Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$ with boundary.  The problem of the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary leads to a nonlinear equation of critical type for the conformal factor. More precisely, if $R=R_g$ and $h=h_g$ denote respectively the scalar curvature of $M$ and the mean curvature of its boundary associated with the metric $g$, the corresponding curvatures $\widetilde R=R_{\widetilde g}$ and $\widetilde h=h_{\widetilde g}$ for the conformal  metric $\widetilde g:=u^{4\over n-2}g$ are given by a positive solution $u$ of the problem
  \begin{equation}\label{problem}
\begin{cases}
 -\Delta u + c_nRu=\widetilde Ru^{n+2\over n-2} &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= \widetilde h u^\frac{n}{n-2} &\text{on \ }\partial M,
\end{cases}
\end{equation}
where $\Delta=\Delta_g$ is the Laplacian, $\nu=\nu_g$ is the outer unit normal with respect to the metric $g$ and $c_n:=\frac{n-2}{4(n-1)}$, $d_n:=\frac{n-2}{2}$.

The problem \eqref{problem} with $\widetilde R=0$ can be can be seen as a higher-dimensional extension of the classical Riemann mapping theorem, which asserts that every simply connected proper subset of $\mathbb{R}^2$ is conformally diffeomorphic to the disk.

When $M$ is   closed   (that is, $\partial M=\emptyset$) the problem \eqref{problem} reduces to the well-known Yamabe problem, a higher-dimensional analogue of the uniformization theorem in complex analysis. The Yamabe problem, which seeks a metric conformal to $g$ with constant scalar curvature, was solved through the combined efforts of Yamabe \cite{yamabe}, Trudinger \cite{trudinger}, Aubin \cite{aubin} and Schoen \cite{schoen}. They proved the existence of a positive solution $u$ to the equation
\begin{equation}\label{eq:yