in \\bar{f}^{-1}(0)$ we have $|\\eta||\\partial_{q_0}\\bar{f}||\\partial_{q_1}\\bar{f}|\\neq 0$. In particular, for all $(q_0,q_1,\\eta)\\in \\bar{f}^{-1}(0)\\setminus (\\Delta\\times\\{0\\})$ the derivative $D\\bar{f}\\neq 0$ and thus, by the inverse function theorem, $\\bar{f}^{-1}(0)\\setminus (\\Delta\\times\\{0\\})$ is a smooth manifold.\n\nNote, that if we consider the function $f\\in C^\\infty(\\R)$ as in \\eqref{Deff} corresponding to a fixed pair $(q_0,q_1)\\in \\R^4$, then\n$$\n\\left( \\bar{f}^{-1}(0)\\setminus \\Delta\\right) \\cap \\left( \\{(q_0,q_1)\\}\\times\\R\\right)=Z(f)\\setminus \\{0\\}.\n$$\nAdditionally, $f'(\\eta)=\\partial_\\eta \\bar{f}(q_0,q_1,\\eta)$ for all $\\eta\\in \\R$. Therefore, if we denote\n\\begin{align*}\n\\mathcal{Q}^c := & \\left\\lbrace (q_0,q_1)\\in \\R^4\\ |\\ \\forall\\ \\eta\\in Z(f)\\setminus\\{0\\}\\quad f'(\\eta)\\neq 0\\right\\rbrace,\\\\\n\\textrm{then}\\qquad \\mathcal{Q}^c =& \\left\\lbrace (q_0,q_1)\\in \\R^4\\ |\\ \\forall\\ (q_0,q_1,\\eta)\\in \\bar{f}^{-1}(0)\\setminus \\Delta\\quad \\partial_\\eta \\bar{f}\\neq 0 \\right\\rbrace.\n\\end{align*}\nConsequently, the set $\\mathcal{Q}^c$ satisfies the assumptions of the theorem. On the other hand, since $\\bar{f}^{-1}(0)\\setminus \\Delta$ is a smooth manifold, the set of regular values of the projection $P: \\bar{f}^{-1}(0)\\setminus \\Delta \\to \\R^4$ is in fact equal to $\\mathcal{Q}^c$. By the Morse-Sard Theorem $\\mathcal{Q}^c$ is residual in $\\R^4$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:Res_c}\nFor a fixed pair $(q_0,q_1)\\in \\R^4, q_0\\neq q_1$ there exists a residual set $\\mathcal{I}_{q_0,q_1}\\subseteq \\R_+$, such that for all $c\\in \\mathcal{I}_{q_0,q_1}$ the corresponding function $f$ as defined in \\eqref{Deff} satisfies property \\eqref{f'neq0}.\n\\end{lem}\n\\begin{proof}\nIn this proof we will follow the same arguments as in the proof of Lemma \\ref{lem:Res_q0q1}. First, we extend the function $f\\in C^\\infty(\\R)$ as in \\eqref{Deff} to a function $\\tilde{f}:\\R_+\\times\\R \\to \\R$ by setting $\\tilde{f}(c,\\eta):=f(\\eta)$. Second, we observe that $\\partial_c \\tilde{f}(c,\\eta)= -\\eta^2 < 0$ for all $\\eta\\neq 0$. By assumption we take $q_0\\neq q_1$, hence $0\\notin\\tilde{f}^{-1}(0)$ and thus $\\partial_c \\tilde{f}(c, \\eta)<0$ for all $(c,\\eta) \\in \\tilde{f}^{-1}(0)$. This allows us to use the inverse function theorem to conclude that $\\tilde{f}^{-1}(0)$ is a smooth manifold. Furthermore, we observe that the set of regular values of the projection $P:\\tilde{f}^{-1}(0) \\to \\R^+$ is equal to\n\\begin{align*}\n\\mathcal{I}_{q_0,q_1} := & \\left\\lbrace c\\in \\R_+\\ |\\ \\forall\\ (c,\\eta)\\in \\tilde{f}^{-1}(0)\\quad \\partial_\\eta \\tilde{f}\\neq 0 \\right\\rbrace\\\\\n = &\\left\\lbrace c\\in \\R_+\\ |\\ \\forall\\ \\eta\\in Z(f)\\quad f'\\neq 0 \\right\\rbrace.\n\\end{align*}\nFinally, by the Morse-Sard Theorem, we conclude that $\\mathcal{I}_{q_0,q_1}$ is residual in $\\R_+$.\n\\end{proof}\n\nIn this final lemma, we show how the condition \\eqref{f'neq0} implies that the function $f$ has an odd number of zeros on each the positive and the negative half line.\n\\begin{lem}\\label{lem:Odd}\nIf $c>0$ and $(q_0, q_1)\\in \\R^4, q_0\\neq q_1$ are such that the corresponding function $f$ satisfies \\eqref{f'neq0}, then $f$ has an odd number of roots on each positive and negative half line.\n\\end{lem}\n\\begin{proof}\nFirst we will show that if $t_0,t_1\\in \\R, t_0<t_1$ are two roots of a smooth function $f\\in C^\\infty(\\R)$, such that $f'(t_0), f'(t_1)\\neq 0$ and $f(t)\\neq 0$ for all\\linebreak $t\\in (t_0,t_1)$, then $f'(t_0)f'(t_1)<0$. Suppose the opposite is true and $f'(t_0)f'(t_1)>0$. Without loss of generality, we can assume that $f'(t_0),f'(t_1)>0$. Then there would exist $\\delta>0$, such that for all $t\\in (t_0, t_0+\\delta),\\ f(t)>0$ and for all $t\\in (t_1-\\delta,t_1),\\ f(t)<0$. Consequently, by the intermediate value theorem there would have to exist $t\\in (t_0+\\delta,t_1-\\delta)$, such that $f(t)=0$. But that brings us a contradiction.\n\nLet now $f$ be the function defined in \\eqref{Deff}. We will show that the set of roots of $f$ is bounded. Observe that $f$ can be bounded from above by\n$$\nf(\\eta) \\leq -c \\eta^2 + |\\eta||q_0||q_1|+\\frac{1}{2}(|q_0|+|q_1|)^2.\n$$\nConsequently, we have\n$f(\\eta) < 0$ for $\\eta \\in (-\\infty, -\\delta_0)\\cup (\\delta_0, +\\infty)$, where\n\\begin{equation}\\label{delta}\n\\delta_0:= \\frac{1}{2c}\\left(\\sqrt{|q_0|^2|q_1|^2+2c(|q_0|+|q_1|)^2}+|q_0||q_1|\\right).\n\\end{equation}\n In particular, the set of roots of $f$ denoted by $Z(f)$ is a subset of $[-\\delta_0,\\delta_0]$. Since $Z(f)$ is bounded and discrete the number $k:=\\#\\{ \\eta \\in Z(f)\\ |\\ \\eta>0\\}\\in \\mathbb{N}$ is well defined. Moreover, \nwe can enumerate the elements of the set $\\{ \\eta \\in Z(f)\\ |\\ \\eta>0\\}=\\{\\eta_i\\}_{i=1}^{k}$, such that $\\eta_i< \\eta_{i+1}$ for $i=1,\\dots k-1$ and\n\\begin{equation}\\label{AssEta}\n\\begin{aligned}\n(\\eta_i, \\eta_{i+1})\\cap Z(f) & = \\emptyset && \\textrm{for} && i=1, \\dots k-1,\\\\\n(0, \\eta_1)\\cap Z(f) & =\\emptyset&& \\textrm{and} && (\\eta_{k}, +\\infty)\\cap Z(f)=\\emptyset.\n\\end{aligned}\n\\end{equation}\nOur aim is to prove that $k$ is an odd number.\n\nWe will show now that if $f$ satisfies \\eqref{f'neq0} then $f'(\\eta_1)<0$. Recall that $f(0)=\\frac{1}{2}|q_1-q_0|^2$, which in the case $q_0\\neq q_1$ means $f(0)>0$. Suppose now that $f'(\\eta_1)>0$. That would imply that there exists $\\delta>0$, such that\\linebreak $f(\\eta)<f(\\eta_1)=0$ for $\\eta \\in (\\eta_1-\\delta, \\eta_1)$. Consequently, by the intermediate value theorem, there would have to exist $\\eta \\in (0, \\eta_1-\\delta)$, such that $f(\\eta)=0$. But that would contradict our assumption \\eqref{AssEta}. Thus $f'(\\eta_1)<0$.\n\nThe last step would be to show that if $f$ satisfies \\eqref{f'neq0} then $f'(\\eta_{k})<0$. Recall that $\\lim_{\\eta \\to +\\infty}f(\\eta)=-\\infty$.\nSuppose now that $f'(\\eta_{k})>0$. That would imply that there exists $\\delta>0$, such that for all $\\eta \\in (\\eta_{k}, \\eta_{k}+\\delta)$, $f(\\eta)>0$. Consequently, by the intermediate value theorem, there would have to exist $\\eta \\in (\\eta_{k}+\\delta,+\\infty)$, such that $f(\\eta)=0$. But that contradicts assumption \\eqref{AssEta}.\n\nLet us recall what we have learned: if $f$ satisfies \\eqref{f'neq0} then $f'(\\eta_1)<0$, $f'(\\eta_i)$ has an opposite sign than $f'(\\eta_{i+1})$ for all $i=1, \\dots k-1$ and finally $f'(\\eta_{k})<0$. This implies that $k$ is an odd number.\n\nUsing the same arguments we prove that $\\#\\{ \\eta \\in Z(f)\\ |\\ \\eta<0\\}$ is an odd number as well.\n\\end{proof}\n\nDespite the fact that the positive Lagrangian Rabinowitz Floer homology has only one generator, the number of positive critical points of the Rabinowitz action functional corresponding to a fixed pair of endpoints $q_0,q_1 \\in \\R^2$ does not necessarily have to be $1$. In fact, its cardinality depends on the value of the energy $c>0$.\nAs shown in Lemma 4.4, the positive critical set of the Rabinowitz action functional is in bijection with the set of positive zeroes of the corresponding function $f$ defined in (4.8). In Figure \\ref{fig:fFunction} we depict the functions $f$ corresponding to the chord endpoints $q_0=(1,0)$, $q_1=(0,1)$ and three different energies $c= \\frac{1}{5}$, $c=\\frac{1}{10}$, and $c=\\frac{1}{20}$, respectively. We see that the number of positive zeroes of the functions increases as the energy $c$ decreases.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.75\\linewidth]{fFunction.png}\n\\caption{The three functions $f$ corresponding to energy $c=\\frac{1}{5}$ (blue), $c=\\frac{1}{10}$ (green), and $c=\\frac{1}{20}$ (magenta) crossing zero in exactly $1$, $3$, and $5$ points, respectively.}\n\\label{fig:fFunction}\n\\end{figure}\n\nThe following proposition shows that the number of positive critical points of the Rabinowitz action functional tends to $\\infty$ as the energy $c$ approaches $0$.\n\n\\begin{prop}\nFor fixed $q_0,q_1 \\in \\R^2$ we have\n$$\n\\lim_{c\\searrow 0}\\# \\left(\\Crit^+ \\A_{q_0,q_1}^{H_0-c}\\right)=+\\infty.\n$$\n\\end{prop}\n\n\\begin{proof}\nBy Lemma 4.4 we know that for any fixed $q_0,q_1 \\in \\R^2$ and $c>0$ we have $ \\#\\left(\\Crit^+ \\A_{q_0,q_1}^{H_0-c}\\right)= \\# \\{ \\eta>0\\ |\\ f(\\eta)=0\\}$, where $f$ is the function defined in (4.8) which depends on $q_0$, $q_1$, and $c$.\nWe express $q_0$ and $q_1$ in polar coordinates as\n\\begin{align*}\nq_0 & = r_0 \\left(\\begin{array}{c}\\cos (\\alpha) \\\\ \\sin (\\alpha)\\end{array}\\right) = r_0 R^T(\\alpha)\\left(\\begin{array}{c} 1 \\\\ 0\\end{array}\\right),\\\\\nq_1 & = r_1 \\left(\\begin{array}{c}\\cos (\\alpha+\\theta) \\\\ \\sin (\\alpha+\\theta)\\end{array}\\right) = r_1 R^T(\\alpha+\\theta)\\left(\\begin{array}{c} 1 \\\\ 0\\end{array}\\right).\n\\end{align*}\nPlugging this into the function $f$ from (4.8), we obtain\n\\begin{align*}\nf(\\eta) & = -c \\eta^2+\\frac{1}{2}\\left(r_0^2+r_1^2\\right)+ r_0 r_1 (1,0)\\left(\\eta R\\left(\\eta+\\theta+\\frac{\\pi}{2}\\right)-R(\\eta+\\theta)\\right)\\left(\\begin{array}{c} 1 \\\\ 0\\end{array}\\right)\\\\\n & = -c \\eta^2-\\eta\\ r_0 r_1 \\sin\\left(\\eta +\\theta\n \\right) +\\frac{1}{2}\\left(r_0^2+r_1^2\\right)-r_0 r_1 \\cos (\\eta +\\theta).\n\\end{align*}\nNote that the function depends only on the relative angle $\\theta$ between the two endpoints. This is not very surprising as the Hamiltonian $H_0$ is invariant under rotations around the origin.\nObserve that\n\\begin{align*}\nf(\\eta) & = - c\\eta^2 +\\frac{1}{2}(r_0-r_1)^2 && \\textrm{for}\\qquad \\eta = 2\\pi n - \\theta, && n \\in \\mathbb{N},\\\\\nf(\\eta) & = - c\\eta^2 +\\frac{1}{2}(r_0+r_1)^2 && \\textrm{for}\\qquad \\eta = \\pi (2n+1) - \\theta, && n \\in \\mathbb{N}.\n\\end{align*}\nConsequently, we have\n\\begin{align*}\nf(\\eta) & <0 && \\textrm{for}\\qquad \\eta = 2\\pi n - \\theta > \\frac{1}{\\sqrt{2c}}|r_0-r_1|, && n\\in\\mathbb{N},\\\\\nf(\\eta) & >0 && \\textrm{for}\\qquad \\eta = \\pi (2n+1) - \\theta < \\frac{1}{\\sqrt{2c}}(r_0+r_1), && n\\in\\mathbb{N}.\n\\end{align*}\nNote that if we assume\n$$\n\\frac{\\theta}{\\pi} +\\frac{1}{\\pi\\sqrt{2c}}|r_0-r_1| < n < \\frac{\\theta}{\\pi} +\\frac{1}{\\pi\\sqrt{2c}}(r_0+r_1)-1,$$\nthen\n$$\n\\frac{\\theta}{\\pi} +\\frac{1}{\\pi\\sqrt{2c}}|r_0-r_1| < n, n+1 < \\frac{\\theta}{\\pi} +\\frac{1}{\\pi\\sqrt{2c}}(r_0+r_1).\n$$\nNow exactly one of $n,n+1$ is even and the other is odd, so\n$$\nf(\\pi n - \\theta)f(\\pi (n+1) - \\theta)<0,\n$$\nand thus by the intermediate value theorem there exists $\\eta\\in (\\pi n-\\theta, \\pi(n+1)-\\theta)$ such that $f(\\eta)=0$.\n\nTherefore, we can estimate\n\\begin{align*}\n\\#\\left(\\Crit^+ \\A_{q_0,q_1}^{H_0-c}\\right) & =\\# \\left\\lbrace \\eta>0\\ |\\ f(\\eta)=0\\right\\rbrace \\\\\n&\\geq \\# \\left( \\frac{\\theta}{\\pi} + \\frac{|r_0-r_1|}{\\pi\\sqrt{2c}}, \\frac{\\theta}{\\pi} +\\frac{(r_0+r_1)}{\\pi\\sqrt{2c}}-1\\right)\\cap \\mathbb{N}.\n\\end{align*}\nThe length of the interval on the right hand side of the inequality is equal to $\\frac{\\sqrt{2}}{\\pi\\sqrt{c}}\\min\\{r_0,r_1\\}-1$. Since $\\lim_{c\\to 0}\\left(\\frac{\\sqrt{2}}{\\pi\\sqrt{c}}\\min\\{r_0,r_1\\}-1\\right)=+\\infty$, we conclude\n$$\n \\lim_{c\\searrow 0} \\#\\left(\\Crit^+ \\A_{q_0,q_1}^{H_0-c}\\right) \\geq \\lim_{c\\searrow 0}\\# \n\\left( \\frac{\\theta}{\\pi} + \\frac{|r_0-r_1|}{\\pi\\sqrt{2c}}, \\frac{\\theta}{\\pi} +\\frac{(r_0+r_1)}{\\pi\\sqrt{2c}}-1\\right)\\cap\\mathbb{N} = +\\infty.\n$$\n\\end{proof}\n\nIn Figures \\ref{fig:1Orbit3Energies}, \\ref{fig:3Orbits} and \\ref{fig:5Orbits} we present the plots of the Reeb chords from $q_0=(0,1)$ to $q_1=(1,0)$ for various energies. More precisely, the graphs depict the projections of the Reeb chords onto the plane of positions. The plots have been obtained using the formula from Lemma \\ref{lem:bijection}.\n\nLet us analyse how the number of Reeb chords depends on the energy $c\\in \\left\\lbrace 1, \\frac{1}{2}, \\frac{1}{5}, \\frac{1}{10}, \\frac{1}{20}\\right\\rbrace$.\nBy Lemma 4.5 we have $\\Crit^+\\A_{q_0,q_1}^{H_0-c}=1$ for $c \\geq \\frac{1}{2}$.\nOn the other hand, from Lemma 4.4 we know that $\\Crit^+\\A_{q_0,q_1}^{H_0-c}$ is in bijection with the zeroes of the corresponding function $f$ in (4.8).\nBy \\eqref{delta} we know that\n$$\n\\left\\lbrace \\eta > 0\\ |\\ f(\\eta)=0\\right\\rbrace \\subseteq [0, \\delta_0],\\qquad\n\\textrm{where}\\qquad\n\\delta_0=\\frac{1}{2c}\\left( \\sqrt{1+8c}+1\\right).\n$$\nCalculating the interval for various energies we obtain\n\\begin{align*}\n\\textrm{For}\\quad c &=\\frac{1}{5},\\quad & \\delta_0 &=\\frac{5}{2}\\left( \\sqrt{1+\\frac{8}{5}}+1\\right)=\\frac{5}{2}\\left( \\sqrt{\\frac{13}{5}}+1\\right)<\\frac{15}{2}=7\\frac{1}{2},\\\\\n\\textrm{For}\\quad c&=\\frac{1}{10},& \\delta_0 &=5\\left( \\sqrt{1+\\frac{4}{5}}+1\\right)=5\\left( \\frac{3}{\\sqrt{5}}+1\\right)< 5\\left( \\frac{3}{2}+1\\right)=12\\frac{1}{2},\\\\\n\\textrm{For}\\quad c&=\\frac{1}{20},& \\delta_0 &=10\\left( \\sqrt{1+\\frac{2}{5}}+1\\right)=10\\left( \\sqrt{\\frac{7}{5}}+1\\right)<10\\left(\\frac{6}{5}+1\\right)=22.\n\\end{align*}\n\nConsequently, Figure \\ref{fig:fFunction}. depicts the three functions $f$ corresponding to energies $c\\in \\left\\lbrace \\frac{1}{5}, \\frac{1}{10}, \\frac{1}{20}\\right\\rbrace$ on the whole interval $[0,\\delta_0]$. Therefore we can deduce that the plot in Figure \\ref{fig:fFunction}. depicts all the zeros of the three functions. Hence\n$$\n\\#\\Crit^+\\A_{q_0,q_1}^{H_0-\\frac{1}{5}}=1, \\qquad \\#\\Crit^+\\A_{q_0,q_1}^{H_0-\\frac{1}{10}}=3\\quad \\textrm{and}\\quad \\#\\Crit^+\\A_{q_0,q_1}^{H_0-\\frac{1}{20}}=5.\n$$\nFrom the discussion above we know that \n$$\n\\#\\Crit^+\\A_{q_0,q_1}^{H_0-1}=\\#\\Crit^+\\A_{q_0,q_1}^{H_0-\\frac{1}{2}}=\\#\\Crit^+\\A_{q_0,q_1}^{H_0-\\frac{1}{5}}=1.\n$$\nIn other words, on each of the level sets corresponding to $c=1$, $c=\\frac{1}{2}$ and $c=\\frac{1}{5}$ there is exactly one Reeb chord from $q_0=(0,1)$ to $q_1=(1,0)$. These Reeb chords are presented in Figure \\ref{fig:1Orbit3Energies}.\n\n\\begin{figure}\n\\begin{minipage}[c]{0.43\\linewidth}\n\\includegraphics[width=\\linewidth]{1Orbit3Energies.png}\n\n\\caption{The unique Reeb chords of energy $c=1$ (green), $c=\\frac{1}{2}$ (blue), and $c=\\frac{1}{5}$ (magenta).}\n\\label{fig:1Orbit3Energies}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[c]{0.54\\linewidth}\n\\vspace*{-.45cm}\n\\includegraphics[width=\\linewidth]{3Orbits.png}\n\\caption{The three Reeb chords of energy $c=\\frac{1}{10}$.}\n\\label{fig:3Orbits}\n\\end{minipage}\\end{figure}\n\nFigure \\ref{fig:3Orbits} depicts the three Reeb chords from $q_0=(0,1)$ to $q_1=(1,0)$ of energy $c=\\frac{1}{10}$, and Figure \\ref{fig:5Orbits} shows the five Reeb chords of energy $c=\\frac{1}{20}$.\n\n\\begin{figure}\\centering\n\\includegraphics[width=.6\\linewidth]{5Orbits.png}\n\\caption{The five Reeb chords of energy $c=\\frac{1}{20}$.}\n\\label{fig:5Orbits}\n\\end{figure}\n\\subsection{Calculating the Maslov index}\n\nIn this subsection we will prove Theorem \\ref{thm:posLRFH}. Let $H_0$ be the Copernican Hamiltonian as in \\eqref{DefH0} and let $\\mathcal{H}$ be the set of perturbations as defined in \\eqref{DefHset}.\nBy Proposition \\ref{prop:posLRFH} we know that for all $h\\in \\mathcal{H}$ and all pairs $q_0,q_1\\in \\R^2$, $q_0\\neq q_1$ the positive Lagrangian Rabinowitz Floer homology of the triple $\\LRFH_*^+(\\A^{H_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*^+(\\A^{H_0-c}_{q_0,q_1})$ for any $c>0$. On the other hand, by Proposition \\ref{prop:CritPM} we know that if $c\\geq \\frac{1}{2}|q_0||q_1|$ then $\\Crit^+\\A^{H_0-c}_{q_0,q_1}$ has only one element. Consequently, $\\LRFH_*^+(\\A^{H_0-c}_{q_0,q_1})$ has only one generator and its boundary operator is $0$. Therefore, in order to calculate the positive Lagrangian Rabinowitz Floer homology explicitly what is left to do is to calculate the Maslov index of $(v,\\eta)\\in \\Crit^+\\A^{H_0-c}_{q_0,q_1}$.\n\n\\begin{lem}\\label{lem:ChordsBijection}\nFor $c>0$ define a Hamiltonian $H_c:T^*\\R^2\\to \\R$ as\n\\begin{equation}\\label{DefHDelta}\nH_\\delta(q,p):=\\frac{|p|^2}{2}+\\delta(p_1q_2-p_2q_1).\n\\end{equation}\nLet $H_0$ be the Copernican Hamiltonian as in \\eqref{DefH0} and let $\\varphi_\\delta$ be a diffeomorphism of $T^*\\R^2$ defined $\\varphi_\\delta(q,p):=\\left(\\sqrt{\\delta}q,\\frac{1}{\\sqrt{\\delta}}p\\right)$. Then\n$$\n(v,\\eta)\\in \\Crit \\A_{q_0,q_1}^{H_\\delta-1}\\qquad \\iff \\qquad \\left(\\varphi_\\delta\\circ v, \\delta\\eta\\right)\\in \\Crit \\A_{\\varphi_\\delta(q_0),\\varphi_\\delta(q_1)}^{H_0-\\frac{1}{\\delta}}.\n$$\n\\end{lem}\n\\begin{proof}\nFirst observe that $\\varphi$ preserves the standard symplectic form, so it is in fact a symplectomorphism. Moreover, the Hamiltonians satisfy the following relation $H_\\delta=\\delta H_0\\circ \\varphi_\\delta$, thus \n$$\nH_0^{-1}\\left(\\frac{1}{\\delta}\\right)=\\varphi_\\delta (H_\\delta^{-1}(1) )\\qquad\\textrm{and}\\qquad X_{H_\\delta}=\\delta D\\varphi_\\delta^{-1}X_{H_0}.\n$$\nNow $(v,\\eta)\\in \\Crit \\A_{q_0,q_1}^{H_\\delta-1}$ if and only if $v(i)=q_i$ for $i=1,2$, $v([0,1])\\subseteq H_\\delta^{-1}(1)$ and $\\partial_t v=\\eta X_{H_\\delta}(v)$. Naturally, the first two conditions are trivially equivalent. It suffices to verify that\n$$\n\\frac{d}{dt}(\\varphi_\\delta\\circ v) = D\\varphi_\\delta (\\partial_t v) = \\eta D\\varphi_\\delta (X_{H_\\delta}(v))=\\delta X_{H_0}(\\varphi_\\delta \\circ v).\n$$\n\\end{proof}\n\\begin{lem}\\label{lem:Convergence}\nLet $\\{H_\\delta\\}_{\\delta>0}$ be the family of Hamiltonians defined in \\eqref{DefHDelta}. Let $H_{\\bullet}:T^*\\R^2\\to\\R$ be the kinetic Hamiltonian $H_{\\bullet}(q,p):=\\frac{1}{2}|p|^2$.\nFix $q_0,q_1\\in \\R^2$, $q_0\\neq q_1$. Then for $\\delta<\\sqrt{\\frac{2}{|q_0||q_1|}}$ (in case $q_0=0$ or $q_1=0$ for any $\\delta>0$) we have $\\#\\Crit^+\\A^{H_\\delta-1}_{q_0,q_1}=1$ and the family of Reeb chords $(v_\\delta,\\eta_\\delta)\\in \\Crit^+\\A^{H_\\delta-1}_{q_0,q_1}$ satisfies\n$$\n\\lim_{\\delta\\to 0}(v_\\delta,\\eta_\\delta)=(v_0,\\eta_0)\\in \\Crit^+\\A^{H_{\\bullet}-1}_{q_0,q_1}.\n$$ \n\\end{lem}\n\n\\begin{proof}\nUsing Lemma \\ref{lem:ChordsBijection} we know that for every $\\delta>0$ there is a bijection between $\\Crit^+\\A_{q_0,q_1}^{H_\\delta-1}$ and $\\Crit^+ \\A_{\\varphi_\\delta(q_0),\\varphi_\\delta(q_1)}^{H_0-\\frac{1}{\\delta}}$. On the other hand, by Proposition \\ref{prop:CritPM} we know that for $|\\varphi_\\delta(q_0)||\\varphi_\\delta(q_1)|=\\delta|q_0||q_1|\\leq \\frac{2}{\\delta}$ we have $1=\\# \\Crit^+ \\A_{\\varphi_\\delta(q_0),\\varphi_\\delta(q_1)}^{H_0-\\frac{1}{\\delta}}=\\# \\Crit^+ \\A_{q_0,q_1}^{H_\\delta-1}$. Consequently, for every $\\delta<\\sqrt{\\frac{2}{|q_0||q_1|}}$ (or in the case either $q_0$ or $q_1$ are zero for any $\\delta>0$) there is a unique solution $(v_\\delta,\\eta_\\delta)\\in \\Crit^+ \\A_{q_0,q_1}^{H_\\delta-1}$.\n\nHaving established that $(v_\\delta,\\eta_\\delta)\\in \\Crit^+\\A^{H_\\delta-1}_{q_0,q_1}$ is uniquely defined for small enough $\\delta$, we can use the function $f$ from Lemma \\ref{lem:bijection} to estimate $\\eta_\\delta$. We simply need to map $(c,q_0,q_1,\\eta)\\mapsto (\\frac{1}{\\delta}, \\sqrt{\\delta}q_0, \\sqrt{\\delta}q_1, \\delta \\eta)$ to obtain that $\\eta_\\delta$ is the unique positive root of the function:\n\\begin{align}\ng_\\delta(\\eta)&:= -\\eta^2+\\delta \\eta q_1^TR\\left(\\delta\\eta+\\frac{\\pi}{2}\\right)q_0 +\\frac{1}{2}\\left(|q_0|^2+|q_1|^2\\right)-q_1^TR(\\delta\\eta)q_0\\nonumber\\\\\n&= \\frac{1}{2}|q_1-q_0|^2-\\eta^2+q_1^T\\left( \\Id-R(\\delta\\eta)-\\delta \\eta R'(\\delta \\eta) \\right)q_0\\label{g_delta}\n\\end{align}\nThis way we know that $(v_\\delta,\\eta_\\delta)\\in \\Crit^+\\A^{H_\\delta-1}_{q_0,q_1}$ if and only if $g_\\delta(\\eta_\\delta)=0$.\n\nBy \\eqref{g_delta} we can use the Taylor expansion to obtain the following estimate\n\\begin{gather*}\n\\frac{1}{2}|q_1-q_0|^2-\\eta^2(1+\\delta^2|q_0||q_1|)\\leq g_\\delta(\\eta) \\leq \\frac{1}{2}|q_1-q_0|^2-\\eta^2(1-\\delta^2|q_0||q_1|),\\\\\ng_\\delta(\\eta_\\delta)=0\\quad \\Rightarrow \\quad \\frac{|q_1-q_0|}{\\sqrt{2(1+\\delta^2|q_0||q_1|)}}\\leq \\eta_\\delta \\leq\\frac{|q_1-q_0|}{\\sqrt{2(1-\\delta^2|q_0||q_1|)}},\n\\end{gather*}\nwhich directly gives us $\\lim_{\\delta\\to 0}\\eta_\\delta=\\frac{1}{\\sqrt{2}}|q_1-q_0|$.\n\nTo show the convergence of $v_\\delta$ we again use Lemma \\ref{lem:bijection} to present the equation for $v_\\delta$ explicitly. By Lemma \\ref{lem:ChordsBijection} is suffices to just use the mapping $(q_0,q_1,\\eta)\\mapsto (\\sqrt{\\delta}q_0, \\sqrt{\\delta}q_1, \\delta \\eta)$ in the formula for $v_\\delta$ from Lemma \\ref{lem:bijection} to obtain:\n\\begin{align*}\n\\varphi_\\delta\\circ v_\\delta(t) & =\\left(\\begin{array}{c c}\n(1-t) R(t\\delta\\eta_\\delta) & t R(\\delta\\eta_\\delta(t-1))\\\\\n -\\frac{1}{\\delta\\eta_\\delta}R(t\\delta\\eta_\\delta) & \\frac{1}{\\delta\\eta_\\delta} R(\\delta\\eta_\\delta(t-1))\n\\end{array}\\right)\n\\left(\\begin{array}{c}\n \\sqrt{\\delta} q_0 \\\\ \\sqrt{\\delta} q_1\n\\end{array}\\right),\\\\\nv_\\delta(t) &= \\left(\\begin{array}{c c}\n(1-t) R(t\\delta\\eta_\\delta) & t R(\\delta\\eta_\\delta(t-1))\\\\\n -\\frac{1}{\\eta_\\delta}R(t\\delta\\eta_\\delta) & \\frac{1}{\\eta_\\delta} R(\\delta\\eta_\\delta(t-1))\n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1\n\\end{array}\\right).\n\\end{align*}\nConsider the functions $t \\mapsto R(t\\delta\\eta_\\delta)$ and $t\\mapsto R((1-t)\\delta\\eta_\\delta)$ on the interval $[0,1]$. Since $\\lim_{\\delta\\to 0}\\delta\\eta_\\delta =0$, we have the uniform convergence $\\lim_{\\delta\\to 0}R(t\\delta\\eta_\\delta)=\\lim_{\\delta\\to 0}R((1-t)\\delta\\eta_\\delta)=\\Id$. Consequently,\n\\begin{equation}\\label{straight}\n\\lim_{\\delta\\to 0}v_\\delta=v_0(t) \\quad \\textrm{with}\\quad v_0(t):= \\left(\\begin{array}{c c}\n(1-t) & t \\\\\n -\\frac{\\sqrt{2}}{|q_1-q_0|} & \\frac{\\sqrt{2}}{|q_1-q_0|} \n\\end{array}\\right)\n\\left(\\begin{array}{c}\nq_0 \\\\ q_1 \\end{array}\\right).\n\\end{equation}\nA straightforward calculation shows that $(v_0,\\eta_0)$ with $\\eta_0:=\\frac{1}{\\sqrt{2}}|q_1-q_0|$ is the unique element of $\\Crit^+\\A^{H_{\\bullet}-1}_{q_0,q_1}$.\n\\end{proof}\n\n\n\n\\vspace*{.25cm}\n\\noindent \\textit{Proof of Theorem \\ref{thm:posLRFH}:}\nBy Proposition \\ref{prop:posLRFH} we know that for all $h\\in \\mathcal{H}$ and all $q_0,q_1\\in \\R^2$ with $q_0\\neq q_1$ the positive Lagrangian Rabinowitz Floer homology $\\LRFH_*^+(\\A^{H_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\\LRFH_*^+(\\A^{H_0-c}_{q_0,q_1})$ for any $c>0$. On the other hand, by Proposition \\ref{prop:CritPM} we know that if $c\\geq |q_0||q_1|/2$ then $\\Crit^+\\A^{H_0-c}_{q_0,q_1}$ has only one element. Consequently, $\\LRFH_*^+(\\A^{H_0-c}_{q_0,q_1})$ has only one generator and its boundary operator is $0$. It remains to calculate the Maslov index of this generator. \n\nLet $\\{H_\\delta\\}_{\\delta>0}$ be the family of Hamiltonians defined in \\eqref{DefHDelta} and let $\\varphi_\\delta$ be the family of symplectomorphisms defined by $\\varphi_\\delta(q,p):=(\\frac{1}{\\sqrt{\\delta}}q,\\sqrt{\\delta}p)$.\nBy Lemma \\ref{lem:ChordsBijection}, for every $c>0$ we have\n$$\n(v,\\eta)\\in \\Crit^+\\A^{H_0-c}_{q_0,q_1}\\qquad\\iff\\qquad(\\varphi_\\frac{1}{c}^{-1} \\circ v, c\\eta)\\in \\Crit^+\\A^{H_{1/c}-1}_{\\frac{1}{\\sqrt{c}}q_0, \\frac{1}{\\sqrt{c}}q_1}.\n$$\nSince the Maslov index is invariant under symplectomorphisms, we get \n$$\n\\mu^{\\rm tr}\\left( v\\left(\\frac{\\cdot}{\\eta}\\right)\\right)=\\mu^{\\rm tr}\\left(\\varphi_\\frac{1}{c}^{-1} \\circ v\\left(\\frac{\\cdot}{c\\eta}\\right)\\right).\n$$\nTherefore, it suffices to calculate $\\mu^{\\rm tr}\\left( v_\\delta\\left(\\frac{\\cdot}{\\eta_\\delta}\\right)\\right)$ of the unique $(v_\\delta, \\eta_\\delta)\\in \\Crit^+\\A^{H_\\delta-1}_{q_0,q_1}$ for $\\delta>0$ small enough.\n\nDenote by $H_{\\bullet}$ the kinetic Hamiltonian $H_{\\bullet}(q,p):=\\frac{1}{2}|p|^2$ and let $(v_0,\\eta_0)$ be the unique element of $\\Crit^+\\A^{H_{\\bullet}-1}_{q_0,q_1}$ as in \\eqref{straight}.\nBy Lemma \\ref{lem:Convergence} we know that $\\lim_{\\delta\\to 0}(v_\\delta, \\eta_\\delta)=(v_0,\\eta_0)$. Continuity of the Maslov index and Example~\\ref{ex:Maslov} implies \n$$\n\\mu^{\\rm tr}\\left( v_\\delta\\left(\\frac{\\cdot}{\\eta_\\delta}\\right)\\right)=\\mu^{\\rm tr}\\left( v_0\\left(\\frac{\\cdot}{\\eta_0}\\right)\\right) = \\frac12\n$$\nfor small enough $\\delta>0$. This concludes the proof of Theorem \\ref{thm:posLRFH}. \n\n\\hfill $\\square$\n\n\n\n\\section{Non-compactly supported potential}\n\nThe aim of this section will be to prove Proposition \\ref{prop:CritH=CritH1}. Let $H_0: T^*\\R^2 \\to \\R$ be the quadratic Hamiltonian defined in \\eqref{DefH0} and let $V:\\R^2 \\to \\R$ be the potential function as in Proposition \\ref{prop:CritH=CritH1}. \n\\begin{equation}\\label{DefH}\n\\textrm{Define}\\qquad H:= H_0 -V.\n\\end{equation}\nThen\n\\begin{equation}\\label{X_H}\n\\begin{aligned}\nX_H & = (p_1+q_2)\\partial_{q_1}+(p_2-q_1)\\partial_{q_2}+\\left(p_2 +\\frac{\\partial V}{\\partial_{q_1}}\\right)\\partial_{p_1}-\\left(p_1-\\frac{\\partial V}{\\partial_{q_2}}\\right)\\partial_{p_2},\\\\\nX_H & = p_r\\partial_r+\\left(1+\\frac{p_\\theta}{r^2}\\right)\\partial_\\theta+\\left( \\frac{p_\\theta^2}{r^3}+ \\frac{\\partial V}{\\partial r}\\right)\\partial_{p_r}+\\frac{\\partial V}{\\partial \\theta}\\partial_{p_\\theta}.\n\\end{aligned}\n\\end{equation}\nThis first step to prove Proposition \\ref{prop:CritH=CritH1} is to show that for positive energy $c>0$ all periodic orbits of $X_H$ on $H^{-1}(c)$ are contained in a compact subset of $T^*\\R^2$:\n\n\\begin{prop}\\label{prop:BoundChord}\nLet $H: T^*\\R^2 \\to \\R$ be the Hamiltonian defined in \\eqref{DefH}. Fix $c>0$ and $q_0,q_1 \\in \\R^2$ and let $\\A^{H-c}_{q_0,q_1}:\\mathscr{H_1}_{q_0,q_1}\\times \\R \\to \\R$ be the corresponding action functional. Then the critical set $\\Crit \\A^{H-c}_{q_0,q_1}$ is bounded in $L^\\infty$.\n\\end{prop}\n\\begin{proof}\nBy an argument similar to the one in the proof of Lemma \\ref{lem:H0Chord} it suffices to show that the set\n\\begin{equation}\\label{H1_r}\nH^{-1}(c)\\cap \\{\\{H, r\\}=0\\}\\cap \\{\\{H,\\{H, r\\}\\}\\leq 0\\}\n\\end{equation}\nis compact. By \\eqref{X_H} we have that\n\\begin{align*}\n\\{H,r\\} & = p_r,\\\\\n\\{H,\\{H,r\\}\\} & = \\frac{p_\\theta^2}{r^3}+ \\frac{\\partial V}{\\partial r}.\n\\end{align*}\nFor $(r, \\theta, p_r, p_\\theta) \\in H^{-1}(c)\\cap \\{\\{H, r\\}=0\\}$ we have\n\\begin{gather}\n \\frac{p_\\theta^2}{2r^2}+p_\\theta =V(r,\\theta)+c\\geq c,\\nonumber\\\\\n \\left(\\frac{p_\\theta}{r}+r\\right)^2 \\geq r^2+2c,\\nonumber\\\\\n \\frac{|p_\\theta|}{r}\\geq \\sqrt{r^2+2c}-r.\\label{p_theta_r}\n\\end{gather}\nConsequently, by our assumption on the potential function $V$, if $\\alpha>2$ and $a>0$ then for $(r, \\theta, p_r, p_\\theta) \\in H^{-1}(c)\\cap \\{\\{H, r\\}=0\\}$, such that \n\\begin{gather*}\nr \\geq r_1:=\\max\\left\\lbrace r_0,\\frac{1}{2}\\sqrt{c}, \\left( \\frac{a\\alpha}{c^2}\\right)^{\\frac{1}{\\alpha-2}}\\right\\rbrace\\\\ \\textrm{we have} \\quad\n\\frac{|p_\\theta|}{r} \\geq \\sqrt{r^2+2c}-r > \\frac{c}{2 r},\\\\\n\\{H,\\{H, r\\}\\} \\geq \\frac{c^2}{4r^3}-\\frac{a\\alpha}{r^{\\alpha+1}}>0.\n\\end{gather*}\n\nOn the other hand, if $\\alpha = 2$ and $a\\in \\left(0 ,\\frac{c^2}{4}\\right)$ then for $(r, \\theta, p_r, p_\\theta) \\in H^{-1}(c)\\cap \\{\\{H, r\\}=0\\}$, such that\n\\begin{gather*}\nr \\geq r_1:=\\max\\left\\lbrace r_0,\\sqrt{\\frac{a}{c-\\sqrt{2a}}}\\right\\rbrace\\\\ \\textrm{we have} \\quad\n\\frac{|p_\\theta|}{r} \\geq \\sqrt{r^2+2c}-r > \\frac{\\sqrt{2a}}{ r},\\\\\n\\{H,\\{H, r\\}\\} \\geq \\frac{2a}{r^3}-\\frac{2a}{r^{3}}>0.\n\\end{gather*}\nConsequently, in both cases the set\n$$\n H^{-1}(c)\\cap \\{\\{H, r\\}=0\\}\\cap \\left\\lbrace \\{H,\\{H, r\\}\\} \\leq 0 \\right\\rbrace\n$$\nis bounded in $r$ and for every $(v, \\eta) \\in \\Crit \\A^{H-c}_{q_0,q_1}$ we have\n$$\n\\sup r \\circ v \\leq R_0 := \\max\\{|q_0|, |q_1|, r_1\\}.\n$$\nBy an argument similar to the one presented in the Lemma \\ref{lem:H0Chord} we obtain that $p_r\\circ v$ and $p_\\theta\\circ v$ are also uniformly bounded in the following way:\n\\begin{align*}\n|p_r| & \\leq \\sqrt{R_0^2+ 2(\\sup_{r\\leq R_0}V+c)},\\\\\n|p_\\theta| & \\leq R_0 \\left( R_0+\\sqrt{R_0^2+ 2(\\sup_{r\\leq R_0}V+c)}\\right).\n\\end{align*}\n\\end{proof}\n\\begin{rem}\nNote that the assertions of Proposition \\ref{prop:BoundChord} hold true also for the potential $V$ satisfying $V(r,\\theta) \\leq \\frac{a}{r^2}$ and $\\frac{\\partial V}{\\partial r} \\geq -\\frac{2 a}{r^3}$ for $r >r_0$ and $a \\in \\left[ \\frac{c^2}{4}, \\frac{c^2}{2} \\right)$. However, we were unable to prove Proposition \\ref{prop:CritH=CritH1} for this class of potentials and that is why we restrict ourselves to $a < \\frac{c^2}{4}$.\n\\end{rem}\n\n\nNow we would like change the Hamiltonian $H$, by multiplying the potential function $V$ with a compactly supported function $\\varphi: T^*\\R^2 \\to [0,1]$, in the following way\n\\begin{equation}\\label{DefH1}\nH_1(q,p):=H_0(q,p)+\\varphi(q,p)V(q).\n\\end{equation}\nThe associated Hamiltonian vector field is\n\\begin{equation}\\label{X_H1}\n\\begin{aligned}\nX_{H_1} & = \\left(p_r-V\\frac{\\partial \\varphi}{\\partial p_r}\\right)\\partial_r+\\left(1+\\frac{p_\\theta}{r^2}-V\\frac{\\partial \\varphi}{\\partial p_\\theta}\\right)\\partial_\\theta\\\\\n& + \\left( \\frac{p_\\theta^2}{r^3}+ \\varphi\\frac{\\partial V}{\\partial r}+V\\frac{\\partial \\varphi}{\\partial r}\\right)\\partial_{p_r}+\\left(\\varphi\\frac{\\partial V}{\\partial \\theta}+V\\frac{\\partial \\varphi}{\\partial \\theta}\\right)\\partial_{p_\\theta}.\n\\end{aligned}\n\\end{equation}\n\n\nIf we choose $\\varphi$, such that for all the Reeb chords $(v,\\eta) \\in \\Crit \\A^{H-c}_{q_0,q_1}$ we would have $v([0,1]) \\subseteq \\varphi^{-1}(1)$, then $\\Crit(\\A^{H-c}_{q_0,q_1}) \\subseteq \\Crit(\\A^{H_1-c}_{q_0,q_1})$. In fact, we would like to choose $\\varphi$, such that $\\Crit(\\A^{H_1-c}_{q_0,q_1})= \\Crit(\\A^{H-c}_{q_0,q_1})$.\n\nLet $\\chi:\\R\\to [0,1]$ be a smooth function, such that $-2<\\chi'< 0$ on $[0,1]$ and\n\\begin{align}\n&&\\chi(x) & =\\begin{cases}\n1 & \\textrm{for} \\quad x\\leq 0,\\\\\n0 & \\textrm{for} \\quad x\\geq 1.\n\\end{cases}\\nonumber\\\\\n&\\textrm{Define}\n&\\chi_0(r) &:=\\chi\\left(\\beta(r-R_1) \\right),\\nonumber\\\\\n&&\\chi_1(r,\\theta, p_r,p_\\theta)& :=\\chi( H_0(r,\\theta,p_r,p_\\theta)-\\sup V-c),\\nonumber\\\\\n&&\\varphi(r,\\theta,p_r,p_\\theta) & := \\chi_0(r)\\chi_1(r,\\theta,p_r,p_\\theta).\\label{defPhi}\n\\end{align}\n\\begin{align}\n&\\textrm{where} & R_1 &:= \\begin{cases}\n\\max\\left\\lbrace |q_0|, |q_1|, r_0, \\frac{1}{2}\\sqrt{c},\\left( \\frac{8a\\alpha}{c^2}\\right)^\\frac{1}{\\alpha-2}\\right\\rbrace & \\textrm{for}\\quad \\alpha>2,\\\\\n\\max\\left\\lbrace |q_0|, |q_1|, r_0, \\frac{c+2\\sqrt{a}}{2\\sqrt{c-2\\sqrt{a}}}\\right\\rbrace & \\textrm{for}\\quad \\alpha=2.\n\\end{cases}\\label{R1}\\\\\n&\\textrm{and} & \\beta &:=\\begin{cases}\n\\frac{2-\\alpha}{2 R_1} & \\textrm{for}\\quad \\alpha>2,\\\\\n\\frac{(c+2\\sqrt{a})^2-16a}{8a R_1} & \\textrm{for}\\quad \\alpha=2.\n\\end{cases}\\label{beta}\n\\end{align}\nNote that in both for $\\alpha>2$ and for $\\alpha=2$ we have $\\beta>0$. In the later case, it follows from the assumption that $a< \\frac{c^2}{4}$.\n\nThis way we have\n\\begin{equation}\n\\chi_0(r) = \\begin{cases}\n1 & \\textrm{for}\\ r\\leq R_1,\\\\\n0 & \\textrm{for}\\ r\\geq R_1+\\frac{1}{\\beta}.\n\\end{cases}\\label{chi0}\n\\end{equation}\nWe will show that for a function $\\varphi$ defined as in \\eqref{defPhi} we have \n$$\n\\Crit (\\A^{H_0-\\varphi V-c}_{q_0,q_1}) = \\Crit(\\A^{H_0-V-c}_{q_0,q_1}).\n$$\nThe first step would be to show that $\\varphi$ has compact support:\n\n\n\\begin{lem}\nThe function $\\varphi: T^*\\R^2\\to \\R$ defined as in \\eqref{defPhi} has compact support.\n\\end{lem}\n\\begin{proof}\nBy \\eqref{defPhi} and \\eqref{chi0} we have\n\\begin{align*}\n\\operatorname{supp}\\varphi & \\subseteq \\{H_0 \\leq \\sup V+c + 1\\}\\cap \\left\\lbrace r \\leq R_1+\\frac{1}{\\beta}\\right\\rbrace\\\\\n& \\subseteq \\left\\lbrace \\frac{1}{2}p_r^2+\\frac{1}{2}\\left( \\frac{p_\\theta}{r}+r\\right)^2 \\leq \\frac{1}{2}r^2+\\sup V+c + 1\\right\\rbrace\\cap \\left\\lbrace r \\leq R_1+\\frac{1}{\\beta}\\right\\rbrace\\\\\n& \\subseteq \\left\\lbrace \\frac{1}{2}p_r^2+\\frac{1}{2}\\left( \\frac{p_\\theta}{r}+r\\right)^2 \\leq \\frac{1}{2} \\left( R_1+\\frac{1}{\\beta}\\right)^2+\\sup V + c+ 1\\right\\rbrace\\cap \\left\\lbrace r \\leq R_1+\\frac{1}{\\beta}\\right\\rbrace.\n\\end{align*}\nSince both of the sets on the right-hand side are compact, their intersection is compact, so $\\operatorname{supp}\\varphi$ is compact as a closed subset of a compact set.\n\\end{proof}\n\n\\begin{lem}\nLet $V:\\R^2\\to \\R$ be a potential function as in Proposition \\ref{prop:CritH=CritH1} and let $\\varphi$ be the corresponding function defined in \\eqref{defPhi}. Then $\\varphi V+c \\in \\mathcal{H}$.\n\\end{lem}\n\n\\begin{proof}\nFirst observe that\n$$\n\\supp d(\\varphi V)=\\supp ( V d\\varphi+ \\varphi dV)\\subseteq \\supp \\varphi,\n$$\nhence $d(\\varphi V)\\in C_c^\\infty(T^*\\R^2)$. On the other hand,\n by definition:\n\\begin{align*}\n\\varphi(p,q) V(q) & +c - d(\\varphi V)(p\\partial_p) = V(q)\\chi_0(q)\\left(\\chi_1(q,p)-d\\chi_1(p\\partial_p)\\right)+c\\\\\n& = V(q) \\chi_0(q) \\left( \\chi_1(q,p)- \\chi'(H_0(q,p) - \\sup V-c) dH_0(p\\partial_p)\\right)+c.\n\\end{align*}\nSince by assumption we the functions $V$, $\\chi_0$, $\\chi_1$ and $-\\chi'$ are non-negative it suffices to prove that $dH_0(p\\partial_p)\\geq 0$ on $\\chi_1^{-1}((0,1))$. Note that on $\\chi_1^{-1}((0,1))$ we have $H_0 > \\sup V+c$. Combining that with \\eqref{H0pdp} we can calculate that on $\\chi_1^{-1}((0,1))$ we have:\n$$ \ndH_0(p\\partial_p)= \\frac{1}{2}\\|p\\|^2+H_0\\geq \\frac{1}{2}\\|p\\|^2 + \\sup V+c > 0.\n$$\n\\end{proof}\n\n\nNow we have proven that $\\varphi$ as in \\eqref{defPhi} is an eligible candidate, we will continue with the proof of Proposition \\ref{prop:CritH=CritH1}. However, we will first start with the proof of a series of lemmas:\n\n\\begin{lem}\\label{lem:vinPhi(1)}\nLet $H$ be the Hamiltonian defined in \\eqref{DefH} and let $\\varphi$ be the function defined in \\eqref{defPhi}.\nThen\n $$\nv([0,1])\\subseteq \\varphi^{-1}(1)\\quad \\textrm{for all}\\quad (v,\\eta)\\in \\Crit\\A^{H-c}_{q_0,q_1}.\n$$\n\\end{lem}\n\n\\begin{proof}\nBy definition of $H$ we have\n$$\nH_ 0 = H+ V \\leq H + \\sup V,\n$$\nso $H^{-1}(c) \\subseteq H_0^{-1}((-\\infty, \\sup V+c])$. On the other hand,\nby Proposition \\ref{prop:BoundChord} for every $(v,\\eta)\\in \\Crit\\A^{H-c}_{q_0,q_1}$ we have\n\\begin{align*}\nv([0,1]) & \\subseteq \\{ r \\leq R_0\\}\\cap H^{-1}(c) \\subseteq \\{r \\leq R_1\\}\\cap H_0^{-1}((-\\infty, \\sup V+c])\\\\\n& = \\chi_0^{-1}(1)\\cap \\chi_1^{-1}(1)=\\varphi^{-1}(1).\n\\end{align*}\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:subsetvarphi(1)}\nLet $H_1$ be the Hamiltonian defined in \\eqref{DefH1} and let $\\varphi$ be the function defined in \\eqref{defPhi}. Then\n$$\nH_1^{-1}(c)\\cap \\{\\{H_1,r\\}=0\\}\\cap \\{\\{H_1,\\{H_1,r\\}\\}\\leq 0\\} \\subseteq \\varphi^{-1}(1).\n$$\n\\end{lem}\n\n\\begin{proof}\nFirst observe that $H_1\\big|_{\\varphi^{-1}(0)}=H_0$, so by \\eqref{supp_h} we have\n$$\nH_1^{-1}(c)\\cap \\{\\{H_1,r\\}=0\\}\\cap \\{\\{H_1,\\{H_1,r\\}\\}\\leq 0\\}\\cap \\varphi^{-1}(0)=\\emptyset.\n$$\nOn the other hand, if $x \\in \\chi_1^{-1}([0,1))$, then $H_0(x)>\\sup V+c$ and\n$$\nH_1(x)=H_0(x)-\\varphi (x) V(x) \\geq H_0(x) - \\sup V>c.\n$$\nHence $\\chi_1^{-1}((0,1])\\cap H_1^{-1}(c)=\\emptyset$. Therefore we can restrict ourselves to the analysis of the set $\\chi_0^{-1}((0,1))$, i.e. when $ R_1 < r < R_1+\\frac{1}{\\beta}$.\n\nLet us calculate\n\\begin{align*}\n\\frac{\\partial\\varphi}{\\partial r} & = \\beta\\chi_0'\\chi_1-\\frac{p_\\theta^2}{r^3}\\chi_0\\chi_1', && \\frac{\\partial \\varphi}{\\partial \\theta}=0,\\\\\n\\frac{\\partial \\varphi}{\\partial p_r} & = p_r\\chi_0\\chi_1', && \\frac{\\partial\\varphi}{\\partial p_\\theta} = \\left(\\frac{p_\\theta}{r^2}+1\\right)\\chi_0 \\chi_1'.\n\\end{align*}\nIn particular, by \\eqref{X_H} we have\n$$\n\\{H_1,r\\} = dr(X_{H_1})= p_r- V\\frac{\\partial \\varphi}{\\partial p_r}=p_r (1- V \\chi_0 \\chi_1').\n$$\nBy assumption $V, \\chi \\geq 0$ and $\\chi'\\leq 0$, thus $\\{H_1,r\\}=0$ implies $p_r=0$.\n\nConsider now $(r, \\theta, p_r, p_\\theta) \\in H_1^{-1}(0)\\cap \\left(\\{H_1, r\\}\\right)^{-1}(0)$. Then\n\\begin{align*}\n\\{H_1,\\{H_1,r\\}\\} & = \\{H_1, p_r\\}- \\frac{\\partial \\varphi}{\\partial p_r}\\{H_1, V\\}-V\\left\\lbrace H_1, \\frac{\\partial \\varphi}{\\partial p_r}\\right\\rbrace\\\\\n& = \\{H_1, p_r\\}-V \\frac{\\partial^2\\varphi}{\\partial p_r^2} \\{H_1, p_r\\}\\\\\n& = \\left(\\frac{p_\\theta^2}{r^3}+ \\varphi\\frac{\\partial V}{\\partial r}+V\\frac{\\partial \\varphi}{\\partial r}\\right)\\left(1-V\\frac{\\partial^2\\varphi}{\\partial p_r^2}\\right)\\\\\n& = \\left(\\frac{p_\\theta^2}{r^3}+ \\chi_0 \\chi_1\\frac{\\partial V}{\\partial r}+V\\left( \\beta\\chi_0'\\chi_1-\\frac{p_\\theta^2}{r^3}\\chi_0\\chi_1'\\right)\\right)\\left(1-\\chi_0\\chi_1'V\\right)\n\\end{align*}\nAgain, since by assumption $V, \\chi \\geq 0$ and $\\chi'\\leq 0$, thus $1-\\chi_0\\chi_1'V\\geq 1$ and $\\{H_1,\\{H_1,r\\}\\}$ has the same sign as \n\\begin{equation}\\label{eq5}\n\\frac{p_\\theta^2}{r^3}+ \\chi_0 \\chi_1\\frac{\\partial V}{\\partial r}+V\\left( \\beta\\chi_0'\\chi_1-\\frac{p_\\theta^2}{r^3}\\chi_0\\chi_1'\\right)\n\\end{equation}\nFurthermore, since by assumption $V, \\chi_0, p_\\theta \\geq 0$ and $\\chi_1' \\leq 0$, consequently $-\\frac{p_\\theta^2}{r^3}\\chi_0\\chi_1'V \\geq 0$ and we can estimate \\eqref{eq5} from below by\n\\begin{equation}\\label{eq6}\n\\frac{p_\\theta^2}{r^3}+ \\chi_0 \\chi_1\\frac{\\partial V}{\\partial r}+ V\\beta \\chi_0'\\chi_1.\n\\end{equation}\nWe will show that \\eqref{eq6} is positive for both cases $\\alpha>2$ and $\\alpha=2$.\n\n\n\\noindent\\textbf{Case $\\alpha>2$:}\\\\\nBy \\eqref{R1} we have that $r > R_1 \\geq \\left(\\frac{8a\\alpha}{c^2}\\right)^{\\frac{1}{\\alpha-2}}$, so $\\frac{c^2}{4r^3}>\\frac{2a\\alpha}{r^{\\alpha+1}}$. On the other hand, $r < R_1+\\frac{1}{\\beta}=R_1\\frac{\\alpha}{\\alpha-2}$. Combining that with the assumptions on the potential $V$ and the fact that $-2\\leq \\chi'\\leq 0$ and $0\\leq \\chi \\leq 1$ we obtain \n\\begin{align*}\n\\frac{p_\\theta^2}{r^3}+ \\chi_0 \\chi_1\\frac{\\partial V}{\\partial r}+\\frac{\\alpha-2}{2R_1}V \\chi_0'\\chi_1 & \\geq \\frac{c^2}{4r^3}-\\frac{a\\alpha}{r^{\\alpha+1}}-\\frac{\\alpha-2}{R_1}\\frac{a}{ r^\\alpha}\\\\\n &> \\frac{a\\alpha}{r^{\\alpha+1}}-\\frac{\\alpha -2}{R_1}\\frac{a}{ r^\\alpha}\\\\\n & = \\frac{a(\\alpha -2)}{R_1 r^{\\alpha+1}}\\left( R_1 \\frac{\\alpha}{\\alpha -2} -r\\right)>0.\n\\end{align*}\n\n\\noindent\\textbf{Case $\\alpha=2$:}\\\\\nBy \\eqref{p_theta_r}, \\eqref{R1} and \\eqref{chi0} we have\n\\begin{gather*}\nr \\geq \\frac{c+2\\sqrt{a}}{2\\sqrt{c-2\\sqrt{a}}}\\qquad\\textrm{and}\\qquad c-2\\sqrt{a} \\geq \\frac{(c+2\\sqrt{a})^2}{4r^2},\\\\\nr^2 + 2c \\geq r^2+c+2\\sqrt{a}+\\frac{(c+2\\sqrt{a})^2}{4r^2}=\\left(r+ \\frac{c+2\\sqrt{a}}{2 r}\\right)^2,\\\\\n\\frac{|p_\\theta|}{r} \\geq \\sqrt{r^2+2c}-r\\geq \\frac{c+2\\sqrt{a}}{2r}.\n\\end{gather*}\nTherefore, by \\eqref{beta} and \\eqref{chi0} we obtain\n\\begin{align*}\n\\frac{p_\\theta^2}{r^3} & + \\chi_0 \\chi_1\\frac{\\partial V}{\\partial r}+V\\beta \\chi_0'\\chi_1 \\geq \\frac{(c+2\\sqrt{a})^2}{4r^3}-\\frac{2a}{r^3}-\\frac{2\\beta a}{r^2}\\\\\n& = \\frac{2a}{r^3}\\left( \\frac{\\left(c+2\\sqrt{a}\\right)^2}{8a}-1-\\beta r\\right)=\\frac{2a}{r^3}\\left( R_1 \\beta + 1 -\\beta r\\right)>0.\n\\end{align*}\n\nConsequently, in both cases we have\n$$\nH_1^{-1}(c)\\cap \\{\\{H_1,r\\}=0\\}\\cap \\{\\{H_1,\\{H_1,r\\}\\}\\leq 0\\}\\cap \\chi_0^{-1}((0,1))=\\emptyset,\n$$\nwhich proves the claim.\n\\end{proof}\n\n\\noindent\\textit{Proof of Proposition \\ref{prop:CritH=CritH1}:}\nBy Lemma \\ref{lem:vinPhi(1)} we know that\n$$\nv([0,1])\\subseteq \\varphi^{-1}(1)\\quad \\textrm{for all}\\quad (v,\\eta)\\in \\Crit\\A^{H-c}_{q_0,q_1}.\n$$\nBut by definition $H_1\\big|_{\\varphi^{-1}(1)}=H\\big|_{\\varphi^{-1}(1)}$. Thus $\\Crit\\A^{H-c}_{q_0,q_1} \\subseteq \\Crit\\A^{H_1-c}_{q_0,q_1}$.\n\nOn the other hand, by an argument as in the proof of Proposition \\ref{prop:BoundChord} we know that for all $(v,\\eta)\\in \\Crit\\A^{H_1-c}_{q_0,q_1}$ if $\\max_{t\\in [0,1]}r \\circ v = r\\circ v(t_0)$, then\n$$\nv(t_0) \\in \\{q_0, q_1\\}\\cup \\left( H_1^{-1}(c)\\cap \\{\\{H_1,r\\}=0\\}\\cap \\{\\{H_1,\\{H_1,r\\}\\}\\leq 0\\}\\right).\n$$\nMoreover, by Lemma \\ref{lem:subsetvarphi(1)} we know that\n\\begin{align*}\nv(t_0) \\in \\{q_0, q_1\\} & \\cup \\left( H_1^{-1}(c)\\cap \\{\\{H_1,r\\}=0\\}\\cap \\{\\{H_1,\\{H_1,r\\}\\}\\leq 0\\}\\right)\\\\\n& \\subseteq H_1^{-1}(c)\\cap \\varphi^{-1}(1) \\subseteq H^{-1}(c)\\cap \\{ r \\leq R_1\\}.\n\\end{align*}\nConsequently, for all $(v,\\eta)\\in \\Crit\\A^{H_1-c}_{q_0,q_1}$ we have $\\max r\\circ v \\leq R_1$. In other words, for all $(v,\\eta)\\in \\Crit\\A^{H_1-c}_{q_0,q_1}$ we have\n$$\nv([0,1]) \\subseteq H_1^{-1}(c)\\cap \\{ r \\leq R_1\\}.\n$$\nOn the other hand, the Hamiltonians satisfy \n$$\nH_1^{-1}(c)=\\{H_0 = \\varphi V+c\\}\\subseteq H_0^{-1}((-\\infty, \\sup V+c]).$$\nConsequently, for all $(v,\\eta)\\in \\Crit\\A^{H_1-c}_{q_0,q_1}$ we have\n$$\nv([0,1]) \\subseteq H_1^{-1}(c)\\cap \\{ r \\leq R_1\\} \\subseteq H_0^{-1}((-\\infty, \\sup V+c])\\cap \\{ r \\leq R_1\\} = \\varphi^{-1}(1).\n$$\nBut on $\\varphi^{-1}(1)$ the two Hamiltonians $H_1$ and $H$ coincide. Therefore, $\\Crit \\A^{H-c}_{q_0,q_1} = \\Crit \\A^{H_1-c}_{q_0,q_1}$.\n\n\\hfill $\\square$\n\n\n\n\\printbibliography\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":97,"cells":{"id":{"kind":"string","value":"2412.08437v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08437v1"},"title":{"kind":"string","value":"Zeta and L functions of Voevodsky motives"},"full_text":{"kind":"string","value":"\\documentclass[12pt]{amsart}\n\\usepackage{verbatim,amssymb,latexsym, 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\\magstep1\n\\DeclareFontFamily{U}{wncy}{}\n\\DeclareFontShape{U}{wncy}{m}{n}{<5>wncyr5<6>wncyr6<7>wncyr7<8>wncyr8<9>wncyr9<10>wncyr10<11>wncyr10<12>wncyr6<14>wncyr7<17>wncyr8<20>wncyr10<25>wncyr10}{}\n\\DeclareMathAlphabet{\\cyr}{U}{wncy}{m}{n}\n\n\\newcommand{\\Sha}{\\cyr{X}}\n\\newcommand{\\sha}{\\cyr{x}}\n\\newcommand{\\Be}{\\cyr{B}}\n\\newcommand{\\be}{\\cyr{b}}\n\n\n\\swapnumbers\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lemma}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\newtheorem{rk}[thm]{Remark}\n\\newtheorem{rks}[thm]{Remarks}\n\\newtheorem{qn}[thm]{Question}\n\\newtheorem{qns}[thm]{Questions}\n\\newtheorem{ex}[thm]{Example}\n\\newtheorem{exs}[thm]{Examples}\n\n\\newcommand{\\prf}{\\noindent{\\bf Proof. }}\n\n\\newcounter{spec}\n\\newenvironment{thlist}{\\begin{list}{\\rm{(\\roman{spec})}}{\\usecounter{spec}\\labelwidth=20pt\\itemindent=0pt\\labelsep=10pt}}{\\end{list}}\n\n\\renewcommand{\\thesubsection}{\\arabic{section}.\\Alph{subsection}}\n\n\\setcounter{tocdepth}{1}\n\n\n\\begin{document}\n\n\\title{Zeta and $L$ functions of Voevodsky motives}\n\\author{Bruno Kahn}\n\\address{IMJ-PRG\\\\Case 247\\\\\n4 place Jussieu\\\\\n75252 Paris Cedex 05\\\\\nFrance}\n\\email{bruno.kahn@imj-prg.fr}\n\\date{Feb. 2012 and Dec. 2024}\n\\subjclass[2020]{11M41, 11G09}\n\\keywords{Zeta functions, motives, six functors formalism}\n\\begin{abstract}\nWe associate an $L$-function $L^\\near(M,s)$ to any geometric motive over a global field $K$ in the sense of Voevodsky. This is a Dirichlet series which converges in some half-plane and has an Euler product factorisation. When $M$ is the dual of $M(X)$ for $X$ a smooth projective variety, $L^\\near(M,s)$ differs from the alternating product of the zeta functions defined by Serre in 1969 only at places of bad reduction; in exchange, it is multiplicative with respect to exact triangles. If $K$ is a function field over $\\F_q$, $L^\\near(M,s)$ is a rational function in $q^{-s}$ and enjoys a functional equation. The techniques use the full force of Ayoub's six (and even seven) operations.\n\\end{abstract}\n\\maketitle\n\n\\hfill Preliminary version\n\n\\tableofcontents\n\n\\section*{Introduction} Let $K$ be a global field, and let $X$ be a smooth projective $K$-variety. In \\cite{serre}, after a long search (see \\cite[letter avril 1963, p. 143]{corr}), Serre succeeded to define local factors of zeta functions associated to the cohomology groups of $X$, generalising both Artin's $L$-functions and the Hasse-Weil zeta function associated to an elliptic curve. When $K$ is a number field, this definition has been generalised by Scholl \\cite{scholl} to `mixed motives', \\emph{i.e.} objects of the abelian category defined by Jannsen and Deligne with systems of realisations in \\cite{jannsen} and \\cite{deligne}, thus allowing for a more conceptual reformulation of Beilinson's conjectures on special values of $L$-functions (see also Fontaine-Perrin Riou \\cite{fpr} and Deninger \\cite{den}).\\footnote{There is no candidate for an abelian category of mixed motives in positive characteristic at present.}\n\n\nThese definitions use $l$-adic cohomology and, unfortunately, depend on a still unproven conjecture: without even mentioning mixed motives, this is already the case in \\cite{serre}. Namely, the local factor from \\cite[\\S 2]{serre} at a finite place $v$ of $K$ is of the form\n\\begin{equation}\\label{eq0}\nL_v^\\Serre(H^i(X),s)=\\det(1-\\phi_v N(v)^{-s}\\mid H^i(\\bar X,\\Q_l)^{I_v})^{-1}\n\\end{equation}\nwhere $\\phi_v$ is a geometric Frobenius at $v$, $l$ is a prime number not dividing $N(v)$ and $I_v$ is the inertia at $v$. Here $N(v)$ is the cardinality of the residue field at $v$. But for this expression to make sense as a complex function of the variable $s$, one needs at the very least that the coefficients of this inverse polynomial be complex numbers, while they are defined as $l$-adic numbers. If $v$ is a place of good reduction, \\eqref{eq0} boils down by smooth and proper base change to $\\det(1-\\phi_v N(v)^{-s}\\mid H^i(\\bar X(v),\\Q_l))^{-1}$ where $X(v)$ is the special fibre of a smooth model of $X$ at $v$, which does not depend on the choice of the prime $l$ and has rational coefficients by Deligne's proof of the ``Riemann hypothesis'' for $X(v)$ \\cite{WeilI}. That this persists when $X$ has bad reduction at $v$ was proven by Terasoma in positive characteristic \\cite{ter} but remains open over number fields in general \\cite[$C_5$]{serre}. See \\cite[5.6.3, 5.6.4]{zetaL} for a detailed discussion. \n\nAnother issue is that the $L$-functions of \\eqref{eq0} seem difficult to manipulate: if $0\\to M'\\to M\\to M''\\to 0$ is a short exact sequence of mixed motives, the equality $L^\\Serre(M,s)=L^\\Serre(M',s)L^\\Serre(M'',s)$ fails in general.\n\nLet us now pass from mixed motives to \\emph{triangulated motives}. Let \\break $\\DM_\\gm(K,\\Q)$ denote Voevodsky's triangulated category of geometric motives over $K$, with rational coefficients \\cite{voetri,mvw}. In this article, we associate to any object $M\\in \\DM_\\gm(K,\\Q)$ a Dirichlet series $L^\\near(M,s)$, called the \\emph{nearby $L$-function} of $M$, which has the following properties:\n\\begin{description}\n\\item[Convergence] $L^\\near(M,s)$ has a finite abscissa of convergence.\n\\item[Multiplicativity] If $M'\\to M\\to M''\\by{+1}$ is an exact triangle, then\n\\[L^\\near(M,s)=L^\\near(M',s)L^\\near(M'',s).\\]\nIn particular, $L^\\near(M[1])=L^\\near(M,s)^{-1}$.\n\\item[Tate twists] $L^\\near(M(1),s) = L^\\near(M,s+1)$.\n\\item[Euler decomposition] There is a factorisation\n\\[L^\\near(M,s)=\\prod_v L^\\near_v(M,s)\\]\nwhere $v$ runs through the finite places of $K$ and $L^\\near_v(M,s)$ is an \\emph{$N(v)$-Euler factor}, i.e. a rational function in $N(v)^{-s}$ with $\\Q$ coefficients whose zeroes and poles are $N(v)$-Weil numbers. \n\\item[Inductivity] Let $L/K$ be a finite extension and $f:\\Spec L\\to \\Spec K$ be the corresponding morphism. Then\n\\[L^\\near(M,s)=L^\\near(f_!M,s)\\]\nfor any $M\\in \\DM_\\gm(L,\\Q)$, where $f_!:\\DM_\\gm(L,\\Q)\\to \\DM_\\gm(K,\\Q)$ is the push-forward functor.\n\\item[``Normalisation''] Let $X$ be a smooth projective $K$-variety. If $v$ is a place of good reduction for $X$, then $L^\\near_v(M(X)^*,s) = \\zeta(X(v),s)$, where $X(v)$ is the special fibre of a smooth projective model of $X$ at $v$ (this zeta function does not depend on the choice of such a model, \\cite[Prop. 5.6]{zetaL}). Here $M(X)^*$ is the dual of the motive $M(X)$ of $X$. This extends to $L^\\near(\\Phi(N)^*,s)$, where $N$ is a Chow motive and $v$ is a place of good reduction for $N$, where $\\Phi:\\Chow(K,\\Q)\\to \\DM_\\gm(K,\\Q)$ is Voevodsky's functor \\cite[Prop. 2.1.4]{voetri}.\n\\item[Rationality and functional equation] If $K$ has positive characteristic with field of constants $k\\simeq \\F_q$, then $L^\\near(M,s)$ is a $q$-Euler factor and enjoys a functional equation\n\\[L^\\near(M^*,1-s) = A(-q)^{-Bs} L^\\near(M,s)\\]\nwhere $A\\in \\Q^*$ and $B\\in \\Z$ are some explicit numbers (Theorem \\ref{t9.2}).\n\\item[Archimedean primes] If $K$ is a number field, to any archimedean place $v$ of $K$ is associated a ``local gamma factor'' $\\Gamma_v(M,s)$ which is multiplicative in $M$ and agrees with that of a smooth projective variety $X$ as in Serre \\cite[\\S 3]{serre} for $M=M(X)^*$.\n\\end{description}\n\nThe definition of $L^\\near(M,s)$ purely rests on formal properties of the rigid $\\otimes$-triangulated category $\\DM_\\gm(K,\\Q)$ and its generalisations over a base, hence, \\emph{in fine}, on algebraic cycles. In particular, it is independent of $l$ \\emph{a priori}. The superscript `near' is there to point out that this definition is not very different from the classical one for smooth projective varieties, but also that it can be computed in terms of Ayoub's specialisation systems (nearby cycles): Theorem \\ref{t9.1}. The quotation marks at normalisation are there because this property is sufficient, together with multiplicaitivity, only to characterise $L^\\near$ up to finite products of Euler factors. It would be nice to find extra properties which make this uniqueness true on the nose. \n\nIf $X$ is smooth projective, $L^\\near_v(M(X)^*,s)$ is in general different from $L_v^\\Serre(X,s)$ when $v$ is a place of bad reduction, where $L_v^\\Serre(X,s)$ is the alternating product of the functions of \\eqref{eq0}: see \\S \\ref{s9.D.2} for the example of an elliptic curve with multiplicative reduction. To explain the idea behind its definition, note that there is a competing local factor, replacing $H^i(\\bar X,\\Q_l)^{I_v}=H^0(I_v,H^i(\\bar X,\\Q_l))$ with $H^1(I_v,H^i(\\bar X,\\Q_l))=\\break H^i(\\bar X,\\Q_l)_{I_v}(-1)$ (compare \\cite[note 164.23]{corr}): $L^\\near(M(X)^*,s)$ is a ``multiplicative average'' of these two alternating products, see Theorem \\ref{t8.2} b) and Definition \\ref{d9.1}. \n\nAfter getting the idea to define triangulated $L$-functions and finding how, I wondered if Grothendieck would himself have had a similar concern. And indeed he raises the issue twice, in \\cite[letter 30.9.1964, top p. 196]{corr} responding to the letter quoted above where Serre formulates the question, and in \\emph{loc. cit.}, letter of 3 and 5 Oct. 1964, c) p. 202. In the first letter, he claims (unless mistaken) that the formula of \\cite{serre} is indeed multiplicative\\footnote{\\emph{Lorsqu'on veut à tout prix une fonction $L$ qui dépende multiplicativement de $M$, il me semble hors de doute que la définition que tu préconises est la meilleure.}}. This looks strange, since taking invariants under inertia is not a (right) exact functor. In the second, he backtracks and prefers a ``définition bébête'' where $L(M,s)$ (for a mixed motive $M$) is defined as the product of $L$-functions of the factors of its semi-simplification. It will turn out, after the fact, that the latter idea is the right one for triangulated motives, see \\eqref{eq9.3}. For mixed motives, however, Grothendieck's first proposal as implemented in the references given at the beginning seems to be the most interesting and the most profound. But it depends on conjectures\\dots\n\nThus I hope that the present construction, its unconditionality and its multiplicative property will be helpful for making progress towards the Beilinson conjectures.\n\nThis version is preliminary because I stopped at a `honest' functional equation in the sense of Grothendieck's first letter quoted above (see \\cite[p. 197]{corr}). To get a nicer one as in \\cite{serre} would involve giving a formula à la Grothendieck-Ogg-\\v Safarevi\\v c \\cite[Exp. X]{SGA5} for the Euler Poincaré characteristic of a motive of the form $f_!M$, where $f=S\\to \\Spec \\F_q$ is a smooth projective curve. While this can obviously be done via an $l$-adic realisation, it is not clear (to me) that the local terms are independent of $l$, see Question \\ref{q10.1}. The same issue arises in higher dimension when using Takeshi Saito's theory of the characteristic cycles \\cite{tsaito}. Joseph Ayoub has suggested to use the Galois action on his ``full'' specialisation systems $\\Psi_x$ \\cite[Déf. 10.14]{ayoubetale}; I hope to come back to this in a further version.\n\n\\subsection*{Strategy} We first associate a zeta function to any motive in $\\DM_\\gm(k,\\Q)$ when $k$ is a finite field, by using categorical traces of powers of Frobenius in this rigid $\\otimes$-category. One key result is that the zeta function of the ``Borel-Moore'' motive of $X$ is the zeta function of $X$, when $X$ is an $\\F_q$-scheme of finite type (Corollary \\ref{c1}). We then extend this to motives over a $\\Z$-scheme of finite type $S$ by the usual product formula over the closed points of $S$. Here a second key result is a trace formula, which allows us to compute these zeta functions as zeta functions of motives over $\\F_q$ when $S$ is an $\\F_q$-scheme (Theorem \\ref{t3.1}): the proof, relying on the previous result, is almost purely motive-theoretic but we cannot avoid using the $l$-adic realisation functor (hence the trace formula of \\cite{SGA5}), because of a problem of idempotents. We also get a functional equation when $S$ is proper over $\\F_q$ (\\emph{ibid.}). The six functors formalism established by Ayoub in his thesis \\cite{ayoub} is central in these definitions and properties.\n\nAll these zeta functions are rational functions of $p^{-s}$ in characteristic $p$. In Corollary \\ref{c6.1}, we show that they converge (in some half-plane) also in characteristic $0$, \\emph{i.e.} when $S$ is dominant over $\\Spec \\Z$.\n\nIn Section \\ref{s9} we arrive at the heart of the matter: the case of motives over a global field $K$. Since $\\Spec K$ is not of finite type over $\\Spec \\Z$, the issue is to get a reasonable definition out of the previous work. We do it in two steps: first define a ``total'' $L$-function (Definition \\ref{l9.1}), and then deduce the ``nearby'' $L$-function from it (Definition \\ref{d9.1}), by applying Lemma \\ref{l2}. In characteristic $p>0$, we get a functional equation in Theorem \\ref{t9.2}. All this latter work uses heavily the six operations again, and even the seventh (the unipotent specialisation system).\n\n\\subsection*{Acknowledgements} This work has been in gestation since 1998. Since then, Science has made progress and much of this progress has been incorporated here. It has been announced at conferences a number of times, the most recent being at Regensburg in 2012 \\cite{zetaLRegensburg}. The reasons of my procrastination are not entirely clear. I thank Joseph Ayoub, Mikhail Bondarko, Fr\\'ed\\'eric D\\'eglise, Luc Illusie and Amílcar Pacheco for helpful exchanges, with a special mention to Ayoub: he kindly wrote up his paper \\cite{ayoubetale} on the $l$-adic realisations upon my request and helped me at a large number of places in this manuscript. As an exercise, the reader may count the number of times when I credit him for a proof or an idea.\n\n\\enlargethispage*{30pt}\n\n\\subsection*{Conventions and notation} As usual, a $\\otimes$-category means a symmetric monoidal unital additive category, and a $\\otimes$-functor is an additive strong symmetric monoidal functor. We refer to \\cite[App. 8.A]{mvw} for tensor triangulated categories.\n\n\n\\numberwithin{equation}{section}\n\n\\section{The rigid $\\otimes$-category $\\DM_\\gm(k,\\Q)$}\\label{s1}\n\nLet $k$ be a field. We consider the category $\\DM_\\gm(k,\\Q)$ defined by Voevodsky in\n\\cite{voetri} (here, with rational coefficients). It is provided with a covariant functor ``motive'' $M:\\Sm(k) \\to \\DM_\\gm(k,\\Q)$, where $\\Sm(k)$ is the category of smooth separated $k$-schemes. In \\cite{voetri}, two properties of\n$\\DM_\\gm(k,\\Q)$ are established when $k$ is of characteristic $0$:\n\n\\begin{itemize}\n\\item It is a rigid tensor pseudo-abelian triangulated category, generated by the motives of smooth projective varieties as such. We write $M^*$ for the dual of a motive $M\\in \\DM_\\gm(k,\\Q)$.\n\\item For any separated $k$-scheme of finite type $X$, there is an associated motive with\ncompact supports $M_c(X)\\in \\DM_\\gm(k,\\Q)$ such that $M_c(X)=M(X)^*(d)[2d]$ if $X$ is smooth\nof pure dimension $d$, $M_c(X)=M(X)$ if $X$ is proper and, if\n$Z\\by{i}X$ is a closed subset with complementary open $U\\by{j} X$, one has an exact triangle of the form\n\\begin{equation}\\label{eq1.1}\nM_c(Z)\\by{i_*} M_c(X)\\by{j^*} M_c(U)\\by{+1}.\n\\end{equation}\n\\end{itemize}\n\n\\begin{thm}\\label{p1.1} These properties hold for any $k$.\n\\end{thm}\n\n\\begin{proof} When $k$ is perfect, see \\cite[App. B]{motiftate} for the first property and \\cite[\\S 5.3]{kelly} for the second. In general, let $k^p$ be the perfect closure of $k$. Then \\cite[Prop. 4.5]{adjoints} or \\cite{suslininsep} show that the base change functor\n\\[\\DM_\\gm(k,\\Q)\\to \\DM_\\gm(k^p,\\Q)\\]\nis an equivalence of categories.\n\\end{proof}\n\n(Theorem \\ref{p1.1} is even true with coefficients $\\Z[1/p]$, where $p$ is the exponential characteristic of $k$, but we won't use this refinement.)\n\n\nWe shall need:\n\n\\begin{lemma}\\label{l1.4} Let $f:X\\to Y$ be a finite, surjective morphism of smooth $k$-schemes of generic degree $d$, where $k$ is assumed to be perfect. Then $f$ induces morphisms $f^*:M_c(Y)\\to M_c(X)$ and $f_*:M_c(X)\\to M_c(Y)$ such that $f_*f^*=d$. Moreover,\n\\begin{itemize}\n\\item If $f$ is radicial, we also have $f^*f_*=d$. In particular $f^*$ is an isomorphism.\n\\item If $f$ is a Galois covering of group $G$, then $f^*$ induces an isomorphism $M_c(Y)\\iso \\epsilon_GM_c(X)$ where $\\epsilon_G$ is the idempotent $\\frac{1}{G}\\sum_{g\\in G} g$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} By duality, we reduce to the same statement for $M(X)$ and $M(Y)$. Then they are already true on the level of finite correspondences.\n\\end{proof}\n\n\\begin{defn}\\label{d1.1} We write $\\DM_\\gm^\\eff(k,\\Q)\\subset \\DM_\\gm(k,\\Q)$ for the full subcategory of effective geometric motives and, if $n\\ge 0$, $d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$ for the thick triangulated subcategory of $\\DM_\\gm^\\eff(k,\\Q)$ generated by motives of smooth varieties of dimension $\\le n$.\n\\end{defn}\n\n\\begin{prop}\\label{p1.2} If $\\dim X\\le n$, then $M_c(X)\\in d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$.\n\\end{prop}\n\n\\begin{proof}We may assume $k$ perfect. Induction on $n$. The case $n=0$ is clear because $M_c(X)=M_c(X_\\rd)$ and $X_\\rd$ is a (proper) étale $k$-scheme. Suppose $n>0$. By closed Mayer-Vietoris, we reduce to $X$ irreducible and then to $X$ a variety. By de Jong's theorem, there exists an alteration $f:\\tilde X\\to X$ with $\\tilde X$ smooth. Choose a smooth open subset $U$ of $X$ over which $f$ is finite, and let $V=f^{-1}(U)$. By induction, $M_c(V)$ is in $d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$ and so is $M_c(U)$ as a direct summand of $M_c(V)$ (Lemma \\ref{l1.4}). By induction again, $M_c(X)\\in d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$.\n\\end{proof}\n\n\\begin{rk} By a similar reasoning, one can see that $d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$ is generated by motives of smooth projective varieties of dimension $\\le n$.\n\\end{rk}\n\n\\section{The case of a finite field}\n\n\\subsection{The ubiquity of Frobenius}\\label{s2.A} Let $k=\\F_q$ be a finite field with $q$ elements.\nConsider the category $\\Sch(k)$ of separated $k$-schemes of finite type, viewed as a symmetric monoidal category for the fibre product over $k$. The identity\nfunctor of $\\Sch(k)$ has a canonical $\\otimes$-endomorphism: the Frobenius endomorphism, namely:\n\\begin{itemize}\n\\item Every object $X\\in \\Sch(k)$ has its ``absolute\" Frobenius endomorphism $F_X$: $F_X$ is\nthe identity on the underlying space of $X$ and is given by $f\\mapsto f^q$ on the structural\nsheaf.\n\\item If $f:X\\to Y$ is a morphism in $\\Sch(k)$, the diagram\n\\[\\begin{CD}\nX@>F_X>>X\\\\\n@V{f}VV @V{f}VV\\\\\nY@>F_Y>>Y\n\\end{CD}\\]\ncommutes.\n\\item One has $F_{X\\times_k Y} = F_X\\times_k F_Y$ for any $X,Y\\in \\Sch$.\n\\end{itemize}\n\nStarting from this situation, we can extend it to other categories associated with $k$. For\nexample, if $A$ is a commutative ring then $\\DM_\\gm^\\eff(k,A)$ is obtained from the full subcategory $\\Sm(k)$ of smooth $k$-schemes\nvia the string of functors\n\\begin{multline}\\label{eq2.2}\n\\Sm(k)\\to \\SmCor(k,A)\\to C^b(\\SmCor(k,A))\\\\\n\\to K^b(\\SmCor(k,A))\\to K^b(\\SmCor(k,A))/(MV+HI)\\to DM_\\gm^\\eff(k,A).\n\\end{multline}\n\nHere, $\\SmCor(k,A)$ is the category of finite correspondences on smooth schemes with coefficients in $A$, $MV$ and $HI$\nare the ``Mayer-Vietoris\" and ``homotopy invariance\" relations and the last step is idempotent\ncompletion. At each step, the Frobenius endomorphism extends: for $\\SmCor(k,A)$ because it\ncommutes with finite correspondences; on the third and fourth for formal reasons; on the fifth\nbecause it acts on the $MV$ and $HI$ relations, and on the last once again for formal reasons.\n\nTo pass from $\\DM_\\gm^\\eff(k,A)$ to $\\DM_\\gm(k,A)$ amounts to $\\otimes$-inverting the Tate object\n$\\Z(1)$, and once again Frobenius passes through this formal operation. Similarly, we may take\ncoefficients in any ring $A$ rather than $\\Z$.\n\nWe also get Frobenius endomorphisms on categories obtained from categories of (pre)sheaves via\nthe following simple trick: let $\\sF$ be a contravariant functor from, say, $\\Sm(k)$ to some\ncategory. We define the Frobenius endomorphism $F_\\sF$ of $\\sF$ by the formula\n\\[F_\\sF(X) = \\sF(F_X)=\\sF(X)\\to \\sF(X).\\]\n\nSuppose that $\\sF$ takes values in the category of sets: in the special case where it is\nrepresentable, say $\\sF=y(Y)$, one finds\n\\[F_{y(Y)} = y(F_Y)\\]\nfor tautological reasons, where $y$ is the Yoneda embedding. \n\n\\begin{rk}\nThis is the \\emph{inverse} of the convention of \\cite[XV.2.1]{SGA5}! Cf. \\emph{loc. cit.} bottom p. 453.\n\\end{rk}\n\nIn this way, categories like $\\DM^\\eff(k,A)$ or $\\DM^\\eff_\\et(k,A)$ carry their own Frobenius\nautomorphism, compatible with the one of $\\DM_\\gm^\\eff(k,A)$ via the comparison functors if need\nbe.\n\n\\begin{ex}\\label{ex2.1} We may describe $\\Z(1)$ as $\\G_m[-1]$. The Frobenius of the group scheme $\\G_m$,\nhence of the abelian sheaf $\\G_m$, coincides with multiplication by $q$. Hence $F_{\\Z(1)}$ is also\nmultiplication by $q$.\n\\end{ex}\n\nLet $E/k$ be a finite extension of degree $m$, and let $f:\\Spec E\\to \\Spec k$. We have a pair of adjoint functors\n\\[\\DM_\\gm(k,\\Q)\\begin{smallmatrix} f^*\\\\\\rightleftarrows\\\\f_*\\end{smallmatrix}\\DM_\\gm(E,\\Q).\\]\n\n\n\n\\begin{lemma}\\label{l2.2a} a) For $M\\in \\DM_\\gm(k,\\Q)$, $F_{f^*M}=f^*F_M^m$.\\\\\nb) For $M\\in \\DM_\\gm(E,\\Q)$, $F_{f_*M}^m = f_*F_M$.\n\\end{lemma}\n\n\\begin{proof} a) Construction \\eqref{eq2.2} shows that any object $N$ of $\\DM_\\gm^\\eff(k,\\Q)$ is isomorphic to a direct summand of an object represented by a bounded complex of finite correspondences. In turn, $M$ is represented by an object of the form $(N,i)$ for $N$ as above and $i\\in \\Z$. The functor $f^*$ respects these constructions. Thus the statement reduces to the case $M=M(X)$ where $X$ is a smooth variety, when it is clear. Same reasoning for b).\n\\end{proof}\n\n\n\n\n\\subsection{A theorem of May} Let $\\sT$ be a tensor triangulated category, i.e. a triangulated category provided with a symmetric monoidal structure which verifies axioms (TC1) -- (TC5) of May \\cite[\\S 4]{may}. Let $\\un$ be the unit object of $\\sT$. As in any symmetric monoidal category, an endomorphism $f$ of a strongly dualisable object $M$ has a \\emph{trace} $\\tr(f)\\in \\End_\\sT(\\un)$, defined as the composition\n\\begin{equation}\\label{eq2.3}\n\\un \\by{\\eta} M\\otimes M^*\\by{f\\otimes 1} M\\otimes M^*\\by{\\sigma} M^*\\otimes M\\by{\\epsilon}\\un.\n\\end{equation}\n\nLet $e$ be an endomorphism of the identity functor of $\\sT$. Then\n \n \\begin{thm}\\label{t2.1} Let $M'\\to M\\to M''\\by{+1}$ be an exact triangle in $\\sT$. We have the equality\n \\[\\tr(e_M)=\\tr(e_{M'})+\\tr(e_{M''}).\\]\n \\end{thm}\n\n\\begin{proof} Although this is only stated in \\cite{may} for $e$ the identity, May's proof for this special case directly generalises. More precisely, one can insert $e$'s in the diagram in the middle of \\cite[p. 55]{may}, just below the occurence of the $\\gamma$'s, without affecting its commutativity.\n\\end{proof}\n\n\\subsection{Zeta functions of endomorphisms} Let $\\sA$ be a rigid additive $\\otimes$-category, and let $N$ be an object of $\\sA$. Let $\\un$ be the unit object of $\\sA$ and let $K=\\End(\\un)$, assumed to be a field of characteristic $0$. In \\cite[Def. 3.1]{modq}, we gave the following definition:\n\n\\begin{defn} \\label{d2a} For $f\\in \\End(M)$, its \\emph{Z function} is\n\\[Z(f,t) = \\exp(\\sum_{n\\ge 1}\n\\tr(f^n)\\frac{t^n}{n})\\in K[[t]]\\]\nwhere $\\tr$ is the categorical trace described in the previous subsection.\n\\end{defn}\n\nFollowing \\cite[Def. 5.1]{modq}, we say that $\\sA$ is \\emph{of homological origin} if it is abelian semi-simple and if it is\n$\\otimes$-equivalent to $\\sA'/\\sN$, where $\\sA'$ is a rigid $\\otimes$-category admitting a strong $\\otimes$-functor with values in the category of $\\Z/2$-graded $L$-vector spaces for some extension $L$ of $K$, and $\\sN$ is the ideal of morphisms universally of trace $0$. By \\cite[Th. 3.2, Rem. 3.3 and Th. 5.6]{modq}, we have\n\n\\begin{thm}\\label{t2.3} Assume that $\\sA$ is of homological origin. Then, for any $(N,f)$ as in Definition \\ref{d2a}, $Z(f,t)\\in K(t)$. If $f$ is invertible, we have the functional equation\n\\[Z(f^{-1},t^{-1})= (-t)^{\\chi(M)} \\det(f) Z(f,t)\\]\nwhere $\\chi(M)=\\tr(1_M)$ and $\\det(f)$ is a certain element of $K^*$ which may be obtained by specialisation from the identity\n\\[\\det(1-ft)= Z(f,t)^{-1}.\\]\n\\end{thm}\n\nIn \\cite[Th. A.41]{zetaL}, a slightly simpler proof of Theorem \\ref{t2.3} is given, as well as a more explicit formula for $\\det(f)$: if $Z(f,t)=\\frac{\\prod_{i=1}^m(1-\\alpha_it)}{\\prod_{j=1}^n(1-\\beta_jt)}$ over the algebraic closure of $K$, then\n\\[\\det(f) = \\frac{\\prod_{i=1}^m \\alpha_i}{\\prod_{j=1}^n \\beta_j}.\\]\n\n\n\\subsection{Number of points and zeta functions of pure motives} In \\cite[pp. 80/81]{kleiman}, Kleiman defines the number of points and the Z function of effective homological motives, hence \\emph{a fortiori} of an effective Chow motive $N=(X,p)$ where $X$ is a smooth projective $k$-variety and $p$ is a projector on $X$, by\n\\[N_s(N)=\\langle F_X^s,{}^t p\\rangle,\\quad Z(N,t)=\\exp(\\sum_{s=1}^\\infty N_s(N)\\frac{t^s}{s})\\]\nwhere $\\langle, \\rangle$ is the intersection product. \n\n\nFor $f=F_N$, the Frobenius endomorphism of $N$, this expression coincides with that of Definition \\ref{d2a} in view of the following lemma, which should have been in \\cite{zetaL}.\n\n\\begin{lemma}\\label{l2.4} a) Let $X,Y$ be two smooth projective varieties, and let $f\\in \\Corr(X,Y)$, $g\\in \\Corr(Y,X)$ be two correspondences. Then\n\\[\\tr(g\\circ f) = \\langle {}^tf,g\\rangle.\\]\nb) For $N=(X,p)$, $N_s(N)=\\tr(F_N^s)$ and $Z(F_N,t)=Z(N,t)$.\n\\end{lemma}\n\n\\begin{proof} a) In any rigid $\\otimes$-category $\\sA$, we have the formula\n\\begin{equation}\\label{eq2.4}\n\\tr(g\\circ f) = {}^t(\\iota_{AB}^{-1}f)\\circ \\iota_{BA}^{-1} g\n\\end{equation}\n\\cite[7.3]{akos}, where $f:A\\to B$, $g:B\\to A$ and $\\iota_{AB}$ is the adjunction isomorphism $\\Hom(\\un,A^*\\otimes B)\\iso \\Hom(A,B)$ (\\emph{loc. cit.}, (6.2)) and similarly for $\\iota_{BA}$. Applying this with $A=h(X)$, $B=h(Y)$ in $\\Chow(k,\\Q)$ and using the fact that $h(X)^*\\simeq h(X)\\otimes \\L^{-\\dim X}$, $h(Y)^*\\simeq h(Y)\\otimes \\L^{-\\dim Y}$ with $\\L$ the Lefschetz motive, we get the usual formulas\n\\[\\Hom(h(X),h(Y))\\osi \\Hom(\\L^{\\dim X},h(X\\times Y)) =CH^{\\dim X}(X\\times Y)\\otimes \\Q\\]\nand similarly for $\\Hom(h(Y),h(X))$. Thus, in the right hand side of \\eqref{eq2.4}, $\\iota_{BA}^{-1} g$ is simply $g$ viewed as a cycle class in $CH^{\\dim Y}(Y\\times X)\\otimes \\Q$ and ${}^t(\\iota_{AB}^{-1}f)$ is $f$ viewed as a cycle class in $CH^{\\dim X}(X\\times Y)\\otimes \\Q$. By the formula for the composition of correspondences, this right hand side is $\\langle {}^tf,g\\rangle$.\n\nb) We apply a) with $X=Y$, $f=p$, $g=F_X^s$, noting that $F_N^s=p\\circ F_X^s$. \n\\end{proof}\n\nBy Theorem \\ref{t2.3} and the existence of a Weil cohomology, we get that $Z(N,t)$ is a rational function of $t$ and satisfies a functional equation of the form of this theorem. Adding the main result of \\cite{WeilI}, we get a more precise result:\n\n\\begin{thm}\\label{t2.4} Assume that $N$ is effective and is a direct summand of $h(X)$ where $X$ is a smooth projective variety of dimension $n$. Then the roots of the numerator and denominator of $Z(N,t)$ are effective Weil $q$-numbers of weights $\\le 2n$.\n\\end{thm}\n\nHere, a Weil $q$-number of weight $i$ is an element $\\alpha\\in \\bar \\Q$ such that $|\\sigma(\\alpha)| =q^i$ for any embedding $\\sigma:\\bar \\Q\\inj \\C$; $\\alpha$ is \\emph{effective} if it is an algebraic integer.\n\n\\subsection{Covariance and contravariance}\\label{s2.E} In \\cite[Prop. 2.1.4]{voetri}, Voevodsky defined a functor\n\\[\\Phi^\\eff:\\Chow^\\eff(k,\\Q)\\to \\DM_\\gm^\\eff(k,\\Q)\\]\nwhich induces similar functors $\\Phi:\\Chow(k,\\Q)\\to \\DM_\\gm(k,\\Q)$ and $\\phi^\\o:\\Chow^\\o(k,\\Q)\\to \\DM_\\gm^\\o(k,\\Q)$ where the last ones are the categories of birational motives defined in \\cite{ks} and \\cite{birat-tri}. These functors are covariant if one takes the \\emph{covariant} convention on Chow motives: the ``graph'' functor is covariant. On the other hand, the computation of zeta functions using traces of powers of Frobenius in $\\Chow(k,\\Q)$ is done in the previous section with the \\emph{contravariant} convention, which might conceivably be an issue.\n\nMore generally, if $\\sA$ is a rigid $\\otimes$-category, then the opposite\\footnote{We prefer the term `opposite' to the older term `dual', which may cause confusion in this and other contexts.} category $\\sA^\\op$ (same objects, change the sense of morphisms) is also a rigid $\\otimes$-category, but changing the sense of morphisms does not preserve the shape of \\eqref{eq2.3}, which seems to create an issue. This is not the case:\n\n\\begin{lemma}\\label{l2.3} Let $f:A\\to A$ be an endomorphism of $A\\in \\sA$. Then $\\tr_\\sA(f)=\\tr_{\\sA^\\op}(f)$.\n\\end{lemma}\n\n\\begin{proof} Consider the covariant strong $\\otimes$-functor $\\sA\\to \\sA^\\op$ given by $A\\mapsto A^*$: it sends $f:A\\to B$ to its transpose ${}^t f:B^*\\to A^*$. Thus, for $B=A$,\n\\[\\tr_\\sA(f) = \\tr_{\\sA^\\op}({}^t f).\\]\n\nBut $\\tr_{\\sA^\\op}({}^t f)=\\tr_{\\sA^\\op}(f)$ \\cite[p. 151]{akos}.\n\\end{proof}\n\nThis lemma shows that the categorical trace commutes with strong $\\otimes$-functors, be they covariant or contravariant.\n\n\\subsection{Number of points of geometric motives} As seen in \\S \\ref{s2.A}, the identity functor of $\\DM_\\gm(k,\\Q)$ has a canonical $\\otimes$-endomorphism: the Frobenius endomorphism, that we denote by $F$.\nFor any $M\\in \\DM_\\gm(k,\\Q)$, we write $F_M$ for the corresponding endomorphism of\n$M$.\n\n\n\n\n\\begin{defn}\\label{d1}For $n\\in \\Z$, $\\sharp_n(M)=\\tr(F_M^n)$; for $n=1$ we set $\\sharp_1=\\sharp$.\n\\end{defn}\n\n\n\\begin{thm}\\label{t2.2} a) For each $n\\in \\Z$, $\\sharp_n$ is an Euler-Poincar\\'e characteristic\nand defines a ring homomorphism\n\\[\\sharp_n:K_0(\\DM_\\gm(k,\\Q))\\to\\Q.\\]\nb) $\\sharp_n$ takes values in $\\Z[1/q]$; for $n\\ge 1$ its restriction to $\\DM_\\gm^\\eff(k,\\Q)$ takes values in $\\Z$, and induces a ring homomorphism\n\\[\\overline\\sharp_n:K_0(\\DM_\\gm^\\o(k,\\Q))\\to \\Z/q^n\\]\nwhere $\\DM_\\gm^\\o(k,\\Q)$ is the category of triangulated birational motives \\cite{ks}.\\\\\nc) We have the identities:\n\\[\\sharp_n(M[1]) = -\\sharp_n(M);\\qquad \\sharp_n(M(1)) = q^n\\sharp_n(M).\\]\nd) If $X$ is smooth projective, then $\\sharp_n(M(X))=|X(\\F_{q^n})|$ for all $n\\ge 1$, and $\\sharp_0(M(X))$ is the Euler characteristic of $X$..\n\\end{thm}\n\nHere the groups $K_0$ considered are those of triangulated categories as in \\cite[VIII.2]{SGA5}.\n\n\\begin{proof} a) Multiplicativity is a general fact for rigid tensor\ncategories (use that $F_{M\\otimes N}=F_M\\otimes F_N$). Additivity follows from Theorem \\ref{t2.1}. \n\nb) Bondarko has proven that the maps\n\\[K_0(\\Chow^\\eff(k,\\Q))\\by{K_0(\\Phi^\\eff)} K_0(\\DM_\\gm^\\eff(k,\\Q))\\]\n\\begin{equation}\\label{eq2.1}\nK_0(\\Chow(k,\\Q))\\by{K_0(\\Phi)} K_0(\\DM_\\gm(k,\\Q))\n\\end{equation}\ninduced by the corresponding functors are bijective \\cite[Th. 6.4.2 and Cor. 6.4.3]{bondarko1}. Here, the left groups are $K_0$ of \\emph{additive} categories. The same holds for the homomorphism\n\\[K_0(\\Chow^\\o(k,\\Q))\\by{K_0(\\Phi^\\o)} K_0(\\DM_\\gm^\\o(k,\\Q))\\]\nby \\cite[Prop. 8.1.1]{bondarko}. The result then follows from \\cite[Th. 8.1 and 9.1]{modq}.\n\nc) The first identity is a special case of a) (consider the exact triangle $M\\to 0\\to M[1]$).\nThe second one follows from multiplicativity and the formula $\\sharp_n(\\Z(1))=q^n$ (Example \\ref{ex2.1}).\n\nd) We use the strong $\\otimes$-functor $\\Phi:\\Chow(k,\\Q)\\to \\DM_\\gm(k,\\Q)$ as in b): it sends the Chow motive $h(X)$ to $M(X)$. The conclusion follows by Lemma \\ref{l2.3}. \\end{proof}\n\n\\begin{lemma}\\label{l2.2} Let $E,f,m$ be as before Lemma \\ref{l2.2a}. For $M\\in \\DM_\\gm(E,\\Q)$ and $n> 0$, one has\n\\[\\tr(f_*F_M^n) =\n\\begin{cases}\n0 &\\text{if $m\\nmid n$}\\\\\n\\tr(F_M^{n/m}) &\\text{if $m\\mid n$.}\n\\end{cases}\n\\]\nFor $n=0$, we have $\\chi(f_*M) = [E:k]\\chi(M)$.\n\\end{lemma}\n\n\\begin{proof} Following a hint of J. Ayoub, we do as for induced representations: as $f^*$ is monoidal, we have $\\tr(f_*F_M^n)=\\tr(f^*f_*F_M^n)$. Write $f^*f_*M=\\bigoplus_{r=0}^{m-1} (\\phi_E^r)^*M$, where $\\phi_E$ is the Frobenius generator of $\\Gamma=Gal(E/k)$. Then the matrix of $f^*f_*F_M$ relatively to this decomposition is\n\\[\\begin{pmatrix}\n0 & 0&\\dots &0& (\\phi_E^{m-1})^*\\phi\\\\\n\\phi & 0& \\dots&0& 0\\\\\n0 & \\phi_E^*\\phi &\\dots&0& 0\\\\\n&&\\ddots\\\\\n0&\\dots& &(\\phi_E^{m-2})^*\\phi&0\n\\end{pmatrix}\\]\nwhere $\\phi:M\\iso \\phi_E^*M$ is a relative Frobenius morphism; moreover we have the relation\n\\[(\\phi_E^{m-1})^*\\phi\\circ\\dots\\circ \\phi_E^*\\phi\\circ \\phi=F_M.\\]\n\nThis can be checked on $M=M(Y)$, $Y$ a smooth $E$-scheme. The conclusion follows. \n\\end{proof}\n\n\n\\subsection{Two twisting lemmas}\\label{s2.F} In the next subsection, we shall need the following generalisation of a well-known lemma (cf. A.\nPacheco \\cite[p. 283]{pacheco}: the idea seems to go very far back). We recall the notion of twisting by a $1$-cocycle from \\cite[I.5.3 and III.1.3]{cg}.\n\nLet $U$ be a quasi-projective $k$-variety, and let $G$ be a finite group of order $m$ acting on $U$ on the right. Then the geometric quotient $V=U/G$ exists. For each $\\sigma\\in G$, we define a $k$-variety $U^{(\\sigma)}$ mapping to $V$ as follows:\n\nLet $E/k$ be ``the'' extension of $k$ of order $m$, and $\\Gamma=Gal(E/k)$. Then $\\Gamma\\times G$ acts on $U_E$. Let $\\phi$ be the arithmetic Frobenius of $k$, $\\phi_E$ its image in $\\Gamma$ and $H_\\sigma = \\langle (\\phi_E^{-1},\\sigma)\\rangle$; we set $U^{(\\sigma)}=U_E/H_\\sigma$. The projection $U_E\\to U\\to V$ is $H_\\sigma$-invariant, which defines $f^{(\\sigma)}:U^{(\\sigma)}\\to V$. Write also $\\pi_\\sigma$ for the projection $U_E\\to U^{(\\sigma)}$. \n\\begin{lemma}\\label{l2.5} We have $U^{(1)}=U$, we write $f:=f^{(1)}$. For a general $\\sigma$, the square\n\\[\\begin{CD}\nU_E@>\\pi_\\sigma>> U^{(\\sigma)}\\\\\n@VVV @VVV\\\\\n\\Spec E@>>> \\Spec k\n\\end{CD}\\]\nis Cartesian; in particular, $\\pi_\\sigma$ is a Galois (étale) covering of group $\\Gamma$.\n\\end{lemma}\n\n\\begin{proof} By faithfully flat descent \\cite[Cor. VIII.5.4]{SGA1}, it suffices to see this after base-changing to $\\Spec E$. Then the left column becomes $\\Gamma\\times U_E \\to \\Gamma\\times \\Spec E$. But, by construction, there exists a $k$-isomorphism $(U^{(\\sigma)})_E\\simeq U_E$ which converts $(\\pi_\\sigma)_E$ into the projection $\\Gamma\\times U_E\\to U_E$ given by the canonical map $\\Gamma\\to *$. Hence the conclusion.\n\\end{proof}\n\n\\begin{lemma}\\label{ltwist} Suppose that $f$ is a $G$-torsor (i.e. an étale Galois covering). Then we have\n\\[\\frac{1}{m}\\sum_{\\sigma\\in G}|U^{(\\sigma)}(k)| = |V(k)|. \\]\n\\end{lemma}\n\n\\begin{proof} \n\n\n\n\nWe have\n\\[V(k)=V(\\bar k)^\\phi.\\]\n\nThe map $f:U(\\bar\nk)\\to V(\\bar k)$ is surjective and $G$ acts simply transitively on its fibres. \n\nLet $x\\in V(k)$. Pick $y\\in U(\\bar k)$ mapping to $x$. Then $\\phi^{-1}y$ is in the fibre of\n$x$, which implies that $y=\\phi y\\sigma $ for a unique $\\sigma\\in G$. Thus $y$\ndefines a $k$-rational point of $U^{(\\sigma)}$. If $y'\\in f^{-1}(x)$ is another point, then\n$y'=y\\tau $ for a unique $\\tau\\in G$, and \n\\[y'=\\phi y\\sigma\\tau =\\phi y'\\tau^{-1}\\sigma\\tau \\]\nso $y'\\in U^{(\\tau^{-1}\\sigma\\tau)}(k)$, and $y'\\in U^{(\\sigma)}(k)$ if and only if $\\tau\\in\nZ_{G}(\\sigma)$.\n\nSummarising: there exists a well-defined conjugacy class $\\bar \\sigma(x)\\subset G$ such that\n$f^{-1}(x)$ is a disjoint union of subsets $f^{-1}(x)_\\sigma$ consisting of $r$ elements of\n$U^{(\\sigma)}(k)$ for $\\sigma$ running through $\\bar \\sigma(x)$, where $r=|Z_{G}(\\sigma)|=|G|/|\\bar \\sigma(x)|$. If $\\bar \\sigma$ is now a given conjugacy class, let \n\\[V(k)_{\\bar\\sigma}=\\{x\\in V(k)\\mid \\bar \\sigma(x)=\\bar \\sigma\\} \\; \\text{(possibly empty!)}.\\] \n\nThen, for every $\\sigma\\in \\bar \\sigma$, the map $U^{(\\sigma)}(k)\\to V(k)_{\\bar \\sigma}$ is a torsor under\n$Z_{G}(\\sigma)$. In particular, $|U^{(\\sigma)}(k)| = r|V(k)_{\\bar \\sigma}|$ and \n\\[\\sum_{\\sigma\\in \\bar\\sigma} |U^{(\\sigma)}(k)| = |\\bar \\sigma|r |V(k)_{\\bar \\sigma}| =\nm|V(k)_{\\bar \\sigma}|. \\] \n\nCollecting over the conjugacy classes of $G$, we get (iii).\n\\end{proof}\n\n\nWe now need a pendant of Lemma \\ref{ltwist} for traces of Frobenius. \n\\begin{lemma}\\label{ltwisttr} With notation and hypotheses as in Lemma \\ref{ltwist}, we have\n\\[\\frac{1}{m}\\sum_{\\sigma\\in G}\\sharp(M_c(U^{(\\sigma)})) = \\sharp(M_c(V)). \\]\n\\end{lemma}\n\n\\begin{proof} Keep the notation in the proof of Lemma \\ref{ltwist}. For $\\sigma\\in G$, let $\\epsilon_\\sigma= \\frac{1}{m}\\sum_{h\\in H_\\sigma} h\\in \\Q[\\Gamma\\times G]$: then, by Lemma \\ref{l1.4},\n\\[M_c(U^{(\\sigma)}) \\simeq \\epsilon_\\sigma M_c(U_E).\\]\n\nWe have $M_c(U_E)= M_c(\\Spec E) \\otimes M_c(U)=M(\\Spec E) \\otimes M_c(U)$, hence $F_{M_c(U_E)}=F_{M(\\Spec E)}\\otimes F_{M_c(U)}$. The endomorphism $F_{M(\\Spec E)}$ coincides with the Frobenius automorphism $\\phi_E\\in Gal(E/k)$ acting on $M(\\Spec E)$. Hence\n\\begin{multline*}\n\\sum_{\\sigma\\in G}\\tr(F_{M_c(U^{(\\sigma)})}) =\\tr\\left(\\sum_{\\sigma\\in G}\\phi_E\\otimes \\epsilon_\\sigma F_{M_c(U)}\\right)\\\\= \\tr\\left(\\sum_{\\sigma\\in G} \\sum_{r=0}^{m-1} \\frac{1}{m}\\phi_E\\otimes (\\phi_E^{-r},\\sigma^r) F_{M_c(U)}\\right)\\\\ = \\frac{1}{m}\\tr\\left(\\sum_{\\sigma\\in G} \\sum_{r=0}^{m-1} \\phi_E^{1-r}\\otimes\\sigma^r F_{M_c(U)}\\right)= \\frac{1}{m}\\left(\\sum_{\\sigma\\in G} \\sum_{r=0}^{m-1})\\tr(\\phi_E^{1-r})\\tr(\\sigma^r F_{M_c(U)})\\right).\n\\end{multline*}\n\nBut $\\tr(\\phi_E^{1-r})=0$ for $1-r\\ne m$ and $\\tr(\\phi_E^{m})=m$ (see Lemma \\ref{l2.2}). Hence the last sum collapses to\n\\[\\frac{1}{m}\\left(\\sum_{\\sigma\\in G} m\\tr(\\sigma^{1-m} F_{M_c(U)})\\right)=m\\tr\\left(\\frac{1}{m}\\sum_{\\sigma\\in G} \\sigma F_{M_c(U)}\\right)=m\\tr(F_{M_c(V)})\\]\nas requested.\n\\end{proof}\n\n\\begin{qn} Does the equality $\\sum_{\\sigma\\in G}[U^{(\\sigma)}] = m[V]$ hold in the Grothendieck group of varieties? It would yield lemmas \\ref{ltwist} and \\ref{ltwisttr} in one gulp.\n\\end{qn}\n\n\n\n\n\n \n\n\n\n\\subsection{More general schemes} Theorem \\ref{t2.2} d) extends to\n\n\\begin{thm}\\label{p1}\nIf $X$ is a separated $k$-scheme of finite type, then\n\\[\\sharp_n(M_c(X))=|X(\\F_{q^n})| \\text{ for all } n\\ge 1.\\]\n\\end{thm}\n\nWe first give a lemma:\n\n\\begin{lemma} \\label{l1.2} Consider an open-closed situation\n\\begin{equation}\nZ\\by{i} X\\yb{j} U\n\\end{equation}\nwhere $Z$ is a closed subset of $X$ with open complement $U$. Then, if Theorem \\ref{p1} is true for two among $X,Z,U$, it is true for the third.\n\\end{lemma} \n\n\\begin{proof} This follows from Theorem \\ref{p1.1} and Theorem \\ref{t2.2} a).\n\\end{proof}\n\nBefore starting the proof, let us explain why it is going to be complicated. Suppose that we know resolution of singularities over $k$. Then we can easily use Lemma \\ref{l1.2} to reduce to $X$ smooth projective. In its absence, we want of course to use de Jong's theorem on alterations. But if we try to do a proof as for Proposition \\ref{p1.2}, we run into the following problem: if $U\\to V$ is an étale covering of smooth varieties and Theorem \\ref{p1} is true for $U$, why should it be true for $V$? The fact that $M_c(V)$ is isomorphic to a direct summand of $M_c(U)$ does not help here because, unlike abelian groups, rational integers do not have direct summands\\dots\\ (In summary: for numbers, the devil is in the idempotents.) Fortunately, the twisting lemmas proven in the previous subsection will help us get around this issue.\n\n\\begin{proof}[Proof of Theorem \\ref{p1}]\n\nWe argue by induction on $\\dim X$. The $0$-dim\\-en\\-sio\\-nal case follows from Theorem \\ref{t2.2} d) (or is trivial). Suppose $\\dim X>0$. By using Lemma \\ref{l1.2}, we first reduce to $X$ a variety and then (by Nagata's theorem) to $X$ proper. By de Jong's equivariant alteration theorem \\cite[Th. 7.3]{dJ}, there exists then a quasi-Galois alteration $f:\\tilde X \\to X$, with $\\tilde X $ smooth projective. Let $V\\subseteq X$ be a smooth open subset over which $f$ is finite, and let $U=f^{-1}(V_0)$. \n\nFor simplicity, write $k_n$ for $\\F_{q^n}$. We have $\\sharp_n(M_c(U))=|U(k_n)|$ by Theorem \\ref{t2.2} d), Lemma \\ref{l1.2} and induction. \n\n Let $G$ be the Galois group of $f$ and $X_G$ be the geometric quotient of $\\tilde X$ by $G$. Write $f':\\tilde X\\to X_G$ for the corresponding factorisation of $f$. If we twist with respect to $f'$ in the style of \\S \\ref{s2.F}, then $\\tilde X^{(\\sigma)}$ is smooth projective for all $\\sigma\\in G$ as a consequence of Lemma \\ref{l2.5}, and we also have $\\sharp_n(M_c(U^{(\\sigma)}))=|U^{(\\sigma)}(k_n)|$. \n\nPutting Lemmas \\ref{ltwist} and \\ref{ltwisttr} together then yields $\\sharp(M_c(U_G))\\allowbreak=|U_G(k)|$. Since $U_G\\to V$ is finite and radicial, Lemma \\ref{l1.4} shows that the same holds for $V$. Finally we get $\\sharp(M_c(X))=|X(k)|$ by induction and Lemma \\ref{l1.2} again.\n\\end{proof}\n\n\n\\begin{rk} It would be more reasonable to give a direct proof of Theorem \\ref{p1}, in the style of Lemma \\ref{l2.4}. Unfortunately I haven't been able to find such a proof.\n\\end{rk}\n\n\n\n\\subsection{The zeta function} Recall the important formula \\cite[I, (3.2.3.6)]{saa}\n\\[F_{M^*}={}^t F_M^{-1}.\\] \n\n\n\nIn view of the above computations, it would be most natural to define the zeta function of $M$ as $Z(F_M,t)$, so that $Z(X,t)=Z(M_c(X),t)$ by Proposition \\ref{p1} b). However this would create awkward functoriality problems in the next section, when we deal with motives over a base: see Theorem \\ref{t3.2} below. One may think of these problems as arising because $S\\mapsto \\DM_\\gm(S,\\Q)$ is a \\emph{homology} theory, whereas the functorial behaviour of zeta and $L$-functions expresses itself naturally in \\emph{cohomological} terms. Our solution to this issue is to give a slightly artificial definition of the zeta function:\n\n\\begin{defn}\\label{d2} $Z(M,t)=Z(F_M^{-1},t)=Z(F_{M^*},t)$.\n\\end{defn}\n\n\n\\begin{thm}\\label{t2} $M\\mapsto Z(M,t)$ is multiplicative on exact triangles, hence defines a homomorphism $K_0(\\DM_\\gm(k,\\Q))\\to 1+t\\Q[[t]]$. For any $M\\in \\DM_\\gm(k,\\Q)$,\\\\\na) $Z(M,t)\\in \\Q(t)$. The degree of this rational function is $-\\chi(M)$, where $\\chi(M)=\\tr(1_M)$.\\\\\nb) We have the functional equation\n\\[Z(M^*,t^{-1}) = (-t)^{\\chi(M)}\\det(F_M)^{-1}Z(M,t)\\] \n where $\\det(F_M)$ is the value at $t=\\infty$ of $(-t)^{\\chi(M)} Z(M,t)^{-1}$. \\\\\n c) We have the identities\n \\[Z(M[1],t) = Z(M,t)^{-1},\\qquad Z(M(1),t) = Z(M,q^{-1}t).\\]\n d) For any $f:X\\to \\Spec k$ of finite type, we have\n \\[Z(M^{BM}(X),t) = Z(X,t).\\]\n\\end{thm}\n\n\\begin{proof} The first fact follows from Theorem \\ref{t2.2}. Using now the surjectivity of \\eqref{eq2.1}, we reduce to the case where $M\\in\n\\Chow(k,\\Q)$. Then we can compute in $\\sM_\\num(k,\\Q)$ and everything follows from Theorem \\ref{t2.3} (see remarks before Theorem \\ref{t2.4}). d) follows from Theorem \\ref{p1}.\n\\end{proof}\n\n\\begin{rk} It is likely that Theorem \\ref{t2} extends to all zeta functions of endomorphisms as\nin Definition \\ref{d2} a), but this seems to require extending Bondarko's theorem to $K_0$ of\nthe categories of endomorphisms as in \\cite[\\S 15]{modq}. (He has done that!)\n\\end{rk}\n\n\\begin{prop} Let $f:\\Spec E\\to \\Spec k$ be a finite extension, and let\n$f_*:\\DM_\\gm(E)\\to \\DM_\\gm(k)$ be the corresponding direct image functor. Then, for\nany $M\\in \\DM_\\gm(E)$, we have\n\\[Z(f_* M,t) = Z(M,t^m)\\]\nwhere $m=[E:k]$.\n\\end{prop}\n\n\\begin{proof} This is clear from Lemma \\ref{l2.2a} b) and the definition of $Z$.\n\\end{proof}\n\n\\begin{defn}\\label{d2.1} We denote by $\\Dir$ the group of formal Dirichlet series beginning with $1$.\n\\end{defn}\n\n\\begin{cor}\\label{c1} For $M\\in \\DM_\\gm(k,\\Q)$, let\n\\[\\zeta(M,s) = Z(M, q^{-s}).\\]\nThen $M\\mapsto \\zeta(M,s)$ is multiplicative on exact triangles, hence defines a homomorphism $K_0(\\DM_\\gm(k,\\Q))\\to \\Dir$. Moreover,\\\\\na) $\\zeta(M,s)$ is a rational function in $q^{-s}$, of degree $-\\chi(M)$.\\\\\nb) We have the functional equation\n\\[\\zeta(M^*,-s) = (-q^{-s})^{\\chi(M)}\\det(F_M)\\zeta(M,s).\\] \nc) We have the identities\n \\[\\zeta(M[1],s) = \\zeta(M,s)^{-1},\\qquad \\zeta(M(1),s) = \\zeta(M,s+1).\\]\n d) For any $f:X\\to \\Spec k$ of finite type, we have\n \\[\\zeta(M^{BM}(X),s) = \\zeta(X,s).\\]\ne) For a finite extension $f:\\Spec E\\to \\Spec k$, we have $\\zeta(M,s)=\\zeta(f_* M,s)$.\\qed\n\\end{cor}\n\nWe shall need the following\n\n\\begin{prop}\\label{p3} Let $M^*\\in d_{\\le n}\\DM_\\gm^\\eff(k,\\Q)$ (see Definition \\ref{d1.1}). Then the zeroes and poles of\n$Z(M,t)$ are effective Weil $q$-numbers of weights $\\in [0,2n]$. In particular, the Dirichlet series $\\zeta(M,s)$ converges absolutely for $\\Re(s)>n$.\n\\end{prop}\n\n\\begin{proof} Consider the diagram\n\\[\\begin{CD}\nK_0(\\Chow^\\eff(k,\\Q))@>\\phi>\\sim> K_0(\\DM_\\gm^\\eff(k,\\Q))\\\\\n@V{\\psi}VV\\\\\nK_0(\\sM_\\num^\\eff(k,\\Q)).\n\\end{CD}\\]\n\nThe dimension filtration induces a filtration on the three $K_0$'s, and $\\phi$ and $\\psi$\nrespect these filtrations. So does the inverse of $\\phi$, because it is induced by Bondarko's\nfunctor\n\\[\\DM_\\gm^\\eff(k,\\Q)\\to K^b(\\Chow^\\eff(k,\\Q))\\]\nwhich obviously respects the dimension filtrations. \n\nSince $M\\mapsto Z(M,t)$ factors through $K_0(\\sM_\\num^\\eff(k,\\Q))$, we are therefore reduced to Theorem \\ref{t2.4}.\n\\end{proof}\n\n\n\\begin{cor} \\label{c2.1} if $\\dim X\\le n$, the zeroes and poles of $Z(X,t)$ are effective Weil $q$-numbers of weights $\\in [0,2n]$.\\qed\n\\end{cor}\n\nOf course, this corollary may also be deduced from \\cite{WeilII}.\n\n\\section{Motives over a scheme of finite type} \n\n\\subsection{Motives over a base} In the sequel, we shall need a theory of triangulated motives over general base schemes, with a formalism of six (even seven) operations as in \\cite{ayoub}. Unfortunately, it is unknown whether the natural generalisation of Voevodsky's construction enjoys such a formalism \\cite[6.2]{ffihes}. This can be corrected by using the subcategory of constructible objects in Ayoub's étale motives without transfers $\\DA^\\et(-,\\Q)$ \\cite{ayoubetale} or Cisinski-Déglise Beilinson motives $\\DM_\\Be(-)$ \\cite{cis-deg}, which coincide anyway \\cite[6.3 and 6.4]{ffihes}: the important point here is that these subcategories are preserved under the six operations (\\cite[Scholie 2.2.34 and Th. 2.2.37]{ayoub} for $f^*,f_*,f_!,f^!$, \\cite[Prop. 2.3.62]{ayoub} for $\\uHom$). We adopt here the viewpoint of \\emph{loc. cit.}, 6.5, using only the \\emph{existence} of a formalism of six operations for some categories $\\D(S)$ which agree with $\\DM_\\gm(k,\\Q)$ when $S=\\Spec k$, and are provided with an $l$-adic realisation functor which commutes with the six operations. Very occasionally, we shall use non-formal properties of such a theory.\n\nWe write $\\Z_S=\\Z$ for the unit object of $\\D(S)$. If $f:X\\to S$ is a smooth $S$-scheme (separated and of finite type), we write $M_S(X)=f_\\# \\Z_X\\in\\D(S)$ for its \\emph{motive}, where $f_\\#$ is the left adjoint of $f^*:\\D(S)\\to \\D(X)$ \\cite[1.4.1, axiom 3]{ayoub}. \n\\subsection{The Borel-Moore motive revisited} This subsection partly answers \\cite[Rem. 6.7.3 3)]{ffihes}.\n\n\\begin{thm}\\label{t3.2} Let $k=\\F_q$ and let $f:X\\to \\Spec k$ be a separated $k$-scheme of finite type. Then we have an isomorphism\n\\[M^{BM}(X)\\simeq f_!\\Z_X\\]\nat least if $X$ is embeddable in a smooth scheme (e.g. if $X$ is quasi-projective). If $f$ is smooth, this isomorphism is canonical and natural for open immersions.\n\\end{thm}\n\n\\begin{proof} Suppose first that $f$ is smooth of (pure) dimension $d$. Applying \\cite[Vol. I, Scholie 1.4.2 3]{ayoub}, we find\n\\begin{multline}\\label{eq3.4}\nf_!\\Z_X\\simeq f_\\# Th^{-1}(\\Omega_f) \\Z_X\\\\\n=f_\\# \\Z_X(-d)[-2d]=M(X)(-d)[-2d]\\simeq M^{BM}(X).\n\\end{multline}\n\nHere $Th(\\Omega_f)$ is the Thom equivalence associated to the module of differentials $\\Omega_f$, which is computed to be the said Tate twist in \\cite[2.4.38 and 2.4.40]{cis-deg}, plus the description of $M_c(X)$ for $X$ smooth given at the beginning of Section \\ref{s1}. This isomorphism clearly commutes with open immersions. \n\nSuppose now that $X$ can be embedded as a closed subscheme of a smooth scheme $Y$. For simplicity, write $\\tilde M^{BM}(X)=f_!\\Z_X$. We then get an isomorphism $\\tilde M^{BM}(X)\\iso M^{BM}(X)$ by completing the isomorphisms in the diagram of exact triangles\n\\[\\begin{CD}\n\\tilde M^{BM}(U)@>>> \\tilde M^{BM}(Y)@>>> \\tilde M^{BM}(X)@>+1>>\\\\\n@V\\wr VV @V\\wr VV\\\\\nM^{BM}(U)@>>> M^{BM}(Y)@>>> M^{BM}(X)@>+1>>\n\\end{CD}\\]\nwhere $U=Y-X$, the top row is the dual of \\eqref{eq1.1} and the bottom row is obtained by applying $g_!$ to the localisation exact triangle \\cite[Vol. 1, p. 77]{ayoub}\n\\begin{equation}\\label{eq3.1}\nj_!j^!\\Z_Y\\to \\Z_Y\\to i_*i^*\\Z_Y\\by{+1}\n\\end{equation}\nwhere $g:Y\\to \\Spec k$ is the structural morphism and $j:U\\to Y$, $i:X\\to Y$ are the open and closed immersion. (Note that $j^!=j^*$ and $i_*=i_!$.)\n\\end{proof}\n\n\\begin{rk}\\label{r1.1} It would be more reasonable and more efficient to define \\emph{a priori} a natural morphism\n\\[M_c(X)= C_*(\\Z_\\tr^c(X))\\to (f_!\\Z_X)^*\\]\nin $\\DM_\\gm(k)$, where $\\Z_\\tr^c(X)(U)=z(U,X)$ is the group of quasi-finite correspondences, and to show that it is an isomorphism by reduction to the smooth (or smooth projective) case. By duality \\cite[Th. 2.3.75]{ayoub}, the right hand side can be written \n\\[(f_!\\Z_X)^*=D_k(f_!\\Z_X)\\simeq f_*D_X(\\Z_X)=f_*f^!\\Z.\\]\nThis amounts to defining a map\n\\[f^*C_*(\\Z_\\tr^c(X))\\to f^!\\Z\\]\nand I don't know how to construct it\\dots\n\\end{rk}\n\n\n\n\n\n\\subsection{Zeta functions}\n\nLet $f:S\\to \\Spec \\Z$ be a scheme of finite type over $\\Z$. For each $x\\in S_{(0)}$, we have a pull-back functor\n\\[i_x^*:\\D(S)\\to \\D(\\kappa(x))\\simeq \\DM_\\gm(\\kappa(x),\\Q).\\]\n\n\\begin{prop}\\label{p3.1} For $M\\in \\D(S)$, the product\n\\[\\zeta(M,s) = \\prod_{x\\in S_{(0)}} \\zeta(i_x^* M,s)\\] \nis convergent in the group $\\Dir$ (Definition \\ref{d2.1}) for the topology given by the order of the first nonzero term. It is multiplicative on exact triangles and we have the identities\n \\[\\zeta(M[1],s) = \\zeta(M,s)^{-1},\\qquad \\zeta(M(1),s) = \\zeta(M,s+1).\\]\n\\end{prop}\n\n\\begin{proof} Since the exponential is continuous on $\\Dir$ for the said topology, it suffices to show that the sum\n\\[\\sum_{x\\in S_{(0)}} \\sum_{n\\ge 1} \\sharp_n(i_x^*M) \\frac{N(x)^{-ns}}{n}\\]\nis convergent. Here, $N(x)$ is the cardinal of the residue field $\\kappa(x)$. Rearranging, we must show that, for any prime $p$ and any $r\\ge 1$, the set\n\\[\\{x\\in S_{(0)}\\mid n[\\kappa(x):\\F_p]=r\\}\\]\nis finite. This is clear, since $S$ has only a finite number of closed points of a given degree over $\\F_p$. The identities follow from the case of finite fields (Corollary \\ref{c1}).\n\\end{proof}\n\n\\subsection{The case of characteristic $p$} In this subsection, we assume that $S$ is an $\\F_q$-scheme. For simplicity we still write $f$ for the structural morphism $X\\to \\Spec \\F_q$.\n\n\\begin{prop}\\label{p3.3} For $n\\ge 0$, let $d^!_{\\le n}\\D(S)$ be the thick triangulated subcategory of $\\D(S)$ generated by the $g_!\\Z_X$, where $g:X\\to S$ runs through the morphisms of relative dimension $\\le n$. Then $(f_!M)^*\\in d_{\\le n+d}\\DM^\\eff(S,\\Q)$ for any $M\\in d^!_{\\le n}\\D(S)$, where $d=\\dim S$. Moreover, for any $M\\in \\D(S)$ there exists $n\\ge 0$ and $r\\in \\Z$ such that $M(r)\\in d^!_{\\le n}\\D(S)$.\n\\end{prop}\n\n\\begin{proof} It suffices to show that $(f_!g_!\\Z_X)^*\\in d_{\\le n+d}\\DM^\\eff(S,\\Q)$ when $\\dim g\\le n$. By Theorem \\ref{t3.2}, this is $M_c(X)$; the claim then follows from Proposition \\ref{p1.2}. It suffices to prove the last statement for generators $M_S(X)(s)$, with $g:X\\to S$ smooth. But $g_!\\Z_X\\simeq M_S(X)(-n)[-2n]$ if $n=\\dim g$ as in \\eqref{eq3.4}, so $M_S(X)(s)(-n-s)\\in d^!_{\\le n}\\D(S)$.\n\\end{proof}\n\n\n\n\n\n\nFor any $x\\in S_{(0)}$, we have a specialisation functor\n\\[\\Sp_x=(f_x)_*i_x^*:\\D(S)\\to \\DM_\\gm(\\F_q,\\Q)\\]\nwhere $f_x:\\Spec \\kappa(x)\\to \\Spec \\F_q$ is the structural morphism. We have\n\\begin{equation}\\label{eq3.3}\n\\zeta(i_x^*M,s)= \\zeta(\\Sp_x M,s)\n\\end{equation}\nby Corollary \\ref{c1}.\n\n\n\n\nFrom Lemma \\ref{l2.2}, we get:\n\n\\begin{lemma}\\label{l3.1} For any $M\\in \\D(S)$ and any $n\\ge 1$, the function\n\\[S_{(0)}\\ni x\\mapsto \\sharp_n(\\Sp_x M) \\]\nis $0$ outside the finite set $\\bigcup_{m\\mid n} S(\\F_{q^m})$.\n\\end{lemma}\n\nThis gives a meaning to the following proposition. For simplicity, we write\n\\begin{equation}\\label{eq3.2}\n\\sharp_n^*(M)=\\sharp_n(M^*) = \\sharp_{-n}(M)\n\\end{equation}\nfor $M\\in \\DM_\\gm(\\F_q)$.\n\n\\begin{prop}[trace formula]\\label{p3.2} We have\n\\[\\sharp^*_n(f_!M) =\\sum_{x\\in S_{(0)}} \\sharp^*_n(\\Sp_x M)\\] \nfor any $M\\in \\D(S)$ and any $n\\ge 1$.\n\\end{prop}\n\n\\begin{proof}We give two proofs, one in a special case and one in general: \n\n1) The case where $M=g_!\\Z_X$ for $g:X\\to S$ a morphism of finite type, where $X$ is embeddable in a smooth $k$-scheme. Then\n\\[\\sharp_n^*(f_!M) = \\sharp_n^*((fg)_!\\Z_X) = \\sharp_n^*(M^{BM}(X)) =\\sharp_n(M_c(X)) = |X(\\F_{p^n})|\\]\nby Theorems \\ref{t3.2} and \\ref{p1}. On the other hand, let $X_x=g^{-1}(x)$. The base change theorem \\cite[scholie 1.4.2 1)]{ayoub} applied to the Cartesian square\n\\begin{equation}\\label{eq3.5}\n\\begin{CD}\nX_x@>I_x>> X\\\\\n@Vg_x VV @Vg VV\\\\\nx@>i_x>> S\n\\end{CD}\n\\end{equation}\nyields\\footnote{This is part of the definition of a crossed functor, for which the reader should consult \\cite[Déf. 1.21.12]{ayoub}.}\n\\[i_x^*g_!\\Z_X\\simeq (g_x)_! I_x^*\\Z_X =(g_x)_! \\Z_{X_x}\\]\nand we get\n\\begin{multline*}\n\\sum_{x\\in S_{(0)}} \\sharp_n^*(\\Sp_x M)=\\sum_{x\\in S_{(0)}} \\sharp_n^*((f_x)_*i_x^* g_! \\Z_X)\\\\\n=\\sum_{x\\in S_{(0)}} \\sharp_n^*((f_x)_!i_x^* g_! \\Z_X)\n=\\sum_{x\\in S_{(0)}} \\sharp_n^*((f_xg_x)_! \\Z_{X_x})\n\\end{multline*}\nwhere, as above\n\\[\\sharp_n^*((f_xg_x)_! \\Z_{X_x}) = |X_x(\\F_{p^n})|.\\]\n\nNow it is clear that $|X(\\F_{p^n})|=\\sum_{x\\in S_{(0)}} |X_x(\\F_{p^n})|$.\n\nFrom 1), one can derive Proposition \\ref{p3.2} for those $M$'s which belong to the smallest triangulated subcategory containing the said $g_!\\Z_X$ and stable under Tate twists. By Proposition \\ref{p3.3}, its pseudo-abelian envelope equals $\\D(S)$; but then we run into the problem of idempotents mentioned before the proof of Theorem \\ref{p1}. This leads us to the second proof:\n\n2) Apply the $l$-adic realisation functor $R_l$.\n\\begin{multline*}\n\\sum_{x\\in S_{(0)}} \\sharp_n^*(\\Sp_x M) = \\sum_{x\\in S_{(0)}} \\sharp_n^*(R_l(\\Sp_x M)) = \\sum_{x\\in S_{(0)}} \\sharp_n^*(\\Sp_x R_l(M)) \\\\\n= \\sharp_n^*(Rf_!R_l(M)) =\\sharp_n^*(R_l(f_!M)) =\\sharp_n^*(f_!M)\n\\end{multline*} \nwhere we used the commutation of $R_l$ with $f_!$, $i_x^*$ and duality, plus \\cite[equation (2) p. 470]{SGA5}.\n\\end{proof}\n\n\\begin{cor}\\label{c3.1} If $S(\\F_{q^n})=\\emptyset$, then $\\sharp_n^*(f_!M)=0$ for any $M\\in \\D(S)$.\n\\end{cor}\n\n\\begin{proof} This follows from Lemma \\ref{l3.1} and Proposition \\ref{p3.2}.\n\\end{proof}\n\n\\begin{qns} 1) Conversely, Corollary \\ref{c3.1} implies Proposition \\ref{p3.2} by the localisation exact triangle \\eqref{eq3.1}, even for $S=\\P^1-\\P^1(\\F_{q^n})$ by the dévissages of \\cite[XV.3.2]{SGA5}. Can one find a proof which avoids the $l$-adic realisation and the trace formula of \\cite{SGA5}?\\\\\n2) The statement breaks down for $n=0$, because the right hand side is generally infinite (\\emph{e.g.} for $M=\\Z_S$). I don't know a formula for $\\chi(f_! M)$, which appears in the functional equation just below. Can one use some renormalisation trick?\n\\end{qns}\n\n\n\\begin{thm} \\label{t3.1} a) One has $\\zeta(M,s)=\\zeta(f_!M,s)$. In particular, there exists $Z(M,t)\\in \\Q(t)$ such that $\\zeta(M,s)=Z(M,q^{-s})$; the zeroes and poles of $Z(M,T)$ are Weil $q$-numbers.\\\\\nb) If $M\\in d^!_{\\le n} \\D(S)$ (see Proposition \\ref{p3.3}), these Weil $q$-numbers are effective of weights $\\le 2(n+d)$, where $d=\\dim S$.\\\\ \nc) If $S$ is projective, one has the functional equation\n\\[\\zeta(D_S(M),-s)=(-q^{-s})^{\\chi(f_!M)}\\det(F_{f_!M})^{-1}\\zeta(M,s)\\]\nwhere $D_S(M)=\\uHom(M,f^!\\Z)$. If $S$ is moreover smooth of dimension $d$, one has the functional equation\n\\[\\zeta(M^*,d-s)=(-q^{-s})^{\\chi(f_!M)}\\det(F_{f_!M})^{-1}\\zeta(M,s)\\]\nwith $M^*:=\\uHom(M,\\Z)$.\n\\end{thm}\n\n\\begin{proof} a) follows from Proposition \\ref{p3.2} and \\eqref{eq3.3}. b) follows from Propositions \\ref{p3.3} and \\ref{p3}. For c), we have the isomorphism $f_!D_S(M)\\simeq (f_*M)^*\\simeq (f_!M)^*$ by \\cite[Th. 2.3.75 and Scholie 1.4.2 4]{ayoub}, hence\n\\begin{multline*}\n\\zeta(D_S(M),-s)= \\zeta((f_!M)^*,-s) = (-q^{-s})^{\\chi(f_!M)}\\det(F_{f_!M})\\zeta(f_!M,s)\\\\\n= (-q^{-s})^{\\chi(f_!M)}\\det(F_{f_!M})\\zeta(M,s)\n\\end{multline*}\n\nThe smooth case follows, since $f^!\\simeq f^*(d)[2d]$ then (the adjoint identity to the one of \\eqref{eq3.4}).\n\\end{proof}\n\n\\begin{rks} 1) Let $\\D^\\eff(S)$ be as in the proof of Proposition \\ref{p3.1}. If $M\\in d_{\\le n} \\D^\\eff(S)$, Theorem \\ref{t3.1} a) may be refined: the zeroes and poles of $\\zeta(M,s)$ are effective Weil numbers of weights $\\le 2n$.\\\\\n2) Theorem \\ref{t3.1} b) extends to $S$ proper by \\cite{cis-deg}.\\\\\n3) I don't know a formula for $\\zeta(D_S(M),-s)$ when $S$ is not proper.\n\\end{rks}\n\n\n\n\\subsection{The general case} We come back to the situation where $f:S\\to \\Spec \\Z$ is an arbitrary $\\Z$-scheme of finite type. We write $d$ for its relative dimension, \\emph{i.e} the maximal dimension of its closed fibres.\n\n\\begin{thm}\\label{t3.3} Let $f:S\\to T$ be a morphism of $\\Z$-schemes of finite type. Then $\\zeta(M,s) = \\zeta(f_!M,s)$ for any $M\\in \\D(S)$ (as formal Dirichlet series).\n\\end{thm}\n\nIn Corollary \\ref{c6.1} below, this Dirichlet series will be shown to be convergent.\n\n\\begin{proof} We immediately reduce to the case $S=\\Spec \\Z$. For a prime number $p$, let $S_p$ be the fibre of $f$ at $p$; we have a Cartesian square similar to \\eqref{eq3.5}:\n\\begin{equation}\\label{eq3.6}\n\\begin{CD}\nS_p@>I_p>> S\\\\\n@Vf_p VV @Vf VV\\\\\n\\Spec \\F_p@>i_p>> \\Spec \\Z.\n\\end{CD}\n\\end{equation}\n\n The equality\n\\[\\zeta(M,s) = \\prod_p \\zeta(I_p^*M,s)\\]\nfollows from the definition. \n By Theorem \\ref{t3.1} a) and proper base change, we have \n\\[\\zeta(I_p^*M,s)=\\zeta((f_p)_!I_p^*M,s)=\\zeta(i_p^*f_!M,s)\\]\nhence\n\\[\\zeta(M,s) = \\prod_p \\zeta(i_p^*f_!M,s)=\\zeta(f_!M,s)\\]\nagain by definition.\n\\end{proof}\n\n\n\\section{Motives over $\\R$ and $\\C$}\n\n\\subsection{Hodge structures} Set\n\\[\\Gamma_\\R(s) = \\pi^{-s/2}\\Gamma(s/2),\\qquad \\Gamma_\\C(s) = 2(2\\pi)^{-s}\\Gamma(s)\\]\nwhere $\\Gamma(s)$ is Euler's Gamma function. (For convenience, we take for $\\Gamma_\\C(s)$ twice the function in Serre \\cite[\\S 3]{serre}.)\n\nIf $V$ is a (pure) complex Hodge structure (i.e., a finite-dimensional $\\C$-vector space\nprovided with a decomposition $V=\\bigoplus_{(p,q)\\in \\Z\\times\\Z} V^{p,q}$), one defines\n\\[\\Gamma(V,s) = \\prod_{(p,q)} \\Gamma_\\C(s-\\inf(p,q))^{h(p,q)}\\]\nwith $h(p,q)=\\dim V^{p,q}$.\n\nIf $V$ is a (pure) real Hodge structure (i.e. a complex Hodge structure plus an involution\n$\\sigma$ such that $\\sigma V^{p,q} = V^{q,p}$), one defines\n\\[\\Gamma(V,s) = \\prod_n \\Gamma_\\R(s-n)^{h(n,+)}\\Gamma_\\R(s-n+1)^{h(n,-)}\\prod_{p<q}\n\\Gamma_\\C(s-p)^{h(p,q)}\\] where $h(p,q)$ is as above and $h(n,\\epsilon) = \\dim(V^{n,n}\\mid \\sigma =\n(-1)^n\\epsilon)$.\n\n\\subsection{Pure motives} If $k=\\R$ or $\\C$, we have a realisation functor\n$H:\\sM_\\hom(k,\\Q)\\to \\Hodge_k$, where $\\Hodge_k$ is the category of pure $k$-Hodge structures.\nFor $M\\in \\sM_\\hom(k)$, we define\n\\[\\Gamma(M,s) = \\Gamma(H(M)^*,s).\\]\n\n\\subsection{Triangulated motives} Using Bondarko's isomorphism \n\\[K_0(\\Chow(k,\\Q))\\iso K_0(\\DM_\\gm(k,\\Q))\\] \nwe may extend the above definition to all objects of $\\DM_\\gm(k,\\Q)$ (alternately, we could go\nthrough the Hodge realisation on $\\DM_\\gm(k,\\Q)$).\n\nThe identities of Corollary \\ref{c1} c) hold for $\\Gamma(M,s)$.\n\n\\begin{rk} One should compare this definition with the much more sophisticated one for mixed motives in \\cite[III.1]{fpr}, using \\emph{mixed} Hodge structures.\n\\end{rk}\n\n\\section{Two elementary lemmas on Dirichlet series}\n\n\\begin{lemma}\\label{l1} Let $(R_p)$ be a sequence of rational functions with complex\ncoefficients, indexed by the prime numbers. We assume that $R_p(0)=1$ for all $p$ and:\n\\begin{thlist}\n\\item There is an integer $w$ such that, for almost all $p$, the inverse zeroes and poles of\n$R_p$ have absolute value $\\le p^{w/2}$.\\\\\n\\item The heights of the $R_p$ are bounded independently of $p$ (N.B.: here, the \\emph{height}\nof a rational function $R=P/Q$ is $\\deg(P)+\\deg(Q)$ if $P$ and $Q$ are coprime polynomials).\n\\end{thlist}\nLet $L(s)=\\prod_p R_p(p^{-s})$. Then $L(s)$ is a Dirichlet series with absolute convergence abscissa $\\le w/2+1$.\n\\end{lemma}\n\n\\begin{proof} Let $\\lambda$ be an inverse pole of $R_p$. Then, for $s=\\sigma + it\\in \\C$:\n\\[\\left|(1-\\lambda p^{-s})^{-1}\\right|=\\left|\\sum_{n= 0}^\\infty \\lambda^n p^{-ns}\\right|\\le \\sum_{n= 0}^\\infty p^{-n(\\sigma-w/2)}\\]\nconverging as soon as $\\sigma > w/2$.\nIf now $\\lambda$ is an inverse zero of $R_p$, we have\n\\[\\left|1-\\lambda p^{-s}\\right|\\le 1+ |p^{w/2-s}|\\le \\sum_{n= 0}^\\infty p^{-n(\\sigma-w/2)}.\\]\n\nThus, if the height of $R_p$ is $\\le H$, we have\n\\[\\left|R_p(p^{-s})\\right|\\le \\left(\\sum_{n= 0}^\\infty p^{-n(\\sigma-w/2)}\\right)^H=\\left(1-p^{-(\\sigma-w/2)}\\right)^{-H}.\\]\n\nCollecting, we find\n\\[\\left|L(s)\\right|\\le\n\\left(\\prod_p\\left(1-p^{-(\\sigma-w/2)}\\right)^{-1}\\right)^H=\\zeta(\\sigma-w/2)^H\\] which\nconverges for $\\sigma-w/2>1$, as is well-known.\n\\end{proof}\n\n\\begin{lemma}\\label{l2} a) Let $f=\\sum_{n= 1}^\\infty a_n n^{-s}$ be a convergent Dirichlet series\nwith complex coefficients, with $a_1=1$. Then the equation $f(s)= g(s)/g(s+1)$ has a unique\nsolution as a convergent Dirichlet series, namely\n\\[g(s) = \\prod_{m= 0}^\\infty f(s+m).\\]\nMoreover, $g$ has the same absolute convergence abscissa as $f$.\\\\\nb) If the coefficients of $f$ belong to a subring $R$ of $\\C$, so do those of $g$. \n\\end{lemma}\n\n\\begin{proof} a) \\emph{Uniqueness:} if $g_1,g_2$ are two solutions, then $h=g_1/g_2$ verifies\n\\[h(s)=h(s+1).\\]\n\nIf $h(s)=\\sum c_n n^{-s}$, this gives the identity\n\\[u(s)=\\sum(c_n-c_n/n) n^{-s} = 0.\\]\n\nSince $g_1,g_2$ are convergent, so are $h$ and $u$, and it is well-known that this implies\n$c_n - c_n/n=0$ for all $n$, hence $c_n=0$ for all $n>1$.\n\n\\emph{Existence:} Let us check first that $g(s)$ converges as a formal Dirichlet series.\nIndeed:\n\\[g(s) = \\prod_{m= 0}^\\infty \\left(\\sum_{n=1}^\\infty \\frac{a_n}{n^m} n^{-s}\\right).\\]\n\nIn this product, the coefficient $b_n$ of $n^{-s}$ is\n\\[b_n = \\sum_{r_1\\dots r_k=n} \\sum_{m_1\\ge 0,\\dots,m_k\\ge 0} \\frac{a_{r_1}}{r_1^{m_1}}\\dots\n\\frac{a_{r_k}}{r_k^{m_k}}=\\sum_{\\substack{r_1\\dots r_k=n\\\\r_1,\\dots,r_k>1}}\n\\frac{a_{r_1}}{1-r_1^{-1}}\\dots \\frac{a_{r_k}}{1-r_k^{-1}} .\\]\n\n Suppose that $f(s)$ converges absolutely for $\\Re(s)>c$. Then $|a_n|=o(n^{c+\\epsilon})$ for\nall $\\epsilon>0$. Therefore\n\\begin{multline*}\n|b_n]\\le \\left|\\sum_{\\substack{r_1\\dots r_k=n\\\\r_1,\\dots,r_k>1}}\n\\frac{a_{r_1}}{1-r_1^{-1}}\\dots \\frac{a_{r_k}}{1-r_k^{-1}}\\right|\\\\ = o(n^{c+\\epsilon})\n\\sum_{\\substack{r_1\\dots r_k=n\\\\r_1,\\dots,r_k>1}} \\frac{1}{1-r_1^{-1}}\\dots\n\\frac{1}{1-r_k^{-1}}=o(n^{c+\\epsilon})g^0_n\n\\end{multline*}\nwhere $g^0_n$ is the $n$-th coefficient of\n\\[g^0(s) =\\zeta(s)\\zeta(s+1)\\dots\\]\n\n To study the absolute convergence of this product, we look at the log of the corresponding\nEulerian product, for $s\\in\\R$:\n\\begin{multline*}\n \\log g^0(s)=\\sum_p\\sum_{m=0}^\\infty -\\log(1-p^{-s-m})=\\sum_p\\sum_{m=0}^\\infty\\sum_{k=1}^\\infty\n\\frac{p^{-k(s+m)}}{k}\\\\ =\\sum_p \\sum_{k=1}^\\infty \\frac{p^{-ks}}{k}\\frac{1}{1-p^{-k}}\\le\n2\\sum_p \\sum_{k=1}^\\infty \\frac{p^{-ks}}{k} \\\\ = 2\\sum_p -\\log(1-p^{-s}) = \\log(\\zeta(s)^2).\n\\end{multline*}\n\nThe $n$-th coefficient of $\\zeta(s)^2$ is the number of divisors of $n$, which is\n$o(n^{\\epsilon})$ for all $\\epsilon>0$. Hence $|b_n|=o(n^{c+\\epsilon})$ for all $\\epsilon >0$.\nIt follows that $g(s)$ converges absolutely for $\\Re(s)>c$. But conversely, if $g(s)$ converges\nabsolutely for $\\Re(s)>c$, so does $f(s)=g(s)/g(s+1)$.\n\nb) is obvious from the formula giving $g(s)$.\n\n\n\\end{proof}\n\n\n\n\\section{Motives over a ring of integers}\n\nLet $K$ be a number field, with ring of integers $O_K$. Consider the category $\\D(O_K)$. For each place $v$ of $K$, we have a homomorphism $\\phi_v:O_K\\to \\kappa(v)$, where $\\kappa(v)$ is\n\\begin{itemize}\n\\item The residue field at $v$ if $v$ is finite;\n\\item The completion of $K$ at $v$ if $v$ is archimediean.\n\\end{itemize}\n\nIn each case, we have a pull-back functor\n\\[\\phi_v^*:\\D(O_K)\\to \\D(\\kappa(v)).\\]\n\nWe shall also use the fact that $\\Z_{\\Spec O_K}$ is a dualising object, of $\\D(O_K)$ which follows from \\cite[Th. 2.3.73]{ayoub} and \\cite[Cor. 5.15]{dJ2}. We write $M\\mapsto M^*$ for the corresponding duality functor.\n\n\n\\begin{thm} \\label{t6.2} Let $M\\in \\D(O_K)$. Then there exists a nonempty open subset $U\\subseteq \\Spec O_K$ such that the cohomology sheaves $H_l^i(M)$ are locally constant constructible for any prime number $l$ invertible on $U$.\n\\end{thm}\n\n\\begin{proof} We are in a case where $\\D(O_K)\\simeq \\DM_\\gm(O_K,\\Q)$, so we can reason as in the proof of Lemma \\ref{l2.2a}; this reduces us to the case where $M=M(X)$ with $f:X\\to \\Spec O_K$ smooth. Then the result is a special case of results of Illusie \\cite[Th. 2.1]{illusie}. (Recall the proof: by Hironaka's resolution of singularities and some spread-out, there exists $U$ and an open immersion $j:X_U\\inj \\bar X$, where $\\bar f:\\bar X\\to U$ is smooth projective and the closed complement $\\bar X-X_U$ is the support of a divisor $D$ with strict normal crossings, relative to $U$. By \\cite[Arcata V.3.1]{SGA412}, the sheaves $R^i\\bar f_*\\Q_l$ are locally constant, and so are the corresponding sheaves for all intersections of the components of $D$. The Leray spectral sequence for $j$ and cohomological purity then show that the same holds for the $R^i(f_{|U})_*\\Q_l$, \\emph{cf.} \\cite[Lemme 3.1]{illusie}.)\n\\end{proof}\n\n\n\n\\begin{cor}\\label{l3} For any $M\\in \\D(O_K[1/l])$, the\nfunction\n\\[\\fp\\mapsto b_\\fp(M)= \\sum_{i\\in\\Z}\\dim H^i_l(M_\\fp)\\]\nfrom primes of $O_K[1/l]$ to $\\N$ is bounded. \\qed\n\\end{cor}\n\n\\begin{thm}\\label{t6.1} The Dirichlet series $\\zeta(M,s)$ has a finite convergence abscissa. More precisely, if\n$M^*\\in d_{\\le n} \\D(O_K)$, then $\\zeta(M,s)$ converges absolutely for $\\Re(s)>n+1$.\n\\end{thm}\n\n\n\\begin{proof} Let $M^*\\in d_{\\le n} \\D(O_K)$. By Proposition \\ref{p3} and Corollary\n\\ref{l3}, the hypotheses of Lemma \\ref{l1} are satisfied for $\\zeta(M,s)$, with $w=2n$. This\nproves the statement for effective motives, hence in general.\n\\end{proof}\n\n\\begin{cor}\\label{c6.1} For any $\\Z$-scheme of finite type $S$ and any $M\\in \\D(S)$, the formal Dirichlet series $\\zeta(M,s)$ has a finite convergence abscissa.\n\\end{cor}\n\n\\begin{proof} This follows from Theorems \\ref{t3.3} and \\ref{t6.1} a).\n\\end{proof}\n\n\\begin{rk} Denis-Charles Cisinski pointed out that there are more motivic and uniform methods to obtain bounds as in Corollary \\ref{l3} (\\cite[Th. 6.3.26]{cis-deg2}, \\cite{schol-etal}, \\cite[Th. 2.4.2]{cisinski}). But all these theorems rest \\emph{in fine} on the smooth and proper base change theorem for étale cohomology \\cite[Arcata, V, Th. 3.1]{SGA412}, so they do not seem to bring something essentially new here as the existence of a bound is sufficient for Theorem \\ref{t6.1} and its corollary.\n\\end{rk}\n\n\n\n\\begin{prop}\\label{p6.1} For $M\\in \\D(O_K)$, let\n\\[\\quad \\xi(M,s)= |d_K|^{s\\chi(M)/2}\\prod_{v\\mid\\infty}\\Gamma(\\phi_v^* M,s)\\cdot \\zeta(M,s) \\]\nwhere $d_K$ is the absolute discriminant of $K$. Then $\\xi(\\pi_* M,s)=\\xi(M,s)$, where $\\pi_*$ is the push-forward functor $\\D(O_K)\\to \\D(\\Z)$. \n\\end{prop}\n\n\\begin{proof} It suffices to check this for the Gamma-discriminant part, and it follows from an elementary computation.\n\\end{proof}\n\n\nTo be complete, we define the completed $\\zeta$ function of a motive over a $\\Z$-scheme of finite type.\n\n\\begin{defn}\\label{d6.1}\nIf $M\\in \\D(S)$ where $f:S\\to \\Spec \\Z$ is a $\\Z$-scheme of finite type, we set $\\xi(M,s)=\\xi(f_!M,s)$.\n\\end{defn}\n\nNote that $\\xi(M,s)=\\zeta(M,s)$ if $f$ is not dominant, and Theorem \\ref{t3.3} still holds when replacing $\\zeta$ by $\\xi$. By Proposition \\ref{p6.1}, this definition does extend the case $S=\\Spec O_K$.\n\n\n\n\\section{A theorem of Serre}\n\nFor $M\\in \\D(O_K)$ and $\\fp\\subset O_K$, define\n\\[N_M(\\fp) = \\sharp(M_\\fp)\\]\nthe number of points of $M$ modulo $\\fp$.\n\n\n\\begin{thm}\\label{t3} Let $M\\in \\D(O_K)$. Suppose that $\\zeta(M,s)$ is\nnot a finite product of Euler factors. Then the set\n\\[\\{\\fp\\mid N_M(\\fp) = 0\\}\\]\nhas a density $1-\\epsilon$, with\n\\[\\epsilon \\ge \\frac{1}{b_\\infty(M)^2}\\]\nwhere $b_\\infty(M)=\\sum_i \\dim H^i_l(M_K)$.\n\\end{thm}\n\n\\begin{proof} It is the same as Serre's \\cite[Th. 6.17]{NX}, which is the special case\n$M=M^{BM}(X)\\oplus M^{BM}(X')[1]$ for $X,X'$ $O_K$-schemes of finite type. For $U$ as in Theorem \\ref{t6.2}, one may compute\nthe traces of the geometric Frobenius $F_{M_\\fp}$ acting on $H_l^*(M_\\fp)$, for $\\fp\\in U$, as\nthe traces of the inverse of the [conjugacy class of the] arithmetic Frobenius $\\phi_\\fp\\in\nGal(\\bar K/K)$ acting on\n$H_l^*(M_K)$. The statement then reduces to the following theorem of Serre \\cite[Th. 5.15]{NX}:\n\n\\begin{thm}\\label{tserre} Let $G$ be a compact group, $K$ be a locally compact field of\ncharacteristic\n$0$ and let $\\rho:G\\to GL_n(K)$, $\\rho':G\\to GL_{n'}(K)$ be two continous $K$-linear\nrepresentations of\n$G$. Then\n\\begin{thlist}\n\\item either $\\tr \\rho= \\tr \\rho'$;\n\\item or the set $\\{g\\in G\\mid \\tr(\\rho)(g)\\ne \\tr(\\rho')(g)\\}$ has a Haar density $\\ge\n\\frac{1}{(n+n')\\sup(n,n')}$.\n\\end{thlist}\n\\end{thm}\n\nTo apply Serre's theorem, we represent $G=G_K$ on $H^{even}_l(M_K)$, yielding $\\rho$, and on\n$H^{odd}_l(M_K)$, yielding $\\rho'$. Here, $K=\\Q_l$. If both those vector spaces are $0$, then\n$\\sH_l(M)$ is supported on $\\Spec O_K[1/l]-U$, where $U$ is the open set as above; then all\nEuler factors of $\\zeta(M,s)$ at $\\fp\\in U$ are equal to $1$. Otherwise, we may apply the\ntheorem. In case (i), we get the same conclusion on the Euler factors as above. Case (ii)\nyields the conclusion of Theorem \\ref{t3}.\n\\end{proof}\n\nMore generally, a large part of Serre's results in \\cite{NX} seem to extend to objects of $\\D(O_K)$ without difficulty. \n\n\\section{Some six functors algebra}\n\n\\subsection{$K_0$ of triangulated categories}\\label{s8.A} Let\n\\[0 \\to\\sT'\\by{i} \\sT\\by{p} \\sT''\\to 0\\]\nbe a short exact sequence of triangulated categories: $i$ is a thick embedding and $\\sT/\\sT'\\iso\\sT''$. We assume that we are in the Verdier situation: $i$ has a right adjoint $\\pi$, hence $p$ has a right adjoint $j$ and any object $M\\in \\sT$ fits in a functorial exact triangle\n\\[i\\pi M\\to M\\to pj M\\by{+1}.\\]\n\nFrom this, it follows:\n\n\\begin{lemma} In the above situation, the map\n\\[K_0(\\sT)\\by{\\left(\\begin{smallmatrix}\\pi\\\\ p \\end{smallmatrix}\\right)} K_0(\\sT')\\oplus K_0(\\sT'')\\]\nis an isomorphism. Alternately, we have an exact sequence\n\\[0\\to K_0(\\sT')\\by{i} K_0(\\sT)\\by{p} K_0(\\sT'')\\to 0\\]\nsplit by $\\pi$ and $j$.\n\\end{lemma}\n\n\\begin{proof} Since $i$ and $j$ are fully faithful, the first map has the right inverse $(i,j)$. It is also a left inverse thanks to the exact triangle above.\n\\end{proof}\n\nLet $(\\sT_\\alpha)_{\\alpha\\in A}$ be an filtered inductive system of triangulated categories, and let $\\sT=\\colim_\\alpha \\sT_\\alpha$. We have an induced homomorphism\n\\begin{equation}\\label{eq8.7}\n\\colim_\\alpha K_0(\\sT_\\alpha)\\to K_0(\\sT).\n\\end{equation}\n\n\\begin{lemma}\\label{l8.1} \\eqref{eq8.7} is an isomorphism.\n\\end{lemma}\n\n\\begin{proof} Write $i_\\alpha:\\sT_\\alpha\\to \\sT$ for the canonical functor. Let $X\\in \\sT$. Then $X\\simeq i_\\alpha X_\\alpha$ for some $(\\alpha,X_\\alpha)$. This shows that \\eqref{eq8.7} is surjective.\n\nLet $\\alpha\\in A$ and $x\\in K_0(\\sT_\\alpha)$ be such that $i_\\alpha x=0$. Writing $x=\\sum_{i\\in I} n_i[X_i]$ for $I$ finite, $n_i\\in\\Z$ and $X_i\\in \\sT_\\alpha$, the hypothesis means that we have an equality\n\\[ \\sum_{i\\in I} n_i[i_\\alpha X_i] = \\sum_{j\\in J} m_j([Y_j]-[Y'_j]-[Y''_j])\\]\nin the free group with generators the isomorphism classes of objects of $\\sT$, where $J$ is finite, $m_j\\in\\Z$ and $Y'_j\\to Y_j\\to Y''_j\\by{+1}$ are exact triangles. All these exact triangles come from exact triangles in $\\sT_\\beta$ for some $\\beta$ dominating $\\alpha$. This shows that \\eqref{eq8.7} is injective.\n\\end{proof}\n\nFor simplicity, we set\n\\[K_0^\\sM(S) = K_0(\\D(S))\\]\nfor any scheme $S$.\n\nLet $K$ be a global field. We write $C$ for $\\Spec O_K$ if $K$ is number field or for the smooth projective model of $K$ in positive characteristic.\n\nLet $j:U\\subseteq V$ be two nonempty open subsets of $C$, and let $i:Z\\to V$ be to complementary closed immersion. We have an exact sequence of triangulated categories\n\\begin{equation}\\label{eq9.4}\n0\\to \\D(Z)\\by{i_*} \\D(V)\\by{j^*}\\D(U)\\to 0\n\\end{equation}\nwhich is split as in the previous section by $i^!$ and $j_*$. Note also that\n\\[\\D(Z)=\\coprod_{v\\in Z} \\D(\\kappa(v)).\\]\n\nHence we get a short exact sequence\n\\[0\\to \\bigoplus_{v\\in Z} K_0^\\sM(\\kappa(v)) \\by{((i_v)_*)} K_0^\\sM(V)\\by{j^*} K_0^\\sM(U)\\to 0\\]\nwhich is split, although we shall not use this.\n\nBy Ivorra \\cite[Prop. 4.16]{ivorra}, we have an equivalence\n\\[2-\\colim_U \\D(U) \\iso \\D(K)\\]\nhence, by Lemma \\ref{l8.1}, a (non split) short exact sequence\n\\begin{equation}\\label{eq8.0}\n0\\to \\bigoplus_v K_0^\\sM(\\kappa(v)) \\by{((i_v)_*)} K_0^\\sM(C)\\by{j^*} K_0^\\sM(K)\\to 0\n\\end{equation}\nwhere $v$ runs through all the closed points of $C$.\n\\subsection{A purity theorem} \nFor the sequel, we note that the theory $\\D$ verifies the axioms of \\cite[Def. 2.3.1]{ayoub} by \\emph{loc. cit.}, Prop. 4.5.31.\n\nLet $S$ be a scheme, $i:Z\\to S$ a closed immersion and $j:U\\to S$ the complementary open immersion. For $M,N\\in \\D(S)$, we have a natural transformation \\cite[\\S 2.3.2]{ayoub}\n\\begin{equation}\\label{eq8.1}\nr_{M,N}:i^*M\\otimes i^!N\\to i^!(M\\otimes N)\n\\end{equation}\nwhich is not an isomorphism in general (\\emph{loc. cit.}, Remark 2.3.13). However,\n\n\\begin{thm}\\label{t8.1} If $M$ is strongly dualisable, \\eqref{eq8.1} is an isomorphism.\n\\end{thm}\n\n\\begin{proof}[Proof (J. Ayoub)] Let $j:U\\to S$ be the complementary open immersion. By the localisation exact triangle\n\\begin{equation}\\label{eq8.6}\ni_*i^!N\\to N\\to j_*j^*N\\by{+1} \n\\end{equation}\nof \\cite[\\S 1.4.4]{ayoub}, we reduce to the cases where $N$ is of the form $j_* N'$ or $i_*N''$.\n\nIn the first case, the left hand side is $0$, and so is the right hand side by the projection formula\n\\[M\\otimes j_* N' \\simeq j_*(j^*M \\otimes N')\\]\nwhich holds because $M$ is dualisable (another lemma of Ayoub, \\emph{cf.} \\cite[Lemma 9.3.1]{ffihes}).\n\nIn the second case, \\eqref{eq8.1} is the composition \n\\[i^*M \\otimes i^!i_*N'' \\simeq i^!i_*(i^*M \\otimes i^!i_*N'') \\simeq\n i^!(M \\otimes i_*i^!i_*N'') \\to i^!(M\\otimes i_*N'')\\]\nby the strong monoidality of $i_*$ \\cite[Lemma 2.3.6]{ayoub}, and the last map is invertible: the counit $i_*i^! \\to Id$ is invertible when applied to an object of the form $i_*N''$.\n\\end{proof}\n\nLet $M\\in \\D(S)$. Applying $i^*$ to \\eqref{eq8.6}, we get another exact triangle\n\\[i^!M\\by{\\tau_M} i^*M\\to i^*j_*j^*M\\by{+1}. \\]\n\n\\begin{cor}\\label{c8.1} Suppose that $\\tau_{\\Z_S}=0$. Then $\\tau_M=0$ for any strongly dualisable $M$.\n\\end{cor}\n\n\\begin{proof}[Proof (J. Ayoub)] We have a commutative diagram\n\\[\\begin{CD}\ni^*M \\otimes i^!N @>r_{M,N}>> i^!(M\\otimes N) \\\\\n@V{1\\otimes \\tau_N}VV @V{\\tau_{M\\otimes N}}VV\\\\\ni^*M \\otimes i^*N @>\\sim >> i^*(M\\otimes N)\n\\end{CD}\\]\nwhere the top map is an isomorphism by Theorem \\ref{t8.1}. The conclusion follows by taking $N=\\Z_S$.\n\\end{proof}\n\n\\begin{rk}\\label{r8.1} The hypothesis of Corollary \\ref{c8.1} is verified in particular when $Z=\\Spec \\kappa$ is a closed point of $S$: then $\\tau_{\\Z_S}\\in \\D(\\kappa)(\\Z_Z(-d)[-2d],\\Z_Z)=H^{2d}(\\kappa,\\Q(d))=0$.\n\\end{rk}\n\n\\begin{cor} \\label{c8.2} Suppose that $S$ and $Z$ are regular and that $i$ is of pure codimension $c$. Then the morphism \\eqref{eq8.1} taken with $N=\\Z_S$ induces an isomorphism \n\\[i^*M(-c)[-2c]\\iso i^!M\\]\nfor any dualisable $M$.\n\\end{cor}\n\n\\begin{proof} This follows from Theorem \\ref{t8.1} and \\cite[Th. 14.4.1]{cis-deg} in the language of Beilinson motives, or from \\cite[Cor. 7.5]{ayoubetale} in the language of $\\DA^\\et$.\n\\end{proof}\n\n\\begin{prop}\\label{p8.1} Let $\\D^\\proj(S)$ be the thick subcategory of $\\D(S)$ generated by the $M(X)(n)$ with $X$ smooth projective over $S$ and $n\\in \\Z$. Suppose that $\\Z_S$ is a dualising object of $\\D(S)$. Then all objects of $\\D^\\proj(S)$ are strongly dualisable. In particular, Theorem \\ref{t8.1}, Corollary \\ref{c8.1} and Corollary \\ref{c8.2} apply to them.\n\\end{prop}\n\n\\begin{proof} The argument is the same as for \\cite[Th. 2.2]{riou}.\n\\end{proof}\n\n\\begin{rk} As Ayoub pointed out, if one wants to prove Corollary \\ref{c8.2} for $M=f_\\#\\Z_X$ with $f$ smooth projective, one can avoid using Theorem \\ref{t8.1} by the following direct computation: $M\\simeq f_!f^!\\Z_S$ and $i^!f_!f^!\\simeq (f_Z)_!f_Z^!i^!$ by the base change isomorphisms, where $f_Z$ is the pull-back of $f$ along $i$.\n\\end{rk}\n\n\n\n\n\n\\section{Zeta and $L$-functions of motives over a global field}\\label{s9}\n\nLet $K$ be a global field. We keep the notation of \\S \\ref{s8.A}: $C$ denotes either $\\Spec O_K$ when $K$ is a number field and $O_K$ is its ring of integers, or the smooth projective model of $K$ over its field of constants $\\F_q$ when $K$ is of positive characteristic.\n\n\\subsection{Zeta functions up to finite Euler products} \n\nLet as above $\\Dir$ denote the group of convergent Dirichlet series, and let $\\Eul$ be the subgroup generated by Euler factors, i.e. Dirichlet series of the form $R(p^{-s})$, where $R\\in\\Q(t)$ and $p$ is a prime number. From this exact sequence we deduce a map\n\\[\\bar\\zeta:K_0^\\sM(K)\\to \\Dir/\\Eul\\]\ninduced by $DM_\\gm(O_K,\\Q)\\ni M\\mapsto \\zeta(M,s)$.\n\nLet $X$ be a smooth projective $K$-variety, and let $\\Sigma_K(X)$ be the set of finite places of $K$ where $X$ has good reduction. Recall that, in \\cite[Prop. 5.6 and Th. 5.7]{zetaL}, we defined an ``approximate zeta function'' by the formula\n\\[\\zeta_{appr}(X,s)=\\prod_{v\\in \\Sigma_K(X)} \\zeta(X(v),s)\\]\nwhere $X(v)$ is the special fibre of a smooth model of $X$ over $\\Spec O_v$. (The point is that $\\zeta(X(v),s)$ does not depend on the choice of $X(v)$.) Then the following is obvious by construction:\n\n\\begin{prop} We have $\\zeta_{appr}(X,s)=\\bar \\zeta(X,s)$\nin $\\Dir/\\Eul$.\\qed\n\\end{prop}\n\n\n\n\n\\subsection{A ``total\" $L$-function for motives over $\\Spec K$}\\label{total}\n\nWe go back to the short exact sequence of triangulated categories\n\\[0\\to \\coprod_v \\D(\\kappa(v))\\by{(i_v)_*} \\D(C)\\by{j^*}\\D(K)\\to 0\\]\nwhich is the $2$-colimit of the exact sequences \\eqref{eq9.4}. It sits fully faithfully into a short exact sequence of larger categories\n\\[0\\to \\coprod_v \\DA^\\et(\\kappa(v),\\Q)\\by{(i_v)_*} \\DA^\\et(C,\\Q)\\by{j^*}\\DA^\\et(K,\\Q)\\to 0.\\]\n\nIn this sequence, $j^*$ has the right adjoint $j_*$ for Brown representability reasons. If $M\\in \\D(K)$, $j_* M$ is of course not constructible in general; nevertheless we would like to define a ``total $L$-function\" of $M$ by the formula\n\\[L^\\tot(M,s)=\\zeta(j_*M,s).\\]\n\nSense can be made of this formula as follows:\n\nWe may write $M=j_U^*M_U$, where $M_U\\in \\D(U)$, for some open subset $U$ of $C$. Write $j=j'_U j_U$. Then $j_*M = (j'_U)_*(j_U)_*j_U^*M_U$. Let $j_{U,V}:V\\to U$ be an open subset of $U$. From the exact triangles\n\\begin{equation}\\label{eq8.2}\n\\bigoplus_{v\\in U-V} (i_v)_*i_v^! M_U \\to M_U\\to (j_{U,V})_*j_{U,V}^*M_U\\by{+1}\n\\end{equation}\nwe deduce in the (co)limit an exact triangle\n\\[\\bigoplus_{v\\in U} (i_v)_*i_v^! M_U \\to M_U\\to (j_{U})_*j_{U}^*M_U\\by{+1}\\footnote{For lightness of notation, we write $v\\in U$ rather than $v\\in U_{(0)}$, and similarly in the sequel.}\\]\nhence, after applying $(j'_U)_*$, an exact triangle\n\\begin{equation}\\label{eq8.3}\n\\bigoplus_{v\\in U} (i_v)_*i_v^! M_U \\to (j'_U)_*M_U\\to j_*M\\by{+1}.\n\\end{equation}\n\nNote that the latter may also be written\n\\[\\bigoplus_{v\\in C} (i_v)_*i_v^! (j'_U)_*M_U \\to (j'_U)_*M_U\\to j_*M\\by{+1}\\]\nas one sees for example by applying the localisation exact triangles to $(j'_U)_*M_U$. (Here we abuse notation by identifying the closed immersions $v\\inj U$ and $v\\inj C$, with the common name $i_v$.) The\nmotive $(i_v)_*i_v^! (j'_U)_*M_U$ is $0$ if $v\\notin U$, because then $i_v^!\n(j'_U)_*=0$.\n\n\n\\begin{prop}\\label{p8.2} \nFor any closed point $v$ of $C$, let $O_v$ be the local ring of $v$ and $j_v:\\Spec K\\inj\\Spec O_v$\nthe corresponding open immersion. Then, with the above notation, we have the relation\n\\[ [i_v^*(j'_U)_*M_U]-[i_v^! (j'_U)_*M_U]= [i_v^*(j_v)_*M]\\in K_0^\\sM(\\kappa(v)).\\]\nIn particular, the left hand side does not depend on the choice of $U$ and $M_U$, and is\ntriangulated in $M$.\n\\end{prop}\n\n\\begin{proof} Let $(V,M_V)$ be another model of $M$. Then $M_U$ and $M_V$ become isomorphic\nafter restricting to some open subset of $U\\cap V$, so we may assume $V\\subseteq U$ and $M_V =\nj_{U,V}^*M_U$. Hence, by \\eqref{eq8.2}, it suffices to show that\n\\[\ni_w^*(i_v)_*= i_w^!(i_v)_*= \n\\begin{cases}\n0& \\text{if $w\\ne v$}\\\\\nId_{\\D(\\kappa(v))}& \\text{if $w\\ne v$.}\n\\end{cases}\n\\]\n\nBoth formulas are obvious, the second because $(i_v)_*$ is fully faithful. This shows that\nthe left hand side of the formula only depends on $M$.\n\nSince $j^*(j'_U)_*M=M$, we may now suppose that $U=C$ in \\eqref{eq8.3}. Let $j'_v:\\Spec\nO_v\\inj C$ be the other inclusion. Applying $(j'_v)^*$ to \\eqref{eq8.3}, we get an exact\ntriangle\n\\[(i_v)_*i_v^! M_C \\to (j'_v)^*M_C\\to (j_v)_*M\\by{+1}\\]\nhence the formula.\n\\end{proof}\n\n\\begin{defn}\\label{l9.1} For any $M\\in \\D(K)$ and any $v\\in C_{(0)}$, we set\n\\[L^\\tot_v(M,s)= \\zeta(i_v^*(j_v)_*M,s).\\]\n\\end{defn}\n\n\\begin{defn} Let $M\\in \\D(K)$ and let $v\\in C_{(0)}$. We say that $M$ \\emph{has good\nreduction at $v$} if $M=j_v^*\\sM$, with $\\sM\\in \\D^\\proj(O_v)$ (see Proposition \\ref{p8.1}). We say that such\nan $\\sM$ is a \\emph{good model} of $M$ at $v$.\n\\end{defn}\n\nLet $M\\in \\D(K)$, and let $\\sM\\in \\D(O_v)$ be such that $j_v^*\\sM=M$. The\nexact triangle\n\\[(i_v)_*i_v^!\\sM\\to \\sM\\to (j_v)_*M\\by{+1}\\]\ngives, after applying $i_v^*$, an exact triangle\n\\[i_v^!\\sM\\to i_v^*\\sM\\to i_v^*(j_v)_*M\\by{+1}.\\]\n\nIf $M$ has good reduction at $v$ and $\\sM$ is a good model, this triangle reads as\n\\[i_v^*\\sM(-1)[-2]\\to i_v^*\\sM\\to i_v^*(j_v)_*M\\by{+1}\\]\nthanks to Theorem \\ref{t8.1} and Proposition \\ref{p8.1}. (The first map is trivial by Corollary \\ref{c8.1} and Remark \\ref{r8.1}, although we shall not need this.) Thus\n\\begin{equation}\\label{eq8.4}\nL^\\tot_v(M,s) = \\frac{\\zeta(i_v^*\\sM,s)}{\\zeta(i_v^*\\sM,s+1)}\n\\end{equation}\nin this case, thanks to Corollary \\ref{c1} c).\n\n\\begin{thm}\\label{t8.2} With the above notation,\\\\\na) The Euler product \n\\[L^\\tot(M,s)= \\prod_{v\\in C} L_v^\\tot(M,s)\\]\nis a Dirichlet series which converges absolutely for $\\Re(s)\\gg 0$.\\\\\nb) If $M=M(X)^*$, where $X$ is smooth projective, we have for $v\\nmid l$\n\\[L^\\tot_v(M,s)=\\prod_{i\\in\\Z} L_v^\\tot(H^i_l(X),s)^{(-1)^i}\\] \nwhere $v$ runs through the maximal ideals of $O_K$ and\n\\[L_v^\\tot(H^i_l(X),s) = \\frac{\\det(1-N(v)^{-s}\\phi_v^{-1}\\mid H^0(I_v,H^i(\\bar X,\\Q_l))}{\\det(1-N(v)^{-s}\\phi_v^{-1}\\mid H^1(I_v,H^i(\\bar X,\\Q_l))}\\]\nwhere $\\phi_v$ and $I_v$ are respectively a Frobenius at $v$ and the inertia group at $v$.\\\\\nc) If $X$ has good reduction at $v$ in b), with special fibre $X_v$, then\n\\[L_v^\\tot(M,s) = \\frac{\\zeta(X_v,s)}{\\zeta(X_v,s+1)}.\\]\n\\end{thm}\n\n\\begin{proof} \na) In view of Theorem \\ref{t6.1} and Proposition \\ref{p8.2}, it suffices to see that\n$i_v^!M_U\\simeq i_v^*M_U(-1)[-2]$ for almost all $v$. But for $V\\subseteq U$ small\nenough, $j_{U,V}^*M_U\\in \\D^\\proj(V)$; hence this follows from Theorem \\ref{t8.1}.\n\nb) We have the isomorphism\n\\[R^l(i_v^*(j_v)_*M)\\simeq i_v^*R(j_v)_* R^l(M)\\]\nfor $M\\in \\D(K)$. \n\nFor $M=M(X)^*$, $f:X\\to \\Spec K$ smooth projective, we have\n\\[R^l(M)= Rf_*\\Q_l\\]\nhence\n\\[H^i(R^l(M)) = H^{i}(\\bar X,\\Q_l).\\]\n\nLet $I_v$ be the absolute inertia group at $v$. For an $l$-adic sheaf $\\sF$ on $\\Spec K$, we have\n\\[i_v^*R^q(j_v)_*\\sF =\n\\begin{cases}\nH^0(I_v,\\sF(\\bar K)) &\\text{if $q=0$}\\\\\nH^1(I_v,\\sF(\\bar K)) &\\text{if $q=1$}\\\\\n0 &\\text{if $q>1$}\n\\end{cases}\\]\nwith Frobenius action induced by the action of $G_K$. \n\nIf $\\sF=H^i(\\bar X,\\Q_l)$ for $X$ smooth projective, we have\n\\[\nL(\\kappa(v),H^i(I_v,\\sF(\\bar K)),s)=\n \\det(1-N(v)^{-s}\\phi_v^{-1}\\mid H^{i}(I_v,H^i(\\bar X,\\Q_l))).\n\\]\nand\n\\[\\frac{L(\\kappa(v),H^0(I_v,\\sF(\\bar K)),s)}{L(\\kappa(v),H^1(I_v,\\sF(\\bar K)),s)}\n=\\frac{\\det(1-N(v)^{-s-1}\\phi_v \\mid H^1(I_v,H^i(\\bar\nX,\\Q_l)))}{\\det(1-N(v)^{-s-1}\\phi_v \\mid H^0(I_v,H^i(\\bar X,\\Q_l)))}.\\]\n\\end{proof}\n\n\\subsection{The nearby $L$-function of a motive over $\\Spec K$}\n\nWe now apply Lemma \\ref{l2} to $L^\\tot(M,s)$. This gives\n\n\\begin{defn}\\label{d9.1} For $M\\in \\D(K)$ and a prime $v$, we define $L_v^\\near(M,s)$ as the\nunique Dirichlet series (with initial coefficient $1$) such that \n\\[\\frac{L_v^\\near(M,s)}{L_v^\\near(M,s+1)} = L_v^\\tot(M,s).\\]\nWe set\n\\[L^\\near(M,s)=\\prod_v L_v^\\near(M,s).\\]\n\\end{defn}\n\n\nClearly,\n\\[\\frac{L^\\near(M,s)}{L^\\near(M,s+1)} = L^\\tot(M,s)\\]\nwhich shows by Lemma \\ref{l2} that $L^\\near(M,s)$ is a convergent Dirichlet series with the\nsame absolute convergence abscissa as $L^\\tot(M,s)$.\n\nIf $M$ has good reduction at $v$, then for any good model $\\sM$ at $v$, one has\n\\begin{equation}\\label{eq9.1}\nL_v^\\near(M,s)= \\zeta(i_v^*\\sM,s)\n\\end{equation}\nthanks to \\eqref{eq8.4}. We shall now handle the general case and relate $L^\\near(M,s)$ with\nAyoub's nearby cycle functor $\\Upsilon$ \\cite{ayoubetale}.\n\n\\begin{thm}\\label{t9.1} We have $L^\\near_v(M,s)=\\zeta(\\Upsilon_v M,s)$, where $\\Upsilon_v:\\D(K)\\allowbreak\\to \\D(\\kappa(v))$ is the ``unipotent\" specialisation functor associated to $O_v$ as in \\cite[Th. 11.13]{ayoubetale}.\n\\end{thm}\n\n\\begin{proof} By \\emph{loc. cit.}, Th. 11.16, there is an exact triangle\n\\begin{equation}\\label{eq9.2}\ni_v^* (j_v)_*M\\to \\Upsilon_v M\\to \\Upsilon_v M(-1)\\by{+1}\n\\end{equation}\nhence the result follows from the uniqueness statement in Lemma \\ref{l2}.\n\\end{proof}\n\n\\begin{cor} $L^\\near_v(M,s)$ is a rational function in $N(v)^{-s}$, whose zeroes and poles are $N(v)$-Weil numbers.\\qed\n\\end{cor}\n\nThis is remarkable because, in Lemma \\ref{l2}, $g(s)$ is in general by no means a rational function of $p^{-s}$ when $f(s)$ is. (Take $f(s)=(1-p^{-s})^{-1}$.)\n\n\\begin{rk}\\label{r9.1} Another argument, which was our initial argument, is to go via the $l$-adic realisation: one has\n\\[L_v^\\tot(M,s) = L_v(i_v^*R(j_v)_*R^l(M),s).\\]\n\nIf $V$ is an $l$-adic representation of $G_K$, we need to show that\n\\[L(i_v^*R(j_x)_* V,s)= f(N(v)^{-s})/f(N(x)^{-s-1})\\]\nfor some $f\\in \\Q(t)$. \n\nWe have\n\\[L(i_v^*R(j_v)_* V,s)= \\frac{\\det(1-\\phi_vN(v)^{-s}\\mid H^1(I_v,V))}{\\det(1-\\phi_vN(v)^{-s}\\mid H^0(I_v,V))}.\\]\n\n\nSince $cd_l(I_v)=1$ this is an Euler-Poincar\\'e characteristic, so we may assume $V$ \\emph{semi-simple}. Then $I_v$ acts through a finite quotient by the $l$-adic monodromy theorem \\cite[Appendix]{serre-tate}, thus\n\\[H^1(I_v,V) = V_{I_v}(-1) \\simeq V^{I_v}(-1)\\]\nand\n\\[L(i_x^*R(j_v)_* V,s)= \\frac{L^\\Serre(V^{ss},s)}{L^\\Serre(V^{ss},s+1)}\\]\nwhere $V^{ss}$ is the semi-simplification of $V$. We thus get another formula for $L^\\near(M,s)$:\n\\begin{equation}\\label{eq9.3}\nL^\\near(M,s)=L^\\Serre(R^l(M)^{ss},s).\n\\end{equation}\n\nBecause of the finiteness of the action of the inertia, one might think of the right hand side of \\eqref{eq9.3} as an \\emph{Artin} $L$-function.\n\\end{rk}\n\n\\begin{qns} Composing with the functor $\\Phi$ of \\S \\ref{s2.E}, we may associate a nearby $L$-function to any Chow motive, and this determines $L^\\near$ by Bondarko's theorem.\\\\ \n1) By the $l$-adic realisation, this definition factors through homological equivalence. Does it even factor through numerical equivalence, as in positive characteristic? (The two $K_0$'s agree under the sign conjecture.)\\\\\n2) Similarly, the operator $\\Upsilon_v$ induces a homomorphism \n\\[\\Upsilon_v:K_0(\\Chow(K))\\to K_0(\\Chow(\\kappa(v)).\\] \nTo what extent can one describe it explicitly?\n\\end{qns}\n\n\\subsection{Examples}\\label{s9.1} We finish with explicit computations. \n\n\\subsubsection{Artin motives} In what comes before, we worked with motives with $\\Q$-coefficients, but everything works just as well for motives with coefficients in a $\\Q$-algebra, for example in a number field. This allows us to consider the nearby $L$-function $L^\\near(\\rho,s)$ attached to a complex Galois representation $\\rho$. Then the action of inertia is semi-simple, thus, by \\eqref{eq9.3} we have $L^\\near(\\rho,s)=L(\\rho,s)$, the Artin $L$-function of $\\rho$. So nothing new happens here.\n\n\\subsubsection{Elliptic curves}\\label{s9.D.2}Let $E$ be an elliptic curve over $K$ with multiplicative reduction at $v$, $V= H^1_l(E)$. By hypothesis, the action of $I_v$ on $V$ is unipotent and nontrivial, hence $\\dim V^{I_v}=1$ and $I_v$ acts trivially on $V/V^{I_v}$; thus $V^{ss}=V^{I_v}\\oplus V/V^{I_v}$ and\n\\begin{align*}\nL_v^\\Serre(h^1(E),s) &= \\det(1-N(v)^{-s}\\phi_x\\mid V^{I_v})^{-1}\\\\\nL_v^\\near(h^1(E),s) &= L_v^\\Serre(H^1(E),s)\\times \\det(1-N(v)^{-s}\\phi_v\\mid V/V^{I_v})^{-1}.\n\\end{align*}\n\nExtra poles thus are explicitly computable: if the multiplicative reduction is split, then $L_v^\\Serre(h^1(E),s)=(1-N(v)^{-s})^{-1}$ and thus the other factor is $(1-N(v)^{1-s})^{-1}$, since the determinant is $H^2_l(E)\\simeq \\Q_l(-1)$. Similarly, if the multiplicative reduction is not split, the other factor is $(1+N(v)^{1-s})^{-1}$. \n\nNote that these factors have a functional equation between $s$ and $2-s$; so, such functional equations for $L^\\Serre(h^1(E),s)$ and $L^\\near(h^1(E),s)$ are equivalent. Similarly, the Beilinson conjectures for the first function easily imply Beilinson-like conjectures for the second, and conversely.\n\n\\section{The functional equation in characteristic $p$}\n\nLet $K=\\F_q(C)$ for $C$ a smooth, projective, geometrically connected curve. \n As usual we abbreviate $\\F_q=:k$, $\\eta =\\Spec K$ and write $j:\\eta\\to C$ for the canonical immersion. If $U\\subseteq C$ is a nonempty open subset, we factor $j$ as a composition\n\\[\\eta\\by{j_U} U\\by{j'_U}C.\\]\n\nLet $M\\in \\DM_\\gm(K,\\Q)$: we want to compare the $L$-functions\n\\[L^\\near(M^*,1-s)\\text{ and } L^\\near(M,s).\\]\n\nWe know that $M\\simeq j^*\\sM$ for some $\\sM\\in \\D(C)$; from the functional equation for $\\zeta(\\sM,s)$ (Theorem \\ref{t3.1}) we can get an approximate functional equation for $L^\\near(M,s)$, but we would like a precise formula.\n\nThe following first approach was suggested by Joseph Ayoub. There exists $U$ and $\\sM_U\\in \\D^\\proj(U)$ such that $M=j_U^*\\sM_U$. Let $\\sM=(j'_U)_* \\sM_U$ with $j'_U:U\\inj C$, so that $M=j^*\\sM$. For $x\\in C$, we have by \\eqref{eq9.1} and Theorem \\ref{t9.1}:\n\\[L^\\near_x(M,s)=\n\\begin{cases}\n\\zeta(i_x^*\\sM_U,s) &\\text{for $x\\in U$}\\\\\n\\zeta(\\Upsilon_x(M),s) &\\text{for all $x\\in C$}\n\\end{cases}\n\\]\nhence\n\\begin{multline*}\nL^\\near(M,s)= \\prod_{x\\in U}\\zeta(i^*_x\\sM_U,s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M),s)\\\\ \n=\\zeta(\\sM_U,s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M),s) \\\\\n= \\zeta(\\sM,s)\\times \\prod_{x\\notin U} \\zeta(i_x^*(j'_U)_*\\sM_U,s)^{-1} \\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M),s)\\\\\n= \\zeta(\\sM,s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M),s-1)\n\\end{multline*}\nby \\eqref{eq9.2}.\n\nOn the other hand, $\\sM^*:=\\uHom(\\sM,\\Z)=(j'_U)_!\\sM_U^*$, hence\n\\begin{multline*}\nL^\\near(M^*,1-s)= \\prod_{x\\in U}\\zeta(i^*_x\\sM_U^*,1-s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M^*),1-s)\\\\ \n=\\zeta(\\sM^*,1-s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M^*),1-s).\n\\end{multline*}\n\nBy \\cite[Th. 11.16]{ayoubetale}, we have\n\\[\\Upsilon_x(M^*)\\simeq \\Upsilon_x(M)^* \\]\nwhere the duality on the right hand side is still relative to $\\Z$, but in $\\D(\\kappa(x))$. Hence\n\\begin{multline*}\nL^\\near(M^*,1-s)=\\zeta(\\sM^*,1-s)\\times \\prod_{x\\notin U} \\zeta(\\Upsilon_x(M)^*,1-s)\\\\\n=(-q^{-s})^{\\chi(f_!\\sM)}\\det(F_{f_!\\sM})^{-1}\\zeta(\\sM,s)\\\\\n\\times \\prod_{x\\notin U} (-q_x^{1-s})^{\\chi(\\Upsilon_x(M))}\\det(F_{\\Upsilon_x(M)})^{-1}\\zeta(\\Upsilon_x(M),s-1)\n\\end{multline*}\nwhere $q_x = |\\kappa(x)|=q^{\\deg(x)}$. We finally get\n\n\\begin{thm}\\label{t9.2} One has\n\\[\\frac{L^\\near(M^*,1-s)}{L^\\near(M,s)}=A (-q)^{-Bs}\n\\]\nwith\n\\begin{align*}\nA&=(-q)^{\\sum\\limits_{x\\notin U} \\deg(x) \\chi(\\Upsilon_x(M))}\\left(\\det(F_{f_!\\sM})\\times \\prod_{x\\notin U} \\det(F_{\\Upsilon_x(M)})\\right)^{-1}\\\\\nB&=\\chi(f_!\\sM))+\\sum_{x\\notin U} \\deg(x) \\chi(\\Upsilon_x(M)).\\qed\n\\end{align*}\n\\end{thm}\n\n\\begin{qn}\\label{q10.1} Comparing with \\cite[Formulas (6) and (7)]{serre}, one would like to relate at least the constant $B$ to a \\emph{conductor} of $M$. This can be defined \\emph{via} the $l$-adic realisation; can one prove that the conductor thus obtained does not depend on $l$?\n\\end{qn}\n\n\n\n\\begin{thebibliography}{SGA4 1/2}\n\\bibitem{akos} Y. André, B. Kahn {\\it Nilpotence, radicauxet structures monoïdales} (with an appendix by P. O'SUllivan), Rendiconti Math. Univ. Padova {\\bf 108} (2002),108--291.\n\\bibitem{ayoub} J. Ayoub Les six op\\'erations de Grothendieck et le formalisme des cycles \\'evanescents dans le monde motivique, I, II, Ast\\'erisque {\\bf 314--315}, SMF, 2008. \n\\bibitem{ayoubetale} J. Ayoub {\\it La r\\'ealisation \\'etale et les op\\'erations de Grothendieck}, Ann. sci. Éc. norm. sup. {\\bf 47} (2014), 1--145.\n\\bibitem{bar-kahn} L. Barbieri-Viale, B. Kahn {\\it On the derived category of $1$-motives},\nAstérisque {\\bf 381} (2016), 1--254.\n\\bibitem{bondarko1} M. V. Bondarko Differential graded motives: weight complex, weight\nfiltrations and spectral sequences for realizations; Voevodsky versus Hanamura. J. Inst. Math.\nJussieu {\\bf 8} (2009), 39--97.\n\\bibitem{bondarko} M. V. Bondarko {\\it Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)}, J. of $K$-theory {\\bf 6} (2010), 387--504.\n\\bibitem{cisinski} D.-C. Cisinski {\\it Cohomological methods in intersection theory}, Homotopy theory and arithmetic geometry--motivic and Diophantine aspects, 49--105, Lect. Notes in Math. {\\bf 2292}, Springer, 2021.\n\\bibitem{cis-deg2} D.-C. Cisinski, F. D\\'eglise {\\it Étale motives}, Compositio Math. {\\bf 152} (2016) 556--666.\n\\bibitem{cis-deg} D.-C. Cisinski, F. D\\'eglise {\\it Triangulated categories of motives},\nSpringer Monographs in Math. {\\bf 2019}, xlii+406 pp.\n\\bibitem{corr} P. Colmez, J.-P. Serre Correspondance Grothendieck-Serre, Documents mathématiques {\\bf 2}, SMF, 2001. \n\\bibitem{WeilI} P. Deligne {\\it La conjecture de Weil, I}, Publ. Math. IH\\'ES {\\bf 43} (1974),\n273--307.\n\\bibitem{WeilII} P. Deligne {\\it La conjecture de Weil, II}, Publ. Math.\nIH\\'ES {\\bf 52} (1980), 137--252.\\bibitem{deligne} {\\it Le groupe fondamental de la droite projective moins trois points}, {\\it in} Galois groups over $\\Q$, Math. Sci. Res. Inst. Publ., {\\bf 16}, Springer, 1989, 79--297.\n\\bibitem{den} C. Deninger {\\it $L$-functions of mixed motives}, {\\it in} Motives (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. Symposia pure Math. {\\bf 55} (I), AMS, 1994, 517--525.\n\\bibitem{Ekedahl} T. Ekedahl {\\it On the adic formalism}, {\\it in} The Grothendieck\nFestschrift (P. Cartier et al., eds.), Vol. II, Birkh\\\"auser 1990, 197--218.\n\\bibitem{fpr} J.-M. Fontaine, B. Perrin-Riou {\\it Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$}, {\\it in} Motives (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. Symposia pure Math. {\\bf 55} (I), AMS, 1994, 599--706.\n\\bibitem{guletskii} V. Guletskii {\\it Zeta functions in triangulated categories}, Mat. Zametki {\\bf 87} (2010), 369--381; translation in\nMath. Notes {\\bf 87} (2010), 345--354.\n\\bibitem{schol-etal} T. Hemo, T. Richarz, J. Scholbach {\\it Constructible sheaves on schemes}, preprint, 2023, \\url{https://doi.org/10.48550/arXiv.2305.18131}.\n\\bibitem{motiftate} A. Huber, B. Kahn {\\it the slice filtration and mixed Tate motives},\nCompositio Math. {\\bf 142} (2006), 907--936.\n\\bibitem{illusie} L. Illusie {\\it Constructibilité générique et uniformité en $\\ell$}. Tunis. J. Math. {\\bf 4} (2022), 159--181.\n\\bibitem{ivorra} F. Ivorra {\\it R\\'ealisation $l$-adique des motifs triangul\\'es g\\'eom\\'etriques. I}, Doc. Math. {\\bf 12} (2007), 607--671.\n\\bibitem{jannsen} U. Jannsen Mixed motives and algebraic $K$-theory, Lect. Notes {\\bf 1400}, Springer, 1990.\n\\bibitem{dJ} A.J. de Jong {\\it Smoothness, semi-stability and alterations}, Publ. Math. IH\\'ES {\\bf 83} (1996), 51--93.\n\\bibitem{dJ2} A.J. de Jong {\\it Families of curves and alterations}, Ann. Inst. Fourier {\\bf 47} (1997), 599--621.\n\\bibitem{modq} B. Kahn {\\it Zeta functions and motives}, Pure Appl. Math. Quarterly {\\bf 5}\n(2009), 507--570 [2008]. \n\\bibitem{zetaLRegensburg} B. Kahn {\\it Zeta and $L$-functions of triangulated motives}, slides of a talk at the workshop \\emph{Algebraic cycles and $L$-functions}, Regensburg, Feb. 27 -- March 2, 2012, \\url{https://webusers.imj-prg.fr/~bruno.kahn/preprints/zetaL-Regensburg.pdf}.\n\\bibitem{ffihes} B. Kahn {\\it The full faithfulnees conjectures in characteristic $p$}, {\\it in} Autour des motifs (II), Panoramas et synthèses {\\bf 41}, SMF, 2013, 187--244.\n\\bibitem{adjoints} B. Kahn {\\it Motifs et adjoints}, Rendiconti Sem. Mat. Univ. Padova {\\bf 139} (2018), 77--128.\n\\bibitem{zetaL} B. Kahn Zeta and $L$-functions of varieties and motives, London Math. Soc. Lect. Notes Ser. {\\bf 462}, Cambridge Univ. press, 2018.\n\\bibitem{ks} B. Kahn, R. Sujatha {\\it Birational motives, I: pure birational motives}, Ann. K-Theory {\\bf 1} (2016), 379--440.\n\\bibitem{birat-tri} B. Kahn, R. Sujatha {\\it Birational motives, II: triangulated birational motives}, IMRN {\\bf 2017}, no. 22, 6778--6831.\n\\bibitem{karp} N. Karpenko {\\it Weil transfer for algebraic cycles}, Indag. Math. (N.S.) {\\bf\n11} (2000), 73--86.\n\\bibitem{kelly} S. Kelly Voevodsky motives and $l$dh-descent, Astérisque {\\bf 391}, SMF, 2017.\n\\bibitem{kleiman} S. Kleiman {\\it Motives}, Algebraic geometry, Oslo 1970, \nProc. Fifth Nordic Summer School in Math., Wolters-Noordhoff, Groningen,\n1972, 53--82.\n\\bibitem{may} J. P. May {\\it The additivity of traces in triangulated categories},\nAdv. Math. {\\bf 163} (2001), 34--73. \n\\bibitem{mvw} C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs {\\bf 2}, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006.\n\\bibitem{pacheco} A. Pacheco {\\it A note on relations between the zeta-functions of Galois coverings of curves over finite fields}, Canad. Math. Bulletin {\\bf 33} (1990), 282--285.\n\\bibitem{raynaud} M. Raynaud {\\it Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes}, Sém. Bourbaki 1964-66, exp. n° {\\bf 286}, 129--147, SMF.\n\\bibitem{riou} J. Riou {\\it Dualité de Spanier-Whitehead en géométrie algébrique}, C. R. Math. Acad. Sci. Paris {\\bf 340} (2005), no. 6, 431–436.\n\\bibitem{saa} N. Saavedra Rivano Cat\\'egories tannakiennes, Lect. Notes\nin Math. {\\bf 265}, Springer, 1972.\n\\bibitem{tsaito} T. Saito {\\it The characteristic cycle and the singular support of a constructible sheaf},\nInvent. Math {\\bf 207} (2) (2017), 597--695. Correction: \\emph{ibid.}, {bf 216} (3) (2019), 1005--1006. Correction 2: \\url{https://www.ms.u-tokyo.ac.jp/~t-saito/CCcor2.pdf}.\n\\bibitem{scholl} A. J. Scholl {\\it Remarks on special values of $L$ functions}, {\\it in} $L$-functions and arithmetic, London Math. Soc. Lect. Notes Ser. {\\bf 153}, Cambridge Univ. Press, 1991, 373--392.\n\\bibitem{cg} J.-P. Serre Cohomologie galoisienne (5th edition), Lect. Notes in Math. {\\bf 5}, Springer, 1994.\n\\bibitem{serre} J.-P. Serre {\\it Facteurs locaux des fonctions z\\^eta des vari\\'et\\'es\nalg\\'ebriques (d\\'efinitions et conjectures)}, S\\'em. DPP 1969/70, exp. {\\bf 19}.\n\\bibitem{NX} J.-P. Serre Lectures on $N_X(p)$, Research Notes in Math. {\\bf 11}, CRC Press, 2012.\n\\bibitem{serre-tate} J.-P. Serre, J. Tate {\\it Good reduction of abelian varieties}, Ann. of Math. {\\bf 88} (1968), 492--517.\n\\bibitem{suslininsep} A. Suslin {\\it Motivic complexes over nonperfect fields}, Ann. of $K$-theory {\\bf 2} (2017), 277--302.\n\\bibitem{ter} T. Terasoma {\\it Monodromy weight filtration is independent of $l$}, preprint, 1998, \\url{https://arxiv.org/abs/math/9802051}.\n\\bibitem{voetri} V. Voevodsky {\\it Triangulated categories of motives over a field}, {\\it in}\nCycles, transfers and motivic cohomology theories, Ann. of Math. Studies. {\\bf 143}, Princeton Univ. press, 2000, 188--238.\n\\begin{center}\n\\sc Acronyms\n\\end{center}\n\\bibitem[SGA1]{SGA1} Revêtements étales et groupe fondamental (SGA 1) (A. Grothendieck et al), revised ed., Doc. Math. {\\bf 3}, SMF, 2003.\n\\bibitem[SGA4 1/2]{SGA412} Cohomologie \\'etale (SGA 4 1/2) (P. Deligne et al),\nLect. Notes in Math. {\\bf 569}, Springer, 1977.\n\\bibitem[SGA5]{SGA5} Cohomologie \\'etale et fonctions L (SGA5) (A. Grothendieck et al), Lect.\nNotes in Math. {\\bf 589}, Springer, 1977.\n\\bibitem[SGA7]{SGA7} Groupes de monodromie en g\\'eom\\'etrie alg\\'ebrique\n(SGA7) (P. Deligne, N. Katz, \\'ed.), Vol. II, Lect. Notes in Math.\n{\\bf 340}, Springer, 1973.\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n"}}},{"rowIdx":98,"cells":{"id":{"kind":"string","value":"2412.08597v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08597v1"},"title":{"kind":"string","value":"Positive co-degree densities and jumps"},"full_text":{"kind":"string","value":"\\documentclass[12pt]{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{cite}\n\n\\usepackage{url}\n\n\n\\usepackage{caption}\n\n\n\n\n\\usepackage{amssymb, amsmath, 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to[out=200,in=330] (4) to[out=330,in=60,looseness=1.5] (1);\n\\draw[hyperedge] (2) to[out=120,in=230,,looseness=1.8] (3) to[out=230,in=300,looseness=1.8] (4) to[out=300,in=120,looseness=1.8] (2);\n\\end{tikzpicture}\n}}\n\n\n\n\\newcommand{\\ah}[1]{{\\color{blue}{Anna: #1}}}\n\\newcommand{\\cp}[1]{{\\color{red}{Cory: #1}}}\n\\newcommand{\\bl}[1]{{\\color{green!70!black}{Bernard: #1}}}\n\n\n\\definecolor{darkpastelgreen}{rgb}{0.01, 0.75, 0.24}\n\n\n\\title{Positive co-degree densities and jumps}\n\\author{J\\'ozsef Balogh \\footnote{Department of Mathematics, University of Illinois Urbana-Champaign, IL, USA\nE-mail: \\texttt{jobal@illinois.edu}.\nResearch supported in part by NSF grants RTG DMS-1937241 and FRG DMS-2152488, the Arnold O. Beckman Research Award (UIUC Campus Research Board RB 24012).} \\and \nAnastasia Halfpap\\footnote{Department of Mathematics, Iowa State University, Ames, IA, USA. E-mail: \\texttt{ahalfpap@iastate.edu}. Research supported by NSF DSM-2152490.} \n\\and \nBernard Lidick\\'y \\footnote{Department of Mathematics, Iowa State University, Ames, IA, USA. E-mail: \\texttt{lidicky@iastate.edu}. Research supported by NSF DSM-2152490 and Scott Hanna Professorship.} \n\\and\nCory Palmer \\footnote{Department of Mathematical Sciences, University of Montana. Email: \\texttt{cory.palmer@umontana.edu}. Research supported by a grant from the Simons Foundation \\#712036.}\n}\n\\date{}\n\n\\newcommand{\\oururl}{\\url{https://lidicky.name/pub/pco/}}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nThe \\textit{minimum positive co-degree} of a nonempty $r$-graph $H$, denoted by $\\delta_{r-1}^+(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S \\subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. Given a family $\\mathcal{F}$ of $r$-graphs, the \\textit{positive co-degree Tur\\'an function} $\\mathrm{co^+ex}(n,\\mathcal{F})$\nis the maximum of $\\delta_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\\mathcal{F}$. The \\textit{positive co-degree density} of $\\mathcal{F}$ is $\\gamma^+(\\mathcal{F}) = \\underset{n \\rightarrow \\infty}{\\lim} \\frac{\\mathrm{co^+ex}(n,\\mathcal{F})}{n}.$ While the existence of $\\gamma^+(\\mathcal{F})$ is proved for all families $\\mathcal{F}$, only few positive co-degree densities are known exactly. \n\nFor a fixed $r \\geq 2$, we call $\\alpha \\in [0,1]$ an \\textit{achievable value} if there exists a family of $r$-graphs $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) = \\alpha$, and call $\\alpha$ a \\textit{jump} if for some $\\delta > 0$, there is no family $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) \\in (\\alpha, \\alpha + \\delta)$. Halfpap, Lemons, and Palmer~\\cite{halfpap2024positive} showed that every $\\alpha \\in [0, \\frac{1}{r})$ is a jump. We extend this result by showing that every $\\alpha \\in [0, \\frac{2}{2r -1})$ is a jump. We also show that for $r = 3$, the set of achievable values is infinite, more precisely, $\\frac{k-2}{2k-3}$ for every $k \\geq 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.\n\n\\end{abstract}\n\n\\noindent \\emph{Keywords:} positive co-degree, hypergraph, Tur\\'an, jump, flag algebra.\\\\\n\\emph{MSC2020:} 05C35, 05C65.\n\n\\section{Introduction}\n\nAn $r$-\\textit{graph} is a hypergraph in which all hyperedges have size $r$. We often refer to the hyperedges of an $r$-graph as $r$-\\textit{edges}. Given a family of $r$-graphs $\\mathcal{F}$, the \\textit{Tur\\'an number} $\\mathrm{ex}(n,\\mathcal{F})$ is the maximum number of $r$-edges possible in an $n$-vertex $r$-graph that contains no member of $\\mathcal{F}$ as a subhypergraph. When $r = 2$, the function $\\mathrm{ex}(n,F)$ is well-studied and relatively well-understood. Given a set of $r$-graphs $\\mathcal{F}$, we define the \\textit{Tur\\'an density} of $\\mathcal{F}$ to be\n\\[ \\pi(\\mathcal{F}) := \\underset{n \\rightarrow \\infty}{\\lim} \\frac{\\mathrm{ex}(n,\\mathcal{F})}{\\binom{n}{r}}.\\]\nThe Erd\\H{o}s-Stone Theorem~\\cite{ErSt} as pointed out by Erd\\H{o}s-Simonovits~\\cite{ErSi} determines the Tur\\'an density of every $2$-graph as a function of its chromatic number.\n\n\\begin{theorem}[Erd\\H{o}s-Stone]\\label{ess}\n\nLet $F$ be a $2$-graph with $\\chi(F) = k$. Then\n$\\pi(F) = 1 - \\frac{1}{k-1}. $\n\n\\end{theorem}\nNote that an extension of Theorem~\\ref{ess} also claims that for every family $\\mathcal{F}$ of $2$-graphs, $\\pi(\\mathcal{F})$ is equal to the minimum of $1-1/(\\chi(F)-1)$ over $F \\in \\mathcal{F}$. While Theorem~\\ref{ess} does not give us perfect information about Tur\\'an numbers (in particular, for bipartite graphs $F$, it only demonstrates that $\\mathrm{ex}(n,F) = o(n^2)$), it yields a good ``approximate'' understanding of Tur\\'an numbers by fully describing Tur\\'an densities.\n\nFor $r \\geq 3$, we do not have an analogue to Theorem~\\ref{ess}, and much less is known about Tur\\'an densities.\nEven $\\pi(\\mathcal{F})$ could be smaller than \n$\\min\\{\\pi(F) : F \\in \\mathcal{F}\\}$, as observed by Balogh~\\cite{jozsituran}.\nNot only do we lack a general theory, but the Tur\\'an densities of many small $r$-graphs remain unknown, despite great effort. Famously, the Tur\\'an density of the tetrahedron $K_4^3$ is still undetermined. The difficulty of determining Tur\\'an densities for hypergraphs has motivated the study of various other hypergraph extremal functions, typically maximizing some variant of the minimum degree. In particular, given an $r$-graph $H$, the \\textit{co-degree} of a set $S \\in \\binom{V(H)}{r-1}$ is the number of $r$-edges containing $S$, and the {\\it minimum co-degree} of $H$, denoted by $\\delta_{r-1}(H)$, is the smallest co-degree realized by an $(r-1)$-set contained in $V(H)$. The \\textit{co-degree Tur\\'an function} of a family of $r$-graphs $\\mathcal{F}$, denoted by $\\mathrm{coex}(n,\\mathcal{F})$, is the largest possible minimum co-degree of an $n$-vertex $r$-graph containing no member of $\\mathcal{F}$ as a subhypergraph.\n\nMubayi and Zhao \\cite{Mubayicode} showed that the \\textit{co-degree density}\n\\[\\gamma(\\mathcal{F}) := \\underset{n \\rightarrow \\infty}{\\lim} \\frac{\\mathrm{coex}(n,\\mathcal{F})}{n}\\]\nexists for every family of $r$-graphs $\\mathcal{F}$, and studied the general behavior of $\\mathrm{coex}(n,\\mathcal{F})$. Note that for every family of $2$-graphs $\\mathcal{F}$ we have $\\gamma(\\mathcal{F}) = \\pi(\\mathcal{F})$; however, for $r \\geq 3$, co-degree Tur\\'an problems are not equivalent to Tur\\'an problems, and co-degree density does not behave in the manner suggested by Theorem~\\ref{ess}. We first define the notion of a \\textit{jump} in density.\n\n\\begin{definition}\nFix $r \\geq 2$. Suppose $\\varphi$ is a function that maps families of $r$-graphs to $[0,1]$. We say that $\\alpha \\in [0, 1)$ is a $\\varphi$-{\\it jump} if there exists $\\delta \\in (0, 1 - \\alpha)$ such that for no family $\\mathcal{F}$ of $r$-graphs, $\\varphi(\\mathcal{F}) \\in (\\alpha, \\alpha + \\delta)$.\n\\end{definition}\n\nWhile Theorem~\\ref{ess} shows that co-degree density (and Tur\\'an density) jumps everywhere when $r = 2$, Mubayi and Zhao showed that co-degree density does not jump when $r \\geq 3$.\n\n\\begin{theorem}[Mubayi-Zhao \\cite{Mubayicode}]\\label{mzjumps}\n\nFor $r \\geq 3$, no $\\alpha \\in [0,1)$ is a $\\gamma$-jump.\n\n\\end{theorem}\n\nThis co-degree phenomenon was further investigated in \\cite{piga2023smallcodegree,ding2023vanishingcodegree}.\n\nWe remark that Theorem~\\ref{mzjumps} suggests a substantial difference in behavior between Tur\\'an and co-degree Tur\\'an problems for hypergraphs. We know for every $r \\geq 3$ that every $\\alpha \\in [0, r!/r^r)$ is a $\\pi$-jump. On the other hand, $\\pi$ is also known to \\textit{not} jump in infinitely many places for $r \\geq 3$; see~\\cite{hg-no-jump, do-jump}.\n\n\n\nThe \\textit{minimum positive co-degree} of an $r$-graph $H$ is the largest integer $k$ such that, whenever $S \\in \\binom{V(H)}{r-1}$ is contained in some $r$-edge of $H$, then $S$ is contained in at least $k$ $r$-edges of $H$. \nThe edgeless $r$-graph is defined to have positive co-degree zero. \nWe denote the minimum positive co-degree of $H$ by $\\delta_{r-1}^+(H)$. \nWe define the \\textit{positive co-degree Tur\\'an number}, denoted by $\\mathrm{co^+ex}(n,\\mathcal{F})$, to be the largest possible minimum positive co-degree of an $n$-vertex $r$-graph containing no member of $\\mathcal{F}$ as a subhypergraph.\n\n\nBalogh, Lemons, and Palmer~\\cite{BLP} introduced the minimum positive co-degree as an alternative notion of minimum degree in $r$-graphs. Since then this parameter has already been studied from several angles. The concept of $\\mathrm{co^+ex}(n,\\mathcal{F})$ was recently introduced by Halfpap, Lemons, and Palmer~\\cite{halfpap2024positive}. The investigation of minimum positive co-degree as an extremal parameter is partially motivated by the admissibility of constructions that mimic the extremal graphs for classical questions. For example, given an $r$-graph $H$, the \\textit{t-blow-up} $H[t]$ of $H$ is the $r$-graph obtained by replacing each vertex $v_i \\in V(H)$ with a class $V_i$ of $t$ vertices, where a set of $r$-vertices spans an $r$-edge if and only if they belong to $r$ distinct classes of $H[t]$ which correspond to an $r$-edge in $H$. In classical Tur\\'an theory, graph blow-ups yield extremal or nearly extremal constructions for all non-bipartite forbidden graphs. Blow-ups also occur as extremal examples for other types of thresholds---for instance, one of the constructions demonstrating the tightness of Dirac's Theorem is a slightly unbalanced blow-up of an edge (i.e., a complete bipartite graph). \n\nFor $r \\geq 3$ and every $r$-graph $H$ a sufficiently large blow-up of $H$ has minimum co-degree $0$, which means that even after adding $o(n^3)$ hyperedges, blow-ups will not provide extremal constructions for minimum co-degree density problems. However, $H[t]$ inherits the \\textit{positive} co-degree properties of $H$. Thus, blow-ups (as well as other constructions with co-degree $0$ sets, such as $r$-graphs containing large strongly independent sets or multiple components) are potential extremal examples for positive co-degree problems. Previous results suggest that extremal constructions for positive co-degree problems in fact \\textit{do} often look analogous to classical extremal constructions. See \\cite{halfpap2024positive} for extremal constructions avoiding some small $3$-graphs, and \\cite{halfpapmagnanspanning} on positive co-degree analogs of Dirac's Theorem, for which the extremal constructions also naturally generalize the graph extremal examples (and have minimum co-degree 0). Due to the expanded range of potential constructions, $\\mathrm{co^+ex}(n,\\mathcal{F})$ and $\\mathrm{coex}(n,\\mathcal{F})$ are generally not equal, and they appear to behave differently. \n\nDefine the \\textit{positive co-degree density} of a family of $r$-graphs $\\mathcal{F}$ as the limit\n\\[\\gamma^+(\\mathcal{F}) := \\underset{n \\rightarrow \\infty}{\\lim} \\frac{\\mathrm{co^+ex}(n,\\mathcal{F})}{n}. \\]\nThe existence of $\\gamma^+(F)$ was established by Halfpap, Lemons, and Palmer~\\cite{halfpap2024positive} via a constructive argument, which can be generalized to finite families $\\mathcal{F}$.\nPikhurko~\\cite{pikhurko2023limitpositiveelldegreeturan} gave a probabilistic argument establishing that $\\gamma^+(\\mathcal{F})$ exists for all families $\\mathcal{F}$. \n\n\\begin{prop}[Halfpap-Lemons-Palmer~\\cite{halfpap2024positive}]\\label{first jump}\n\nFix $r \\geq 2$ and let $\\mathcal{F}$ be a family of $r$-graphs. Then\n\\[\\gamma^+(\\mathcal{F}) \\in \\{0\\} \\cup \\left[\\frac{1}{r}, 1\\right].\\]\n\\end{prop}\n\nIn other words, every $\\alpha \\in [0,1/r)$ is a $\\gamma^+$-jump. \nProposition~\\ref{first jump} describes behavior similar to that of the classical Tur\\'an density. Every $r$-graph $F$ that is contained in some blow-up of an $r$-edge can be shown to be ``degenerate'', having $\\mathrm{co^+ex}(n,F) = o(n)$, so $\\gamma^+(F) = 0$. On the other hand, if $F$ is not contained in any blow-up of an $r$-edge, then $\\gamma^+(F) \\geq \\frac{1}{r}$, since the balanced $n$-vertex blow-up of an $r$-edge has minimum positive co-degree approximately $\\frac{n}{r}$. \nAn $r$-graph $F$ is \\emph{$k$-partite} if there is a partition of $V(F)$ into $k$ classes such that each edge intersects each part at most once. \n\n\nAlthough Proposition~\\ref{first jump} suggests that $\\gamma$ and $\\gamma^+$ exhibit fundamentally different beha\\-viors, it is not clear what behavior to expect from $\\gamma^+$. Currently, we know the exact value $\\gamma^+(F)$ only for very few $3$-graphs $F$. Halfpap, Lemons, and Palmer~\\cite{halfpap2024positive} determined the values of $\\gamma^+(F)$ for many small $3$-graphs $F$, and bounded $\\gamma^+(F)$ in some other instances. \nVarious authors have reported improvements on several of these initial bounds, with the current best known values summarized in Table~\\ref{previous bounds}. For comparison, the best-known bounds on $\\pi$ and $\\gamma$ for these $3$-graphs are also provided.\nGraphs not defined in Table~\\ref{fig-3graphs} are\ndefined by their edge sets as\n\\begin{align*}\nK_4^{3-} &= \\{123, 124, 134\\}, \\quad F_5 = \\{123, 124, 345\\},\\quad\n\\mathbb{F} = \\{ 123, 345, 156, 246, 147, 257, 367 \\},\\\\\nC_\\ell &= \\{123, 234, 345,\\ldots, (\\ell-2)(\\ell-1)\\ell, (\\ell-1)\\ell 1, \\ell 12\\},\n\\quad\nC_\\ell^- = C_\\ell - \\{\\ell 12\\}.\n\\end{align*}\nSee~\\cite{halfpap2024positive,l2norm} for more details about $3$-graphs in Table~\\ref{previous bounds}. Note that the $3$-graph denoted here as $K_4^{3-}$ is often called $K_4^{-}$ in the literature; we adopt the notation $K_4^{3-}$ to distinguish this $3$-graph from the 2-graph obtained by deleting an edge from $K_4$, which we denote by $K_4^-$.\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l||c|c|c|c|c|c|}\n\\hline\n $F$ & $\\leq \\pi(F) $ & $\\pi(F) \\leq $ & $\\leq \\gamma(F) $ & $\\gamma(F) \\leq $ & $\\leq \\gamma^+(F) $ & $\\gamma^+(F) \\leq $ \\\\\n \\hline\n \\hline\n{$K_4^{3-}$} & 2/7 \\cite{K43-extremalfrankl} & 0.28689 \\cite{flagmatic} & $1/4$ \\cite{codegreeconj} & $1/4$ \\cite{FalgasK4-} & $ 1/3 $ \\cite{halfpap2024positive}& $ 1/3 $ \\cite{halfpap2024positive}\\\\\n{$F_5$} & 2/9 \\cite{Bollobascancellative} & 2/9 \\cite{F5Frankl} & 0 \\cite{l2norm} & 0 \\cite{l2norm} & $1/3$ \\cite{halfpap2024positive}& $1/3$ \\cite{halfpap2024positive}\\\\\n{$F_{3,2}$} & 4/9 \\cite{F32Mubayi} & 4/9 \\cite{F32furedi} & 1/3 \\cite{codF32falgas} & 1/3 \\cite{codF32falgas} & $1/2$ \\cite{halfpap2024positive} & $1/2$ \\cite{halfpap2024positive}\\\\\n{$\\mathbb{F}$} & 3/4 \\cite{VeraSos} & 3/4 \\cite{FanoFuredi} & 1/2 \\cite{MubayiFano} & 1/2 \\cite{MubayiFano} & 2/3 \\cite{halfpap2024positive} & 2/3 \\cite{halfpap2024positive}\\\\\n{$K_4^3$} &5/9 \\cite{MR177847} & 0.5615 \\cite{BaberTuran} & 1/2 \\cite{MR1829685} & 0.529 \\cite{l2norm} & $1/2$ \\cite{halfpap2024positive} & 0.543 \\cite{volec} \\\\ \n\n{$F_{3,3}$} & 3/4 \\cite{F32Mubayi} & 3/4 \\cite{F32Mubayi} & 1/2 \\cite{l2norm} & 0.604 \\cite{l2norm} & 3/5 \\cite{halfpap2024positive}& 0.616 \\\\\n{$C_{5}$} & $2\\sqrt{3} - 3$ \\cite{F32Mubayi} & 0.46829 \\cite{flagmatic} & 1/3 \\cite{l2norm} & 0.3993 \\cite{l2norm} & $1/2$ \\cite{halfpap2024positive} & 1/2 \\cite{Wu} \\\\\n{$C_{7}$} &$2\\sqrt{3} - 3$ \\cite{F32Mubayi} &\n0.464186 & 1/3 \\cite{l2norm} & 0.371 & 1/2 \\cite{halfpap2025} & 1/2 \\cite{halfpap2025} \\\\\n{$C_{5}^-$} &1/4 \\cite{F32Mubayi} & 1/4\\cite{lidicky2024c5-} & 0 \\cite{l2norm} & 0 \\cite{piga} & 1/3 \\cite{halfpap2024positive} & 1/3 \\cite{Wu} \\\\\n\n\n{$J_4$} & $1/2$ \\cite{DaisyBollobas} & $0.50409$ \\cite{flagmatic} & $1/4$ \\cite{l2norm} & $0.473$ \\cite{l2norm} & $4/7$ \\cite{halfpap2024positive} & 4/7 \\\\\n\n{$F_{4,2}$} & $4/9$ & $0.4933328$ & $1/3$ & $0.4185 $ & $3/5$ & $3/5$ \\\\\n\\hline\n\\end{tabular}\n\\captionof{table}{Best-known density bounds for $\\pi, \\gamma$, and $\\gamma^+$.}\\label{previous bounds}\n\\end{center}\n\\end{table}\n\nKamčev, Letzter, and Pokrovskiy~\\cite{Kamcev2024} proved that the Turán density of longer tight cycles $C_\\ell$ is $2\\sqrt{3} - 3$, when $\\ell$ is not multiple of three and sufficiently large (when $\\ell $ is divisible by three, then $C_\\ell$ is $3$-partite, hence its Tur\\'an density is $0$).\nSimilarly, Balogh and Luo~\\cite{balogh2024turandensitylongtight}\nproved for $\\ell$ sufficiently large and not divisible by $3$ that the Tur\\'an density of $C_\\ell^-$ is $1/4$.\nRecently, this was proved for every $\\ell\\ge 5$ by Lidick\\'y, Mattes, and Pfender~\\cite{lidicky2024c5-}.\nThat the co-degree density is $1/3$ for tight cycles of length at least $10$ and not divisible by $3$ was proved by Piga, Sanhueza-Matamala, and Schacht~\\cite{piga2024codegreeturandensity3uniform} and Ma~\\cite{ma2024codegreeturandensity3uniform}.\nOn the other hand, the fact that the co-degree density of $C_\\ell^-$ is $0$ is due to Piga, Sales, and Sch\\\"ulke~\\cite{piga2023smallcodegree}.\n\n\nGiven $r \\geq 2$, we call $\\alpha \\in [0,1]$ an \\textit{achievable value} (for $\\gamma^+$) if there exists a family $\\mathcal{F}$ of $r$-graphs such that $\\gamma^+(\\mathcal{F}) = \\alpha$. \nBefore our article, the only known achievable values of $\\gamma^+$ for $r=3$ were $0,1/3, 1/2,$ and $2/3$. \n\nOur goal is to understand the positive co-degree density by demonstrating additional $\\gamma^+$-jumps for every $r$, as well as by expanding the known list of achievable values of $\\gamma^+$ for $r =3$.\nOur paper is organized as follows. In Section~\\ref{new results}, we summarize our main results. In Section~\\ref{prelims}, we state some additional definitions and lemmas which will be used in our proofs. In Section~\\ref{proofs}, we demonstrate further $\\gamma^+$-jumps for every $r$, and for $r =3$ establish an infinite set of achievable values for $\\gamma^+$. In Section~\\ref{J4}, we use flag algebras to exactly determine $\\gamma^+$ for two additional $3$-graphs, hence adding two values to the list of achievable values of $\\gamma^+$. Finally, in Section~\\ref{conclusion}, we have some concluding remarks and list a wide variety of open problems in the area.\n\n\\section{New results}\\label{new results}\n\nOur first main theorem extends the range of jumps described in Proposition~\\ref{first jump}. We will need the following definition.\nAn \\emph{$r$-triangle}, denoted by $T^r$, is an $r$-graph with $r+1$ vertices and three $r$-edges. Notice that $T^r$ can be obtained from the $2$-graph triangle $T^2$ by adding $r-2$ vertices and including them to each of the three edges.\nSuch hypergraphs are sometimes called daises in the literature.\nThe $T^2$ part of a $T^r$ is called the \\emph{base} of the $T^r$.\nNotice that $T^3$ is the same as $K_4^{3-}$.\n\n\\begin{restatable}{theorem}{secondjump}\\label{general jumps}\nLet $\\mathcal{F}$ be a family of $r$-graphs for $r \\geq 2$. Then \n\n\\[ \\gamma^+(\\mathcal{F}) \\in \\left\\{0, \\frac{1}{r}\\right\\} \\cup \\left[\\frac{2}{2r-1}, 1\\right]. \\]\nThus, every $\\alpha \\in[0, \\frac{2}{2r-1})$ is a $\\gamma^+$-jump.\n\nMoreover, $\\gamma^+(\\mathcal{F}) = 0$ if and only if some member of $\\mathcal{F}$ is $r$-partite, and $\\gamma^+(\\mathcal{F}) = \\frac{1}{r}$ if and only if no member of $\\mathcal{F}$ is $r$-partite, but some member of $\\mathcal{F}$ is contained in a blow-up of some $T^r$. \n\\end{restatable}\n\nFor $r=3$, we also provide an infinite set of achievable values for $\\gamma^+$, based on forbidden families involving the following $3$-graphs. For $k \\geq 3$, let $J_k$ be the $(k+1)$-vertex $3$-graph, on vertex set $[k+1]$, with $3$-edges of the form $1ij$ for every $i , j \\in \\{2, \\dots, k+1\\}$. Let $K_4^3$ denote the complete $4$-vertex $3$-graph, and let $F_{3,2}$ denote the $5$-vertex $3$-graph, on vertex set $\\{1,2,3,4,5\\}$, with edge set $\\{123, 124, 125, 345\\}$. See Table~\\ref{fig-3graphs} for an illustration of these and other relevant $3$-graphs. \n\n\n\\begin{figure}\n\\begin{center}\n \\begin{tabular}{ | l|| c | Sc | Sc | }\n \\hline\n $K_4^3$ \n & \n\\vc{\\begin{tikzpicture}\n\\outercycle{5}{4}\n\\drawhyperedge{0}{4}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{4}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{4}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{2}{2}\n\\drawhypervertex{3}{2}\n\\drawhyperedge{3}{4}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\end{tikzpicture} \n} \n &\n \\vc{\n\\begin{tikzpicture}[scale=0.55]\n\\foreach \\i in {1,2,3,4}{\n\\draw (90*\\i-45:1.5) coordinate(\\i);\n}\n\n\\foreach \\r in {0-45,90-45,180-45,270-45}{\n\\begin{scope}[rotate=\\r]\n\\draw[hyperedge] \n(0:1.5) to[out=140,in=275,looseness=1.2] (90:1.5) to[out=265,in=40,looseness=1.2] (180:1.5) to[out=35,in=145,looseness=1.2] (0:1.5) \n;\n\\end{scope}\n}\n\n\\draw \n(1) node[vtx,label=right:{\\tiny 1}]{}\n(2) node[vtx,label=left:{\\tiny 2}]{}\n(3) node[vtx,label=left:{\\tiny 3}]{}\n(4) node[vtx,label=right:{\\tiny 4}]{}\n;\n\\end{tikzpicture}\n} \n \\\\\n \\hline\n$F_{3,2}$ &\n\\vc{\\begin{tikzpicture}\\outercycle{6}{5}\n\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{3}{1}\n\\drawhypervertex{4}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{3}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhypervertex{4}{3}\n\\end{tikzpicture} \n}\n&\n\\vc{\n\\begin{tikzpicture}\n\\draw\n(0,-0.7) coordinate(1) node[vtx,label=left:{\\tiny $1$}](a){}\n(0,0) coordinate(2) node[vtx,label=below:{\\tiny $2$}](b){}\n(0,0.7) coordinate(3) node[vtx,label=left:{\\tiny $3$}](c){}\n(1,-0.5) coordinate(4) node[vtx,label=right:{\\tiny $5$}](d){}\n(1,0.5) coordinate(5) node[vtx,label=right:{\\tiny $4$}](d){}\n;\n\\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend right] (1);\n\\draw[hyperedge] (1) to[out=0,in=190] (4) to[out=190,in=250] (5) to[out=250,in=0] (1);\n\\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (2);\n\\draw[hyperedge] (3) to[out=0,in=120,looseness=1.2] (4) to[out=120,in=170,looseness=1.2] (5) to[out=170,in=0,looseness=1.2] (3);\n\\end{tikzpicture}\n}\n\\\\\n\\hline\n$F_{4,2}$ &\n\\vc{ \\begin{tikzpicture}[flag_pic]\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{2}{2}\n\\drawhypervertex{3}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{4}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{1}{4}\n\\drawhypervertex{4}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{4}{5}\n\\drawhypervertex{5}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{3}{6}\n\\drawhypervertex{4}{6}\n\\drawhypervertex{5}{6}\n\\end{tikzpicture} } \n&\n\\vc{\n\\begin{tikzpicture}\n\\draw\n(0,-0.7) coordinate(1) node[vtx,label=below:{\\tiny $1$}](a){}\n(0,-0.2) coordinate(2) node[vtx,label=below:{\\tiny $2$}](b){}\n(0, 0.2) coordinate(6) node[vtx,label=above:{\\tiny $3$}](b){}\n(0,0.7) coordinate(3) node[vtx,label=left:{\\tiny $4$}](c){}\n(1,-0.5) coordinate(4) node[vtx,label=right:{\\tiny $6$}](d){}\n(1,0.5) coordinate(5) node[vtx,label=right:{\\tiny $5$}](d){}\n;\n\\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend right] (1);\n\\draw[hyperedge] (1) to[bend left=40] (6) to[bend left=40] (3) to[bend right=40] (1);\n\\draw[hyperedge] (1) to[bend left=70,looseness=2] (2) to[bend left=70,looseness=2] (6) to[bend right=70,looseness=2] (1);\n\\draw[hyperedge] (1) to[out=0,in=190] (4) to[out=190,in=250] (5) to[out=250,in=0] (1);\n\\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (2);\n\\draw[hyperedge] (6) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (6);\n\\draw[hyperedge] (3) to[out=0,in=120,looseness=1.2] (4) to[out=120,in=170,looseness=1.2] (5) to[out=170,in=0,looseness=1.2] (3);\n\\end{tikzpicture}\n} \n\\\\\\hline\n$F_{3,3}$\n &\n\\vc{\\begin{tikzpicture}\\outercycle{7}{6}\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{3}{1}\n\\drawhypervertex{4}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{3}{2}\n\\drawhypervertex{5}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{4}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{1}{4}\n\\drawhypervertex{3}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{3}{5}\n\\drawhypervertex{5}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{4}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{2}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{2}{8}\n\\drawhypervertex{3}{8}\n\\drawhypervertex{5}{8}\n\\drawhyperedge{9}{6}\n\\drawhypervertex{2}{9}\n\\drawhypervertex{4}{9}\n\\drawhypervertex{5}{9}\n\\end{tikzpicture} \n}\n &\n\\vc{\n\\begin{tikzpicture}\n\\draw\n(0,-0.7) coordinate(1) node[vtx,label=left:{\\tiny $3$}](a){}\n(0,0) coordinate(2) node[vtx,label=left:{\\tiny $2$}](b){}\n(0,0.7) coordinate(3) node[vtx,label=left:{\\tiny $1$}](c){}\n(1,-0.7) coordinate(4) node[vtx,label=right:{\\tiny $6$}](d){}\n(1,0.0) coordinate(5) node[vtx,label=right:{\\tiny $5$}](d){}\n(1,0.7) coordinate(6) node[vtx,label=right:{\\tiny $4$}](d){}\n;\n\\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend right] (1);\n\\draw[hyperedge] (1) to[out=0,in=190] (4) to[out=190,in=250] (5) to[out=250,in=0] (1);\n\\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (2);\n\\draw[hyperedge] (3) to[out=-60,in=120,looseness=1.2] (4) to[out=120,in=170,looseness=1.2] (5) to[out=170,in=-60,looseness=1.2] (3);\n\\draw[hyperedge] (1) to[out=60,in=190] (5) to[out=190,in=250] (6) to[out=250,in=60] (1);\n\\draw[hyperedge] (2) to[out=0,in=140] (5) to[out=140,in=210,looseness=1.5] (6) to[out=210,in=0] (2);\n\\draw[hyperedge] (3) to[out=0,in=120,looseness=1.2] (5) to[out=120,in=170,looseness=1.2] (6) to[out=170,in=0,looseness=1.2] (3);\n\\draw[hyperedge] (1) to[out=10,in=170,looseness=1.5] (4) to[out=170,in=190] (6) to[out=190,in=10] (1);\n\\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (6) to[out=210,in=0] (2);\n\\draw[hyperedge] (3) to[out=-10,in=120,looseness=1.2] (4) to[out=120,in=190,looseness=1.2] (6) to[out=190,in=-10,looseness=1.2] (3);\n\\end{tikzpicture}\n}\n \\\\\n \\hline\n $J_4$\n &\n\\vc{\\begin{tikzpicture}\\outercycle{6}{5}\n\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhyperedge{4}{5}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{5}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{3}{5}\n\\drawhypervertex{4}{5}\n\\end{tikzpicture} \n} \n &\n\\vc{\n\\begin{tikzpicture}[scale=0.75]\n\\clip (-2.5,-1.1) rectangle (1.5,1.1);\n\\draw\n(-2,0) coordinate(1) node[vtx,label=left:{\\tiny $1$}](b){}\n(45:1) coordinate(2) node[vtx,label=right:{\\tiny $2$}](c){}\n(135:1) coordinate(3) node[vtx,label=left:{\\tiny $3$}](d){}\n(225:1) coordinate(4) node[vtx,label=left:{\\tiny $4$}](d){}\n(315:1) coordinate(5) node[vtx,label=right:{\\tiny $5$}](d){}\n;\n\\draw[hyperedge] (1) to[out=70,in=140,looseness=0.9] (2) to[out=140,in=60,looseness=0.8]\n(3) to[out=120,in=70,looseness=0.8] (1);\n\\draw[hyperedge] (1) to[out=0,in=250,looseness=1.3] (3) to[out=250,in=110] \n(4) to[out=110,in=0,looseness=1.3] (1);\n\\draw[hyperedge] (1) to[out=-70,in=240,looseness=0.8] (4) to[out=300,in=220,looseness=0.8] \n(5) to[out=220,in=-70,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=0,in=110,looseness=0.7] (5) to[out=110,in=250,looseness=1.1] \n(2) to[out=250,in=0,looseness=0.7] (1);\n\\draw[hyperedge] (1) to[out=-25,in=205,looseness=0.8] (2) to[out=205,in=110,looseness=0.8] \n(4) to[out=110,in=-25,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=25,in=250,looseness=0.9] (3) to[out=250,in=155,looseness=0.8] \n(5) to[out=155,in=25,looseness=0.8] (1);\n\\draw[line width = 0.5pt]\n(3) to[out=250,in=155,looseness=0.8] (5)\n(2) to[out=205,in=110,looseness=0.8] (4)\n(5) to[out=110,in=250,looseness=1.1] (2)\n(4) to[out=300,in=220,looseness=0.8] (5)\n(3) to[out=250,in=110] (4)\n(2) to[out=140,in=60,looseness=0.8] (3)\n;\n\\end{tikzpicture}\n} \n \\\\\n \\hline\n $J_5$\n &\n\\vc{\\begin{tikzpicture}\\outercycle{7}{6}\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{3}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{4}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{0}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{0}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{0}{8}\n\\drawhypervertex{3}{8}\n\\drawhypervertex{5}{8}\n\\drawhyperedge{9}{6}\n\\drawhypervertex{0}{9}\n\\drawhypervertex{4}{9}\n\\drawhypervertex{5}{9}\n\\end{tikzpicture} \n}\n&\n\\vc{\n\\begin{tikzpicture}\n\\draw\n(-2,0) coordinate(1) node[vtx,label=left:{\\tiny $1$}](1){}\n\\foreach \\x in {2,...,6}\n{\n(-144+72*\\x:1) coordinate(\\x) node[vtx](v\\x){}\n}\n;\n\\draw[hyperedge,opacity=0.2] (1)--(2)--(3)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(2)--(4)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(2)--(5)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(2)--(6)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(3)--(4)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(3)--(5)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(3)--(6)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(4)--(5)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(4)--(6)--(1);\n\\draw[hyperedge,opacity=0.2] (1)--(5)--(6)--(1);\n\\draw [line width = 0.5pt] (2)--(3)--(4)--(5)--(6)--(2)--(4)--(6)--(3)--(5)--(2) ;\n\\draw (-2,0) coordinate(1) node[vtx](1){}\n\\foreach \\x in {2,...,6}\n{\n(-144+72*\\x:1) coordinate(\\x) node[vtx](v\\x){}\n};\n\\end{tikzpicture}\n}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\captionof{table}{Small 3-uniform hypergraphs.}\\label{fig-3graphs}\n\\end{figure}\n\n\n\n\n\\begin{restatable}{theorem}{densities}\\label{densities}\n\nFor every $k \\geq 4$,\n\\[ \\gamma^+(\\{K_4^3, F_{3,2}, J_k\\}) = \\frac{k-2}{2k-3}.\\]\n\n\n\\end{restatable}\n\nWe also investigate whether the densities exhibited by Theorem~\\ref{densities} can be achieved by forbidding a single $3$-graph. For $k = 4$, we have $\\gamma^+(\\{K_4^3, F_{3,2}, J_4\\}) = \\frac{2}{5}$.\nLet $F_1$ be a 7-vertex 3-graph \nwith edges $\\{125, 135, 235, 126, 146, 246, 347 \\}$,\nsee Figure~\\ref{F1F2}.\nWe show that $F_1$ has positive co-degree density $\\frac{2}{5}$.\n\n\n\\begin{restatable}{theorem}{singlegraph}\\label{single graph}\n\n$\\gamma^+(F_1) = \\frac{2}{5}$.\n\n\\end{restatable}\n\n\n\n\nFinally, we utilize flag algebras to determine the positive co-degree densities of two graphs, which exhibit new achievable values of $\\gamma^+$ larger than $\\frac{1}{2}$. We determine $\\gamma^+(J_4)$, and show that the (asymptotic) extremal construction is the blow-up of the complement of the Fano plane. \nNotice that $J_4$ is a 3-daisy; see~\\cite{ellis2024daisies} for recent progress on Tur\\'an densities of $r$-daisies.\n\\begin{restatable}{theorem}{singlegraphJfour}\\label{J4}\n$\\gamma^+(J_4) = \\frac{4}{7}$.\n\\end{restatable}\n\nWe introduce another $3$-graph, which will have a different new density. \nLet $F_{4,2}$ be the $6$-vertex $3$-graph with edges $\\{123, 124, 134, 156, 256, 356, 456 \\}$, depicted in Table~\\ref{fig-3graphs}.\nNote that in $F_{4,2}$ the common neighborhood of 5 and 6 is $\\{1,2,3,4\\}$, and $\\{1,2,3,4\\}$ spans a $K_4^{3-}$. We determine $\\gamma^+(F_{4,2})$, and show that its (asymptotic) extremal construction is the balanced blow-up of $K_5^3$.\n\n\\begin{restatable}{theorem}{Fdensity}\\label{F42density}\n$\\gamma^+(F_{4,2}) = \\frac{3}{5}$.\n\\end{restatable}\n\nWe also include some non-tight results obtained using flag algebras. \n\\begin{restatable}{theorem}{FlagDensity}\\label{thm:FlagDensity}\nThe following bounds hold.\n\\[\n\\pi(F_{4,2}) \\leq 0.4933327,\n\\quad\n\\gamma(F_{4,2}) \\leq 0.4185,\n\\quad\n\\gamma^+(F_{3,3}) \\leq 0.616.\n\\]\n\\end{restatable}\n\nThe lower bounds in Table~\\ref{previous bounds} for $\\pi(F_{4,2})$ and $\\gamma(F_{4,2})$ come from the lower bound constructions for $F_{3,2}$.\n\n\\section{Preliminaries}\\label{prelims}\n\nWe begin by stating some results related to supersaturation and a hypergraph removal lemma. The hypergraph removal lemma states that an $r$-graph containing only few copies of some subhypergraph $F$ can be made $F$-free by the deletion of only few $r$-edges. For a discussion of removal lemmas, including the below formulation, see~\\cite{ConlonFox}.\n\n\\begin{lemma}\\label{hypergraph removal} \n Fix $\\alpha > 0$ and let $F$ be an $r$-graph. There exists $\\delta > 0$ such that if $H$ is an $n$-vertex $r$-graph containing at most $\\delta n^{|V(F)|}$ copies of $F$, then there exists $E' \\subset E(H)$ such that $|E'| \\leq \\alpha n^r$ and $H - E'$ is $F$-free.\n\n\\end{lemma}\n\nAlthough an $r$-graph with $o(n^{|V(F)|})$ copies of $F$ can be made $F$-free by deleting $o(n^r)$ $r$-edges, it is not obvious that the deletion would change the minimum positive co-degree by only $o(n)$. The following ``clean-up'' lemma due to Halfpap, Lemons, and Palmer~\\cite{halfpap2024positive} allows us to apply the hypergraph removal lemma to minimum positive co-degree problems. Roughly, this lemma guarantees that any positive co-degree drop arising from the deletion of a small set of $r$-edges can be mitigated by the deletion of another set of $r$-edges.\n\n\n\\begin{lemma}[Halfpap-Lemons-Palmer~\\cite{halfpap2024positive}]\\label{positive codegree cleanup} \n\nLet $H$ be an $n$-vertex $r$-graph and fix $0< \\varepsilon <1$ small enough that $(r+1)! \\varepsilon^{1/{2^{r-1}}}n^r < |E(H)|$. Let $H_1$ be a subhypergraph of $H$ obtained by the deletion of at most $\\varepsilon n^r$ $r$-edges. Then $H_1$ has a subhypergraph $H_2$ \nwith $\\delta_{r-1}^+(H_2) \\geq \\delta_{r-1}^+(H) - 2^r r! \\varepsilon^{1/{2^{r-1}}}n$.\n\n\\end{lemma}\n\nIn practice it is not difficult to fulfill\nthe condition in Lemma~\\ref{positive codegree cleanup} that $(r+1)! \\varepsilon^{1/{2^{r-1}}}n^r < |E(H)|$, since $\\delta_{r-1}^+(H)$ can be used to give a lower bound on $|E(H)|$.\n\n\n\n\n\n\\begin{lemma}[Halfpap-Lemons-Palmer~\\cite{halfpap2024positive}]\\label{edge approx}\nFix $c>0$ and\nsuppose ${H}$ is an $r$-graph with $\\delta_{r-1}^+(H) \\geq cn$. Then, for $n$ large enough, $|E( {H})| \\geq \\frac{1}{2}\\frac{c^r}{r!} n^r$.\n\\end{lemma}\n\nThus, for an $n$-vertex $r$-graph $H$ with $n$ sufficiently large and $\\delta_{r-1}^+(H) \\geq cn$, we can choose $\\varepsilon$ in Lemma~\\ref{positive codegree cleanup} as a function of $c$ and $r$ alone.\n\nLemmas~\\ref{hypergraph removal} and \\ref{positive codegree cleanup} can be used to prove supersaturation and related properties for minimum positive co-degree problems. In particular, we have the following basic formulation.\n\n\\begin{theorem}[Halfpap-Lemons-Palmer~\\cite{halfpap2024positive}]\\label{supersaturation}\n\nFix $\\varepsilon > 0$ and let $F$ be an $r$-graph. Then there exists $\\delta > 0$ such that, if $H$ is an $n$-vertex $r$-graph with \n\\[\\delta_{r-1}^+(H) >\\mathrm{co^+ex}(F) + \\varepsilon n,\\]\nthen $H$ contains at least $\\delta n^{|V(F)|}$ copies of $F$. \n\\end{theorem}\n\n\nBy a standard argument (see, e.g., \\cite{Keevashsurvey}), if $\\delta > 0$ and $t \\in \\mathbb{N}$ are fixed and $n$ is sufficiently large, then every $n$-vertex $r$-graph $H$ containing $\\delta n^{|V(F)|}$ copies of $F$ must contain \\acopyof $F[t]$. \nThus, as an immediate consequence of Theorem~\\ref{supersaturation}, we have blow-up invariance for $\\mathrm{co^+ex}(n,F)$.\n\n\\begin{cor}[Halfpap-Lemons-Palmer~\\cite{halfpap2024positive}]\\label{blow up invariance}\n\nLet $F$ be an $r$-graph and $t$ a positive integer. Then\n$$\\mathrm{co^+ex}(n,F) \\leq \\mathrm{co^+ex}(n,F[t]) \\leq \\mathrm{co^+ex}(n,F) + o(n).$$\n\\end{cor}\n\nWe remark that Corollary~\\ref{blow up invariance} is useful because it implies that if $H$ and $F$ are $r$-graphs such that $F$ is contained in $H[t]$ for some $t$, then $\\gamma^+(F) \\leq \\gamma^+(H)$. For example, when paired with the facts that a single $3$-edge $e$ has $\\gamma^+(e) = 0$ and $\\gamma^+(C_5) = \\frac{1}{2}$, and with an appropriate lower bound construction, Corollary~\\ref{blow up invariance} implies that $\\gamma^+(C_{\\ell}) = 0$ if $\\ell \\equiv 0 \\,\\,(\\textrm{mod } 3)$ and $\\gamma^+(C_{\\ell}) = \\frac{1}{2}$ if $\\ell \\not\\equiv 0 \\,\\,(\\textrm{mod } 3)$ and $C_{\\ell}$ is contained in a blow-up of $C_5$. In fact, this resolves the positive co-degree density of every tight cycle except for $C_4 = K_4^3$ and $C_7$.\nHalfpap~\\cite{halfpap2025} proved $\\gamma^+(C_7)=1/2$. \n\nIn Section~\\ref{proofs}, we will apply essentially the same idea, finding one construction whose blow-up contains another and then relating their positive co-degree densities. However, for our purposes, a somewhat different formulation from the above statements will be desirable. The proof ideas for Theorem~\\ref{supersaturation} and Corollary~\\ref{blow up invariance} are used to derive the following lemma.\n\n\\begin{lemma}\\label{family removal}\n\nLet $F$ be a fixed $r$-graph on $f$ vertices and $\\mathcal{F} = \\{F_1, F_2, \\dots ,F_k\\}$ a finite family of $r$-graphs such that $F_i[f]$ contains $F$ for every $i \\in [k]$. For every $d \\in [0,1)$, if $\\gamma^+(F) > d$, then for some $\\beta > 0$ and $n$ sufficiently large there exists an $n$-vertex, $\\{F\\} \\cup \\mathcal{F}$-free $r$-graph $H$ with $\\delta_{r-1}^+(H) > \\left( d + \\beta \\right)n$.\n\\end{lemma}\n\n\n\n\n\n\\begin{proof}\n\nGiven $d$ as stated, choose $\\beta > 0$ so that $\\gamma^+(F) \\geq d + 3\\beta$, and $\\alpha > 0$ such that $2^rr!\\alpha^{1/{2^{r-1}}} < \\beta$ and $ (r+1)!\\alpha^{1/{2^{r-1}}} n^r < \\frac{1}{2}\\frac{(2\\beta)^r}{r!} n^r$.\nFor each $i \\in [k]$, take $\\delta_i > 0$ as guaranteed by Lemma~\\ref{hypergraph removal} such that any hypergraph $H$ containing fewer than $\\delta_i n^{|V(F_i)|}$ copies of $F_i$ can be made $F_i$-free by the deletion of at most $\\frac{\\alpha}{k} |E(H)|$ $r$-edges. Choose $N \\in \\mathbb{N}$ sufficiently large such that for all $n \\geq N$, we have: \n\\begin{itemize}[itemsep=1pt, parsep=0pt]\n\\item $\\mathrm{co^+ex}(n,F) \\geq (d + 2\\beta)n$;\n\\item for each $i \\in [k]$, if $H$ is an $n$-vertex graph containing $\\delta_i n^{|V(F_i)|}$ copies of $F_i$, then $H$ contains \\acopyof $F_i[f]$.\n\\end{itemize}\n\nFix $n \\geq N$ and let $H$ be an $n$-vertex $r$-graph with $\\delta_{r-1}^+(H) = \\mathrm{co^+ex}(n,F)$. Then $H$ contains fewer than $\\delta_i n^{|V(F_i)|}$ copies of $F_i$ for every $i \\in [k]$, and thus can be made $\\mathcal{F}$-free by deletion of at most $\\alpha |E(H)|$ edges by repeated application of Lemma~\\ref{hypergraph removal}. Since we have \n\\[(r+1)!\\alpha^{1/{2^{r-1}}} n^r < \\frac{1}{2}\\frac{(2\\beta)^r}{r!} n^r \\leq |E(H)|\\] \nby Lemma~\\ref{edge approx} and the definition of $\\alpha$, we can now apply Lemma~\\ref{positive codegree cleanup} to delete an additional set of $r$-edges, resulting in an $\\{F\\} \\cup \\mathcal{F}$-free, $n$-vertex $r$-graph $H'$ with \n\\[\\delta_{r-1}^+(H') \\geq \\delta_{r-1}^+(H) - 2^rr!\\alpha^{1/{2^{r-1}}}n >\\delta_{r-1}^+(H) - \\beta n \\geq (d + \\beta)n. \\qedhere\n\\] \n\\end{proof}\n\n\nLemma~\\ref{family removal} is an important tool in finding positive co-degree densities as it essentially allows us to expand our list of forbidden configurations. \n\n\nWe conclude this section with some relevant definitions and notation. \nWe often consider $4$-vertex cliques with one edge removed; these may be $2$-uniform or $3$-uniform. To avoid ambiguity, we denote by $K_4^-$ the $2$-graph on $4$ vertices and $5$ edges, and by $K_4^{3-}$ the $3$-graph on $4$ vertices and three $3$-edges.\n\n\n\n\nSome of the hypergraphs we consider can be naturally described as arising from lower-uniformity hypergraphs. We define the following operation, which increases the uniformity of a hypergraph by one.\nGiven an $r$-graph $H$, the \\textit{suspension} $\\widehat{H}$ is the $(r+1)$-graph with vertex set consisting of $V(H)$ and one new vertex $v$, and $(r+1)$-edges\n\\[ \nE(\\widehat{H}) = \\{ e \\cup \\{v\\} : e \\in E(H) \\}.\n\\]\n\n\n\n\nWe call $V(H)$ the \\textit{$r$-graph vertices} and $v$ the \\textit{spike vertex}.\nNotice that the $(r+1)$-triangle $T^{r+1}$ is a suspension of the $r$-triangle $T^{r}$.\n\n\nLet $H$ be an $r$-graph. For a subset $X=\\{x_1,\\ldots,x_{r-1}\\}$ of size $r-1$ of vertices of $H$, denote by $N(X)$ the set of all vertices $v$ such that $X\\cup\\{v\\} \\in E(H)$. We use $d(X) := |N(X)|$, which is the co-degree of $X$. \nTo simplify notation we use $N(x_1,\\ldots,x_{r-1}) := N(X)$ and $d(x_1,\\ldots,x_{r-1}) := d(X)$. We call $N(X)$ the \\textit{neighborhood} of $X$.\n\nIn a $3$-graph $H$, the \\textit{link graph} $L(v)$ of a vertex $v \\in V(H)$ is the auxiliary $2$-graph on $V(H) - \\{v\\}$ where $xy$ is an edge if and only if $vxy$ is a $3$-edge of $H$.\n\nFor 3-graphs $G$ and $H$ the \\emph{density} of $G$ in $H$, denoted by $d(G,H)$, is the number of subgraphs of $H$ isomorphic to $G$ divided by $\\binom{|V(H)|}{|V(G)|}$. Notice that the density is always in $[0,1]$. \n\n\n\\section{Jumps and positive co-degree densities below $\\frac{1}{2}$}\\label{proofs}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{general jumps}]\nLet $\\mathcal{F}$ be a family of $r$-graphs. \nBy Proposition~\\ref{first jump}, $\\gamma^+(\\mathcal{F}) \\in \\{0\\} \\cup [\\frac{1}{r}, 1]$, so it is sufficient to show that there is no family $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) \\in (\\frac{1}{r}, \\frac{2}{2r-1})$. \n\n\nFirst assume that forbidding $\\mathcal{F}$ implies that some blow-up $T^r[t]$ of $T^r$ is also forbidden (recall, $T^r$ denotes the triangle with three edges on $r+1$ vertices).\nCorollary~\\ref{blow up invariance} gives $\\gamma^+(T^r[t]) = \\gamma^+(T^r)$.\nAs \n\\[\n\\gamma^+(\\mathcal{F}) \\leq \\gamma^+(T^r[t]) = \\gamma^+(T^r),\n\\]\nit is sufficient to show that $\\gamma^+(T^r) \\leq \\frac1r$.\n\nSuppose that $H$ is an $n$-vertex $r$-graph with $\\delta_{r-1}^+(H) > \\frac{n}{r}$, and let $v_1v_2\\dots v_r$ be an $r$-edge of $H$. Consider the $r$ vertex sets, each of size $r-1$, contained in the $r$-edge $v_1v_2\\dots v_r$. Each of them is a set with positive co-degree, hence each has neighborhood of size greater than $\\frac{n}{r}$. Since there are $r$ such sets, there must be a vertex, say $v_{r+1}$, which is contained in at least two such neighborhoods. Relabeling if needed, we may assume that \n\\[\nv_{r+1} \\in N(v_1, \\dots, v_{r-2}, v_{r-1}) \\cap N(v_1, \\dots, v_{r-2}, v_{r}).\n\\]\n\nThe three $r$-edges $v_1v_2\\dots v_{r-2}v_{r-1}v_r$, $v_1v_2\\dots v_{r-2}v_{r-1}v_{r+1}$, and $v_1v_2\\dots v_{r-2}v_{r}v_{r+1}$ form \\acopyof $T^r$. Thus, $\\frac{1}{r} \\geq \\gamma^+\\left(T^r\\right) \\geq \\gamma^+\\left(\\mathcal{F}\\right)$. This implies that blow-ups of $T^r$ are $\\mathcal{F}$-free.\n\n\nNow assume that forbidding $\\mathcal{F}$ does not exclude any blow-up of $T^r$.\nConsider the following $n$-vertex blow-up of $T^r$. The three vertices corresponding to the base triangle $T^2$ are blown up to classes of size $\\frac{n}{2r-1}$. All other vertices are blown up to classes of size $\\frac{2n}{2r-1}$. In total, we have $3$ classes of size $\\frac{n}{2r-1}$ and $r-2$ classes of size $\\frac{2n}{2r -1}$, for a total of $n$ vertices, as desired. A set of $r-1$ vertices whose intersection with any class has at least two vertices will have co-degree $0$. A set of $r-1$ vertices which intersects all three classes of size $\\frac{n}{2r-1}$ will also have co-degree $0$. All other sets of size $r-1$ have neighborhood of size exactly $\\frac{2n}{2r-1}$, corresponding either to two classes of size $\\frac{n}{2r-1}$ or one class of size $\\frac{2n}{2r-1}$.\nThis construction implies $\\gamma^+(\\mathcal{F}) \\geq \\frac{2}{2r-1}$.\n\nNow, we can characterize families of $r$-graphs $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) \\in \\{0, \\frac{1}{r}\\}$. Corollary~\\ref{blow up invariance} establishes that if $F \n\\in \\mathcal{F}$ is $r$-partite, then $\\gamma^+(F) = 0$, so $\\gamma^+(\\mathcal{F}) = 0$ as well. If no $F \\in \\mathcal{F}$ is $r$-partite, then any blow-up of an $r$-edge is $\\mathcal{F}$-free, so $\\gamma^+(\\mathcal{F}) \\geq \\frac{1}{r}$. Thus, $\\gamma^+(\\mathcal{F}) = 0$ if and only if some $F \\in \\mathcal{F}$ is $r$-partite. Similarly, if an $F \\in \\mathcal{F}$ is contained in a blow-up of $T^r$, then \n\\[\n \\gamma^+(\\mathcal{F}) \n\\leq \\gamma^+(F) \\leq \\gamma^+(T^r) \\leq \\frac{1}{r}.\\]\nThus, if some $F \\in \\mathcal{F}$ is contained in some $T^r$ blow-up but no member of $\\mathcal{F}$ is $r$-partite, then we have $\\gamma^+(F) = \\frac{1}{r}$. On the other hand, if no member of $\\mathcal{F}$ is contained in any blow-up of $T^r$, then the above-described blow-up of $T^r$ establishes that $\\gamma^+(\\mathcal{F}) \\geq \\frac{2}{2r - 1}$.\n\\end{proof}\n\n\\begin{remark}\nThe characterization in Theorem~\\ref{general jumps} implies the positive co-degree densities for a variety of natural $3$-graphs. It is straightforward to verify that $C_{\\ell}^-$ (the $\\ell$-vertex (tight) cycle with one edge deleted) is contained in a sufficiently large blow-up of $K_4^{3-}$. Moreover, $C_{\\ell}^-$ is $3$-partite if and only if $\\ell \\equiv 0 \\pmod 3$. Thus, $\\gamma^+(C_{\\ell}^-) = 0$ when $\\ell \\equiv 0 \\pmod 3$, and $\\gamma^+(C_{\\ell}^-) = \\frac{1}{3}$ otherwise. This generalizes the result of Wu~\\cite{Wu} that $\\gamma^+(C_5^-) = \\frac{1}{3}$.\n\\end{remark}\n\nThe next natural question is whether Theorem \\ref{general jumps} is best possible. That is, does there exist some family $\\mathcal{F}$ of $r$-graphs with $\\gamma^+(\\mathcal{F}) = \\frac{2}{2r - 1}$? We answer this question in the affirmative when $r = 3$. Furthermore, we show that an infinite number of densities in the interval $[\\frac{2}{5}, \\frac{1}{2}]$ are achievable when $r = 3$. We begin by exhibiting a family $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) = \\frac{2}{5}$. We first define $F_{3,2}^{++1}$ and $F_{3,2}^{++2}$, two superhypergraphs of $F_{3,2}$. Each is created by adding two edges to $F_{3,2}$. Let $F_{3,2}$ have vertex set $\\{a,b,c,d,e\\}$ and edge set $\\{abc, abd, abe, cde\\}$. Then we create $F_{3,2}^{++1}$ by adding $acd$ and $ace$, and we create $F_{3,2}^{++2}$ by adding $acd$ and $bce$, see Figure~\\ref{F32}.\n\n\\begin{figure}[h]\n \\centering\n\n\\begin{tikzpicture}\n\n\n\\draw (0,0) node[vtx](a1){};\n\\draw (1,0) node[vtx](b1){};\n\\draw (-1,2) node[vtx](c1){};\n\\draw (0.5, 2) node[vtx](d1){}; \n\\draw (2, 2) node[vtx](e1){}; \n\\draw[fill=pink,opacity=0.5]\n(a1)to[bend left = 15] (b1)to[bend left=5] (c1)to[bend left=5] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(a1)to[bend left = 15] (b1)to[bend left=5] (d1)to[bend left=5] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(a1)to[bend left = 15] (b1)to[bend left=5] (e1)to[bend left=5] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(c1)to[bend left = 10] (d1)to[bend left=10] (e1)to[bend right=15] cycle\n;\n\\draw (0,0) node[vtx,label=below:$a$](a1){};\n\\draw (1,0) node[vtx,label=below:$b$](b1){};\n\\draw (-1,2) node[vtx,label=above:$d$](c1){};\n\\draw (0.5, 2) node[vtx,label=above:$c$](d1){}; \n\\draw (2, 2) node[vtx,label=above:$e$](e1){}; \n\n\n\\draw (4,0) node[vtx](a2){};\n\\draw (5,0) node[vtx](b2){};\n\\draw (3,2) node[vtx](c2){};\n\\draw (4.5, 2) node[vtx](d2){}; \n\\draw (6, 2) node[vtx](e2){}; \n\\draw[fill=pink,opacity=0.5]\n(a2)to[bend left = 15] (b2)to[bend left=5] (c2)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a2)to[bend left = 15] (b2)to[bend left=5] (d2)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a2)to[bend left = 15] (b2)to[bend left=5] (e2)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(c2)to[bend left = 10] (d2)to[bend left=10] (e2)to[bend right=15] cycle\n;\n\n\\draw[fill=blue,opacity=0.5]\n(c2)to[bend right = 15] (d2)to[bend right=15] (a2)to[bend right=15] cycle\n;\n\n\\draw[fill=blue,opacity=0.5]\n(d2)to[bend right = 15] (e2)to[bend right=12] (a2)to[bend right=12] cycle\n;\n\n\\draw (4,0) node[vtx, label=below:$a$](a2){};\n\\draw (5,0) node[vtx, label=below:$b$](b2){};\n\\draw (3,2) node[vtx, label=above:$d$](c2){};\n\\draw (4.5, 2) node[vtx, label=above:$c$](d2){}; \n\\draw (6, 2) node[vtx, label=above:$e$](e2){}; \n\n\n\n\\draw (8,0) node[vtx](a3){};\n\\draw (9,0) node[vtx](b3){};\n\\draw (7,2) node[vtx](c3){};\n\\draw (8.5, 2) node[vtx](d3){}; \n\\draw (10, 2) node[vtx](e3){}; \n\\draw[fill=pink,opacity=0.5]\n(a3)to[bend left = 15] (b3)to[bend left=5] (c3)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a3)to[bend left = 15] (b3)to[bend left=5] (d3)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a3)to[bend left = 15] (b3)to[bend left=5] (e3)to[bend left=5] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(c3)to[bend left = 10] (d3)to[bend left=10] (e3)to[bend right=15] cycle\n;\n\n\\draw[fill=blue,opacity=0.5]\n(c3)to[bend right = 15] (d3)to[bend right=15] (a3)to[bend right=15] cycle\n;\n\n\\draw[fill=blue,opacity=0.5]\n(d3)to[bend right = 15] (e3)to[bend right=15] (b3)to[bend right=15] cycle\n;\n\n\\draw (8,0) node[vtx, label=below:$a$](a3){};\n\\draw (9,0) node[vtx, label=below:$b$](b3){};\n\\draw (7,2) node[vtx, label=above:$d$](c3){};\n\\draw (8.5, 2) node[vtx, label=above:$c$](d3){}; \n\\draw (10, 2) node[vtx, label=above:$e$](e3){}; \n\n\n\\end{tikzpicture}\n\n \\caption{$F_{3,2}$ and two superhypergraphs, $F_{3,2}^{++1}$ and $F_{3,2}^{++2}$.}\n \\label{F32}\n\\end{figure}\n\n\\begin{prop}\\label{2/5 family}\nFor $\\mathcal{F} = \\{K_4^3, F_{3,2}^{++1}, F_{3,2}^{++2}, J_4\\}$ and $\\mathcal{F'} = \\{K_4^3, F_{3,2}, J_4\\}$ we have\n\\[ \n\\gamma^+(\\mathcal{F}) = \\gamma^+(\\mathcal{F'}) = \\frac{2}{5}.\n\\]\n\n\\end{prop}\n\n\n\\begin{proof}\nSince $F_{3,2}$ is a subgraph of both $F_{3,2}^{++1}, F_{3,2}^{++2}$, hence\n$\\gamma^+(\\mathcal{F}) \\geq \\gamma^+(\\mathcal{F'})$.\nAs noted in the proof of Theorem~\\ref{general jumps}, an appropriately balanced $n$-vertex blow-up of $T^3=K_4^{3-}$ has minimum positive co-degree $\\frac{2n}{5} + O(1)$. \nAny blow-up of $K_4^{3-}$ is $J_4$-free and $F_{3,2}$-free since $K_4^{3-}$ is $4$-partite and neither of $F_{3,2}$ and $J_4$ are. \nSince blow-ups $K_4^{3-}$ are also $K_4^{3}$-free, we have\n$\\frac{2}{5} \\leq \\gamma^+(\\mathcal{F'}) \\leq \\gamma^+(\\mathcal{F})$.\n\nNext we show $\\gamma^+(\\mathcal{F}) \\leq \\frac{2}{5}$, which completes the proof.\nFix an $\\varepsilon > 0$ and suppose that $H$ is an $n$-vertex $3$-graph with $\\delta_2^+\n(H) \\geq \\left( \\frac{2}{5} + \\varepsilon \\right)n$ for sufficiently large $n$. \nSince $\\gamma^+(K_4^{3-})=\\frac13$, $H$ contains \\acopyof $K_4^{3-}$, say with vertex set $\\{a,b,c,d\\}$ and edge set $\\{abc, abd, acd\\}$. \n\nConsider the following five positive co-degree pairs: $ab, ac, ad, bc,$ and $bd$. Since $\\delta_2^+(H) \\geq \\left(\\frac{2}{5} + \\varepsilon \\right)n,$ there exists some vertex $e$ that is in the neighborhood of at least three of these pairs. Note that $c$ and $d$ are symmetric; up to this symmetry, we have six cases based on which three of these five pairs form an edge with $e$.\n\n\n\n\n\\begin{enumerate}[label=(\\arabic*),itemsep=0.9pt, parsep=0pt]\n\n\\item If $abe, ace, ade \\in E(H)$ then $abc, abd, acd, abe, ace, ade$ form a $J_4$.\n\n\\item If $abe, ace, bce \\in E(H)$ then $abc, abe, ace, bce$ form a $K_4^3$.\n\n\\item If $abe, ace, bde \\in E(H)$ then $acb$, $acd$, $ace$, $bde$ form an $F_{3,2}$. Moreover, $abd$ and $abe$ are $3$-edges, so $\\{a,b,c,d,e\\}$ spans an \\acopyof $F_{3,2}^{++1}$.\n\n\n\\item If $ace, ade, bce \\in E(H)$ then $adb, adc, ade, bce$ form an $F_{3,2}$. Moreover, $abc$ and $ace$ are $3$-edges, so $\\{a,b,c,d,e\\}$ spans an \\acopyof $F_{3,2}^{++1}$. \n\n\\item If $abe, bce, bde \\in E(H)$ then $bea, bec, bed, acd$ form an $F_{3,2}$. Moreover, $bac$ and $bad$ are $3$-edges, so $\\{a,b,c,d,e\\}$ spans an \\acopyof $F_{3,2}^{++1}$.\n\n\\item If $ace, bce, bde \\in E(H)$ then $acb, acd, ace, bde$ form an $F_{3,2}$. Moreover, $abd$ and $cbe$ are $3$-edges, so $\\{a,b,c,d,e\\}$ spans an \\acopyof $F_{3,2}^{++2}$.\n\\end{enumerate}\n\nThis implies $\\gamma^+(\\mathcal{F}) \\leq \\frac{2}{5}$.\n\\end{proof}\n\n\nWe use Proposition~\\ref{2/5 family} as the base of an inductive argument, showing that an infinite number of positive co-degree densities in $[\\frac{2}{5}, \\frac{1}{2}]$ are achievable by families of $3$-graphs.\n\n\\begin{proof}[Proof of Theorem~\\ref{densities}]\nFor a lower bound, observe that any blow-up of $J_{k-1}$ is $\\{K_4^3, F_{3,2}, J_k\\}$-free, and that an appropriately balanced $n$-vertex blow-up of $J_{k-1}$ (with class sizes $\\frac{k-2}{2k-3},$ $ \\frac{1}{2k-3},\\ldots, \\frac{1}{2k-3}$) has minimum positive co-degree $\\left(\\frac{k-2}{2k-3}\\right)n + O(1)$. Thus, $\\gamma^+(\\{K_4^3, F_{3,2}, J_k\\}) \\geq \\frac{k-2}{2k-3}$ for all $k \\geq 4$.\n\n\nTo show that this lower bound is best-possible, we use induction on $k$. \nProposition~\\ref{2/5 family} yields the statement for $k=4$.\nWe assume the statement holds for $k \\geq 4$, and prove it for $k + 1$. Suppose that $H$ is an $n$-vertex $3$-graph with minimum positive co-degree $\\delta_2^+(H) > \\frac{k-1}{2k-1} n$ where is $n$ sufficiently large. If $H$ contains one of $K_4^3$ or $F_{3,2}$, then we are done, so suppose not. Then by the inductive hypothesis, $\\gamma^+(\\{K_4^3, F_{3,2}, J_k\\}) = \\frac{k-2}{2k-3} < \\frac{k-1}{2k -1}$, so $H$ contains a $J_k$, say $J$ (here we use that $n$ is large enough). Let $V(J) = \\{v_1, \\dots, v_{k+1}\\}$, where $v_1$ is the universal vertex of $J$ (i.e., $E(J) = \\{v_1v_iv_j : 2 \\leq i < j \\leq k+1\\}$). Define \n\\[ S = \\{(v_1, v_i) : 2 \\leq i \\leq k+1 \\} \\cup \\{(v_i,v_{i+1}): 2 \\leq i \\leq k\\}.\\]\n\nObserve that $|S| = 2k - 1$ and every vertex pair in $S$ has positive co-degree in $H$. Since $\\delta_2^+(H) > \\frac{k-1}{2k-1} n$, there exists a vertex $u \\in V(H)$ which is in the neighborhood of at least $k$ pairs $(v_i,v_j) \\in S$. Note that $u$ may be an element of $V(J)$. However, if $(v_i,v_j) \\in S$ is a pair such that $u \\in N(v_i,v_j)$, then $u \\not\\in \\{v_i,v_j\\}$. \n\n \nIf $u \\in N(v_1, v_i)$ for every $i \\in \\{2, \\dots, k+1\\}$ then $u \\not\\in V(J)$, and $V(J) \\cup \\{u\\}$ spans \\acopyof $J_{k+1}$, with universal vertex $v_1$.\n\nHence, we may assume that there is some $i \\geq 2$ for which $u \\in N(v_i, v_{i+1})$.\nOur goal in this case is to find $F_{3,2}$. Observe that there must be a $j$ such that $u \\in N(v_1, v_j)$. If there is such a $j \\not \\in \\{i, i+1\\}$, then $u \\not\\in \\{v_1,v_i, v_{i+1}, v_j\\}$, and $\\{v_1, v_i, v_{i+1}, v_j, u\\}$ will span an $F_{3,2}$, using $3$-edges $v_1v_jv_i, v_1v_jv_{i+1}, v_1v_ju, v_iv_{i+1}u$. Hence, we may assume that every such $j$ is in $\\{i, i+1\\}$. In particular, $u$ is in at most two neighborhoods of the form $N(v_1,v_j)$. Since $k \\geq 4$ and $u$ is in the \nneighborhood of at least $k$ pairs from $S$, we have $u \\in N(v_{\\ell},v_{\\ell + 1})$ for some $\\ell \\neq i$. Now, if $u \\in N(v_1, v_i) $ and $u \\in N(v_1, v_{i+1})$, then $\\{v_1,v_i,v_{i+1},u\\}$ spans a $K_4^3$. If not, then without loss of generality $u \\in N(v_1, v_i) $ and $u$ is in the neighborhood of the three pairs $(v_i, v_{i+1}), (v_{\\ell}, v_{\\ell + 1}), (v_m, v_{m+1})$ for some $\\ell,m$. One of these pairs, say $(v_{\\ell}, v_{\\ell + 1})$, must be disjoint from $(v_1, v_i)$, in which case we observe that $\\{v_1,v_i, v_\\ell, v_{\\ell+1},u\\}$ will span an $F_{3,2}$. \\end{proof}\n\nAlthough infinitely many positive co-degree densities in $[\\frac{2}{5}, \\frac{1}{2}]$ can be achieved by forbidden families of $3$-graphs, it remains unclear whether all of these densities can be achieved by forbidding a single $3$-graph. Using Proposition~\\ref{2/5 family}, we now show that $\\frac{2}{5}$ is indeed achievable by a single $3$-graph. We begin by defining two new $3$-graphs $F_1$ and $F_2$ as follows.\n\n\n\n\n\\begin{align*}V(F_1) & = \\{a,b,c,d,e,f,g\\}, \\quad & E(F_1) & = \\{abe, ace, bce, abf, adf, bdf, cdg\\},\\\\\nV(F_2) & = \\{a,b,c,d,e,g\\}, \\quad &E(F_2) & = \\{abe, ace, bce, ade, bde, cdg\\}.\n\\end{align*}\n\nWe depict $F_1$ and $F_2$ in Figure~\\ref{F1F2}.\nWe remark that each of $F_1,F_2$ can be viewed as a partial identification of two $K_4^{3-}$ copies, along with an extra $3$-edge (using $g$) which ensures that $c$ and $d$ have positive co-degree.\n\n\n\n\\begin{figure}[h]\n \\centering\n\n\\begin{tikzpicture}\n \n\\draw (5,0.4) node[vtx,label=below:$c$](x1){};\n\\draw (7,0) node[vtx,label=below:$a$](z1){};\n\\draw (7.5,0.8) node[vtx,label=below:$b$](w1){};\n\\draw (9, 0.4) node[vtx,label=below:$d$](y1){}; \n\\draw (7, 2) node[vtx,label=above:$e$](f){}; \n\n\\draw (7, -1) node[vtx,label=below:$g$](g1){}; \n\n\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend left = 25] (f) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw (7.5,0.8) node[vtx,label = below:$b$](w1){};\n\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (f) to[bend right=25] cycle\n;\n\n\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend right=45] (y1) to[bend left=25] (g1) to[bend left=25] cycle\n;\n\n\\draw (5,0.4) node[vtx,label=below:$c$](x1){};\n\\draw (7,0) node[vtx,label=below:$a$](z1){};\n\n\\draw (9, 0.4) node[vtx,label=below:$d$](y1){}; \n\\draw (7, 2) node[vtx,label=above:$e$](f){}; \n\n\n\\draw (0,0.4) node[vtx,label=below:$c$](x2){};\n\\draw (2,0) node[vtx,label=below:$a$](z2){};\n\\draw (2.5,0.8) node[vtx,label=below:$b$](w2){};\n\\draw (4, 0.2) node[vtx,label=below:$d$](y2){}; \n\\draw (1, 2) node[vtx,label=above:$e$](f){}; \n\\draw (3, 2) node[vtx,label=above:$f$](c){}; \n\n\\draw (2, -1) node[vtx,label=below:$g$](g2){}; \n\n\\draw[fill=pink,opacity=0.5]\n(x2) to[bend left=15] (w2)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x2) to[bend left=15] (z2)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z2) to[bend left=1] (w2)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y2) to[bend right=15] (w2)to[bend right = 15] (c) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z2) to[bend right=1] (w2)to[bend left=1] (c) to[bend right=10] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(x2) to[bend right=45] (y2)to[bend left=25] (g2) to[bend left=25] cycle\n;\n\n\\draw (2.5,0.8) node[vtx,label=below:$b$](w2){};\n\n\\draw[fill=pink,opacity=0.5]\n(y2) to[bend right=15] (z2)to[bend right=35] (c) to[bend right=25] cycle\n;\n\n\\draw (0,0.4) node[vtx,label=below:$c$](x2){};\n\\draw (2,0) node[vtx,label=below:$a$](z2){};\n\n\\draw (4, 0.2) node[vtx,label=below:$d$](y2){}; \n\\draw (1, 2) node[vtx,label=above:$e$](f){}; \n\\draw (3, 2) node[vtx,label=above:$f$](c){}; \n\n\n\n\n\n\\draw (7, -2.2) node{$F_2$}; \n\\draw (2, -2.2) node{$F_1$}; \n\n\n\\end{tikzpicture}\n \\caption{$F_1$ and $F_2$.}\n \\label{F1F2}\n\\end{figure}\n\nNote that in both $F_1$ and $F_2$, the vertex $g$ has very little structural interaction with the other vertices.\nSuppose $H$ is isomorphic to the subhypergraph of $F_1$ (resp.\\ $F_2$) induced on $V(F_1) \\setminus \\{g\\}$ (resp.\\ $V(F_2) \\setminus \\{g\\}$). Then $H$ is guaranteed to extend to \\acopyof $F_1$ (resp.\\ $F_2$) if $d(c,d) > 4$.\nWe will be working in $3$-graphs with minimum positive co-degree much larger than $4$, so to find copies of $F_1$ or $F_2$, it will suffice to find $F_1 \\setminus \\{g\\}$ or $F_2 \\setminus \\{g\\}$ and to demonstrate that the pair $\\{c,d\\}$ has positive co-degree.\n\n\nNote that $F_2$ is $5$-partite, so $\\gamma^+(F_2) \\geq \\frac{1}{2}$. However, $F_1$ is $4$-partite, and is contained in a blow-up of $J_4$, but it is not contained in a blow-up of $J_3=K_4^{3-}$. Thus, $\\gamma^+(F_1) \\geq \\frac{2}{5}$. \nWe now prove that $\\gamma^+(F_1) = \\frac{2}{5}$.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{single graph}]\nFor a contradiction, assume that there exists $\\varepsilon > 0$ such that $\\gamma^+(F_1) \\geq \\frac{2}{5} + 3\\varepsilon$. We begin with a claim that will allow us to expand our forbidden family. \n\n\n\\begin{claim}\nThe $3$-blow-up of each member of the following family\n\\[\\mathcal{F} = \\{K_4^3, J_4, F_{3,2}^{++1}, F_2 \\}\\]\n contains $F_1$.\n \\end{claim}\n\n \\begin{proof}\nObserve that $F_2$ can be obtained from $F_1$ by identifying $e$ and $f$.\nUsing the notation\n\\begin{align*}\nV(K_4^3) &=\\{1,2,3,4\\}, &E&(K_4^3) \\quad =\\{123,124,134,234\\},\\\\\nV(J_4) &=\\{1,2,3,4,5\\}, &E&(J_4)\\quad \\ =\\{123, 124, 125, 134, 135, 145\\},\\\\\nV(F_{3,2}^{++1})&=\\{1,2,3,4,5\\}, &E&(F_{3,2}^{++1})=\\{123, 124, 125, 345, 134, 135\\}\n\\end{align*}\nthe following maps prove the claim for $K_4^3$, $J_4$, and $F_{3,2}^{++1}$.\n\n\n\n$f_1: F_1\\to K_4^3:\\ a\\to 1,\\ b\\to 2,\\ c \\to 3, \\ d \\to 4, \\ e \\to 4,\\ f \\to 3, \\ g \\to 1$.\n\n$f_2: F_1\\to J_4: \\ \\ a\\to 2, \\ b\\to 3,\\ c \\to 4,\\ d \\to 5, \\ e \\to 1, \\ f \\to 1, \\ g \\to 1$.\n\n\n$f_3: F_1\\to F_{3,2}^{++1}: a\\to 2,\\ b\\to 3,\\ c \\to 4, \\ d \\to 5,\\ e \\to 1, \\ f \\to 1, \\ g \\to 3$.\n\n\\end{proof} \n \n \n Thus, by Lemma~\\ref{family removal}, for $n$ large enough there exists a $3$-graph $H$ with $\\delta_2^+(H) \\geq \\left(\\frac{2}{5} + \\varepsilon\\right)n$ which is $\\{F_1\\} \\cup \\mathcal{F}$-free. By Proposition~\\ref{2/5 family}, \n\\[\n\\gamma^+(\\{K_4^3, J_4, F_{3,2}^{++1}, F_{3,2}^{++2}\\}) = \\frac{2}{5},\n\\]\nso $H$ contains \\acopyof $F_{3,2}^{++2}$, which we shall call $F$. Put \n\\[V(F) = \\{a,b,c,d,e\\},\\quad \\text{ with } \\quad E(F) = \\{abc, abd, abe, cde, acd, bce\\}.\\] \n\nObserve that every pair of vertices in $V(F)$ has positive co-degree. Since $\\delta_2^+(H) \\geq \\left( \\frac{2}{5} + \\varepsilon \\right)n$, there exists some vertex $f$ that is in the neighborhood of at least five pairs from $V(F)$.\nLet $G_f$ be the link graph of vertex $f$ induced on vertex set $V(F)$ (i.e.\\ $G_f = L(f)[V(F)]$). \n\n\n\\begin{claim}\\label{VFclaims}\nThe following statements hold:\n\\begin{enumerate}[label=(\\roman*),itemsep=1pt, parsep=0pt]\n \\item For every $xyz \\in E(H)$, at most two of $xy, xz, yz$ are in $E(G_f)$. \n \n\\item There is no isolated vertex in $G_f$.\n\n \\item At most one of $ad, cd$ is in $E(G_f)$ and\n at most one of $be, ce$ is in $E(G_f)$.\n\n\\item At most one of $cd, de$ is in $E(G_f)$ and at most one of $ce, de$ is in $E(G_f)$. \n\n\\item At most one of $ab, ac$ is in $E(G_f)$ and at most one of $ab, bc$ is in $E(G_f)$.\n\n\\item At most one of $ab, ad$ is in $E(G_f)$ and at most one of $ab, be$ is in $E(G_f)$.\n\n\\item At most one of $ac, ad$ is in $E(G_f)$ and at most one of $bc, be$ is in $E(G_f)$.\n\n\\item $G_f$ is triangle-free.\n\n\\end{enumerate}\n\n\n\\end{claim}\n\\begin{proof}\nTo prove each statement, we shall assume that the statement does not hold and use this assumption to find some forbidden structure in $H$. \nNotice that the two parts in each of the statements (iii)--(vii) are symmetric so it is sufficient to prove only the first part of each of those claims. Here the symmetry is coming from the fact that $a\\to b,\\ b\\to a,\\ c\\to c,\\ d\\to e, \\ e\\to d$ is an automorphism of $F$.\n\nFor (i), observe that if $xyz \\in E(H)$ and $xy, xz, yz \\in E(G_f)$, then $\\{x,y,z,f\\}$ spans a $K_4^3$ in $H$.\n\n\nFor (ii), assume that $G_f$ contains a vertex of degree 0. Then the other four vertices in $V(F)$ span (at least) five edges of $G_f$, so $G_f$ contains a \\acopyof $K_4^-$. As illustrated in Figure~\\ref{k4-link}, a \\acopyof $K_4^-$ \n\nin $G_f$ implies that $H$ contains $F_2$ if the appropriate pair of vertices ($x$ and $y$, in the figure) has positive co-degree.\n Since all pairs of vertices in $F$ have positive co-degree, \\acopyof a $K_4^-$ in $G_f$ indeed implies that $H$ contains \\acopyof $F_2$.\n\n\n\n\\begin{figure}[h]\n\\centering \n\n\\begin{tikzpicture}\n\n\\draw (0,0) node[vtx,label=below:$x$](x){};\n\\draw (2,0) node[vtx,label=below:$z$](z){};\n\\draw (0,2) node[vtx,label=above:$w$](w){};\n\\draw (2,2) node[vtx,label=above:$y$](y){};\n\n\\draw \n (x) to (z)\n (z) to (y)\n (z) to (w)\n (w) to (x)\n (w) to (y);\n\n\\draw (1, -1) node{$G_f$};\n\n\n\\draw (5,0.4) node[vtx,label=below:$x$](x1){};\n\\draw (7,0) node[vtx,label=below:$z$](z1){};\n\\draw (7.5,0.8) node[vtx,label=below:$w$](w1){};\n\\draw (9, 0.4) node[vtx,label=below:$y$](y1){}; \n\\draw (7, 2) node[vtx,label=above:$f$](f){}; \n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend left = 25] (f) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw (7.5,0.8) node[vtx,label = below:$w$](w1){};\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (f) to[bend right=25] cycle\n;\n\n\\draw (5,0.4) node[vtx,label=below:$x$](x1){};\n\\draw (7,0) node[vtx,label=below:$z$](z1){};\n\n\\draw (9, 0.4) node[vtx,label=below:$y$](y1){}; \n\\draw (7, 2) node[vtx,label=above:$f$](f){}; \n\n\\draw (7,-1) node{$H$};\n\n\\end{tikzpicture}\n\\caption{\\acopyof $K_4^-$ \nin $G_f$ and the resulting structure in $H$.} \\label{k4-link}\n\n\\end{figure}\n\nFor (iii), observe that if $ad, cd \\in E(G_f)$, then $\\{a,c,d,f\\}$ spans \\acopyof $K_4^{3-}$ \nwith spike vertex $d$, and $\\{a, b, c, e\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $b$, so $\\{a,b,c,d,e, f\\}$ spans \\acopyof $F_1$ (since $d(e,f) > 0$ by (ii)).\n\n\nFor (iv), if $cd, de \\in E(G_f)$, then $\\{c,d,e,f\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $d$, and $\\{a,b,c,e\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $b$, so $\\{a,b,c,d,e,f\\}$ spans \\acopyof $F_1$ (since $d(a,f) > 0$ by (ii)).\n\n\n\nFor (v), if $ab, ac \\in E(G_f)$, then $\\{a,b,c,d\\}$ and $\\{a, b, c, f\\}$ span copies of $K_4^{3-}$ with spike vertex $a$, so $\\{a,b,c,d,f\\}$ spans \\acopyof $F_2$ (since $d(d,f) > 0$ by (ii)).\n\n\n\nFor (vi), if $ab, ad \\in E(G_f)$, then $\\{a,b,c,d\\}$ and $\\{a, b, d, f\\}$ span copies of $K_4^{3-}$ with spike vertex $a$, so $\\{a,b,c,d,f\\}$ spans \\acopyof $F_2$ (since $d(c,f) > 0$ by (ii)).\n\n\nFor (vii), if $ac, ad \\in E(G_f)$, then $\\{a,b,c,d\\}$ and $\\{a, c, d, f\\}$ span copies of $K_4^{3-}$ with spike vertex $a$, so $\\{a,b,c,d,f\\}$ spans \\acopyof $F_2$ (since $d(b,f) > 0$ by (ii)).\n\n\nFinally, we show (viii). From (i), we know that $\\{c,d,e\\}$ does not form a triangle in $G_f$, and from (v) and (vi) it follows that $ab$ is not contained in a triangle in $G_f$. Thus, any triangle in $G_f$ must include one edge spanned by $\\{c,d,e\\}$, and two edges with one vertex in $\\{a,b\\}$ and the other in $\\{c,d,e\\}$. By (i), $\\{a,c,d\\}$ does not span a triangle (nor does $\\{b, c, e\\}$). Up to symmetry, there are two other potential triangles: $\\{a, d, e\\}$, and $\\{a,c,e\\}$. Observe that if $\\{a,d,e\\}$ spans a triangle in $G_f$, then $\\{a,d,e, f\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $f$, and $\\{a, b, c, e\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $b$, so $\\{a,b,c,d,e,f\\}$ spans \\acopyof $F_1$ (since $d(c,d) > 0$). \n\nNext, suppose $\\{a,c,e\\}$ spans a triangle in $G_f$. By (iii), (iv), (v), and (vii), none of $be, de, ab, ad$ are in $E(G_f)$. Thus, two of $bc, bd, cd$ are in $E(G_f)$. We shall show that this is impossible. Observe that if $bc \\in E(G_f)$, then $\\{a,b,c,f\\}$ and $\\{b,c,e,f\\}$ each spans a $K_4^{3-}$ with spike vertex $c$, so $\\{a,b,c,e,f\\}$ spans \\acopyof $F_2$ (since $d(a,e) > 0$). Moreover, if $cd \\in E(G_f)$, then $\\{a,c,d,f\\}$ and $\\{c,d,e,f\\}$ span $K_4^{3-}$ with spike vertex $c$, so $\\{a,c,d,e,f\\}$ spans \\acopyof $F_2$ (since $d(a,e) > 0$). Thus, $G_f$ is triangle-free.\n\\end{proof}\n\n\nWith Claim~\\ref{VFclaims} established, we are ready to determine the structure of $G_f$. By (i), at most two out of five edges of $G_f$ are spanned by $\\{c,d,e\\}$, so one of $a,b$ has degree at least $2$. Without loss of generality, $d(a) \\geq d(b)$ and $d(a) \\geq 2$. \n\n\\begin{observation}\n $ab \\not\\in E(G_f)$. \n\\end{observation}\n\n\\begin{proof}\nSuppose to the contrary that $ab \\in E(G_f)$. By (v) and (vi), $ac$ and $ad$ are not in $E(G_f)$, so we must have $ae \\in E(G_f)$. By (v) and (vi), we also have $bc, be \\not\\in E(G_f)$. Since $G_f$ has five edges, at least three of $bd, cd, ce, de$ are in $E(G_f)$. By (iv), at most one of $ce, de$ is in $E(G_f)$, so $bd, cd \\in E(G_f)$. Also by (iv), if $cd \\in E(G_f)$, then $de \\not \\in E(G_f)$, so $ce \\in E(G_f)$. \nSince $ae \\in E(G_f)$, there exists a vertex $x$ not in $V(F)$ that is a common neighbor of $a$ and $e$.\nThe set of edges $\\{adb,dce, fab, fbd, fdc, fce, aex\\}$ forms $F_1$. See Figure~\\ref{noab} for an illustration of $G_f$, $F$, and $F_1$.\n\\end{proof}\n\n\n\n\\begin{figure}[h]\n \\centering\n\n\\begin{tikzpicture}\n \n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (c) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (d) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (e) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(d) to[bend left=15] (c)to[bend left=50] (a) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(e) to[bend right=15] (c)to[bend right=50] (b) to[bend left=15] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(c) to[bend left=15] (d)to[bend right=20] (e) to[bend left=15] cycle\n;\n\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw[thick, blue] (a) to[bend right = 25] (b);\n\n\\draw[thick, blue] (b) to[bend right = 20] (d);\n\n\\draw[thick, blue] (a) to[bend left = 20] (e);\n\n\\draw[thick, blue] (c) to[bend right = 10] (e);\n\n\\draw[thick, blue] (d) to[bend right = 10] (c);\n\n\\draw (-1, -4) node{$G_f$ and $F$};\n\n\n\\begin{scope}[yshift = 1cm]\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$d$](z1){};\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\\draw (9, -2.8) node[vtx,label=below:$e$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$b$](f){}; \n\\draw (8, -1) node[vtx,label=above:$c$](c){}; \n\n\\draw (7, -4) node[vtx,label=below:$x$](g1){}; \n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend right=45] (y1) to[bend left=25] (g1) to[bend left=25] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend right = 15] (c) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend right=1] (w1)to[bend left=1] (c) to[bend right=10] cycle\n;\n\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (c) to[bend right=25] cycle\n;\n\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$d$](z1){};\n\\draw (9, -2.8) node[vtx,label=below:$e$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$b$](f){}; \n\\draw (8, -1) node[vtx,label=above:$c$](c){}; \n\\end{scope}\n\n\\draw (7,-4) node{$F_1$};\n\n\\end{tikzpicture}\n\n \\caption{The configuration when $ab \\in E(G_f)$, and the resulting copy of $F_1$.}\n \\label{noab}\n\\end{figure}\n\n\n\n Since $d(a) \\geq 2$, two of $ac, ad, ae$ are in $E(G_f)$. By (vii), at most one of $ac, ad$ is in $E(G_f)$, so $E(G_f)$ contains $ae$ and exactly one of $ac, ad$. \nSuppose first that $ad \\in E(G_f)$. By (iii), $cd\\not\\in E(G_f)$, and by (viii), $de \\not\\in E(G_f)$. Since $d(b) \\leq d(a) = 2$, we must have $ce \\in E(G_f)$. Now, by (iii), $be \\not\\in E(G_f)$, so we must have $bc, bd \\in E(G_f)$. \nSince $ae \\in E(G_f)$, there exists a vertex $x$ not in $V(F)$ that is a common neighbor of $a$ and $e$.\nThe set of edges $\\{adb,bce, fad, fbd, fbc, fce, aex\\}$ forms $F_1$. See Figure~\\ref{noad} for an illustration of $G_f$, $F$, and $F_1$.\n\n\n\\begin{figure}[h]\n \\centering\n\n\\begin{tikzpicture}\n \n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (c) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (d) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (e) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(d) to[bend left=15] (c)to[bend left=50] (a) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(e) to[bend right=15] (c)to[bend right=50] (b) to[bend left=15] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(c) to[bend left=15] (d)to[bend right=20] (e) to[bend left=15] cycle\n;\n\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw[thick, blue] (a) to[bend left = 20] (e);\n\n\\draw[thick, blue] (b) to[bend right = 20] (d);\n\n\\draw[thick, blue] (a) to[bend left = 15] (d);\n\n\\draw[thick, blue] (c) to[bend right = 10] (e);\n\n\\draw[thick, blue] (c) to[bend right = 55] (b);\n\n\\draw (-1, -4) node{$G_f$ and $F$};\n\n\n\\begin{scope}[yshift = 1cm]\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$b$](z1){};\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\\draw (9, -2.8) node[vtx,label=below:$e$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$d$](f){}; \n\\draw (8, -1) node[vtx,label=above:$c$](c){}; \n\n\\draw (7, -4) node[vtx,label=below:$x$](g1){}; \n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend right=45] (y1) to[bend left=25] (g1) to[bend left=25] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend right = 15] (c) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend right=1] (w1)to[bend left=1] (c) to[bend right=10] cycle\n;\n\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (c) to[bend right=25] cycle\n;\n\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$b$](z1){};\n\\draw (9, -2.8) node[vtx,label=below:$e$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$d$](f){}; \n\\draw (8, -1) node[vtx,label=above:$c$](c){}; \n\\end{scope}\n\n\\draw (7,-4) node{$F_1$};\n\n\\end{tikzpicture}\n\n \\caption{The configuration when $ad, ae \\in E(G_f)$, and the resulting copy of $F_1$.}\n \\label{noad}\n\\end{figure}\n\n\nThus, we have $ac, ae \\in E(G_f)$ and $ad \\not\\in E(G_f)$. By (viii), $ce \\not\\in E(G_f)$, and by (iv), at most one of $cd, de$ is in $E(G_f)$. Thus, $d(b) = 2$. \nBy (vii), at most one of $bc$, $be$ is in $E(G_f)$,\nhence $bd \\in E(G_f)$. Suppose for a contradiction that $be \\in E(G_f)$. Then by (viii), $de\\not\\in E(G_f)$. Since $G_f$ has at least 5 edges, $dc \\in E(G_f)$.\nSince $bd \\in E(G_f)$, there exists a vertex $x$ not in $V(F)$ that is a common neighbor of $b$ and $d$.\nThe set of edges $\\{abe,adc, fbe, fae, fac, fcd, bdx\\}$ forms $F_1$. See Figure~\\ref{fig:nobd} for an illustration of $G_f$, $F$, and $F_1$.\n\n\n\\begin{figure}[h]\n \\centering\n\n\\begin{tikzpicture}\n \n\n\n\\draw (0,0) node[vtx,label=above:$a$](b){};\n\\draw (-2,0) node[vtx,label=above:$b$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$d$](e){};\n\\draw (-3,-3) node[vtx,label=below:$e$](d){};\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (c) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (d) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (e) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(d) to[bend left=15] (c)to[bend left=50] (a) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(e) to[bend right=15] (c)to[bend right=50] (b) to[bend left=15] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(c) to[bend left=15] (d)to[bend right=20] (e) to[bend left=15] cycle\n;\n\n\n\n\\draw[thick, blue] (a) to[bend left = 20] (e);\n\n\\draw[thick, blue] (b) to[bend right = 20] (d);\n\n\\draw[thick, blue] (a) to[bend left = 15] (d);\n\n\\draw[thick, blue] (c) to[bend right = 10] (e);\n\n\\draw[thick, blue] (c) to[bend right = 55] (b);\n\n\\draw (c) node[vtx,label=below:$c$](c){};\n\n\n\\draw (-1, -4) node{$G_f$ and $F$};\n\n\n\\begin{scope}[yshift = 1cm]\n\\draw (5,-2.6) node[vtx,label=below:$b$](x1){};\n\\draw (7,-3) node[vtx,label=below:$a$](z1){};\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\\draw (9, -2.8) node[vtx,label=below:$d$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$e$](f){}; \n\\draw (8, -1) node[vtx,label=above:$c$](c){}; \n\n\\draw (7, -4) node[vtx,label=below:$x$](g1){}; \n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend right=45] (y1) to[bend left=25] (g1) to[bend left=25] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (f) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (f) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (f) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend right = 15] (c) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend right=1] (w1)to[bend left=1] (c) to[bend right=10] cycle\n;\n\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (c) to[bend right=25] cycle\n;\n\n\n\\end{scope}\n\n\\draw (7,-4) node{$F_1$};\n\n\\end{tikzpicture}\n\n \\caption{The configuration when $be\\in E(G_f)$, and the resulting copy of $F_1$.}\n \\label{fig:nobd}\n\\end{figure}\n\n\n\nThus, $bc \\in E(G_f)$. By (viii), $cd \\not\\in E(G_f)$, so we must have $de \\in E(G_f)$.\nHence $E(G_f) = \\{ ae,ac,bc,bd,de \\}$; see Figure~\\ref{goodC5} for an illustration.\n\n\n\n\n\\begin{figure}[h]\n\n\\centering\n\\begin{tikzpicture}\n\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (c) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (d) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (e) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(d) to[bend left=15] (c)to[bend left=50] (a) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(e) to[bend right=15] (c)to[bend right=50] (b) to[bend left=15] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(c) to[bend left=15] (d)to[bend right=20] (e) to[bend left=15] cycle\n;\n\n\n\n\\draw[thick, blue] (a) to[bend left = 20] (c);\n\n\\draw[thick, blue] (b) to[bend right = 20] (d);\n\n\\draw[thick, blue] (a) to[bend left = 20] (e);\n\n\\draw[thick, blue] (d) to[bend right = 20] (e);\n\n\\draw[thick, blue] (b) to[bend right = 20] (c);\n\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\n\n\n\\end{tikzpicture}\n\\caption{$F$ and $G_f$ when $ac, ae \\in E(G_f)$.}\\label{goodC5}\n\n\\end{figure}\n\nUnlike in the previous cases, we cannot immediately find \\acopyof $F_1$ (or any other forbidden hypergraph) in Figure~\\ref{goodC5}. \nHowever, we now have that the subhypergraph of $H$ induced on $\\{a,b,c,d,e,f\\}$ has edge set $\\{abc, abd, abe, adc, bce, cde, fac, fcb, fbd, fde, fea \\}$. \nWe call this subhypergraph $F'$, and we shall use $F'$ to find \\acopyof some forbidden hypergraph. \n\nObserve first that each of the $15$ pairs of vertices in $F'$ have positive co-degree. Thus, there exists some $g \\in V(H)$ that is in the neighborhood of at least $15\\left( \\frac{2}{5} + \\varepsilon \\right) > 6$ pairs. Let $G_g$ be the link graph of vertex $g$ induced on $V(F')$ (i.e.\\ $G_g = L(g)[V(F')]$). \nAgain, we begin with some observations on $G_g$. \n\n\\begin{claim}\\label{Gg}\n$\\delta(G_g) \\geq 1$. Moreover, in $G_g$,\n$N(f) = \\{c,d,e\\}$.\n\\end{claim}\n\n\\begin{proof}\nAs in the proof of Claim~\\ref{VFclaims}, since all pairs from $\\{a,b,c,d,e,f\\}$ have positive co-degree, $G_g$ cannot contain \\acopyof $K_4^-$.\n\n If $G_g$ has an isolated vertex $v$, then $G_g - v$ is a component on $5$ vertices and $7$ edges that necessarily contains a \\acopyof $K_4^-$. Thus, every vertex of $G_g$ has positive degree. \n\n\nNext, observe that the previous analysis of $G_f$ in fact shows that if some vertex $v \\in V(H) \\setminus V(F)$ has positive co-degree with at least $5$ pairs from $V(F)$, then the link graph of $v$ induced on $V(F)$ must be equal to $G_f$. In particular, either $G_g$ contains $G_f$ or $g$ has positive co-degree with at most $4$ pairs from $V(F)$. Observe that if $G_g$ contains $G_f$, then $\\{a,b,c,f\\}$ and $\\{a,b,c,g\\}$ span copies of $K_4^{3-}$ with spike vertex $c$. This implies that $\\{a,b,c,f,g\\}$ spans \\acopyof $F_2$, since we have argued that $d(f,g) > 0$. Thus, $g$ has positive co-degree with at most $4$ pairs from $V(F)$, which implies that $f$ has degree at least $3$ in $G_g$, since $|E(G_g)| \\geq 7$.\n\nFinally, to determine $N(f)$ in $G_g$, we consider the interaction between $G_g$ and $G_f$. Suppose that $x,y$ are neighbors of $f$ in $G_g$ such that $xy \\in E(G_f)$. Then $\\{x,y,f,g\\}$ forms \\acopyof $K_4^{3-}$ with spike vertex $f$. We consider the possible values of $x,y$; we know $xy \\in \\{ac,ae, bc, bd, de\\}$. Since $\\{a,b,c,d\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $a$, and we have established that $g$ has positive co-degree with every vertex in $V(F)$, we can find \\acopyof $F_1$ in $H$ on vertex set $\\{a,b,c,d,f,g\\}$ if $xy \\in \\{bc, bd\\}$. Similarly, if $xy \\in \\{ac, ae\\}$, then we can find \\acopyof $F_1$ on $\\{a,b,c,e,f,g\\}$. Thus, in $G_g$, either $N(f)$ is an independent set or $N(f)$ contains precisely the edge $de$. Recall that $|N(f)| \\geq 3$, so the first outcome is impossible, as the independence number of $G_f$ is $2$. The second outcome occurs only if \n$N(f) = \\{c,d,e\\}$.\n\\end{proof}\n\n\nWith Claim~\\ref{Gg} established, we conclude that $|E(G_g)| = 7$, with exactly 4 edges of $G_g$ spanned by $\\{a,b,c,d,e\\}$. Given that $cf, df, ef \\in E(G_g)$, we shall prove that this is not possible.\n\nFirst, observe that none of $cd,ce,de$ are in $E(G_g)$. Indeed, if $de \\in E(G_g)$, then $\\{d,e,f,g\\}$ spans \\acopyof $K_4^3$ (recall that $de \\in E(G_f)$). If either $cd$ or $ce$ is in $E(G_g)$, then either $\\{c,d,f,g\\}$ or $\\{c,e,f,g\\}$ spans \\acopyof $K_4^{3-}$ with spike vertex $g$; if $\\{c,d,f,g\\}$ spans $K_4^{3-}$, then $\\{a,b,c,d,f,g\\}$ spans an \\acopyof $F_1$, while if $\\{c,e,f,g\\}$ spans $K_4^{3-}$, then $\\{a,b,c,e,f,g\\}$ spans an \\acopyof $F_1$.\n\nNotice that Claim~\\ref{VFclaims} applies also to $G_g$ since Claim~\\ref{Gg} proves (ii) and it is the only statement that used the number of edges of $G_f$.\nNext, we prove that $ab \\not\\in E(G_g)$. Indeed, by (v) and (vi), \nif $ab \\in E(G_g)$, then $ac,bc,ad,be$ are not in $E(G_g)$. \nRecall $cd,ce,de$ are not in $E(G_g)$.\nSo if $ab \\in E(G_g)$, then at most three edges of $G_g$ are spanned by $\\{a,b,c,d,e\\}$. However, by Claim~\\ref{Gg}, $d(f) = 3$ in $G_g$, so we have $|E(G_g)| \\leq 6$, a contradiction. Thus, $ab \\not\\in E(G_g)$.\n\n\nFinally, by (vii), at most two of $ac, ad, bc, be$ are in $E(G_g)$. So in order to have $|E(G_g)| = 7$, both $ae$ and $bd$ must be in $E(G_g)$. However, this results in \\acopyof $F_1$ on $\\{a,b,d,e,g,f\\}$ with edges $\\{age, afe, abc, gfe, gfd, gbd, bdf \\}$ as depicted in Figure~\\ref{finalconfig}. \n\\end{proof}\n\n\n\n\n\n\\begin{figure}[h!]\n\n\\centering\n\\begin{tikzpicture}\n\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below left:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\\draw (-1,-5) node[vtx,label=below:$f$](f){};\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (c) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (d) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(a) to[bend right=15] (b)to[bend right=15] (e) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(d) to[bend left=15] (c)to[bend left=50] (a) to[bend right=15] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(e) to[bend right=15] (c)to[bend right=50] (b) to[bend left=15] cycle\n;\n\n\n\\draw[fill=pink,opacity=0.5]\n(c) to[bend left=15] (d)to[bend right=20] (e) to[bend left=15] cycle\n;\n\n\n\n\n\\draw[line width=1pt, blue] \n (a) to[bend left = 20] (c)\n (b) to[bend right = 20] (d)\n (a) to[bend left = 20] (e)\n (d) to[bend right = 20] (e)\n (b) to[bend right = 20] (c);\n\n\\draw[line width=1.5pt, darkpastelgreen, dashed] \n(d) to[bend right = 20] (f)\n(e) to[bend left = 20] (f)\n(c) to (f)\n(b) to[bend right = 35] (d)\n(a) to[bend left = 35] (e);\n\n\\draw (0,0) node[vtx,label=above:$b$](b){};\n\\draw (-2,0) node[vtx,label=above:$a$](a){};\n\\draw (-1,-3) node[vtx,label=below left:$c$](c){};\n\\draw (1,-3) node[vtx,label=below:$e$](e){};\n\\draw (-3,-3) node[vtx,label=below:$d$](d){};\n\n\\draw (-1,-5) node[vtx,label=below:$f$](f){};\n\n\n\n\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$g$](z1){};\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\\draw (9, -2.8) node[vtx,label=below:$b$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$e$](s1){}; \n\\draw (8, -1) node[vtx,label=above:$d$](t1){}; \n\n\\draw (7, -4) node[vtx,label=below:$c$](g1){}; \n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend right=45] (y1) to[bend left=25] (g1) to[bend left=25] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (w1)to[bend left=15] (s1) to[bend left=2] cycle\n;\n\\draw[fill=pink,opacity=0.5]\n(x1) to[bend left=15] (z1)to[bend left=15] (s1) to[bend left=15] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend left=1] (w1)to[bend left=1] (s1) to[bend right=1] cycle\n;\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (w1)to[bend right = 15] (t1) to[bend right=2] cycle\n;\n\n\\draw[fill=pink,opacity=1]\n(z1) to[bend right=1] (w1)to[bend left=1] (t1) to[bend right=10] cycle\n;\n\n\\draw (7.5,-2.2) node[vtx,label=below:$f$](w1){};\n\n\\draw[fill=pink,opacity=0.5]\n(y1) to[bend right=15] (z1)to[bend right=35] (t1) to[bend right=25] cycle\n;\n\n\\draw (5,-2.6) node[vtx,label=below:$a$](x1){};\n\\draw (7,-3) node[vtx,label=below:$g$](z1){};\n\\draw (9, -2.8) node[vtx,label=below:$b$](y1){}; \n\\draw (6, -1) node[vtx,label=above:$e$](s1){}; \n\\draw (8, -1) node[vtx,label=above:$d$](t1){}; \n\n\n\n\n\\end{tikzpicture}\n\\caption{$F'$ (edges involving $f$ indicated by $G_f$ in thick blue) and a configuration in $G_g$ (in dashed green) yielding $F_1$.}\\label{finalconfig}\n\\end{figure}\n\n\\section{Positive co-degree densities from flag algebras}\\label{sec:J4}\n\n\nThis section contains calculations using flag algebras, as introduced by Razborov~\\cite{Raz07}. \nFor an introduction to flag algebras and formal definitions, see~\\cite{FalgasK4-,Raz07,FALGAS-RAVRY_VAUGHAN_2013}. \nComputer code is available at \\oururl.\n\nWe use flag algebras to prove Theorem~\\ref{J4} that $\\gamma^+(J_4) = 4/7$, with the asymptotically unique construction being the balanced blow-up of the complement of the Fano plane. In Figure~\\ref{fig:Fano}, we illustrate (a blow-up of) the Fano plane, with a particular labeling of classes to which we will later refer. We denote the complement of the Fano plane by $\\overline{\\mathbb{F}}$ and the $n$-vertex, balanced blow-up of the complement of the Fano plane by $\\overline{\\mathbb{F}}_n$. \n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[scale=5,line width=6pt]\n(0,0) coordinate (x7) -- ++(1,0) coordinate[pos=0.5](x1) coordinate(x2)\n(x7) -- ++(60:1) coordinate[pos=0.5](x3) coordinate(x4)\n(x2) -- ++(120:0.5) coordinate(x5)\n(x7) -- ++(30:0.57735) coordinate(x6) (x4)--(x5) \n(x2)--(x3)\n(x6)--(x5)\n(x4)--(x1)\n(x6) circle (0.2886751345 cm) ;\n\\draw\n\\foreach \\x/\\y in {1/1,2/2,3/6,4/7,5/3,6/4,7/5}{\n(x\\x) node[circle,draw,fill=white]{$X_\\y$}\n}\n;\n\\end{tikzpicture}\n\\end{center}\n \\caption{A blow-up of the Fano plane. 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} \n}\n\n\\scalebox{\\ourscale}{\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n \\draw(\\xshift,\\yshiftC) node{$F_9$};\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_1$}\n(x5) node{\\tiny$X_1$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{2}{2}\n\\drawhypervertex{3}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{4}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{1}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{3}{5}\n\\drawhypervertex{4}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{3}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{2}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{2}{8}\n\\drawhypervertex{3}{8}\n\\drawhypervertex{5}{8}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n \\draw(\\xshift,\\yshiftC) node{$F_{10}$};\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_3$}\n(x5) node{\\tiny$X_3$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n \\draw(\\xshift,\\yshiftC) node{$F_{11}$};\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_3$}\n(x5) node{\\tiny$X_2$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{3}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{4}{5}\n\\drawhypervertex{5}{5}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n \\draw(\\xshift,\\yshiftC) node{$F_{12}$};\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_2$}\n(x5) node{\\tiny$X_1$}\n;\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{2}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{3}{3}\n\\drawhypervertex{4}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{1}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{3}{5}\n\\drawhypervertex{5}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{4}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhypervertex{5}{7}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n \\draw(\\xshift,\\yshiftD) node{$F_{13}$};\n \\drawhyperedge{0}{6}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_1$}\n(x2) node{\\tiny$X_1$}\n(x3) node{\\tiny$X_1$}\n(x4) node{\\tiny$X_1$}\n(x5) node{\\tiny$X_2$}\n;\n\\end{tikzpicture} }\n}\n\\end{center}\n\\caption{The family $\\mathcal{F}$ of thirteen $6$-vertex $3$-graphs.}\\label{J4 positive density set}\n\\end{figure}\n\nLet $\\mathcal{F}$ be the family of thirteen 3-graphs 6-vertex induced subgraphs of $\\overline{\\mathbb{F}}_n$, depicted in Figure~\\ref{J4 positive density set}. We include labels to indicate which classes of $\\overline{\\mathbb{F}}_n$ each vertex belongs to, corresponding to the labeling in Figure~\\ref{fig:Fano}. Note that due to the symmetries of $\\overline{\\mathbb{F}}_n$, the indicated labelings are not the unique ways a subgraph can be obtained.\n\n\nLet $\\mathcal{F}_6$ be the family of $J_4$-free $6$-vertex $3$-graphs. \n\nThe following claim shows that if a $J_4$-free $n$-vertex $3$-graph $G$ has minimum positive co-degree at least $\\frac{4}{7}n$ then every $3$-graph on six vertices with positive density in $G$ is in $\\mathcal{F}$.\n\n\n\\begin{claim}\\label{claim:fa47}\n For every fixed $\\delta > 0$, there exists $n_0$ such that for every $n \\geq n_0$, if $G_n$ is a $J_4$-free $n$-vertex 3-graph with $\\delta_2^+(G_n) \\geq \\frac{4n}{7}$, then\n\\[\n\\sum_{H \\in \\mathcal{F}_6 \\setminus \\mathcal{F}} d(H,G_n) \\leq \\delta.\n\\] \n\\end{claim}\n\\begin{proof}\nThe claim is proved by a standard application of flag algebras. \nThe notable part is encoding the condition $\\delta_2^+(G_n) \\geq \\frac{4n}{7}$ into flag algebras by \\eqref{eq:f47}.\n \\begin{align}\n 0 \\leq\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{3}{2}\n\\draw (x0) node[labeled_vertex]{};\\draw (x1) node[labeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\n\\drawhyperedge{0}{3}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\end{tikzpicture} } \\times \\Bigg( 7 \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{3}{2}\n\\draw (x0) node[labeled_vertex]{};\\draw (x1) node[labeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\drawhyperedge{0}{3}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\end{tikzpicture} } - 4 \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{2}{2}\n\\draw (x0) node[labeled_vertex]{};\\draw (x1) node[labeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\end{tikzpicture} } \\Bigg). \n\\label{eq:f47}\n\\end{align}\nIf two labeled vertices 1 and 2 have zero co-degree, the right-hand side of \\eqref{eq:f47} is zero because of the first term. If they have positive co-degree, both terms in the product are non-negative. \n\nThe calculation is computer-assisted and too large to fit in this paper. The details are available at \\oururl.\n\\end{proof}\n\n\n\n \nIn the proof of Theorem~\\ref{J4} we will use an induced removal lemma.\n\n\n\n\\begin{theorem}[Induced Removal Lemma, \\cite{MR2299696, MR2604289}]\\label{thm:removal}\nLet $r, C \\in \\mathbb{Z}$ and $\\varepsilon >0$ be fixed.\nFor every family of $r$-graphs $\\mathcal{F}$ on at most $C$ vertices, there exists $\\delta > 0$ such that every sufficiently large $n$, every $r$-graph on $n$ vertices,\nwhich contains at most $\\delta n^{|V(F)|}$ induced copies of $F$ for every $F \\in \\mathcal{F}$, can be made induced $\\mathcal{F}$-free by adding and/or deleting at most $\\varepsilon n^r$ hyperedges.\n\\end{theorem}\n\n\n\nClaim~\\ref{claim:fa47} together with Theorem~\\ref{thm:removal} forces 3-graphs with positive co-degree at least $\\frac{4}{7}n$ to be highly structured, which is the core of the proof of Theorem~\\ref{J4}.\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{J4}]\n\nRecall that $\\overline{\\mathbb{F}}_n$ is the blow-up of the complement of the Fano plane (see Figure~\\ref{fig:Fano}) on vertices $x_1,\\ldots,x_7$, where each vertex $x_i$ is blown-up to\n$\\frac{n}{7}$ vertices into the set $X_i$.\nLet $x \\in X_i$ and $y \\in X_j$ for some $i,j \\in [7]$. If $i=j$ then $d(x,y) = 0$. If $i \\neq j$ then \n$d(x,y) = \\frac{4}{7}n$. \nAs $J_4 \\not\\subseteq \\overline{\\mathbb{F}}_n$, $\\mathrm{co^+ex}(n,J_4) \\geq \\frac{4}{7}n + o(n)$, which implies $\\gamma^+(J_4) \\geq 4/7$.\n\nNext, we show that $\\gamma^+(J_4) \\leq 4/7$. Fix $\\beta > 0$ and $\\varepsilon > 0$ small enough such that $24 \\varepsilon^{1/4} < \\frac{1}{12} \\left(\\frac{4}{7}\\right)^3 \\approx 0.015$ and $48\\varepsilon^{1/4} < \\beta$. \nWe shall fix $n$ sufficiently large such that Claim~\\ref{claim:fa47} and Theorem~\\ref{thm:removal} imply that every $J_4$-free, $n$-vertex $3$-graph $G$ has a subgraph $G'$ at edit distance at most $\\varepsilon n^3$ from $G$ such that \nevery 6-vertex induced subgraph of $G'$ is in $\\mathcal{F}$.\nWe will also take $n$ sufficiently large that Lemma~\\ref{edge approx} may be applied with $c = \\frac{4}{7}$ and any $n$-vertex $3$-graph $H$ with $\\delta_2^+(H) \\geq \\left(\\frac{4}{7} - 48 \\varepsilon^{1/4}\\right)n$ contains a $K_4^3$. This last condition is possible because $\\gamma^+(K_4^3) \\leq 0.543$ (see Table~\\ref{previous bounds}), and $0.543 <\\frac{4}{7} - 48 \\varepsilon^{1/4}$ by the choice of $\\varepsilon$.\n\n\nFix an $n$-vertex $J_4$-free $3$-graph $G_n$ with $\\delta_2^+(G_n) \\geq \\frac{4}{7}n$. Our goal is to show that $\\delta_2^+(G_n) < \\left( \\frac{4}{7} + \\beta\\right)n$, which will establish $\\gamma^+(J_4) \\leq \\frac{4}{7}$. The proof will also imply that in fact, the balanced blow-up of the complement of the Fano plane is the asymptotically unique extremal construction.\n\n\nBy Lemma \\ref{edge approx} and the choice of $\\varepsilon$, $G_n$ contains more than $24 \\varepsilon^{1/4} n^3$ hyperedges. \nWe can thus apply Lemma~\\ref{positive codegree cleanup} to conclude that $G'_n$ contains an $n$-vertex subgraph $G''_n$ with $\\delta_2^+(G''_n) \\geq \\left(\\frac{4}{7} - 48 \\varepsilon^{1/4}\\right)n$. Since $G'_n$ may not be a subgraph of $G_n$, we apply Lemma~\\ref{positive codegree cleanup} to the subgraph of $G'_n$ obtained by deleting any $3$-edges that were added to $G_n$ by the application of Theorem~\\ref{thm:removal}.\nThus, $G''_n$ contains $K_4^3$, say on vertices $v_1,v_2,v_3,v_4$.\nHence $\\{v_1,v_2,v_3,v_4\\}$ also spans $K_4^3$ in $G'_n$.\nA search through $\\mathcal{F}$ shows that there are only two possible subgraphs on $5$ vertices containing $K_4^3$, which we label $A$ and $B$ below.\n\\begin{align*}\nA&:=\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{5}{0}\n\\draw (x0) node[unlabeled_vertex]{};\\draw (x1) node[unlabeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\\draw (x3) node[unlabeled_vertex]{};\\draw (x4) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\labelvertex{2}\\labelvertex{3}\\labelvertex{4}\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhyperedge{4}{5}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{5}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{3}{5}\n\\drawhyperedge{6}{5}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{4}{6}\n\\end{tikzpicture} } \n&\nB&:=\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{5}{0}\n\\draw (x0) node[unlabeled_vertex]{};\\draw (x1) node[unlabeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\\draw (x3) node[unlabeled_vertex]{};\\draw (x4) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\labelvertex{2}\\labelvertex{3}\\labelvertex{4}\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhyperedge{4}{5}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{5}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{3}{5}\n\\drawhyperedge{6}{5}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{3}{6}\n\\drawhypervertex{4}{6}\n\\drawhyperedge{7}{5}\n\\drawhypervertex{2}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\end{tikzpicture} } \n\\end{align*}\n\nWe now partition $V(G'_n)$ into 7 sets $X_1,\\ldots,X_7$ as follows. \nPut $v_i \\in X_i$ for $i \\in [4]$.\nFor $v \\in V(G'_n) \\setminus \\{v_1,v_2,v_3,v_4 \\}$, define $G_v := G_n[v_1,v_2,v_3,v_4,v]$.\nIf $G_v$ is isomorphic to $A$ then put $v \\in X_i$ where $d_{G_v}(v,v_i) = 0$ for some $v_i$.\nIf $G_v$ is isomorphic to $B$ then put \n$v \\in X_5$ if $v_1v_2v, v_3v_4v \\not\\in E(G_v)$ or\n$v \\in X_6$ if $v_1v_3v, v_2v_4v \\not\\in E(G_v)$ or \n$v \\in X_7$ if $v_1v_4v, v_2v_3v \\not\\in E(G_v)$.\nThis addresses every vertex $v$, so we have a partition of $V(G'_n)$ into $X_1,\\ldots,X_7$.\nThis labeling matches that in Figure~\\ref{fig:Fano}.\n\nNow, we show that $G'_n$ is in fact a blow-up of the complement of the Fano plane with classes $X_1, \\dots, X_7$. \n\n\n\n\\begin{claim}\\label{no bad edges}\n$G'_n$ satisfies the following conditions.\n\n(i) For every $i \\in [7]$, no edge of $G'_n$ intersects $X_i$ in more than one vertex.\n\n(ii) If $a \\in X_i, b \\in X_j, c \\in X_k$ for classes $X_i,X_j,X_k$ which correspond to an edge of the Fano plane as labeled in Figure~\\ref{fig:Fano}, then $abc \\not\\in E(G'_n)$.\n\n(iii) If $a \\in X_i, b \\in X_j, c \\in X_k$ for distinct classes $X_i,X_j,X_k$ which do not correspond to an edge of the Fano plane as labeled in Figure~\\ref{fig:Fano}, then $abc \\in E(G'_n)$.\n\n\nIn particular, $G'_n$ is a blow-up of the complement of the Fano plane.\n\n\\end{claim}\n\n\n\\begin{proof}\nWe prove conditions (i), (ii), and (iii) one by one; in each case, we will argue that if the condition is not satisfied, then $G'_n$ must contain some subgraph that is not in $\\mathcal{F}$. Throughout, refer to Figure~\\ref{J4 positive density set} for the labeled members of $\\mathcal{F}$.\nWe implemented this check by a computer to reduce the number of cases needed to be done by hand. \n\nFor (i), suppose for a contradiction that there exists an edge $abc$ such that $a,b$ are in the same class of $G'_n$. First note that by the definition of $X_1, \\dots, X_7$, if $abc$ either intersects $\\bigcup_{i=1}^4 X_i$ in at most two vertices or $abc$ is contained in $X_i$ for some $i \\in [4]$, then there exists $\\{i,j,k\\} \\subset [4]$ such that two of $N(x_ix_j)$, $N(x_ix_k)$, and $N(x_jx_k)$ contain $a,b,c$. Up to symmetry, we may assume $a,b,c \\in N(x_1x_2)$ and $N(x_1x_3)$. \nThe the set of edges of an induced subgraph $H$ of $G'_n$ on $\\{x_1,x_2,x_3,a,b,c\\}$\nincludes \n\\[E = \\{x_1x_2x_3,\\, x_1x_2a,\\, x_1x_2b,\\, x_1x_2c,\\, x_1x_3a,\\, x_1x_3b,\\, x_1x_3c,\\, abc \\}.\\]\nSince no graph of $\\mathcal{F}$ contains a subset of edges isomorphic to $E$, we have $H \\not\\in \\mathcal{F}$, a contradiction. We use an analogous claim repeatedly. We use computer for verification as well as arguments by hand.\n\nThus, if edge $abc$ exists, we must have $a,b \\in X_i$ and $c \\in X_j$ for some $i \\neq j$ with $i,j \\in [4]$. \nNote that at most one of $a,b,c$ is in $\\{x_1,x_2,x_3,x_4\\}$, since by the definition of the classes, no edge containing $x_i$ and $x_j$ for $\\{i,j\\} \\in [4]$ intersects $X_i$ or $X_j$.\nWe have (up to symmetry of the classes) two cases.\n\n \\textbf{Case 1:} $a,b \\in X_3$, $c \\in X_4$, and $x_3 \\not\\in \\{a,b\\}$.\n Define $H=G'_n[x_1,x_2,x_3, a, b,c]$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_1x_2x_3,\\, x_1x_2a,\\, x_1x_2b,\\, x_1x_2c,\\, x_1x_3c,\\, abc \\}\\] \nand avoids edges $N=\\{x_1x_3a,\\, x_1x_3b,\\, x_2x_3a,\\, x_2x_3b\\}$.\n Since no graph in $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ and avoids $N$, we have $H \\not\\in \\mathcal{F}$, a contradiction. \n\n \n \\textbf{Case 2:} $a = x_3$, $b \\in X_3$, and $c \\in X_4$.\n Consider the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3,x_4, b,c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_1x_2x_3,\\, x_1x_2x_4,\\, x_1x_3x_4,\\, x_2x_3x_4,\\, x_1x_2b,\\, x_1x_4b,\\, x_2x_4b,\\, x_1x_2c,\\, x_1x_3c,\\, x_2x_3c,\\, abc \\}.\\] \n Since no graph in $\\mathcal{F}$ contains a subset of edges isomorphic to $E$, we have $H \\not\\in \\mathcal{F}$, a contradiction. \n \n\n\nFor (ii), suppose for a contradiction that there exists an edge $abc$ spanning three classes that correspond to an edge in Figure~\\ref{fig:Fano}. Up to symmetry, there are two cases.\n\n\n\\textbf{Case 1:} The edge $abc$ intersects $\\bigcup_{i = 1}^4 X_i$ in two vertices. Without loss of generality, $a \\in X_1, b\\in X_4$, and $c \\in X_7$.\nObserve that we cannot have both $a = x_1$ and $b = x_4$ as otherwise $c \\not\\in X_7$ by definition of the classes, $N(x_1,x_4)$ is disjoint from $X_7$. \nWithout loss of generality, $b \\neq x_4$. We consider the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_2x_3x_4,\\, x_2x_3a,\\, x_2x_4a,\\, x_3x_4a,\\, x_2x_3b,\\, x_2x_4c,\\, x_3x_4c,\\, abc \\}\\] \nand avoids edges containing both $x_4$ and $b$.\n Since no graph in $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ while avoiding edges containing both $x_4$ and $b$, we have $H \\not\\in \\mathcal{F}$, a contradiction.\n\n\n\n\n\\textbf{Case 2:} The edge $abc$ does not intersect $\\bigcup_{i = 1}^4 X_i$. Without loss of generality, $a \\in X_5, b \\in X_6,$ and $c \\in X_7$.\nWe define the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3, a,b,c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_1x_2x_3,\\, x_1x_2b,\\, x_1x_2c,\\, x_1x_3a,\\, x_1x_3c,\\, x_2x_3a,\\, x_2x_3b,\\, abc \\}\\] \nand avoids edges $N=\\{x_1x_2a$, $x_2x_3c$, $x_1x_3b\\}$.\n Since no graph in $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ while avoiding $N$ we have $H \\not\\in \\mathcal{F}$, a contradiction.\n\n\n\nFinally, for (iii), suppose for a contradiction that there exist vertices $a \\in X_i, b\\in X_j, c \\in X_k$ such that $X_i,X_j,X_k$ do not correspond to an edge in Figure~\\ref{fig:Fano} and $abc \\not\\in E(G'_n)$. Up to symmetry of classes, there are three cases.\n\n\n\\textbf{Case 1:}\nWe have $i,j,k \\in [4]$. Without loss of generality, $a \\in X_1, b \\in X_2,$ and $ c \\in X_3$.\nBy the definition of $X_1,X_2,X_3$, note that at most one of $a,b,c$ is in $\\{x_1,x_2,x_3\\}$. Without loss of generality, $b \\neq x_2$ and $c \\neq x_3$. We define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_2x_3x_4,\\, x_2x_3a,\\, x_2x_4a,\\, x_3x_4a,\\, x_3x_4b,\\, x_2x_4c \\}\\]\nand avoids edges $N = \\{abc,\\, x_2x_3b,\\, x_2x_4b,\\, x_2x_3c,\\, x_3x_4c \\}$.\n Since, no graph of $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ while avoiding edges in $N$, $H \\not\\in \\mathcal{F}$, a contradiction.\n \n\n\\textbf{Case 2:}\nPrecisely two of $i,j,k$ are in $[4]$. Without loss of generality, $a \\in X_1, b\\in X_2$, and $c \\in X_6$.\nObserve that by the definition of $X_6$, at most one of $a,b$ is in $\\{x_1,x_2\\}$; without loss of generality, $b \\neq x_2$. We define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4,a,b,c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_2x_3x_4,\\, x_2x_3a,\\, x_2x_4a,\\, x_3x_4a,\\, x_3x_4b,\\, x_2x_3c,\\, x_3x_4c \\}\\] \nand avoids edges $N = \\{abc,\\, x_2x_3b,\\, x_2x_4b,\\, x_2x_4c\\}$.\n Since no graph in $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ while avoiding edges in $N$, we have $H \\not\\in \\mathcal{F}$, a contradiction.\n \n\n\\textbf{Case 3:}\nExactly one of $i,j,k$ is in $[4]$. Without loss of generality, $a \\in X_1$, $b \\in X_5$, and $c \\in X_6$.\n\nWe define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. \n The set of edges of $H$ \nincludes \n\\[E = \\{x_2x_3x_4,\\, x_2x_3a,\\, x_2x_4a,\\, x_3x_4a,\\, x_2x_3b,\\, x_2x_4b,\\, x_3x_4b,\\, x_2x_3c,\\, x_3x_4c \\}\\] \nand avoids edges $N = \\{abc,\\, x_2x_4c,\\, x_3x_4b \\}$.\n Since, no graph of $\\mathcal{F}$ contains a subset of edges isomorphic to $E$ while avoiding edges in $N$, $H \\not\\in \\mathcal{F}$, a contradiction.\n\n\nWe conclude that all conditions (i), (ii), and (iii) hold, i.e., $G'_n$ is a blow-up of $\\overline{\\mathbb{F}}$.\n\\end{proof}\n\n\nFinally, we show that $G'_n$ is almost balanced. Recall that $G'_n$ contains a subgraph $G''_n$\nwith $\\delta_2^{+}(G''_n) \\geq (\\frac{4}{7} - 48 \\varepsilon^{1/4})n$. Fix $\\alpha \\geq 0$ so that a largest class in $G'_n$ has size at least $\\left(\\frac{1}{7} + \\alpha\\right) n$. Without loss of generality, $X_1$ is a largest class. We bound the co-degree in $G''_n$ of pairs containing vertices in $X_1$. Observe that there are three sets of classes disjoint from $X_1$ that appear as neighborhoods of vertex pairs in $G''_n$. Namely,\n\\[N(x_1, x_2) \\subseteq X_3 \\cup X_4 \\cup X_6 \\cup X_7;\\]\n\\[N(x_1, x_3) \\subseteq X_2 \\cup X_4 \\cup X_5 \\cup X_7;\\]\n\\[N(x_1, x_4) \\subseteq X_2 \\cup X_3 \\cup X_5 \\cup X_6.\\]\nThus, in $G''_n$, we have\n\\[d(x_1,x_2) + d(x_1,x_3) + d(x_1,x_4) \\leq \\sum_{i = 2}^7 2|X_i| \\leq \\left(\\frac{12}{7} - 2\\alpha\\right)n.\\]\nBy averaging, one of $d(x_1,x_2), d(x_1,x_3), d(x_1,x_4)$ is at most $\\left( \\frac{4}{7} - \\frac{2\\alpha}{3} \\right)n$. Thus, $\\delta_2^+(G''_n) \\leq \\frac{4n}{7}$, a contradiction if $\\delta_2^{+}(G_n) \\geq \\left( \\frac{4}{7} + \\beta\\right)n$. We conclude that $\\gamma^+(J_4) \\leq \\frac{4}{7}$.\nTo see that $G''_n$ should be approximately balanced, note that by the minimum positive co-degree condition on $G''_n$, we thus must have $\\frac{2\\alpha}{3} \\leq 48\\varepsilon^{1/4}$, i.e., $G''_n$ (and $G'_n$) contains no class of size larger than $\\left(\\frac{1}{7} + 72 \\varepsilon^{1/4}\\right)n$. This upper bound implies that $G'_n$ contains no class of size smaller than $\\left(\\frac{1}{7} - 432 \\varepsilon^{1/4}\\right)n$.\n\\end{proof}\n\nWe determine the positive co-degree density and the asymptotically unique extremal construction for $F_{4,2}$. \nSince the proof is analogous to (but simpler than) the proof of Theorem~\\ref{J4}, we only include a sketch.\n\n\n\\begin{proof}[Sketch of the proof of Theorem~\\ref{J4}]\nObserve first that $F_{4,2}$ is $6$-partite, so it is not contained in the balanced blow-up of $K_5^3$, which implies $\\gamma^+(F_{4,2}) \\geq \\frac{3}{5}$. We now show that $\\gamma^+(F_{4,2}) \\leq \\frac{3}{5}$. Let $\\mathcal{F}$ be the family of seven $6$-vertex $3$-graphs depicted in Figure~\\ref{seven-six-graphs}.\n\n\\begin{figure}\n\n\\begin{center}\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{6}{0}\n\\draw[opacity=0](0,0)--(0,-3.4);\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_4$}\n(x4) node{\\tiny$X_5$}\n(x5) node{\\tiny$X_5$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{3}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{4}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{0}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{0}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{0}{8}\n\\drawhypervertex{3}{8}\n\\drawhypervertex{5}{8}\n\\drawhyperedge{9}{6}\n\\drawhypervertex{1}{9}\n\\drawhypervertex{2}{9}\n\\drawhypervertex{3}{9}\n\\drawhyperedge{10}{6}\n\\drawhypervertex{1}{10}\n\\drawhypervertex{2}{10}\n\\drawhypervertex{4}{10}\n\\drawhyperedge{11}{6}\n\\drawhypervertex{1}{11}\n\\drawhypervertex{2}{11}\n\\drawhypervertex{5}{11}\n\\drawhyperedge{12}{6}\n\\drawhypervertex{1}{12}\n\\drawhypervertex{3}{12}\n\\drawhypervertex{4}{12}\n\\drawhyperedge{13}{6}\n\\drawhypervertex{1}{13}\n\\drawhypervertex{3}{13}\n\\drawhypervertex{5}{13}\n\\drawhyperedge{14}{6}\n\\drawhypervertex{2}{14}\n\\drawhypervertex{3}{14}\n\\drawhypervertex{4}{14}\n\\drawhyperedge{15}{6}\n\\drawhypervertex{2}{15}\n\\drawhypervertex{3}{15}\n\\drawhypervertex{5}{15}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n \\draw[opacity=0](0,0)--(0,-3.4);\n\\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_4$}\n(x4) node{\\tiny$X_4$}\n(x5) node{\\tiny$X_4$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{3}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{4}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{0}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{1}{7}\n\\drawhypervertex{2}{7}\n\\drawhypervertex{3}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{1}{8}\n\\drawhypervertex{2}{8}\n\\drawhypervertex{4}{8}\n\\drawhyperedge{9}{6}\n\\drawhypervertex{1}{9}\n\\drawhypervertex{2}{9}\n\\drawhypervertex{5}{9}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n\\draw[opacity=0](0,0)--(0,-3.4);\n \\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_4$}\n(x5) node{\\tiny$X_4$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{3}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{4}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{0}{6}\n\\drawhypervertex{3}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{0}{7}\n\\drawhypervertex{4}{7}\n\\drawhypervertex{5}{7}\n\\drawhyperedge{8}{6}\n\\drawhypervertex{1}{8}\n\\drawhypervertex{2}{8}\n\\drawhypervertex{3}{8}\n\\drawhyperedge{9}{6}\n\\drawhypervertex{1}{9}\n\\drawhypervertex{2}{9}\n\\drawhypervertex{4}{9}\n\\drawhyperedge{10}{6}\n\\drawhypervertex{1}{10}\n\\drawhypervertex{3}{10}\n\\drawhypervertex{5}{10}\n\\drawhyperedge{11}{6}\n\\drawhypervertex{1}{11}\n\\drawhypervertex{4}{11}\n\\drawhypervertex{5}{11}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n \\draw[opacity=0](0,0)--(0,-3.4);\n \\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_1$}\n(x2) node{\\tiny$X_2$}\n(x3) node{\\tiny$X_2$}\n(x4) node{\\tiny$X_3$}\n(x5) node{\\tiny$X_3$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{2}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{3}{3}\n\\drawhypervertex{4}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{1}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{3}{5}\n\\drawhypervertex{5}{5}\n\\drawhyperedge{6}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{4}{6}\n\\drawhypervertex{5}{6}\n\\drawhyperedge{7}{6}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhypervertex{5}{7}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n \\draw[opacity=0](0,0)--(0,-1.4);\n\\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_3$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_3$}\n(x5) node{\\tiny$X_3$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{1}{3}\n\\drawhypervertex{5}{3}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n \\draw[opacity=0](0,0)--(0,-1.4);\n \\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_2$}\n(x2) node{\\tiny$X_2$}\n(x3) node{\\tiny$X_3$}\n(x4) node{\\tiny$X_3$}\n(x5) node{\\tiny$X_3$}\n;\n\\drawhyperedge{0}{6}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{6}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{6}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{6}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{5}{3}\n\\drawhyperedge{4}{6}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{3}{4}\n\\drawhypervertex{5}{4}\n\\drawhyperedge{5}{6}\n\\drawhypervertex{0}{5}\n\\drawhypervertex{4}{5}\n\\drawhypervertex{5}{5}\n\\end{tikzpicture} } \n \\vc{ \\begin{tikzpicture}[flag_pic]\n \\draw[opacity=0](0,0)--(0,-1.4);\n \\outercycle{6}{0}\n\\draw\n(x0) node{\\tiny$X_1$}\n(x1) node{\\tiny$X_1$}\n(x2) node{\\tiny$X_1$}\n(x3) node{\\tiny$X_2$}\n(x4) node{\\tiny$X_2$}\n(x5) node{\\tiny$X_2$}\n;\n\\drawhyperedge{0}{6}\n\\end{tikzpicture} \n} \n\\end{center}\n\\caption{The family $\\mathcal{F}$ of seven $6$-vertex $3$-graphs.}\\label{seven-six-graphs}\n\\end{figure}\n\nObserve that $\\mathcal{F}$ consists of the empty $3$-graph and the $6$-vertex blow-ups of $K_3^3$, $K_3^4$, and $K_5^3$. Thus, $\\mathcal{F}$ is exactly the set of $6$-vertex induced subgraphs of the balanced blow-up of $K_5^3$.\n\nUsing flag algebras, we show that for sufficiently large $n$, if $G_n$ is a $F_{4,2}$-free $n$-vertex 3-graph with $\\delta_2^+(G_n) \\geq \\frac{3n}{5}$, then\nthe 6-vertex 3-graphs with positive density in $G_n$ are in $\\mathcal{F}$.\nThe details of the calculations by computer are available at \\oururl.\nUsing the induced removal lemma (Theorem~\\ref{thm:removal}), $G_n$ has a small edit distance to $G'_n$, where every 6-vertex subgraph vertices belongs to $\\mathcal{F}$.\n\nRecall that $\\gamma^+(K_4^3) \\leq 0.543 < \\frac{3}{5}$, so by Lemma~\\ref{positive codegree cleanup}, $G'_n$ contains a subgraph with minimum positive co-degree larger than $(0.543 + \\varepsilon)n$ for some $\\varepsilon > 0$. Thus, we can assume $G'_n$ contains $K_4^3$, say on vertices $v_1,v_2,v_3,v_4$.\nA search through $\\mathcal{F}$ shows that there are only two possible subgraphs on $5$ vertices containing $K_4^3$.\n\\begin{align*}\nA&:=\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{5}{0}\n\\draw (x0) node[unlabeled_vertex]{};\\draw (x1) node[unlabeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\\draw (x3) node[unlabeled_vertex]{};\\draw (x4) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\labelvertex{2}\\labelvertex{3}\\labelvertex{4}\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhyperedge{4}{5}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{5}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{3}{5}\n\\drawhyperedge{6}{5}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{2}{6}\n\\drawhypervertex{4}{6}\n\\end{tikzpicture} } \n&\nB&:=\n \\vc{ \\begin{tikzpicture}[flag_pic]\\outercycle{5}{0}\n\\draw (x0) node[unlabeled_vertex]{};\\draw (x1) node[unlabeled_vertex]{};\\draw (x2) node[unlabeled_vertex]{};\\draw (x3) node[unlabeled_vertex]{};\\draw (x4) node[unlabeled_vertex]{};\n\\labelvertex{0}\\labelvertex{1}\\labelvertex{2}\\labelvertex{3}\\labelvertex{4}\n\\drawhyperedge{0}{5}\n\\drawhypervertex{0}{0}\n\\drawhypervertex{1}{0}\n\\drawhypervertex{2}{0}\n\\drawhyperedge{1}{5}\n\\drawhypervertex{0}{1}\n\\drawhypervertex{1}{1}\n\\drawhypervertex{3}{1}\n\\drawhyperedge{2}{5}\n\\drawhypervertex{0}{2}\n\\drawhypervertex{1}{2}\n\\drawhypervertex{4}{2}\n\\drawhyperedge{3}{5}\n\\drawhypervertex{0}{3}\n\\drawhypervertex{2}{3}\n\\drawhypervertex{3}{3}\n\\drawhyperedge{4}{5}\n\\drawhypervertex{0}{4}\n\\drawhypervertex{2}{4}\n\\drawhypervertex{4}{4}\n\\drawhyperedge{5}{5}\n\\drawhypervertex{1}{5}\n\\drawhypervertex{2}{5}\n\\drawhypervertex{3}{5}\n\\drawhyperedge{6}{5}\n\\drawhypervertex{1}{6}\n\\drawhypervertex{3}{6}\n\\drawhypervertex{4}{6}\n\\drawhyperedge{7}{5}\n\\drawhypervertex{2}{7}\n\\drawhypervertex{3}{7}\n\\drawhypervertex{4}{7}\n\\drawhyperedge{8}{5}\n\\drawhypervertex{1}{8}\n\\drawhypervertex{2}{8}\n\\drawhypervertex{4}{8}\n\\drawhyperedge{9}{5}\n\\drawhypervertex{0}{9}\n\\drawhypervertex{3}{9}\n\\drawhypervertex{4}{9}\n\\end{tikzpicture} } \n\\end{align*}\nNote that $A$ is the (unique) $5$-vertex blow-up of $K_4^3$ and $B = K_5^3$. We now partition $V(G'_n)$ into five sets $X_1,\\ldots,X_5$ as follows. \nPut $v_i \\in X_i$ for $i \\in [4]$.\nFor $v \\in V(G'_n) \\setminus \\{v_1,v_2,v_3,v_4 \\}$, define $G_v := G_n[v_1,v_2,v_3,v_4,v]$.\nIf $G_v$ is isomorphic to $A$, then put $v \\in X_i$, where $d_{G_v}(v,v_i) = 0$.\nIf $G_v$ is isomorphic to $B$, then put $v \\in X_5$.\n\nAn inspection of cases establishes the following claim.\n\n\\begin{claim}\n$G'_n$ satisfies the following conditions.\\\\\n(i) For every $i \\in [5]$, no edge of $G'_n$ intersects $X_i$ in more than one vertex.\\\\\n(ii) For every $a \\in X_i, b \\in X_j,$ and $c \\in X_k$ with $i,j,k$ pairwise distinct, $abc \\in E(G'_n)$.\n\\end{claim}\n\nIn particular, $G'_n$ is a blow-up of $K_5^3$. From here, the positive co-degree condition can be used to establish that the vertex-partition of $G_n'$ is essentially balanced.\n\\end{proof}\n\nFinally, Theorem~\\ref{thm:FlagDensity} was proved using flag algebras. The certificates for the proofs are available at \\oururl. These bounds are unlikely to be tight.\n\n\n\n\n\\section{Concluding remarks}\\label{conclusion}\n\n\nWhile we significantly expand the known sets of jumps and achievable values for $\\gamma^+$, the general behavior of $\\gamma^+$ remains mysterious, even for $r = 3$. It is unclear whether $\\gamma^+$ has a jump everywhere, though we conjecture that more jumps exist than are characterized in Theorem~\\ref{general jumps}.\n\n\\begin{ques}\\label{jump question}\nFor $r \\geq 3$, which values of $\\alpha \\in [\\frac{2}{2r - 1}, 1]$ are $\\gamma^+$-jumps? \nDoes there exist an $\\alpha \\in [\\frac{2}{2r - 1}, 1]$ which is \\textit{not} a $\\gamma^+$-jump?\n\\end{ques}\n\n\nMany more concrete questions could be asked when $r=3$. \nFor example, it is unclear, how far Theorem~\\ref{densities} is from completely characterizing achievable densities in the range $[0,\\frac{1}{2}]$ when $r=3$.\n\n\\begin{ques}\\label{turan suspension}\nAre there achievable values of $\\gamma^+$ in $[\\frac{2}{5}, \\frac{1}{2}]$ that are not of the form $\\frac{k-2}{2k - 3}$, when $r=3$? \n\\end{ques}\n\nA negative answer to Question~\\ref{turan suspension} would suggest some similarity between $\\gamma^+$ and $\\pi$, since the extremal constructions in Theorem~\\ref{densities} are a fairly natural analogue of Tur\\'an graphs. However, note that there is no hope for an Erd\\H{o}s-Stone-Simonovits-type result giving values of $\\gamma^+$. \nIndeed, since a balanced blow-up of $K_4^3$ has positive co-degree density $\\frac{1}{2}$, every $3$-graph $F$\nwith $0 < \\gamma^+(F) < \\frac{1}{2}$ is $4$-partite. \n\nTo begin addressing either Question~\\ref{jump question} or \\ref{turan suspension}, it seems natural to start with the next interval between known achievable values. Even this next case seems difficult.\n\n\\begin{ques}\nFor $r = 3$, is every $\\alpha \\in [\\frac{2}{5}, \\frac{3}{7})$ a $\\gamma^+$-jump? \nIs there a family $\\mathcal{F}$ with $\\gamma^+(\\mathcal{F}) \\in \\left(\\frac{2}{5}, \\frac{3}{7}\\right)$?\n\\end{ques}\n\n\n\nThe use of flag algebras introduces the potential for new approaches to positive co-degree questions, particularly when combined with proving the existence of $\\gamma^+$-jumps. As illustrated by Theorems~\\ref{J4} and \\ref{F42density}, flag algebra calculations have the potential to directly determine values of $\\gamma^+$, and even inexact bounds (e.g., $\\gamma^+(K_4^3) \\leq 0.543$) given by flag algebras are sometimes substantially better and more useful than what seems tractable by hand. When combined with known jumps of the function $\\gamma^+$, flag algebra bounds also have the potential to produce exact results. For instance, any $3$-graph $F$ that can be shown via a flag calculation to have $\\gamma^+(F) < \\frac{2}{5}$ must have $\\gamma^+(F) \\in \\left\\{0, \\frac{1}{3}\\right\\}$. Thus, obtaining estimates via flags and ``rounding down'' via known jumps is a very efficient way to determine the densities of many small $3$-graphs. Since we can directly characterize those $3$-graphs with positive co-degree density in $\\left\\{0, \\frac{1}{3}\\right\\}$, it is now also possible to directly determine by inspection whether a fixed $3$-graph $F$ has $\\gamma^+(F) \\in \\left\\{0, \\frac{1}{3}\\right\\}$; however, it seems unlikely that the set \n\\[\\mathcal{F}(r,d) := \\{F \\text{ an $r$-graph} : \\gamma^+(F) = d\\}\\]\ncan be as simply characterized for other values of $d$, even when $r = 3$. We are interested to see whether further understanding of jumps will include characterizations of this type. If they do not, the potential combination of estimated densities with the theory of jumps is an appealing approach to determining densities exactly.\n\n\n\nIt is also open whether every achievable value of $\\gamma^+$ can be achieved as the density of a single $r$-graph. Theorem~\\ref{single graph} shows that $\\frac{2}{5}$ is achievable by a single $3$-graph, as is every achievable density known outside the interval $\\left(\\frac{2}{5}, \\frac{1}{2}\\right)$.\n\n\\begin{ques}\nFor a fixed $k \\geq 5$, is there a $3$-graph $F_k$ such that $\\gamma^+(F_k) = \\frac{k-2}{2k-3}$? More generally, if $\\mathcal{F}$ is a family of $r$-graphs with $\\gamma^+(\\mathcal{F}) = \\alpha$, does there always exist a single $r$-graph $F$ with $\\gamma^+(F) = \\alpha$?\n\\end{ques}\n\n\n\nEvery $3$-graph that is not the subgraph of a (blow-up of a) suspension will have positive co-degree density at least $\\frac{1}{2}$. The complete list of known achievable densities at least $\\frac{1}{2}$ is as follows: $\\frac{1}{2}$ (achieved by $F_{3,2}$), $\\frac{4}{7}$ (achieved by $J_4$), $\\frac{3}{5}$ (achieved by $F_{4,2}$), and $\\frac{2}{3}$ (achieved by the Fano plane). \n\n\\begin{ques}\nFor $r= 3$, find other achievable values for $\\gamma^+$ larger than $\\frac{1}{2}$. \nIs there an $\\alpha \\in \\left[\\frac{1}{2}, 1\\right)$ which is a $\\gamma^+$-jump? Is there an $\\alpha \\in \\left[\\frac{1}{2}, 1\\right)$ for which we can characterize the $3$-graphs with $\\gamma^+(F) = \\alpha$?\n\\end{ques}\n\nVery little is known about the $\\gamma^+$ function for $r$-graphs when $r \\geq 4$.\nA natural starting point would be to study extensions of $3$-graphs whose positive co-degree densities are known. For example, the \\textit{r-daisy} $\\mathcal{D}_r$ is the $6$-edge $r$-graph on $r+2$ vertices whose all six edges contain the same $r-2$ vertices and each pair of the remaining 4 vertices is in one edge. \nThere was a recent breakthrough on the Tur\\'an density of $r$-daisies~\\cite{ellis2024daisies}.\nNote that $J_4$ is the $3$-daisy. \n\n\\begin{ques}\nWhat is $\\gamma^+(\\mathcal{D}_r)$ for $r \\geq 4$?\n\\end{ques}\n\n\n\n\\section{Acknowledgments}\nThis work used computing resources at the Center for Computational Mathematics, University of Colorado Denver, including the Alderaan cluster, supported by the National Science Foundation award OAC-2019089.\nThe authors thank Jan Volec for a discussion on Theorem~\\ref{J4}.\n\nPart of the work was done while the first and third authors were at AIM.\n\n\n\n\n\\bibliographystyle{abbrvurl}\n\\bibliography{references.bib}\n\n\\end{document}\n\n\\begin{tikzpicture}[scale=0.8]\n\\draw\n(-2,0) coordinate(1) node[vtx,label=left:{\\tiny $1$}](b){}\n(45:1) coordinate(2) node[vtx,label=right:{\\tiny $2$}](c){}\n(135:1) coordinate(3) node[vtx,label=left:{\\tiny $3$}](d){}\n(225:1) coordinate(4) node[vtx,label=left:{\\tiny $4$}](d){}\n(315:1) coordinate(5) node[vtx,label=right:{\\tiny $5$}](d){}\n;\n\\draw[hyperedge] (1) to[out=70,in=140,looseness=0.9] (2) to(3) to[out=120,in=70,looseness=0.8] (1);\n\\draw[hyperedge] (1) to[out=0,in=250,looseness=1.3] (3) to(4) to[out=110,in=0,looseness=1.3] (1);\n\\draw[hyperedge] (1) to[out=-70,in=240,looseness=0.8] (4) to(5) to[out=220,in=-70,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=0,in=110,looseness=0.7] (5) to(2) to[out=250,in=0,looseness=0.7] (1);\n\\draw[hyperedge] (1) to[out=-25,in=205,looseness=0.8] (2) to(4) to[out=110,in=-25,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=25,in=250,looseness=0.9] (3) to(5) to[out=155,in=25,looseness=0.8] (1);\n\\draw[line width = 0.5pt]\n(3)--(4)--(5)--(2)--(4) (2)--(3)--(5)\n;\n\\end{tikzpicture}\n\n\n\\begin{tikzpicture}[scale=0.8]\n\\draw\n(-2,0) coordinate(1) node[vtx,label=left:{\\tiny $1$}](b){}\n(45:1) coordinate(2) node[vtx,label=right:{\\tiny $2$}](c){}\n(135:1) coordinate(3) node[vtx,label=left:{\\tiny $3$}](d){}\n(225:1) coordinate(4) node[vtx,label=left:{\\tiny $4$}](d){}\n(315:1) coordinate(5) node[vtx,label=right:{\\tiny $5$}](d){}\n;\n\\draw[hyperedge] (1) to[out=70,in=140,looseness=0.9] (2) to[out=140,in=60,looseness=0.8]\n(3) to[out=120,in=70,looseness=0.8] (1);\n\\draw[hyperedge] (1) to[out=0,in=250,looseness=1.3] (3) to[out=250,in=110] \n(4) to[out=110,in=0,looseness=1.3] (1);\n\\draw[hyperedge] (1) to[out=-70,in=240,looseness=0.8] (4) to[out=300,in=220,looseness=0.8] \n(5) to[out=220,in=-70,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=0,in=110,looseness=0.7] (5) to[out=110,in=250,looseness=1.1] \n(2) to[out=250,in=0,looseness=0.7] (1);\n\\draw[hyperedge] (1) to[out=-25,in=205,looseness=0.8] (2) to[out=205,in=110,looseness=0.8] \n(4) to[out=110,in=-25,looseness=0.9] (1);\n\\draw[hyperedge] (1) to[out=25,in=250,looseness=0.9] (3) to[out=250,in=155,looseness=0.8] \n(5) to[out=155,in=25,looseness=0.8] (1);\n\\draw[line width = 0.5pt]\n(3) to[out=250,in=155,looseness=0.8] (5)\n(2) to[out=205,in=110,looseness=0.8] (4)\n(5) to[out=110,in=250,looseness=1.1] (2)\n(4) to[out=300,in=220,looseness=0.8] (5)\n(3) to[out=250,in=110] (4)\n(2) to[out=140,in=60,looseness=0.8] (3)\n;\n\\end{tikzpicture}\n\n\n\\end{document}\n\n\n\\section*{Appendix}\n\n\\begin{claim}\\label{no bad edges-OLD}\n$G'_n$ satisfies the following conditions.\n\n\\begin{enumerate}[label=(\\roman*),itemsep=1pt, parsep=0pt]\n\n\\item For every $i \\in [7]$, no edge of $G'_n$ intersects $X_i$ in more than one vertex.\n\n\\item If $a \\in X_i, b \\in X_j, c \\in X_k$ for classes $X_i,X_j,X_k$ that correspond to an edge of the Fano plane as labeled in Figure~\\ref{fig:Fano}, then $abc \\not\\in E(G'_n)$.\n\n\\item If $a \\in X_i, b \\in X_j, c \\in X_k$ for distinct classes $X_i,X_j,X_k$ that do not correspond to an edge of the Fano plane as labeled in Figure~\\ref{fig:Fano}, then $abc \\in E(G'_n)$.\n\n\\end{enumerate}\nIn particular, $G'_n$ is a blow-up of the complement of the Fano plane.\n\n\\end{claim}\n\n\n\\begin{proof}\nWe consider conditions (i), (ii), and (iii) one by one; in each case, we will argue that if the condition is not satisfied, then $G'_n$ must contain some subgraph that is not in $\\mathcal{F}$. Throughout, refer to Figure~\\ref{J4 positive density set} for the labeled members of $\\mathcal{F}$. \n\nFor (i), suppose for a contradiction that there exists an edge $abc$ such that $a,b$ are in the same class of $G'_n$. First note that by the definition of $X_1, \\dots, X_7$, if $abc$ either intersects $\\bigcup_{i=1}^4 X_i$ in at most two vertices or $abc$ is contained in $X_i$ for some $i \\in [4]$, then there exists $\\{i,j,k\\} \\subset [4]$ such that two of $N(x_ix_j)$, $N(x_ix_k)$, and $N(x_jx_k)$ contain $a,b,c$. Up to symmetry, we may assume $a,b,c \\in N(x_1x_2)$ and $N(x_1x_3)$. We observe two properties of the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3,a,b,c\\}$: every pair of vertices in $H$ has positive co-degree in $H$, and the pairs $x_1,x_2$ and $x_1,x_3$ both have co-degree $4$ in $H$. \nThe only member of $\\mathcal{F}$ in which all pairs of vertices have positive co-degree is $F_4$, so if $H \\in \\mathcal{F}$, then $H = F_4$. However, observe that in $F_4$, each vertex is contained in precisely one pair with co-degree $4$, while $x_1$ is contained in two pairs with co-degree $4$ in $H$. \nThus, $H \\not\\in \\mathcal{F}$, a contradiction.\n\nThus, if edge $abc$ exists, we must have $a,b \\in X_i$ and $c \\in X_j$ for some $i \\neq j$ with $i,j \\in [4]$. We have (up to symmetry of the classes) two cases.\n\n \\textbf{Case 1:} $a,b \\in X_3$, $c \\in X_4$, and $x_3 \\not\\in \\{a,b\\}$.\n Define the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3, a, b,c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. Observe that by the definition of $X_1, X_2, X_3,X_4$, it follows that every pair from $\\{x_1,x_2,a,b,c\\}$ has positive co-degree in $H$ and that $x_ix_3a$, $x_ix_3b$ are not edges for any $i \\in [4]$. \n The only members of $\\mathcal{F}$ which contain five vertices with pairwise positive co-degree are $F_1, F_3,$ and $F_4$. \n In each of these subgraphs, every vertex has positive co-degree with either 4 or 5 vertices, and every pair with positive co-degree has co-degree at least 2. However, observe that both $d(x_3,a) \\leq 1$ and $d(x_3, b) \\leq 1$, so either $x_3$ has positive co-degree with only $3$ vertices in $H$, or $H$ contains a pair with co-degree exactly 1. In either case, we conclude that $H \\not\\in \\mathcal{F}$. \n \n\n \\textbf{Case 2:} $a = x_3$, $b \\in X_3$, and $c \\in X_4$.\n Consider the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3,x_4, b,c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. Note that by the definition of $X_1, X_2, X_3,X_4$, it follows that every pair from $\\{x_1,x_2,x_3,x_4,b\\}$ has positive co-degree in $H$ and that $b,x_3$ have co-degree exactly $1$ in $H$. Thus, as in the previous case, we conclude that $H \\not\\in \\mathcal{F}$, a contradiction.\n\nFor (ii), suppose for a contradiction that there exists an edge $abc$ spanning three classes that correspond to an edge in Figure~\\ref{fig:Fano}. Up to symmetry, there are two cases.\n\n\n\\textbf{Case 1:} The edge $abc$ intersects $\\bigcup_{i = 1}^4 X_i$ in two vertices. Without loss of generality, $a \\in X_1, b\\in X_4$, and $c \\in X_7$.\nObserve that we cannot have both $a = x_1$ and $b = x_4$ as otherwise $c \\not\\in X_7$. Without loss of generality, $b \\neq x_4$. We consider the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. By (i), $x_4$ and $b$ have co-degree $0$ in $H$; by the definition of the $X_is$ and the fact that $abc \\in E(G'_n)$, all other pairs of vertices in $H$ have positive co-degree.\nObserve also that $\\{x_2,x_3,x_4, a, b\\}$ spans a blow-up of $K_4^3$, with $x_4$ and $b$ in the same blow-up class. \nThe unique member of $\\mathcal{F}$ that contains both five vertices with pairwise positive co-degree and a 5-vertex blow-up of $K_4^3$ is $F_1$. \nWe claim that $F_1$ is not isomorphic to $H$.\nIndeed, consider the link graph of $c$ in $H$. Since $c \\in X_7$, we have $x_2x_3c \\not \\in E(G'_n)$. If $H = F_1$, then the link graph of $c$ in $H$ avoids a matching between the classes $\\{ \\{x_2,b\\},\\{x_3\\}, \\{x_4 \\}, \\{a\\} \\}$. Thus, it must be that $ax_2c$ and $abc$ are not $3$-edges, a contradiction, as we assume $abc \\in E(G'_n)$.\n\n\n\n\n\\textbf{Case 2:} The edge $abc$ does not intersect $\\bigcup_{i = 1}^4 X_i$. Without loss of generality, $a \\in X_5, b \\in X_6,$ and $c \\in X_7$.\nWe define the subgraph $H$ of $G'_n$ induced on $\\{x_1,x_2,x_3, a,b,c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. It follows from the definitions of $X_5,X_6,X_7$, and the fact that $x_1x_2x_3, abc \\in E(G'_n)$, that all pairs of vertices in $H$ have positive co-degree. As noted above, $F_4$ is the unique subgraph of $\\mathcal{F}$ in which all 6 vertices have pairwise positive co-degree. Moreover, in $F_4$, the vertices can be partitioned into three pairs so that each pair has co-degree 4, and any pair of vertices not included in this partition has co-degree 3. We attempt to find such a partition of $V(H)$.\n\nBy the definitions of $X_5,X_6,X_7$, we know that $x_1x_2a, x_1x_3b$, and $x_2x_3c$ are not edges of $G'_n$. It follows that the only possible partition of $V(H)$ is $\\{\\{x_1,c\\}, \\{x_2,b\\}, \\{x_3,a\\} \\}$. In particular, $d(a,b) = 3$. Since $abc \\in E(G'_n)$ and $d(x_2,b) = d(x_3,a) = 4$ implies that $abx_2, abx_3 \\in E(G'_n)$, we must have $abx_1 \\not \\in E(G'_n)$. However, this implies that $d(x_1,a) \\leq 2$, a contradiction.\n\nFinally, for (iii), suppose for a contradiction that there exist vertices $a \\in X_i, b\\in X_j, c \\in X_k$ such that $X_i,X_j,X_k$ do not correspond to an edge in Figure~\\ref{fig:Fano} and $abc \\not\\in E(G'_n)$. Up to symmetry of classes, there are three cases.\n\n\n\\textbf{Case 1:}\nWe have $i,j,k \\in [4]$. Without loss of generality, $a \\in X_1, b \\in X_2,$ and $ c \\in X_3$.\nBy the definition of $X_1,X_2,X_3$, note that at most one of $a,b,c$ is in $\\{x_1,x_2,x_3\\}$. Without loss of generality, $b \\neq x_2$ and $c \\neq x_3$. We define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. Observe that $H$ contains a $K_4^3$ (on $\\{a,x_2,x_3,x_4\\}$) and by (i), both $d(x_2,b)$ and $d(x_3,c)$ are $0$. \nObserve that every graph $F \\in \\mathcal{F}$ is a blow-up with the property that if two vertices $x,y \\in V(F)$ have co-degree 0, then they are clones of one another, i.e., for $u,v \\in V(F)$, $xuv$ is an edge if and only if $yuv$ is an edge.\nThus, if $H \\in \\mathcal{F}$, then $b$ is a clone of $x_2$ in $H$ and $c$ is a clone of $x_3$ in $H$. However, since $ax_2x_3 \\in E(G'_n)$ since $a \\in X_1$, this implies that $abc \\in E(G'_n)$, a contradiction. \n\n\\textbf{Case 2:}\nPrecisely two of $i,j,k$ are in $[4]$. Without loss of generality, $a \\in X_1, b\\in X_2$, and $c \\in X_6$.\nObserve that by the definition of $X_6$, at most one of $a,b$ is in $\\{x_1,x_2\\}$; without loss of generality, $b \\neq x_2$. We define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4,a,b,c\\}$. Suppose for a contradiction that $H \\in \\mathcal{F}$. In the previous case we established that all possible edges spanning three classes from $X_1,X_2,X_3,X_4$ are present, so $\\{x_2,x_3,x_4,a,b\\}$ spans a blow-up of $K_4^3$. Note that $d(x_2,b) = 0$ by (i); as noted in the previous case, this implies that if $H \\in \\mathcal{F}$, then $b$ is a clone of $x_2$ in $H$. In particular, $ax_2c \\not \\in E(G'_n)$. By (ii), $x_3ac \\not \\in E(G'_n)$, thus, $d(a,c) \\leq 1$ in $H$. If $d(a,c) = 0$, then $a$ is a clone of $c$, a contradiction since $x_2x_4a \\in E(G'_n)$ since $a \\in X_1$ and $x_2x_4c \\not \\in E(G'_n)$ since $c \\in X_6$. If $d(a,c) = 1$, then each pair from $\\{x_2,x_3,x_4,a,c\\}$ has positive co-degree. Observe that in every member of $\\mathcal{F}$ containing five vertices with pairwise positive co-degree, the minimum positive co-degree is at least 2, a contradiction.\n\n\\textbf{Case 3:}\nExactly one of $i,j,k$ is in $[4]$. Without loss of generality, $a \\in X_1$, $b \\in X_5$, and $c \\in X_6$.\n\nWe define the subgraph $H$ of $G'_n$ induced on $\\{x_2,x_3,x_4, a, b, c\\}$. By the previous two cases, every pair of vertices from $\\{x_2,x_3,x_4, a, b\\}$ has positive co-degree, and $c$ must have positive co-degree with every vertex in $\\{x_2,x_3,x_4,a\\}$. Thus, either every pair of vertices in $H$ has positive co-degree, of $d(b,c) = 0$ and $c$ is a clone of $b$. By (ii) and the previous case, $c$ cannot be a clone of $b$, because $x_2ab \\not\\in G'_n$ but $x_2 a c \\in E(G'_n)$. Thus, all pairs of vertices in $H$ have positive co-degree. However, as previously noted, in $F_4$ (the unique element of $\\mathcal{F}$ containing 6 vertices with pairwise positive co-degree) the minimum co-degree is 3. By (ii), we have $x_2ab \\not\\in E(G'_n)$ and we assume $abc \\not\\in E(G'_n)$, so $d(a,b) \\leq 2$, a contradiction. \n\n\nWe conclude that all conditions (i), (ii), and (iii) hold, i.e., $G'_n$ is a blow-up of $\\overline{\\mathbb{F}}$.\n\\end{proof}\n\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":99,"cells":{"id":{"kind":"string","value":"2412.08595v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.08595v1"},"title":{"kind":"string","value":"Numerical Analysis of HiPPO-LegS ODE for Deep State Space Models"},"full_text":{"kind":"string","value":"\\documentclass{article}\n\n\n\n\\usepackage{arxiv}\n\n\\usepackage[utf8]{inputenc} \\usepackage[T1]{fontenc} \\usepackage[hidelinks]{hyperref} \\usepackage{url} \\usepackage{booktabs} \\usepackage{amsfonts} \\usepackage{nicefrac} \\usepackage{microtype} \\usepackage{lipsum}\t\t\\usepackage{graphicx}\n\\usepackage{natbib}\n\\usepackage{doi}\n\n\\usepackage{amsfonts}\n\\usepackage{stmaryrd}\n\\usepackage{amsmath,amssymb}\n\\usepackage{mathtools}\n\\usepackage{amsthm}\n\\usepackage{enumitem} \n\\usepackage{cancel}\n\n\\usepackage{xcolor}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{proposition}[theorem]{Proposition}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}{Definition}[section]\n\\newtheorem{conjecture}{Conjecture}[section]\n\\newtheorem{example}{Example}[section]\n\\newtheorem{aim}{Aim}[section]\n\\newtheorem{question}{Question}[section]\n\\newtheorem{task}{Task}\n\\newtheorem{observation}{Observation}[section]\n\n\\newcommand{\\pr}[1]{ \\left( #1 \\right) }\n\\newcommand{\\inner}[2]{\\langle #1, #2 \\rangle}\n\\newcommand{\\derivative}[1]{#1^{\\prime}}\n\n\n\n\\title{Numerical Analysis of HiPPO-LegS ODE for\\\\\nDeep State Space Models\n}\n\n\n\\author{\n\\normalfont\n\\begin{tabular}{@{}c@{\\hspace{1em}}c@{\\hspace{-1.3em}}c@{}} \\textbf{Jaesung R. Park} & \\textbf{Jaewook J. Suh} & \\textbf{Ernest K. Ryu} \\\\\n Mathematical Sciences &\n Comp.\\ Applied Math.\\ \\& Oper.\\ Res.\\ & Mathematics \\\\\n Seoul National University & Rice University & University of California, Los Angeles \\\\\n \\texttt{ryanpark7@snu.ac.kr} & \\texttt{jacksuh@rice.edu} & \\texttt{eryu@math.ucla.edu}\n\\end{tabular}\n}\n\n\\date{}\n\n\\renewcommand{\\headeright}{ }\n\\renewcommand{\\shorttitle}{ }\n\n\\hypersetup{\npdftitle={LegS arXiv},\npdfsubject={ },\npdfauthor={Jaesung R. Park, Jaewook J. Suh, Ernest K. Ryu},\npdfkeywords={ },\n}\n\n\\begin{document}\n\\maketitle\n\n \n\\begin{abstract}\nIn deep learning, the recently introduced state space models utilize HiPPO (High-order Polynomial Projection Operators) memory units to approximate continuous-time trajectories of input functions using ordinary differential equations (ODEs), and these techniques have shown empirical success in capturing long-range dependencies in long input sequences. However, the mathematical foundations of these ODEs, particularly the singular HiPPO-LegS (Legendre Scaled) ODE, and their corresponding numerical discretizations remain unexplored. In this work, we fill this gap by establishing that HiPPO-LegS ODE is well-posed despite its singularity, albeit without the freedom of arbitrary initial conditions, and by establishing convergence of the associated numerical discretization schemes for Riemann-integrable input functions. \n\\end{abstract}\n\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{sec:intro}\n\n\n\n\nState-space representation is a classical mathematical framework widely used to model physical systems in diverse fields such as control engineering, signal processing, and computational neuroscience, and there has been much recent interest in using state-space representations for sequence modeling in deep learning \\citep{gu_efficiently_2022, dao2024transformers, zhu2024vision, nguyen2022s4nd, goel_its_2022}.\nThe foundation of these models could be understood as the combination of classical state space representation \\citep{williams2007linear, zak2003systems} and the HiPPO (high-order polynomial projection operators) theory \\citep{gu2020hippo}.\n\n\nHiPPO is a framework using an $N$-dimensional ordinary differential equations (ODEs) to approximate the continuous-time history of an input function $f$.\nIn particular, the HiPPO-LegS (Legendre Scaled) ODE is\n\\[\n\\derivative{c}(t) = -\\frac{1}{t}Ac(t) + \\frac{1}{t}Bf(t),\n\\] \nfor $t\\in [0,T]$, where $T>0$ is some terminal time and $f \\colon [0,t] \\to \\mathbb{R}$ is an input function.\nWith a specific choice of $A \\in \\mathbb R^{N \\times N}$ and $B \\in \\mathbb R^{N \\times 1}$, the solution $c\\colon [0,T]\\rightarrow \\in \\mathbb{R}^N$ encodes the continuous-time history of $f$ via\n\\begin{align*} \n c_j(t) &= \\Big\\langle f(\\cdot ), P_{j-1}\\big(\\tfrac{2\\,\\cdot} {t}-1\\big) \\Big\\rangle_{L^2([0,t])}\n =\n \\int f(s ) P_{j-1}\\big(\\tfrac{2s} {t}-1\\big) \\;ds,\n\\end{align*}\nwhere $P_{j-1}$ is the $j-1$-th Legendre polynomial, for $j=1,2,\\dots$ \\citep{gu2020hippo}.\nThe input function $f$ could be reconstructed by the formula\n\\begin{align*}\n f(s) \\approx \\sum_{j=1}^Nc_j(t) P_{j-1}\\big(\\tfrac{2s}{t}-1\\big)\\qquad\\text{ for }0\\le s\\le t.\n\\end{align*}\nSince $\\{P_n(\\frac{2\\,\\cdot}{t}-1)\\}_{n=0}^\\infty$ forms an orthogonal basis of $L^2([0, t])$, this approximation can be viewed as the projection of $f$ on to the $N$-dimensional subspace spanned by $\\{P_n(\\frac{2\\,\\cdot}{t}-1)\\}_{n=0}^{N-1}$.\n\n\n\nUnlike previous linear time-invariant (LTI) methods such as Legendre Memory Units (LMUs) \\citep{voelker2019legendre} where the measure of projection is of fixed length, the LegS \nallows the model to keep track of the whole trajectory of $f(t)$. This distinctive property makes the LegS formulation powerful in many practical applications.\n\nHowever, despite receiving much attention for state space models in deep learning, the careful mathematical foundation of the LegS ODE is missing. To begin with, the singularity at $t=0$ renders the question of existence and uniqueness of the solution $c(t)$ a non-obvious matter. Moreover, it is unclear if the numerical methods used on the work of \\cite{gu2020hippo} are mathematically justified in the sense of consistency: in the limit of small stepsizes, do the discrete simulations converge to the true continuous-time solution?\n\n\n\\paragraph{Contributions.} \nIn this work, we provide the careful mathematical foundations of the HiPPO-LegS ODE formulation and its discretization. Specifically, we show that (i) the solution to the LegS ODE exists and is unique, but the initial condition is fixed to a predetermined value depending on $f(0)$ and (ii) the commonly used discretization schemes LegS converges to the exact continuous-time solution for all Riemann-integrable $f$.\n\n\n\n\n\n\n\\subsection{Related works} \\label{sec:related_works}\n\n\\paragraph{State Space Models for Deep Learning.}\nThe use of state-space models(SSMs) in deep learning has gained significant traction due to their ability to process sequential and temporal data efficiently.\nWhile the transformers architecture \\citep{vaswani2017attention} has become the golden standard for language models, recent SSM models such as mamba \\citep{gu_mamba_2023} has been reported to achieve state-of-the-art results, especially in the long-sequence regime. \n\nLarge-scale SSMs deploy an initialization scheme motivated by the HiPPO theory. \\citep{gu2023train} One distinctive characteristic of state-of-the-art SSMs is that the computation cost displays a linear growth with respect to sequence length, unlike the quadratic growth of transformers. S4 \\citep{gu_efficiently_2022} uses Fast Fourier Transform to attain the linear dependency, whereas mamba leverages hardward-aware computation techniques. The SSM architecture has been applied to or motivated numerous model structures \\citep{fu2022hungry, hasani2022liquid, sun2023retentive, peng2023rwkv}, used across various modalities \\citep{zhu2024vision, li2025videomamba, shams2024ssamba}. \n\n\n\\paragraph{Legendre Memory Units for LSTMs.}\nA fundamental challenge in training recurrent neural networks(RNNs) is the vanishing gradient problem, which causes long-range dependencies in temporal data to be lost during training \\citep{bengio1994learning, le2015simple}. While LSTMs \\citep{schmidhuber1997long} alleviate this problem by incorporating nonlinear gating mechanisms, modeling very long sequences remains challenging.\nMotivated by applications in computational neuroscience, LMUs \\citep{voelker2019dynamical, voelker2019legendre} introduced a novel approach to extend LSTM's capability to `remember' the sequence information by using a $N$-dimensional ODE. Using Pad\\'{e} approximants \\citep{pade1892representation}, the ODE is constructed so that the solution is the projection of the input function on the orthonormal basis of measure $\\mathbb{I}_{[t-\\theta, t]}$, where $\\theta$ is a hyperparameter. \nThe HiPPO framework could be understood as a generalization of LMUs. \nWhile LMU and its variants has proven to be effective for long sequence modeling \\citep{liu_lmuformer_2024,zhang2023performance, chilkuri_parallelizing_2021}, their scope is limited to LTI methods.\n\n\n\n\n\n\\paragraph{ Convergence Analysis from Control Theory.}\nState-space models have been extensively studied in control theory \\citep{kalman1960new, zabczyk2020mathematical}, with significant research dedicated to discretization schemes and their analysis \\citep{kowalczuk1991discretization}. However, these results are not directly applicable to the LegS ODE due to their exclusive focus on LTI systems \\citep{karampetakis2014error} or their assumption of discrete-time inputs \\citep{meena2020discretization}. Furthermore, the objectives of state-space models in deep learning applications differ fundamentally from those in classical control theory, where controlling or statistically estimating the state is usually the focus. This divergence makes it challenging to directly adapt these results to modern deep learning contexts.\n\n\n\n\n\n\\section{Problem setting and preliminaries} \\label{sec:problem setting}\n\n\\paragraph{LegS ODE.}\nIn this work, we consider the LegS ODE\n\\begin{align} \n \\derivative{c}(t) &= -\\frac{1}{t}Ac(t) + \\frac{1}{t}Bf(t) \\label{eq:legs}\n\\end{align}\nfor $t\\in [0,T]$, where $T>0$ is some terminal time and $c\\colon [0,T]\\rightarrow \\in \\mathbb{R}^N$ is the state vector that encodes the continuous-time history information of input function $f \\colon [0,t] \\to \\mathbb{R}$. The matrices $A \\in \\mathbb R^{N \\times N}$ and $B \\in \\mathbb R^{N \\times 1}$ are defined as\n\\[\n \\begin{aligned}\n A_{ij} &= \\begin{cases}\n (2i-1)^{1/2}(2j-1)^{1/2} & \\text{if} \\quad i > j \\\\\n i & \\text{if} \\quad i=j \\\\\n 0 & \\text{if} \\quad i < j ,\n \\end{cases} \\\\\n B_j &= (2j-1)^{1/2}.\n\\end{aligned}\n\\]\nSince $A$ is lower-triangular, we immediately recognize that $A$ is diagonalizable with non-repeated eigenvalues $\\{1, 2, \\dots, N\\}$. \nSo, $A$ is diagonalizable, and we use the eigendecomposition\n\\[\nA=V DV^{-1}\n\\]\nwhere $V\\in\\mathbb{R}^{N\\times N}$ is an invertible matrix and\n\\[\n D = \\mathrm{diag}\\,(1,2,\\dots,N).\n\\]\nIn the indexing of $A_{ij}$, $B_j$, and $c_i(t)$, we have $i,j\\in\\{1,\\dots,N\\}$, i.e., we use $1$-based indexing. Note that the prior HiPPO paper \\citep{gu2020hippo} uses $0$-based indexing.\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Shifted Legendre polynomials.}\nWe write $P_j(x) \\colon [-1,1] \\to [-1,1]$ to denote the $j$-th Legendre polynomial, normalized such that $P_j(1)=1$, for $j=0,1,\\dots$.\nHowever, we wish to operate on the domain $[0,1]$, so we perform the change of variables $x \\to 2x-1$.\nThis yields, \n\\[\n\\tilde{P}_j(x)= P_j(2x-1) = \\sum_{k=0}^j (-1)^j \\binom{j}{k} \\binom{j+k}{k} (-x)^k,\n\\]\nthe $j$-th shifted Legendre polynomial, for $j=0,1,\\dots$. The shifted Legendre polynomials satisfy the recurrence relation\n\\[\nx\\tilde{P}_j'(x) = j\\tilde{P}_j + \\sum_{k=1}^{j-1}(2k+1)\\tilde{P}_j(x)\n\\]\nwhich can be derived by combining the following well-known results \\citep{arfken2011mathematical}.\n\\begin{align*}\n (2n + 1)P_j(x) &= P_{j+1}'(x) - P_{j-1}'(x) \\\\\n P_{j+1}'(x) &= (n+1)P_j(x) + xP_j'(x)\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Numerical discretization methods.} \nIn this work, we analyze the numerical solution methods of the LegS ODE used in the prior work \\citep{gu2020hippo}. For all the discretization schemes, we consider a meshgrid with initial time $t_0=0$ and stepsize $h = \\left\\lfloor \\frac{T}{n} \\right\\rfloor$. Starting with $c^0=c(0)$, we iteratively apply the discretization scheme and denote the numerical solution at $n$-th step as $c^n$, and $f(nh)$ as $f^n$. \n\nThe \\textbf{\\underline{backward Euler}} method\n\\begin{align*}\n c^{n+1} &= \\Big(I + \\frac{1}{n+1}A\\Big)^{-1}c^n + \\Big(I + \\frac{1}{n+1}A\\Big)^{-1} \\frac{1}{n+1} B f^{n+1},\n \\qquad\\text{ for }n=0,1,2,\\dots\n\\end{align*}\nis well defined as is.\nHowever, the \\textbf{\\underline{forward Euler}} method\n\\begin{align*}\n c^{n+1} &= \\Big(I - \\frac{1}{n}A\\Big) c^n + \\frac{1}{n}Bf^n\n \\qquad\\text{ for }n=?\n\\end{align*}\nand the \\textbf{\\underline{bilinear (trapezoidal)}} method\n\\begin{align*}\n c^{n+1} &= \\Big(I + \\frac{1}{2(n+1)}A\\Big)^{-1} \\Big(I - \\frac{1}{2n}A\\Big)c^n + \\Big(I + \\frac{1}{2(n+1)}A\\Big)^{-1} \\Big(\\frac{1}{2n}f^n + \\frac{1}{2(n+1)}f^{n+1}\\Big)B,\n \\qquad\\text{ for }n=?\n\\end{align*}\nare not well defined for $n=0$. One remedy would be to use the identity $\\dot{c}(0) = \\pr{A+I}^{-1} f'(0)$, which we derive in Lemma~\\ref{lem:t=0 calc}. However, if $f$ is not differentiable at $t=0$, then even this remedy is not possible.\nHence, in Section~\\ref{sec:error_analysis} where we consider general $f$, we simply perform the ill-defined iteration letting $f'(0)=0$, e.g. $c^1 = c^0$ for forward Euler methods. As will be shown soon, this modification does not affect the convergence results. \n\n\n\n\n\nIn the prior work \\cite{gu2020hippo}, the authors simply sidestep this issue by shifting the $n$ index up by $1$, leading to the \n\\textbf{\\underline{approximate bilinear}} method\n\\begin{align*}\n c^{n+1} &= \\pr{I + \\frac{1}{n+1}A/2}^{-1} \\pr{I - \\frac{1}{n+1}A/2}c^n + \\pr{I + \\frac{1}{n+1}A/2}^{-1} \\pr{\\frac{1}{n+1}f^{n+1}}B,\\qquad\\text{ for }n=0,1,2,\\dots.\n\\end{align*}\nIn this work, we establish convergence of the approximate method, despite the shift in the $n$ index.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Convergence of numerical discretization methods.} \nTo numerically solve a first-order initial value problem of the form \n\\begin{align*} \n c'(t) = g(t,c(t)) \\label{eq:IVP} \n\\end{align*}\nfor $t\\ge 0$ with initial condition $c(0) = c^0$, we consider one-step methods of the form\n\\[\nc^{i+1} = c^i + h\\Phi(t_i, t_{i+1}, c^i, c^{i+1}; h), \\quad i=0,1,...,n-1\n\\]\nwith stepsize $h = T/n$ and $t_i=t_0+hi$ for $i=0,\\dots,n-1$.\nHere, $\\Phi$ is a numerical integrator intended to make the following approximation\n\\[\n\\Phi(t_i, t_{i+1}, c^i, c^{i+1}; h)\\approx c(t_{i+1})-c(t_i)= \\int^{t_i+1}_{t_i}g(s,c(s))\\;ds.\n\\]\n\nTo analyze such methods, one often estimates the \\emph{local truncation error}\n\\begin{align*}\nT_n \n&= \\frac{c(t_{n+1}) - c(t_n)}{h} - \\Phi(t_n, t_{n+1}, c(t_n), c(t_{n+1}; h)\n\\end{align*}\nand then estimates its accumulation to bound the \\emph{global error}\n\\[\ne_n = c(t_n) - c^n.\n\\]\n\n\n\n\nClassical ODE theory states that if the right-hand-side $g$ in the initial value problem is Lipschitz continuous with respect to $c$, the solution exists and is unique in an interval including the initial point $t_0=0$. Moreover, under the same conditions, the global error can be bounded with the local truncation error \\citep{ascher_petzold_1998,Süli_Mayers_2003}. \nHowever, this standard theory does not apply to the LegS ODE due to the singularity at $t=0$.\n\n\n\n\n \n\n\n\\section{LegS is well-posed} \\label{sec:well-posed}\nIn this section, we show that the LegS ODE is well-posed despite the singularity. Crucially, however, we show that there is no freedom of choosing the initial condition.\n\n\n\n\n\\begin{theorem}[Existence and uniqueness] \\label{thm:exist and unique}\n For $T>0$ and $c_0 \\in \\mathbb{R}^N$, we say $c\\colon [0,T]\\rightarrow \\mathbb R^N$ is a solution (in the extended sense) of the LegS ODE if $c$ is absolutely continuous on $(0,T]$, $c$ satisfies \\eqref{eq:legs} for almost all $t \\in (0,T]$, and $c(0) = c_0$.\n Assume $f\\colon [0,T]\\rightarrow \\mathbb R$ is Lebesgue measurable and locally integrable. \n Further, assume $f$ is continuous at $t=0$.\n Then, the solution exists and is unique if $c_0 = f(0)e_1$, where $e_1\\in \\mathbb{R}^N$ is the first standard basis vector.\n Otherwise, if $c_0 \\ne f(0)e_1$, a solution does not exist.\n\\end{theorem}\n\n\\begin{proof}\n Since $A$ is diagonalizable, the problem can be effectively reduced to $N$ $1$-dimensional problems. \n Recall $A = VDV^{-1}$ where $D=\\mathrm{diag}\\,(1,2,\\dots,N)$. \n We see the LegS ODE \\eqref{eq:legs} could be rewritten as \n \\begin{align*}\n \\derivative{c}(t) \n &= -\\frac{1}{t} A c(t) + \\frac{1}{t} B f(t) = -\\frac{1}{t} VDV^{-1} c(t) + \\frac{1}{t} B f(t). \n \\end{align*}\n Multiply both sides by $V^{-1}$ and denote $\\tilde{c}(t) = V^{-1}c(t)$. Then, the ODE becomes\n \\begin{align*}\n \\derivative{\\tilde{c}}(t) = V^{-1}\\derivative{c}(t) &= -\\frac{1}{t} D V^{-1} c(t) + \\frac{1}{t} V^{-1}B f(t) = -\\frac{1}{t} D\\tilde{c}(t) + \\frac{1}{t} V^{-1}B f(t),\n \\end{align*}\n which is a decoupled ODE with respect to $\\tilde{c}$. Recalling $D_{jj} = j$, we see that the $j$-th component of the above equation is\n \\begin{align} \n \\derivative{\\tilde{c}}_j(t)&= -\\frac{j}{t}\\tilde{c}_j(t) + \\frac{d_j}{t} f(t) \\label{eq:1d_legs}\n \\end{align}\n where $d_j = (V^{-1}B)_j$ and $\\tilde{c}_j$ is $j$-th component function of $\\tilde{c}$. Since $V$ and $V^{-1}$ are absolutely continuous bijective map from $\\mathbb{R}^N$ to $\\mathbb{R}^N$, \n the existence and uniqueness of the solution of the LegS ODE is satisfied if and only the existence and uniqueness of the solution of the ODE \\eqref{eq:1d_legs} is satisfied for all \n $j \\in \\{1, 2, ..., N\\}$. \n\n We now proceed by examining the existence and uniqueness of the solution \n of the ODE \\eqref{eq:1d_legs}. \n We first establish existence by presenting the explicit form of the solution. \n Define $\\tilde{c}_j\\colon[0,T]\\to\\mathbb{R}$ as\n \\begin{equation*}\n \\tilde{c}_j(t) = \\begin{cases}\n \\frac{d_j}{t^j} \\int^t_0 s^{j-1} f(s) ds & \\text{if} \\quad t \\in (0,T] \\\\\n \\frac{d_j}{j} f(0) & \\text{if} \\quad t = 0 .\n \\end{cases}\n \\end{equation*}\n By fundamental theorem of calculus, $\\frac{d}{dt} \\pr{ \\int^t_0 s^{j-1} f(s) ds } = t^{j-1} f(t)$ holds for almost all $t\\in(0,T]$ and thus $\\tilde{c}_j$ is differentiable for almost all $t\\in(0,T]$. \n Therefore,\n \\[\n t^j \\pr{ \\derivative{\\tilde{c}}_j(t) + \\frac{j}{t} \\tilde{c}_j(t) } \n = \\frac{d}{dt} (t^j\\tilde{c}_j(t)) \n = \\frac{d}{dt} \\pr{ d_j \\int^t_0 s^{j-1} f(s) ds }\n = d_j t^{j-1} f(t) \n \\]\n holds for almost all $t\\in(0,T]$. \n Dividing both sides by $t^j$, we conclude $\\tilde{c}_j$ satisfies \\eqref{eq:1d_legs} for almost all $t\\in(0,T]$.\n \n We now show $\\tilde{c}_j$ is continuous on $[0,T]$. \n It is sufficient to check $\\tilde{c}_j$ is continuous at $t=0$. \n Since $f$ is continuous at $t=0$, from L'H\\^{o}pital's rule we have\n \\begin{equation*}\n \\begin{aligned}\n \\lim_{t\\to0^+} \\tilde{c}_j(t)\n &= \\lim_{t\\to0^+} \\frac{d_j}{t^j} \\int^t_0 s^{j-1} f(s)ds \\\\ \n &= \\lim_{t\\to0^+} \\frac{d_j }{j t^{j-1}} \\pr{ t^{j-1} f(t) }\n = \\frac{d_j}{j} \\lim_{t\\to0^+} f(t) \n = \\frac{d_j}{j} f(0) = \\tilde{c}_j(0).\n \\end{aligned}\n \\end{equation*}\n Therefore, $\\tilde{c}_j$ is continuous at $t=0$. \n Lastly, $\\tilde{c}_j$ is absolutely continuous on $(0,T]$, since for every $[t_0,t] \\subset (0,T]$, \n both $\\frac{1}{t^j}$ and $\\int^t_0 s^{j-1} f(s) ds$ are absolutely continuous on $[t_0,t]$ and therefore their product is also absolutely continuous on $[t_0,t]$. \n As a result, we conclude $\\tilde{c}_j$ becomes the solution of the ODE \\eqref{eq:1d_legs}. \n\n We now establish the uniqueness. \n Suppose $\\hat{c}_j$ is another solution of the ODE \\eqref{eq:1d_legs}. \n Multiplying both sides of \\eqref{eq:1d_legs} by $t^j$ and reorganizing, for almost all $t\\in(0,T]$ we have\n \\[\n \\frac{d}{dt} (t^j\\hat{c}_j(t)) \n = t^j \\pr{ \\derivative{\\hat{c}}_j(t) + \\frac{j}{t} \\hat{c}_j(t) } \n = d_j t^{j-1} f(t) .\n \\]\n Since $\\hat{c}_j$ is a solution, it is absolutely continuous on $(0,T]$, therefore $t^j\\hat{c}_j(t)$ is absolutely continuous on $(0,T]$. \n Thus for $[t_0,t] \\subset (0,T]$, \n by fundamental theorem of calculus we obtain\n \\[\n t^j \\hat{c}_j(t) - t_0^j \\hat{c}_j(t_0) = d_j \\int_{t_0}^t s^{j-1} f(s) ds.\n \\]\n Taking limit $t_0 \\to 0^+$, since $\\hat{c}_j$ is a solution it is continuous at $0$, we have\n \\[\n t^j \\hat{c}_j(t) = d_j \\int_{0}^t s^{j-1} f(s) ds.\n \\]\n Dividing both sides by $t^j$ we conclude\n \\begin{equation*}\n \\hat{c}_j(t) = \\frac{d_j}{t^j} \\int^t_0 s^{j-1} f(s)ds = \\tilde{c}_j(t) \\end{equation*}\n for all $t \\in (0,T]$. \n It remains to check $\\hat{c}_j(0)=\\tilde{c}_j(0)$. \n Since $\\hat{c}$ is continuous at $0$, we know $\\hat{c}_j(0)=\\lim_{t\\to0^+}\\hat{c}_j(t)$. Thus\n \\begin{equation*}\n \\hat{c}_j(0)\n = \\lim_{t\\to0^+} \\hat{c}_j(t)\n = \\lim_{t\\to0^+} \\tilde{c}_j(t)\n = \\tilde{c}_j(0). \n \\end{equation*}\n Therefore, we conclude $\\hat{c}_j(t) = \\tilde{c}_j(t)$ for all $t\\in[0,T]$, \n the solution of the ODE \\eqref{eq:1d_legs} is unique. \n \n\n \n\n \n\n As a result, we conclude the solution of the LegS ODE uniquely exists if $\\tilde{c}_j(0) = \\frac{d_j}{j}f(0)$, and it is given by $c = V \\tilde{c}$. \n Finally, we show the unique solution $c = V \\tilde{c}$ should satisfy $c(0) = f(0)e_1$. \n From \\[\n V^{-1}c_j(0) = \\tilde{c}_j(0) = \\frac{d_j}{j}f(0) = \\frac{1}{j}(V^{-1}B)_j f(0),\n \\]\n we see\n \\[\n V^{-1}c(0) = \\text{diag} \\pr{ 1, \\frac{1}{2}, \\frac{1}{3}, ... , \\frac{1}{N} } (V^{-1}B) f(0) = D^{-1} V^{-1} B f(0).\n \\]\n Multiplying both sides by $V$, we conclude\n \\begin{align*}\n c(0) \n = V D^{-1} V^{-1} B f(0)\n = ( V^{-1} D V )^{-1}\n = A^{-1}B f(0) \n = f(0)e_1\n \\end{align*}\n where $e_1 = [1, 0, \\dots, 0] ^{t}$. \n Therefore if $c_0 = f(0)e_1$, the solution exists and is unique, and otherwise, there is no solution.\n \\end{proof}\n\n\n\\begin{lemma} [Behavior at $t=0$] \\label{lem:t=0 calc}\nConsider the setup of Theorem~\\ref{thm:exist and unique}, and further assume that $f$ is one-sided differentiable at $t=0$.\nThen, the one-sided derivative $c'(0)$ exists and is given by\n \\begin{align}\n c'(0) \n &= (A+I)^{-1}B f'(0).\\label{eq:c'(0)}\n \\end{align}\n\\end{lemma}\n\\begin{proof} \nWe examine the differentiability of $V^{-1}c(t) = \\tilde{c}(t) = (\\tilde{c}_j(t))_{j=1}^N$ by checking for each component. Recall that for the $j$-th component we have $\\tilde{c}_j(t) = \\frac{d_j}{t^j}\\int_0^t s^{j-1}f(s)ds$ and $\\tilde{c}_j(0) = \\frac{d_j}{j}f(0)$. \nWe will denote the one-sided derivative of $c(t)$ and $f(t)$ at $t=0$ as $c'(0)$ and $f'(0)$, respectively. \nThen,\n\\begin{align*}\n \\lim_{t \\to 0^+} \\frac{\\tilde{c}\n_j(t) - \\tilde{c}_j(0)}{t} \n &= \\lim_{t \\to 0^+} d_j \\frac{\\frac{1}{t^j} \\int_0^t s^{j-1} f(s)ds - \\frac{1}{j}f(0)}{t} \\\\\n &= \\lim_{t \\to 0^+} d_j \\frac{\\int_0^t s^{j-1} f(s)ds - \\frac{t^j}{j}f(0)}{t^{j+1}} \\\\\n &\\overset{(1)}{=} \\lim_{t \\to 0^+} d_j \\frac{t^{j-1}f(t) - t^{j-1} f(0)}{(j+1)t^{j}} \\\\\n &= \\lim_{t \\to 0^+} \\frac{d_j}{j+1} \\frac{f(t) - f(0)}{t} \\\\\n &= \\frac{d_j}{j+1} f'(0)\n\\end{align*}\nwhere LH\\^{o}pital's rule was used for $(1)$. \n Folding back to vector form, recalling $V^{-1}c'_j(0) = \\tilde{c}'_j(t) = \\frac{d_j}{j+1}f'(0) = \\frac{1}{j+1}(V^{-1}B)_j f'(0)$, we obtain\n \\begin{align*}\n V^{-1}c'(0) \n &= \\text{diag} \\pr{ \\frac{1}{2}, \\frac{1}{3}, ... , \\frac{1}{N+1} } (V^{-1}B) f'(0) \n \\end{align*}\nand hence $c'(0) = \\pr{A+I}^{-1}B$. \nNote that $(A+I)^{-1}B = [1/2, 1/(2\\sqrt{3}), 0, ..., 0]^{t}$.\n\\end{proof}\n\n\n\n\n\\paragraph{Ill-posedness with negative eigenvalues.}\nThe well-posedness argument of Theorem~\\ref{thm:exist and unique} crucially relies on the fact that all eigenvalues of $A$ are positive.\nTo see what happens when $A$ has negative eigenvalues, consider the case $N=1$ and $A=-1$. \nThis leads to the ODE\n\\begin{align*}\n \\frac{d}{dt}c(t) &= \\frac{1}{t} c(t) + \\frac{1}{t} f(t),\\qquad c(0)=c_0,\n\\end{align*}\nwhich has solutions\n\\begin{align*}\n c(t) = t \\int_0^t \\frac{1}{s^2}f(s)ds + Ct\n\\end{align*}\nfor any $C\\in \\mathbb{R}$. Since the initial condition does not determine the value of $C$, the solution is not unique.\n\n\n\n\n\n\n\n\n\n\\section{Convergence of LegS discretization schemes} \\label{sec:error_analysis}\n\n\n\nIn this section, we address whether the numerical discretization methods introduced in Section~\\ref{sec:problem setting} are convergent, i.e., do the methods produce numerical solutions $c^n$ that converge to the exact continuous-time solution $c(t)$ as $h\\rightarrow 0$?\n\nThe answer is yes, but, as we discuss in Section~\\ref{subsec:smooth_f}, the standard analysis based on local truncation error does not lead to a convergence guarantee for all of the schemes under consideration, and such approaches would require certain regularity conditions on $f$, such as Lipschitz continuity, anyway. Rather, in Section~\\ref{subsec:legs is quadrature}, we identify the numerical schemes as quadrature rules on the input function $f$. Using this insight, in Section~\\ref{subsec:riemann_f}, we show that the discretization schemes are convergent under the general assumption of Riemann integrability of $f$.\n\n\n\nExtending the framework to accommodate general Riemann-integrable functions $f$ is important, given the nature of the application. The HiPPO memory unit is used in deep learning to analyze sequence data, such as audio signals. For such data, there is no inherent expectation of smoothness, and discontinuities are to be expected. Therefore, we aim to guarantee that the mathematics remains sound for such data.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Convergence for smooth $f$} \\label{subsec:smooth_f}\nDiscretization schemes ODEs with well-behaved right-hand-sides have a well-established theory based on the local truncation error (LTE). This line of analysis can be applied to the LegS ODE despite the singularity at $t=0$. For example, forward Euler method, the LTE $T_n$ satisfies\n\\[\n|T_n| \\leq \\frac{1}{2} h M_2, \\qquad M_2 = \\max_{t \\in [t_0, t_N]} |c''(t)|.\n\\]\n\nThe proof follows from a relatively straightforward extension of the standard arguments. However, when $f$ is not differentiable, then $c''(t)$ may not be bounded, and this approach, as is, fails to yield a convergence guarantee.\n\n\n\n\nAnother issue is that the LTE for the approximated bilinear method does not converge to $0$. \nFor $N=1$, the approximated bilinear method reduces to \n\\begin{align*}\n c^{n+1}\n &= \\frac{2n+1}{2n+3}c^n + \\frac{2}{2n+3}f^{n+1} = c^n - \\frac{2}{2n+3}(c^n - f^{n+1}).\n\\end{align*}\nUsing the exact solution is $c(t) = \\frac{1}{t}\\int_0^t f(s)ds$, the exact value of the LTE is\n\\begin{align*}\n hT_n\n &= c(t_{n+1}) - c(t_{n}) + \\frac{2}{2n+3} (c(t_n) - f^{n+1}) \\\\\n &= \\frac{1}{(n+1)h}\\int_0^{(n+1)h}f(s)ds - \\frac{2n+1}{n(2n+3)h}\\int_0^{nh}f(s)ds - \\frac{2}{2n+3}f^{n+1} .\n\\end{align*}\nWith the particular choice linear function $f(x) = ax$, we obtain \n\\begin{align*}\n T_n &= \\frac{a}{2}(n+1) - \\frac{n(2n+1)}{2(2n+3)}a - \\frac{2a}{2n+3}(n+1) = -\\frac{a(n+1)}{2(2n+3)}\\rightarrow -\\frac{a}{4}\\ne 0.\n\\end{align*}\nThus, the LTE of the approximated bilinear method does not vanish as $h\\rightarrow 0$, implying that the method is not even consistent. Consequently, a global error analysis based on the LTE will not guarantee convergence. In Section~\\ref{subsec:riemann_f}, however, we employ an alternative proof technique to establish that the approximated bilinear method is indeed convergent.\n\n\n\n\n\n\n\n\n\n\\subsection{LegS is a quadrature of $f$} \\label{subsec:legs is quadrature}\n\nIn this section, we provide the key insight that we can identify the numerical schemes as quadrature rules on the input function $f$. Recall that the LegS equation ODE was proposed for online approximation of the input function $f(t)$ on the interval $[0,t]$.\nSpecifically, the matrix $A$ and $B$ was specifically chosen so that the $j$-th component of $c(t) \\in \\mathbb{R}^N$ is given by\n\\begin{align*}\nc_j(t) &= \\frac{\\sqrt{2j-1}}{t} \\int_0^t P_{j-1} \\Big(\\frac{2s}{t} - 1 \\Big) f(s) ds,\n\\end{align*}\nwhere $P_{j-1}$ denotes the $(j-1)$-th Legendre polynomial, for $j=1,2,\\dots,N$ \\citep{gu2020hippo}.\nSo $c_j(t)$ is a (signed) weighted integral of $f(\\cdot)$ on $[0,t]$, and the following lemma shows that the numerical schemes of Section~\\ref{sec:problem setting} are, in fact, quadrature rules approximating this integral.\n\n\n\n\\begin{lemma} \\label{lem:numsol_linear}\nConsider applying any of the discretization schemes introduced in Section~\\ref{sec:problem setting}\n (forward Euler, backward Euler, bilinear, or approximate bilinear)\n to the LegS ODE \\eqref{eq:legs}, with initial time $t_0=0$ and timestep $h = T/n$. Then, the numerical solution $c^n$ at step $n$ satisfies\n \\begin{align*}\n c^n &= \\frac{1}{n} \\sum_{l=0}^n \\alpha^{(n)}_l f^l,\n \\end{align*}\n where $\\alpha^{(n)}_l \\in \\mathbb{R}^N$ only depends on $l$ and $n$. \n\\end{lemma}\n\n\n\\begin{proof}\n\n For notational simplicity, we define the following quantities for $n \\in \\mathbb{N}$, with $n \\geq 2$ for $Q_n$.\n \\begin{align*}\n Q_n &:= \\prod_{j=1}^{n-1} \\pr{I - \\frac{1}{j+1}A} \\\\\n \\tilde{Q}_n &:= \\prod_{j=1}^n \\pr{I - \\frac{1}{2j}A}\\\\\n R_n &:= \\prod_{j=1}^n \\pr{I + \\frac{1}{j}A}^{-1} \\\\\n \\tilde{R}_n &:= \\prod_{j=1}^n \\pr{I + \\frac{1}{2j}A}^{-1}\n \\end{align*}\n We use the $\\prod$ notation when the multiplications are commutative. Note that all $Q_n, \\tilde{Q}_n, R_n, \\tilde{R}_n$ are invertible. \n We start by proving for the forward Euler method. \n Recall that the forward Euler method yields the following recurrence relation at step $n$:\n\\begin{align*}\n c^{n+1} &= \\pr{ I - \\frac{1}{n}A}c^n + \\frac{1}{n} B f^n.\n\\end{align*}\nRepeating this procedure, we can obtain an exact formula for the numerical solution obtained by applying forward Euler method to the LegS ODE. By induction we obtain,\n\\begin{align*}\n c^{n+1} \n &= \\pr{ I - \\frac{1}{n}A }c^n + \\frac{1}{n}Bf^n \\\\\n &= \\pr{ I - \\frac{1}{n}A} \\pr{ I - \\frac{1}{n-1}A} c^{n-1} + \\pr{ I - \\frac{1}{n}A} \\frac{1}{n-1}Bf^{n-1} + \\frac{1}{n}Bf^n \\\\\n &= Q_n \\pr{c^1 + Bf^1} + Q_n \\sum_{l=2}^n \\frac{1}{l} Q_{l}^{-1} B f^l.\n\\end{align*}\nAs explained in Section~\\ref{sec:problem setting}, we `zero out' the ill-defined iteration, thereby letting $c^1=c^0$. Hence we have\n\\begin{align} \\label{eq:forward_numsol}\n c^{n} \n &= Q_{n-1} e_1 f^0 + Q_{n-1} B f^1 + Q_{n-1} \\sum_{l=1}^{n-1} \\frac{1}{l} Q_l^{-1} B f^l.\n\\end{align}\nImmediately one can see that $c^n$ is expressed as $\\frac{1}{n} \\sum_{l=0}^n \\alpha_l^{(n)}$, where $\\alpha_l^{(n)}$ depends only on $l$ and $n$. \n\nFor the backward Euler method, we start from\n\\begin{equation*}\n\\begin{aligned}\n c^{n+1} \n &= \\pr{ I + \\frac{1}{n+1} A }^{-1} c^{n}\n + \\pr{ I + \\frac{1}{n+1} A }^{-1} \\frac{1}{n+1} B f^{n+1}.\n\\end{aligned}\n\\end{equation*}\n\nThen, we can derive inductively \n\\begin{equation*}\n \\begin{aligned}\n c^{n+1} \n &= \\pr{ I + \\frac{1}{n+1}A }^{-1} c^n + \\pr{ I + \\frac{1}{n+1}A }^{-1} \\frac{1}{n+1}Bf^{n+1} \\\\\n &= \\pr{ I + \\frac{1}{n+1}A }^{-1} \\pr{ I + \\frac{1}{n}A }^{-1} c^{n-1} + \\pr{ I + \\frac{1}{n+1}A }^{-1} \\pr{ I + \\frac{1}{n}A }^{-1} \\frac{1}{n}Bf^n + \\pr{ I + \\frac{1}{n+1}A }^{-1} \\frac{1}{n+1}Bf^{n+1} \\\\\n &= R_{n+1} (c^0 + Bf^1) + R_{n+1} \\sum_{l=1}^{n} \\frac{1}{l+1} R_{l}^{-1} B f^{l+1}.\n \\end{aligned}\n\\end{equation*}\nThus we can verify $c^n$ has the desired form with the specific expression\n\\begin{equation} \\label{eq:back_numsol}\n \\begin{aligned}\n c^n &=\n R_{n} (e_1f(0) + Bf^1) + R_{n} \\sum_{l=1}^{n-1} \\frac{1}{l+1} R_{l}^{-1} B f^{l+1}.\n \\end{aligned}\n\\end{equation}\n\nFor the bilinear method, we start from\n\\begin{align*}\n c^{n+1} &= (I + \\frac{1}{n+1}A/2)^{-1}(I - \\frac{1}{n}A/2)c^n + (I + \\frac{1}{n+1}A/2)^{-1} \\frac{1}{2} (\\frac{1}{n}f^n + \\frac{1}{n+1}f^{n+1})B.\n\\end{align*}\n\nSimilarly by induction we obtain\n\\begin{equation*}\n \\begin{aligned}\n c^{n+1} \n &= \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I - \\frac{1}{n}A/2 } c^n + \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\frac{1}{2} \\pr{\\frac{1}{n}Bf^{n} + \\frac{1}{n+1}Bf^{n+1} }\\\\\n &= \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I + \\frac{1}{n}A/2 }^{-1} \\pr{ I - \\frac{1}{n}A/2 } \\pr{ I - \\frac{1}{n - 1}A/2 } c^{n-1}\\\\\n &+ \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I + \\frac{1}{n}A/2 }^{-1}\\pr{ I - \\frac{1}{n - 1}A/2 } \\frac{1}{2} \\pr{\\frac{1}{n-1}Bf^{n-1} + \\frac{1}{n}Bf^{n} }\\\\\n &+ \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\frac{1}{2} \\pr{\\frac{1}{n}Bf^{n} + \\frac{1}{n+1}Bf^{n+1} }\\\\\n &= \\tilde{Q}_n \\tilde{R}_{n+1} \\pr{I+A/2} c^1 + \n \\tilde{Q}_{n} \\tilde{R}_{n+1} \\sum_{l=1}^{n} \\tilde{Q}_l^{-1} \\tilde{R}_l^{-1} \\frac{1}{2}\\pr{\\frac{1}{l}Bf^l + \\frac{1}{l+1}Bf^{l+1}}.\n \\end{aligned}\n\\end{equation*}\n\nAs for the forward Euler case, we `zero out' the ill-defined term in the first iteration. This yields $c^1 = \\pr{I+\\frac{A}{2}}^{-1}c^n + \\pr{I+\\frac{A}{2}}^{-1}\\pr{\\frac{1}{2}f^1}$. \nThen,\n\\begin{equation*} \n \\begin{aligned} \n c^n &=\n \\tilde{Q}_{n-1}\\tilde{R}_{n}e_1f^0\n +\\tilde{Q}_{n-1}\\tilde{R}_{n}\\frac{f^1}{2}\n +\\tilde{Q}_{n-1}\\tilde{R}_{n} \\sum_{l=1}^{n-1} \\tilde{Q}_l^{-1} \\tilde{R}_l^{-1} \\frac{1}{2}\\pr{\\frac{1}{l}Bf^l + \\frac{1}{l+1}Bf^{l+1}}.\n \\end{aligned}\n\\end{equation*}\nRearranging terms,\n\\begin{align}\\label{eq:bilin_numsol}\n c^n &=\n \\tilde{Q}_{n-1}\\tilde{R}_{n}e_1f^0\n + \\tilde{Q}_{n-1}\\tilde{R}_{n}\n \\sum_{l=1}^{n-1} \\frac{1}{2l}\n \\pr{\n \\tilde{Q}_l^{-1} \\tilde{R}_l^{-1} \n + \\tilde{Q}_{l-1}^{-1} \\tilde{R}_{l-1}^{-1} \n }Bf^l\n\\end{align}\nwhere we define $\\tilde{Q}_0=\\tilde{R}_0=I$. Thus we recover the desired form for $c^n$. \n\nFor the approximate bilinear method, we start from\n\\begin{align*}\n c^{n+1} &= (I + \\frac{1}{n+1}A/2)^{-1}(I - \\frac{1}{n+1}A/2)c^n + (I + \\frac{1}{n+1}A/2)^{-1} (\\frac{1}{n+1}f^{n+1})B.\n\\end{align*}\n\nBy induction, we obtain\n\\begin{equation*}\n \\begin{aligned}\n c^{n+1} \n &= \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I - \\frac{1}{n+1} A/2 } c^n + \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{\\frac{1}{n+1}Bf^{n+1} }\\\\\n &= \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I + \\frac{1}{n}A/2 }^{-1} \\pr{ I - \\frac{1}{n+1}A/2 } \\pr{ I - \\frac{1}{n}A/2 } c^{n-1}\\\\\n &+ \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{ I + \\frac{1}{n}A/2 }^{-1}\\pr{ I - \\frac{1}{n+1}A/2 } \\pr{\\frac{1}{n}Bf^{n} }\\\\\n &+ \\pr{ I + \\frac{1}{n+1}A/2 }^{-1} \\pr{\\frac{1}{n+1}Bf^{n+1} }\\\\\n &= \\tilde{Q}_{n+1} \\tilde{R}_{n+1} c^0 + \n \\tilde{Q}_{n+1} \\tilde{R}_{n+1} \\sum_{l=1}^{n} \\tilde{Q}_{l+1}^{-1} \\tilde{R}_l^{-1} \\pr{\\frac{1}{l+1}Bf^{l+1}}.\n \\end{aligned}\n\\end{equation*}\n\nThus we can verify $c^n$ has the desired form with the specific expression\n\\begin{equation} \\label{eq:apbil_numsol}\n \\begin{aligned}\n c^n &=\n \\tilde{Q}_{n}\\tilde{R}_{n}c^0 + \\tilde{Q}_{n}\\tilde{R}_{n} \\sum_{l=1}^{n-1} \\tilde{Q}_{l+1}^{-1} \\tilde{R}_l^{-1} \\pr{\\frac{1}{l+1}Bf^{l+1}}.\n \\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\n\n\\subsection{Convergence for Riemann integrable $f$} \\label{subsec:riemann_f}\n\nIn this section, we prove the convergence of all discretization methods of interest for Riemann integrable $f$'s. \nIn particular, we prove the convergence of the approximated bilinear method, justifying its use in the HiPPO paper \\citep{gu2020hippo}. \nThe results are summarized in the following theorem:\n\n\n\\begin{theorem} \n\\label{thm:conv_riemann}\n Consider the LegS equation \\eqref{eq:legs} with dimension $N$, defined on $t \\in [0,t]$. Assume that $f$ is Riemann integrable on $[0,t]$. \n Given a discretization method, assume that one performs the iterations defined by \n \\begin{align*}\n c^{n+1} &= c^n + h \\Phi(t_n,t_{n+1},c^n,c^{n+1}; h) \\\\\n c^0_j &= \\delta_{1j} f(0)\n \\end{align*}\n using mesh points $t_n = nh$ for $n = 1, 2, \\dots, \\left\\lfloor T/h \\right\\rfloor$, where $\\Phi$ is the one-step integrator defined by the given discretization method. Denote the exact solution at step $n$ as $c(t_n) = c(nh) \\in \\mathbb{R}^N$. Then, for all the discretization methods defined introduced in Section \\ref{sec:problem setting}, \n we have the convergence of the numerical solution to the exact solution in the sense that\n \\begin{align*}\n \\lim_{n \\to \\infty} c^n = c(t) \n \\quad\n \\text{as}\n \\quad\n t_n \\to t \\in [0, T]\n \\end{align*}\n when $h \\to 0$ and $n \\to \\infty$.\n\\end{theorem}\n\n\nTo prove the theorem, we first denote the the numerical solution obtained by the application of a discretization method to the LegS ODE as $c^n = \\frac{1}{n} \\sum_{l=0}^n \\alpha^{(n)}_l f^l$, in light of Lemma \\ref{lem:numsol_linear}. If we could show for all Riemann integrable $f$\n\\[\nc^n_j = \\frac{1}{n} \\sum_{l=0}^n \\pr{\\alpha^{(n)}_l}_jf^l \\to c_j(t) = \\frac{\\sqrt{2j-1}}{t}\\int_0^t P_{j-1}\\pr{\\frac{2s}{t}-1}f(s)ds,\n\\]\nwith $t_n \\to t$ as stepsize $h \\to 0$ and $n \\to \\infty$, and where $P_j$ is the $j$-th Legendre polynomial, we are done. \n\n\nInstead of directly working on the coefficients $\\alpha^{(n)}_l$, the key idea is to consider a function sequence defined on $[0,1]$ that interpolates those points. This significantly reduces the complexity of analyzing the asymptotic behavior of the numerical solution as the number of mesh points $n$ goes to infinity. \nWe start by proving an elementary lemma that enables this approach.\n\n\\begin{lemma} \\label{lem:converge_condition}\n Assume that $f : [0,t] \\to \\mathbb{R}$ is a Riemann integrable function, and $\\{G^{(n)}\\}_{n \\in \\mathbb{N}}$ is a sequence of continuous functions defined on $[0,1]$. Assume further that for some $G \\in C[0,1]$,\n \\begin{align*}\n G^{(n)}(x) \\to G(x)\n \\end{align*}\n uniformly on $x \\in (0,1)$ as $n \\to \\infty$. \n Then, for $h = \\left\\lfloor\\frac{t}{n}\\right\\rfloor$,\n \\begin{align*}\n \\frac{1}{n} \\sum_{l=1}^n G^{(n)} \\pr{\\frac{l}{n}} f\\pr{lh}\n \\to\n \\frac{1}{t}\\int_0^t G \\pr{\\frac{s}{t}}f(s)ds\n \\end{align*}\n as $h \\to 0$ and $n \\to \\infty$. \n\\end{lemma}\n\n\\begin{proof}\n Since $f(x)$ is Riemann integrable and $G$ is continuous, it is straightforward to show that $G\\pr{\\frac{x}{t}}f(x)$ is Riemann integrable for $x \\in [0,t]$. From basic properties of Riemann integrable functions, one could easily show that the Riemann sum $\\frac{1}{n} \\sum_{l=1}^n G \\pr{\\frac{l}{n}}f\\pr{lh}$ converges to $\\frac{1}{t}\\int_0^t G \\pr{\\frac{s}{t}}f\\pr{s} ds$ as $n \\to \\infty$. Since $f$ is bounded, let $\\sup_{x \\in [0,t]} |f(x)| \\leq M$ and construct a triangular inequality.\n \\begin{align*}\n \\left| \\frac{1}{t}\\int_0^t G \\pr{\\frac{s}{t} }f(s)ds - \\frac{1}{n}\\sum_{l=1}^n G^{(n)} \\pr{\\frac{l}{n}} f \\pr{lh} \\right| \n &\\leq \\left| \\frac{1}{t}\\int_0^t G \\pr{\\frac{s}{t}} f(s)ds - \\frac{1}{n} \\sum_{l=1}^n G \\pr{\\frac{l}{n}}f\\pr{lh} \\right| \\\\\n &+ \\frac{1}{n} \\sum_{l=1}^n \\left| G \\pr{\\frac{l}{n}}f\\pr{lh} - G^{(n)} \\pr{\\frac{l}{n}} f\\pr{lh} \\right| \\\\\n &\\leq \\frac{\\epsilon}{2} + \\sup_{x \\in [0,t]} \\left| f(x) \\right| \\max_{l \\in \\{0, \\dots, n\\}} \\left| G\\pr{\\frac{l}{n}} - G^{(n)} \\pr{\\frac{l}{n}} \\right| \\\\\n &\\leq \\frac{\\epsilon}{2} + M \\| G - G^{(n)} \\|_{\\sup} \\\\\n &\\leq \\epsilon\n \\end{align*}\n\\end{proof}\n\n\nThis result is relevant since if we interpret the shifted Legendre polynomials as $G\\pr{\\frac{s}{t}}$ in the lemma above, we can obtain a sufficient condition for a numerical solution to converge to the exact solution of the ODE. This observation is specified in the next corollary.\n\n\\begin{corollary} \\label{cor:conv_to_quad}\n Consider a vector $c^n \\in \\mathbb{R}^N$ such that $c^n_j = \\frac{1}{n} \\sum_{l=0}^n \\pr{\\alpha^{(n)}_l}_j f(lh)$ where $h = \\left\\lfloor\\frac{t}{n}\\right\\rfloor$ and $f : [0,t] \\to \\mathbb{R}$ a Riemann integrable function. Assume $\\sup_{l} \\pr{\\alpha^{(n)}_l}_j \\leq C$. Denote the $j$-th Legendre polynomial as $P_j$. \n \n Now, consider a uniformly bounded function sequence $\\{F^{(n)}\\}_n$ such that $F \\in C([0,1]; \\mathbb{R}^N)$ for all $F \\in \\{F^{(n)}\\}_{n \\in \\mathbb{N}}$. Further, assume that for all $F \\in \\{F^{(n)}\\}_{n \\in \\mathbb{N}}$, the $j$-th component satisfies for all $j \\in J$ and $l \\in \\{1, 2, \\dots, n-1\\}$\n \\begin{align} \\label{eq:F_condition}\n \\pr{F \\pr{\\frac{l}{n}}}_j &= \\pr{\\alpha^{(n)}_l}_j.\n \\end{align} \n Then, if\n \\begin{align*}\n \\pr{F^{(n)}(x)}_j \\to \\sqrt{2j-1} P_{j-1} \\pr{2x-1} \n \\end{align*}\n uniformly on $x\\in [0,1]$ as $n \\to \\infty$ for all $j \\in J$, \n \\begin{align*}\n \\pr{c^n}_j \\to \\frac{\\sqrt{2j-1}}{t}\\int_0^t P_{j-1}\\pr{\\frac{2s}{t}-1}f(s)ds\n \\end{align*}\n as $h \\to 0$ and $n \\to \\infty$ for all $j \\in J$. \n\\end{corollary}\n\n\\begin{proof}\n Since the sequence $\\{F^{(n)}\\}$ is uniformly bounded, we let $F_j \\leq M_j$ for all $F \\in \\{F^{(n)}\\}$. Also, $|f| \\leq M$ from the definition of Riemann integrable functions. \n Now fix component index $j$. Since $\\pr{F^{(n)}\\pr{\\frac{l}{n}}}_j$ is equal to $\\pr{\\alpha^{(n)}_l}_j$ for $l \\in \\{1, 2, \\dots, n-1\\}$, we can write\n \\begin{align*}\n \\pr{c^n}_j &= \\frac{1}{n} \\sum_{l=1}^n \\pr{F^{(n)}\\pr{\\frac{l}{n}}}_j f(lh) + \\frac{1}{n} \\pr{\\alpha^{(n)}_0 f(0) + \\alpha^{(n)}_n f(nh) - \\pr{F^{(n)}(1)}_j f(nh)}\n \\end{align*}\n Using this equality, we can construct the following triangular inequality:\n \n \\begin{align*}\n \\left| \\pr{c^n}_j - \\frac{2j-1}{t}\\int_0^t P_{j-1}\\pr{\\frac{2s}{t}-1}f(s)ds \\right| \n & \\leq \\underbrace{\\left| \\frac{1}{n} \\sum_{l=1}^n \\pr{F^{(n)}\\pr{\\frac{l}{n}}}_j f(lh) - \\frac{2j-1}{t}\\int_0^t P_{j-1}\\pr{\\frac{2s}{t}-1}f(s)ds \\right|}_{(1)} \\\\\n &+ \\frac{1}{n} \\left| \\pr{\\alpha^{(n)}_0}_j f(0) + \\pr{\\alpha^{(n)}_n}_j f(nh) \\right| + \\frac{1}{n} \\left| \\pr{F^{(n)}(1)}_j f(nh) \\right|\\\\\n &\\leq \\frac{\\epsilon}{2} + \\frac{M}{n}\\pr{2C+M_j}\\\\\n &\\leq \\epsilon\n \\end{align*}\n where the convergence of $(1)$ follows from Lemma \\ref{lem:converge_condition}.\n\\end{proof}\n\nThe result of this corollary implies that instead of directly proving the convergence of the numerical solution $c^n$ to the exact solution, it would suffice to find a function sequence $\\{F^{(n)}\\}_{n \\in \\mathbb{N}}$ satisfying \\eqref{eq:F_condition} that converges to the shifted Legendre polynomial. \nA natural choice to construct such a sequence would be to interpolate the $n-1$ points using polynomials so that the function would satisfy \\eqref{eq:F_condition}.\nHowever, the degree of the interpolating polynomials in the sequence could diverge as $n \\to \\infty$, complicating the analysis of their limiting behavior. \nSurprisingly, the following lemma shows that for all discretization methods of interest, the sequence $\\{F^{(n)}\\}_{n \\in \\mathbb{N}}$ is a polynomial sequence of fixed degree. \nNote that if we interpolate $n+1$ points, i.e. including $\\pr{0, \\pr{\\alpha^{(n)}_0}_j}, \\pr{1, \\pr{\\alpha^{(n)}_n}_j}$ as interpolating points, the following lemma does not work. \n\n\n\n\\begin{lemma} \\label{lem:degree_bound}\n Denote the $j$-th index of the numerical solution obtained by a given discretization method as $\\pr{c^n}_j = \\frac{1}{n} \\sum_{l=0}^n \\pr{\\alpha^{(n)}_l}_j f^l$ where $h = \\left\\lfloor\\frac{t}{n}\\right\\rfloor$. \n Consider the following $n-1$ points \n \\begin{align} \\label{inter_points}\n \\pr{\\frac{1}{n}, \\pr{\\alpha^{(n)}_1}_j}, \\pr{\\frac{2}{n}, \\pr{\\alpha^{(n)}_2}_j}, \\dots, \\pr{1 - \\frac{1}{n}, \\pr{\\alpha^{(n)}_{n-1}}_j}.\n \\end{align}\n Then, for all $n \\in \\mathbb{N}$ and $j \\in J$, there exists a degree $j-1$ polynomial $F^{(n)}_j$ that interpolates the above $n-1$ points obtained by \n any discretization method introduced in Section~\\ref{sec:problem setting}.\n Moreover, define $\\{F^{(n)}\\}_{n \\in \\mathbb{N}}$ as a vector-valued function sequence that for each $n$, the $j$-th component is $F^{(n)}_j$. \n Then, given the eigenvalue decomposition $A=VDV^{-1}$, the sequence $\\{F^{(n)}(x)\\}_{n \\in \\mathbb{N}}$ pointwise converges to \n \\begin{align*}\n F(x) &= V \\text{diag} \\pr{ 1, 2x, 3x^2, \\dots, Nx^{N-1}} V^{-1} e_1\n \\end{align*}\n as $n \\to \\infty$ for all $x \\in [0,1]$.\n\\end{lemma}\n\n\\begin{proof} \n We start with the forward Euler case. Fix $n$. Let $p^{(n)} : [0,1] \\to \\mathbb{R}^N$ where the $j$-th component is a polynomial interpolating the $n-1$ points in \\eqref{inter_points} with the lowest possible degree.\n Referring to \\eqref{eq:forward_numsol}, $p^{(n)}$ by construction satisfies\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} \n = \\frac{n}{l}\\prod_{k=l+1}^{n-1}\\pr{ I - \\frac{1}{k}A } A e_1\n \\end{align*}\n for $l \\in \\{1, 2, ..., n-1\\}$. Let $A = VDV^{-1}$ where $D \\in \\mathbb{R}^{N \\times N}$ is the diagonal matrix with entries $(D)_{jj}=j$. Then, \n \\begin{align*}\n \\prod_{k=l+1}^{n-1}\\pr{ I - \\frac{1}{k}A }\n &= \\prod_{k=l+1}^{n-1} V \\pr{ I - \\frac{1}{k}D }V^{-1} \\\\\n &= V \\pr{ \\prod_{k=l+1}^{n-1} \\pr{ I - \\frac{1}{k}D } }V^{-1} \\\\\n &= V \\pr{ \\text{diag}\\pr{ \\prod_{k=l+1}^{n-1}\\pr{ 1 - \\frac{1}{k} }, \\prod_{k=l+1}^{n-1}\\pr{ 1 - \\frac{2}{k} }, \\dots, \\prod_{k=l+1}^{n-1}\\pr{ 1 - \\frac{j}{k} } } }V^{-1}\n \\end{align*}\n Now, since we are interested in the limiting behavior of $n \\to \\infty$, we can assume that $n$ is considerably larger than $N$. \n Then we can cancel out terms in the denominator and the numerator to calculate the $i$-th term in the diagonal matrix, \n \\begin{align*}\n \\frac{1}{l}\\prod_{k=l+1}^{n-1}\\pr{ 1- \\frac{i}{k} } \n &= \\frac{1}{l} \\frac{1}{\\prod_{k=l+1}^{n-1} k} \\prod_{k=l+1}^{n-1}\\pr{ k - i }\n = \\frac{\\prod_{k=1}^{i-1}\\pr{ l - k }}{\\prod_{k=1}^{i}\\pr{ n - k }}\n \\end{align*}\n where $\\prod_{k=1}^{0}\\pr{ l - k } = 1$. Therefore we arrive at\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} \n = n V \\text{diag}\\pr{ \\frac{1}{n-1}, \\frac{ \\pr{ l-1}}{(n-1)(n-2)}, \\dots, \\frac{\\prod_{k=1}^{N-2}\\pr{ l - k } }{{\\prod_{k=1}^{N-1}\\pr{ n - k }}}, \\frac{\\prod_{k=1}^{N-1}\\pr{ l - k } }{{\\prod_{k=1}^{N}\\pr{ n - k }}}} D V^{-1} e_1\n \\end{align*}\n for $l \\in \\{1, 2, \\dots, n-1\\}$. \n Now let $l = nx$ so that $x \\in [0,1]$ and define $F_j^{(n)} : [0,1] \\to \\mathbb{R}$ so that \n \\begin{align*}\n F_j^{(n)}(x) &= n e_j^{t} V \\text{diag}\\pr{ \\frac{1}{n-1}, \\frac{ \\pr{ nx-1}}{(n-1)(n-2)}, \\dots, \\frac{\\prod_{k=1}^{N-2}\\pr{ nx - k } }{{\\prod_{k=1}^{N-1}\\pr{ n - k }}}, \\frac{\\prod_{k=1}^{N-1}\\pr{ nx - k } }{{\\prod_{k=1}^{N}\\pr{ n - k }}}} D V^{-1} e_1.\n \\end{align*}\n Notice in the above expression of $F_j^{(n)}(x)$ that the $i$-th term in the diagonal matrix is a $i-1$ degree polynomial of $x$. Since $V$ and $V^{-1}$ are both lower triangular, we conclude that $F_j^{(n)}$ is a $j-1$ degree polynomial.\n Moreover, for any fixed $x \\in [0,1]$, taking the limit $n \\to \\infty$ yields\n \\begin{align} \\label{eqref:F^n_limit}\n \\lim_{n \\to \\infty} F^{(n)}(x) &= V \\text{diag} \\pr{ 1, 2x, 3x^2, \\dots, Nx^{N-1}} V^{-1} e_1.\n \\end{align}\n This concludes the proof for the forward Euler method. \n\n We use a similar approach for other methods. For the backward Euler method, referring to \\eqref{eq:back_numsol}, the interpolating function satisfies\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} &= \\frac{n}{l}\\prod_{k=l}^n \\pr{I + \\frac{1}{k}A}^{-1}Ae_1\n \\end{align*}\n for $l \\in \\{1, 2, \\dots, n-1\\}$. Then the product term in the RHS is equal to\n \\begin{align*}\n \\prod_{k=l}^{n}\\pr{ I + \\frac{1}{k}A }^{-1}\n &= \\prod_{k=l+1}^{n} V \\pr{ I + \\frac{1}{k}D }^{-1} V^{-1}\n = V \\pr{ \\prod_{k=l}^{n} \\pr{ I + \\frac{1}{k}D } }^{-1} V^{-1} \\\\\n &= V \\pr{ \\text{diag}\\pr{ \\prod_{k=l}^{n}\\pr{ 1 + \\frac{1}{k} }^{-1}, \\prod_{k=l}^{n}\\pr{ 1 + \\frac{2}{k} }^{-1}, \\dots, \\prod_{k=l}^{n}\\pr{ 1 + \\frac{j}{k} }^{-1} } } V^{-1}\n \\end{align*}\n and by canceling out terms assuming $n$ is large,\n \\begin{align*}\n \\frac{1}{l}\\prod_{k=l}^{n}\\pr{ 1 + \\frac{i}{k} }^{-1} \n &= \\frac{1}{l} \\frac{\\prod_{k=l}^{n} k}{\\prod_{k=l}^{n} \\pr{ k + i }}\n = \\frac{\\prod_{k=1}^{i-1}\\pr{ l + k }}{\\prod_{k=1}^{i}\\pr{ n + k }}.\n \\end{align*}\n Similar as in the forward Euler case, we can define the interpolating polynomial $F_j^{(n)}$ as \n \\begin{align*}\n F_j^{(n)}(x) &= n e_j^{t} V \\text{diag}\\pr{ \\frac{1}{n+1}, \\frac{ \\pr{ nx+1}}{(n+1)(n+2)}, \\dots, \\frac{\\prod_{k=1}^{N-2}\\pr{ nx + k } }{{\\prod_{k=1}^{N-1}\\pr{ n + k }}}, \\frac{\\prod_{k=1}^{N-1}\\pr{ nx + k } }{{\\prod_{k=1}^{N}\\pr{ n + k }}}} D V^{-1} e_1.\n \\end{align*}\n which converges to \\eqref{eqref:F^n_limit} pointwise as $n \\to \\infty$ for $x \\in [0,1]$.\n\n For the bilinear method, referring to \\eqref{eq:bilin_numsol}, the interpolating polynomial satisfies\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} &= \\frac{n}{2l} \\pr{I+\\frac{1}{2n}A}^{-1} \n \\pr{\n \\prod_{k=l}^{n-1} \n \\pr{I - \\frac{1}{2k}A}\\pr{I + \\frac{1}{2k}A}^{-1}\n + \\prod_{k=l+1}^{n-1} \n \\pr{I - \\frac{1}{2k}A}\\pr{I + \\frac{1}{2k}A}^{-1}\n }Ae_1 \\\\\n &= \\frac{n}{2l} \\pr{I+\\frac{1}{2n}A}^{-1} \n \\pr{I + \\pr{I-\\frac{1}{2l}A}\\pr{I+\\frac{1}{2l}A}^{-1}}\n \\prod_{k=l+1}^{n-1} \n \\pr{I - \\frac{1}{2k}A}\\pr{I + \\frac{1}{2k}A}^{-1}\n Ae_1\n \\end{align*}\n for $l \\in \\{1, 2, \\dots, n-1\\}$. \n For the RHS, the last term simplifies to \n \\begin{align*}\n \\prod_{k=l+1}^{n-1} \n \\pr{I - \\frac{1}{2k}A}\\pr{I + \\frac{1}{2k}A}^{-1} \n &= V \\prod_{k=l+1}^{n-1} \\pr{I - \\frac{1}{2k}D}\\pr{I + \\frac{1}{2k}D}^{-1} V^{-1} \\\\ \n &= V \\pr{ \\text{diag}\\pr{ \\prod_{k=l+1}^{n-1}\\pr{\\frac{k-1/2}{k+1/2} }, \\prod_{k=l+1}^{n-1}\\pr{\\frac{k-1}{k+1} }, \\dots, \\prod_{k=l+1}^{n-1} \\pr{ \\frac{k-j/2}{k+j/2} }}}V^{-1}\n \\end{align*}\n and the prefix terms simplify to \n \\begin{align*}\n &\\frac{n}{2l} \\pr{I+\\frac{1}{2n}A}^{-1} \n \\pr{I + \\pr{I-\\frac{1}{2l}A}\\pr{I+\\frac{1}{2l}A}^{-1}}\\\\\n &= \\frac{n}{2l} V \\pr{I+\\frac{1}{2n}D}^{-1} \n \\pr{I + \\pr{I-\\frac{1}{2l}D}\\pr{I+\\frac{1}{2l}D}^{-1}} V^{-1} \\\\\n &= V \\pr{\\text{diag} \\pr{ \n \\pr{\\frac{n^2}{(n+1/2)(l+1/2)}}, \\pr{\\frac{n^2}{(n+1)(l+1)}}, \\dots, \\pr{\\frac{n^2}{(n+1/2j)(l+1/2j)}}}} V^{-1}.\n \\end{align*}\n Combining these two terms and simplifying the denominators and numerators, we obtain \n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} \n = V \\text{diag}\\pr{ \\frac{4n^2}{(2n-1)(2n+1)}, \\frac{ ln}{(n-1)(n+1)}, \\dots, \\frac{\\prod_{k=1}^{N}\\pr{ l - N/2 + k } }{{\\prod_{k=1}^{N}\\pr{ n - N/2 + k - 1}}} \n \\cdot \\frac{n^2}{(l+j/2)(n+j/2)} } D V^{-1} e_1\n \\end{align*}\n\n Similar as in the forward Euler case, we can define $F_j^{(n)}$ as \n \\begin{align*}\n F_j^{(n)}(x) \n = P^{-1} \\text{diag}\\pr{ \\frac{4n^2 }{(2n-1)(2n+1)}, \\dots,\n \\frac{\\prod_{k=1}^{N}\\pr{ nx - N/2 + k } }{{\\prod_{k=1}^{N}\\pr{ n - N/2 + k - 1}}} \\cdot\n \\frac{n^2}{(nx+N/2)(n+N/2)}} D P e_1.\n \\end{align*}\n which converges to \\eqref{eqref:F^n_limit} pointwise for $x \\in (0,1)$.\n\n For the approximated bilinear method, \n referring to \\eqref{eq:apbil_numsol}, the interpolating function satisfies\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} &= \\frac{n}{l}\n \\prod_{k=l+1}^n \n \\pr{I - \\frac{1}{2k}A}\n \\prod_{k=l}^n \n \\pr{I + \\frac{1}{2k}A}^{-1}\n Ae_1 \\\\\n &= \\frac{n}{l}\n \\pr{ I + \\frac{1}{2l}A}^{-1}\n \\prod_{k=l+1}^n \n \\pr{I - \\frac{1}{2k}A} \\pr{I + \\frac{1}{2k}A}^{-1}\n Ae_1\n \\end{align*}\n for $l \\in \\{1, 2, \\dots, n-1\\}$. \n Assuming $n$ is large, the product term in RHS simplifies as\n \\begin{align*}\n \\prod_{k=l+1}^n \n \\pr{I - \\frac{1}{2k}A} \\pr{I + \\frac{1}{2k}A}^{-1}\n &= \\prod_{k=l+1}^n V \n \\pr{I - \\frac{1}{2k}D} \\pr{I + \\frac{1}{2k}D}^{-1} V^{-1} \\\\\n &= V \\pr{ \\text{diag}\n \\pr{ \\prod_{k=l+1}^{n}\\pr{ \\frac{k-1/2}{k+1/2} }, \\prod_{k=l+1}^{n}\\pr{ \\frac{k-1}{k+1} }, \\dots, \\prod_{k=l+1}^{n}\\pr{ \\frac{k - j/2}{k + j/2}}}}\n V^{-1}\n \\end{align*}\n and the prefix term simplifies to\n \\begin{align*}\n \\frac{1}{l}\\pr{ I + \\frac{1}{2l}A}^{-1} \n &= \\frac{1}{l} P^{-1} \\pr{ I + \\frac{1}{2l}D}^{-1} P \n = V \\pr{ \\text{diag} \\pr{\\frac{1}{l+1/2}}, \\pr{\\frac{1}{l+3/2}} \\dots,\n \\pr{\\frac{1}{l+j/2}} }V^{-1}.\n \\end{align*}\n Assuming that $n$ is large and canceling out terms, we obtain\n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} \n = V \\text{diag}\n \\pr{ \\frac{2n}{2n+1}, \\frac{ln}{n(n+1)}, \\dots, \\frac{\\prod_{k=1}^{j}\\pr{ l - j/2 + k } }{{\\prod_{k=1}^{j}\\pr{ n - j/2 + k}}} \n \\cdot \\frac{n}{l+j/2} } D V^{-1} e_1\n \\end{align*}\n \n Similar as in the forward Euler case, we can define $F_j^{(n)}$ as \n \\begin{align*}\n p^{(n)}\\pr{\\frac{l}{n}} \n = V \\text{diag}\n \\pr{ \\frac{2n}{2n+1}, \\frac{n^2x}{n(n+1)}, \\dots, \\frac{\\prod_{k=1}^{j}\\pr{ nx - j/2 + k } }{{\\prod_{k=1}^{j}\\pr{ n - j/2 + k}}} \n \\cdot \\frac{n}{nx+j/2} } D V^{-1} e_1\n \\end{align*}\n which pointwise converges to \\eqref{eqref:F^n_limit}.\n \n\\end{proof}\n\nCombining the results, we can prove Theorem \\ref{thm:conv_riemann}.\n\n\n\n\\begin{proof} [Proof of Theorem \\ref{thm:conv_riemann}]\n Given a discretization method, we can express the numerical solution as $c^n = \\frac{1}{n} \\sum_{l=0}^n \\alpha^{(n)}_l f^l$ by Lemma \\ref{lem:numsol_linear}.\n Then, define the function sequence $\\{F^{(n)}\\}_{n \\in \\mathbb{N}}$ as in Lemma \\ref{lem:degree_bound}. Then for all $n \\in \\mathbb{N}$, the $j$-th component of $F^{(n)} : [0,1] \\to \\mathbb{R}^N$ is a $j-1$ degree polynomial with pointwise limit $F(x)$ for all $x \\in [0,1]$. This implies that all the coefficients of the polynomials in the sequence converge to the coefficients of $F$, and hence we can conclude that $\\{F^{(n)}\\}_{n \\in \\mathbb{N}}$ converges to $F$ uniformly. \n Note that we also have that all the coefficients $\\alpha^{(n)}_l$ of $c^n$ are uniformly bounded, since the limit is well-defined.\n\n Now it suffices to show that the $j$-th component of the limit function $F(x) : [0,1] \\to \\mathbb{R}^N$ is equal to $\\sqrt{2j-1}$ times the $j-1$-th shifted Legendre polynomial, $\\sqrt{2j-1}P_{j-1}(2x-1)$. \n Once this is shown, we can apply the result of Corollary \\ref{cor:conv_to_quad} to conclude the proof.\n \n Recall that the exact form of $F$ is:\n \\begin{align*}\n F(x) &= V \\text{diag}\\pr{1, 2x, 3x^2, \\dots, jx^{j-1}, \\dots, Nx^{N-1}} V^{-1} e_1.\n \\end{align*}\n\n Differentiating both sides, we get\n \\begin{align*}\n F'(x) &= V \\text{diag}\\pr{0, 2, 6x, \\dots, j(j-1)x^{j-2}, \\dots, N(N-1)x^{N-2}} V^{-1} e_1.\n \\end{align*}\n\n Combining these two equations, we obtain the following differential equation that holds for all $x \\in [0,1]$:\n \\begin{align} \\label{eq:F_recur}\n xF'(x) &= (A - I) F(x)\n \\end{align}\n\n Since we know the exact form of $A$, we can derive a recurrence relation for $F$ for arbitrary dimension $N$. \n Rewriting the $j$-th component $F$ as\n \\begin{align*}\n F_j(x) &= \\sqrt{2j-1}f_{j-1}(x),\n \\end{align*}\n we obtain the recurrence relation \n \\begin{align} \\label{eq:f_recurrence}\n xf'_j(x) = jf_j(x) + \\sum_{l=1}^{j-1} (2l+1)f_{l}(x).\n \\end{align}\n Notice that \\eqref{eq:f_recurrence} is exactly the recurrence relation satisfied the $j$-th shifted Legendre polynomial $\\tilde{P}_j$. \n Matching the initial condition $F(1) = Ae_1 = B$, we have $f_j(1) = 1$ for all $j \\in J$. Then by induction, we can prove that $f_j(x) = \\tilde{P}_j(x) = P_j(2x-1)$.\n Finally, utilizing the uniqueness of the solution for the IVP defined with ODE \\eqref{eq:F_recur} and initial condition at $x=1$, we arrive at the conclusion:\n \\begin{align*}\n F_j(x) &= \\sqrt{2j-1}P_{j-1}(2x-1).\n \\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{unsrtnat}\n\\bibliography{ref}\n\n\n\\include{appendix}\n\n\\end{document}\n\n\n"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":591,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3ODU0OTE4NCwic3ViIjoiL2RhdGFzZXRzL2V0aHotc3B5bGFiL21hdGhfbGF0ZXgiLCJleHAiOjE3Nzg1NTI3ODQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.daWkArWXKVkglqQq63PsbkQCxRvosm8ZuVSTtIYZKcssH81PFOrwhMvpzXE_AZXKGjGz58OQQ3_veWW2465VAw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":591,"numBytesParquet":27690937}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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2412.02620v1 | http://arxiv.org/abs/2412.02620v1 | The Dimension of the Disguised Toric Locus of a Reaction Network |
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2412.02679v1 | http://arxiv.org/abs/2412.02679v1 | On z-Superstable and Critical Configurations of Chip Firing Pairs | \documentclass[11pt,twoside]{amsart}
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2412.04389v1 | http://arxiv.org/abs/2412.04389v1 | Hypergraph burning, matchings, and zero forcing | \documentclass[12pt]{amsart}
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2412.04579v1 | http://arxiv.org/abs/2412.04579v1 | Solvable families of random block tridiagonal matrices | \documentclass[12pt]{article}
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2412.12130v1 | http://arxiv.org/abs/2412.12130v1 | On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers | \documentclass[11pt,a4paper]{article}
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2412.04740v1 | http://arxiv.org/abs/2412.04740v1 | Estimates for the first eigenvalue of the one-dimensional $p$-Laplacian | \documentclass[a4paper,12pt]{article}
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2412.04848v2 | http://arxiv.org/abs/2412.04848v2 | Quadratic Modelings of Syndrome Decoding | \documentclass[runningheads]{llncs}
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2412.04932v1 | http://arxiv.org/abs/2412.04932v1 | Trickle groups | \pdfsuppresswarningpagegroup=1
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2412.04973v1 | http://arxiv.org/abs/2412.04973v1 | On the Laplace equation with non-local dynamical boundary conditions | \documentclass[10pt,a4paper]{article}
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2412.04962v1 | http://arxiv.org/abs/2412.04962v1 | Divisible design graphs from symplectic graphs over rings with precisely three ideals | \documentclass[10pt]{elsarticle}
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2412.04952v1 | http://arxiv.org/abs/2412.04952v1 | Non-isomorphic maximal function fields of genus $q-1$ | \documentclass[12pt,a4]{amsart}
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2412.05068v1 | http://arxiv.org/abs/2412.05068v1 | On constant mean curvature surfaces satisfying integrable boundary conditions | \documentclass[12pt]{amsart}
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2412.05067v1 | http://arxiv.org/abs/2412.05067v1 | Exotic newforms constructed from a linear combination of eta quotients | \documentclass[
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2412.05238v1 | http://arxiv.org/abs/2412.05238v1 | Some splitting and rigidity results for sub-static spaces | \documentclass[10pt,a4paper]{amsart}
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2412.06829v2 | http://arxiv.org/abs/2412.06829v2 | Stably unactivated neurons in ReLU neural networks | \documentclass[11pt]{amsart}
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2412.05521v1 | http://arxiv.org/abs/2412.05521v1 | Random attractors for the stochastic Nernst-Planck-Navier-Stokes system with multiplicative white noise | \documentclass[a4paper,twoside]{article}
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2412.05513v1 | http://arxiv.org/abs/2412.05513v1 | Distributional Solution and Spectral Shift Function of Heun Differential Equation | \documentclass[10pt,reqno,titlepage]{amsart}
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2412.05503v1 | http://arxiv.org/abs/2412.05503v1 | Critical scaling profile for trees and connected subgraphs on the complete graph | \documentclass[11pt,hidelinks]{article}
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2412.05500v2 | http://arxiv.org/abs/2412.05500v2 | Syzygies of canonical ribbons on higher genus curves | \documentclass[11pt,twoside]{amsart}
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2412.05494v1 | http://arxiv.org/abs/2412.05494v1 | On a bipartite graph defined on groups | \documentclass[12pt]{amsart}
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2412.05612v1 | http://arxiv.org/abs/2412.05612v1 | The buckling and clamped plate problems on differential forms | \documentclass[11pt,a4paper,twoside]{article}
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2412.05588v2 | http://arxiv.org/abs/2412.05588v2 | Prime rings having nontrivial centralizers of (skew) traces of Lie ideals | \documentclass[12pt]{article}
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2412.05682v1 | http://arxiv.org/abs/2412.05682v1 | On the Rank of a bicomplex matrix | \documentclass[16pt,oneside,reqno]{amsart}
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2412.05698v1 | http://arxiv.org/abs/2412.05698v1 | Controlled rough SDEs, pathwise stochastic control and dynamic programming principles | \documentclass[11pt]{amsart}
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2412.06848v1 | http://arxiv.org/abs/2412.06848v1 | Application of Random Matrix Theory in High-Dimensional Statistics | \documentclass[12pt]{article}
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2412.05805v1 | http://arxiv.org/abs/2412.05805v1 | Exact Hausdorff dimension of some sofic self-affine fractals | \documentclass[reqno, a4paper,12pt]{amsart}
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2412.05902v1 | http://arxiv.org/abs/2412.05902v1 | Long-Time behavior of the tangential surface Navier-Stokes equation | \documentclass[a4paper]{amsart}
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2412.05945v3 | http://arxiv.org/abs/2412.05945v3 | The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals | \documentclass[12pt]{amsart}
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2412.06123v1 | http://arxiv.org/abs/2412.06123v1 | An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group | \documentclass{baustms}
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2412.06109v1 | http://arxiv.org/abs/2412.06109v1 | Permutation clones that preserve relations |
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2412.06246v1 | http://arxiv.org/abs/2412.06246v1 | The smallest singular value for rectangular random matrices with Lévy entries |
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2412.06331v1 | http://arxiv.org/abs/2412.06331v1 | The maximum forcing numbers of quadriculated tori | \documentclass[12pt, a4paper]{article}
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2412.06423v1 | http://arxiv.org/abs/2412.06423v1 | Equilibrium States for Piecewise Weakly Convex Interval Maps | \documentclass[letterpaper,10pt,reqno]{amsart}
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2412.06406v1 | http://arxiv.org/abs/2412.06406v1 | Invariant Probability Measures under $p$-adic Transformations | \documentclass{amsart}
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2412.06402v2 | http://arxiv.org/abs/2412.06402v2 | VC-dimensions for set familes between partially ordered set and totally ordered set | \documentclass[a4paper,11pt]{article}
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2412.06437v2 | http://arxiv.org/abs/2412.06437v2 | Minimization of the first eigenvalue for the Lamé system | \documentclass[11pt]{article}
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2412.06527v1 | http://arxiv.org/abs/2412.06527v1 | Higher genus Gromov-Witten theory of one-parameter Calabi-Yau threefolds II: Feynman rule and anomaly equations | \documentclass[10pt]{amsart}
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2412.06572v2 | http://arxiv.org/abs/2412.06572v2 | Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry | \documentclass{amsart}
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2412.06635v1 | http://arxiv.org/abs/2412.06635v1 | Motivic cohomology of mixed characteristic schemes | \documentclass[10pt,twoside]{article}
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2412.06595v2 | http://arxiv.org/abs/2412.06595v2 | Totally nonnegative matrices, chain enumeration and zeros of polynomials | \documentclass[reqno]{amsart}
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2412.06659v1 | http://arxiv.org/abs/2412.06659v1 | Ischebeck's formula, Grade and quasi-homological dimensions | \documentclass[12pt]{amsart}
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2412.06705v3 | http://arxiv.org/abs/2412.06705v3 | Matroids and amplitudes | \documentclass{amsart}
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2412.07054v1 | http://arxiv.org/abs/2412.07054v1 | Transcendence of ${}_3F_2(1)$ Hypergeometric Series and $L$-values of Modular Forms | \documentclass[12pt]{amsart}
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2412.07092v2 | http://arxiv.org/abs/2412.07092v2 | Linear and Sublinear Diversities | \documentclass[11pt]{article}
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2412.07090v1 | http://arxiv.org/abs/2412.07090v1 | The maximum sturdiness of intersecting families | \documentclass[11pt,a4paper]{article}
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2412.07164v1 | http://arxiv.org/abs/2412.07164v1 | Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive | \documentclass[reqno, 12pt]{article}
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2412.07314v1 | http://arxiv.org/abs/2412.07314v1 | A measure with small support and p-summable Fourier transform | \documentclass[a4paper,10pt]{article}
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2412.07308v2 | http://arxiv.org/abs/2412.07308v2 | Iwasawa theory and ranks of elliptic curves in quadratic twist families | \documentclass[table]{amsart}
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2412.07410v1 | http://arxiv.org/abs/2412.07410v1 | KMS states on quantum Cuntz-Krieger algebras | \documentclass[a4paper,reqno]{amsart}
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2412.07396v1 | http://arxiv.org/abs/2412.07396v1 | Processus aléatoires et applications -- Algorithmes MCMC et vitesse de convergence | \documentclass[11pt,a4paper,titlepage,twoside
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2412.07394v2 | http://arxiv.org/abs/2412.07394v2 | Numerical approximations for a hyperbolic integrodifferential equation with a non-positive variable-sign kernel and nonlinear-nonlocal damping | \documentclass[11pt]{article}
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2412.07452v1 | http://arxiv.org/abs/2412.07452v1 | On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature | \documentclass[12pt,twoside]{article} \usepackage{geometry} \geometry{left=3.5cm,right=3.5cm,top=3.5cm,bottom=3.5cm}
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2412.07510v1 | http://arxiv.org/abs/2412.07510v1 | Roman domination number of zero-divisor graphs over commutative rings | \begin{filecontents*}{example.eps}
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2412.07483v1 | http://arxiv.org/abs/2412.07483v1 | Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map | \documentclass[a4paper,11pt]{amsart}
\usepackage[a4paper,top=3cm,bottom=3cm,
left=3cm,right=3cm,marginparwidth=60pt]{geometry}
\usepackage{anysize}
\marginsize{1.3in}{1.3in}{1in}{1in}
\usepackage{comment}
\usepackage{xcolor}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage[all]{xy}
\usepackage[utf... |
2412.07561v1 | http://arxiv.org/abs/2412.07561v1 | The $L_q$ Minkowski problem for $\mathbf{p}$-harmonic measure | \documentclass[12pt]{article}
\usepackage{amsmath, graphicx, amsfonts,amssymb, calrsfs}
\usepackage{amsfonts,mathrsfs, color, amsthm}
\addtolength{\topmargin}{-0.1\textheight}
\addtolength{\textheight}{0.2\textheight}
\addtolength{\textwidth}{0.2\textwidth}
\setlength{\oddsidemargin}{-0.08in}
\setlength{\evensidemarg... |
2412.07613v1 | http://arxiv.org/abs/2412.07613v1 | Discontinuous Galerkin methods for the complete stochastic Euler equations | \documentclass[10pt]{amsart}
\usepackage{enumitem}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{comment}
\usepackage{hyperref}
\usepackage{mathrsfs}
\usepackage{breqn}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{stmaryrd}
\usepackage{geometry}
\geometry{margin= 1in}
\let\vec\m... |
2412.07667v1 | http://arxiv.org/abs/2412.07667v1 | A symbolic computational approach to the generalized gambler's ruin problem in one and two dimensions | \documentclass[reqno]{amsart}
\usepackage[utf8]{inputenc}
\usepackage{xcolor,fullpage,hyperref,cleveref,enumitem}
\usepackage{tikz,amsmath,amssymb,color,mathtools}
\usetikzlibrary{calc}
\usepackage{float}
\usepackage{pgf}
\usepackage[backrefs]{amsrefs}
\usepackage{amsthm}
\usepackage{listings}
\lstset{frame=tb,
la... |
2412.07644v1 | http://arxiv.org/abs/2412.07644v1 | Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations | \documentclass[11pt]{article}
\usepackage[centertags]{amsmath}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{upquote}
\usepackage{cases}
\usepackage{hyperref}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{scalerel}
\usepackage{scalerel}
\newcommand\reallywi... |
2412.07886v1 | http://arxiv.org/abs/2412.07886v1 | Convergence estimates for the Magnus expansion IE. Finite dimensional Banach algebras | \documentclass[a4paper,11pt]{amsart}
\usepackage[utf8]{inputenc}
\usepackage{CJKutf8}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage{float}
\usepackage{cancel}
\usepackage{slashed}
\usepackage{comment}
\newcommand{\mychoice}[3]{#1
}
\mychoice{
\newcommand{\plabel}[1]{ \lab... |
2412.07928v1 | http://arxiv.org/abs/2412.07928v1 | Renormalization for Bruin-Troubetzkoy ITMs | \documentclass[english]{amsart}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{csquotes, xpatch}
\usepackage{microtype}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{mathtools}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{caption}
\usepackage{sub... |
2412.08075v2 | http://arxiv.org/abs/2412.08075v2 | When entropy meets Turán: new proofs and hypergraph Turán results | \documentclass[reqno,11pt]{amsart}
\usepackage{amsthm, amsmath, amssymb, stmaryrd}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[hyphenbreaks]{breakurl}
\usepackage[hyphens]{url}
\usepackage{systeme}
\usepackage[shortlabels]{enumitem}
\usepackage[hidelinks]{hyperref}
\usepackage{microtype}
\usepac... |
2412.08183v1 | http://arxiv.org/abs/2412.08183v1 | States and IR divergences in factorization algebras |
\documentclass[preprint]{ptephy_v1}
\preprintnumber{YITP-24-168, KUNS-3030} \usepackage{hyperref}
\usepackage{bm}
\usepackage[all]{xy}
\usepackage{tcolorbox}
\usepackage{tcolorbox}
\tcbuselibrary{breakable, skins, theorems}
\usepackage{mathrsfs}
\newtheorem{theorem}{Theorem}
\newtheorem{condition}{Condition}
\newth... |
2412.08334v2 | http://arxiv.org/abs/2412.08334v2 | Maker-Breaker on Galton-Watson trees | \pdfminorversion=5
\documentclass{article}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{bbm}
\usepackage{enumerate}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage{hyperref}
\usepackage{color}
\usepackage{curves}
\usepackage{tikz}
\... |
2412.08400v1 | http://arxiv.org/abs/2412.08400v1 | Combinatorial Characterization of Exponential Families of Lumpable Stochastic Matrices | \documentclass[a4paper,11pt,oneside]{article}
\pdfoutput=1
\RequirePackage{amsmath, amsthm, amssymb, mathtools, mathrsfs}
\RequirePackage{natbib}
\RequirePackage{tikz,tikz-cd}
\RequirePackage{tabularray}
\RequirePackage{algorithm2e}
\RequirePackage{nicematrix}
\NiceMatrixOptions{
code-for-first-row = \tiny ,
code-for-l... |
2412.08415v2 | http://arxiv.org/abs/2412.08415v2 | The two-boost problem and Lagrangian Rabinowitz Floer homology |
\documentclass[a4paper]{amsart}
\usepackage{hyperref, enumitem}
\usepackage[dvipsnames]{xcolor} \usepackage{amsfonts,amsmath, amssymb,mathrsfs,mathtools,amsthm} \usepackage[doi=false, url=false, eprint=false, maxbibnames=10, firstinits=true]{biblatex}
\addbibresource{bibliography.bib}
\usepackage{todonotes}
\usepacka... |
2412.08437v1 | http://arxiv.org/abs/2412.08437v1 | Zeta and L functions of Voevodsky motives | \documentclass[12pt]{amsart}
\usepackage{verbatim,amssymb,latexsym, amscd}
\usepackage[T1]{fontenc}
\usepackage[all]{xy}
\usepackage{times}
\usepackage[colorlinks,linkcolor=blue,citecolor=blue,urlcolor=red]{hyperref}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\sA}{\mathcal{A}}
\newco... |
2412.08597v1 | http://arxiv.org/abs/2412.08597v1 | Positive co-degree densities and jumps | \documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{cite}
\usepackage{url}
\usepackage{caption}
\usepackage{amssymb, amsmath, amsthm}
\usepackage{tikz}
\usepackage{thm-restate}
\usepackage{cellspace}
\setlength{\cellspacetoplimit}{6pt}
\setlength{\cellspacebottomlimit}{6pt}
\usepackage[pdfte... |
2412.08595v1 | http://arxiv.org/abs/2412.08595v1 | Numerical Analysis of HiPPO-LegS ODE for Deep State Space Models | \documentclass{article}
\usepackage{arxiv}
\usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[hidelinks]{hyperref} \usepackage{url} \usepackage{booktabs} \usepackage{amsfonts} \usepackage{nicefrac} \usepackage{microtype} \usepackage{lipsum} \usepackage{graph... |
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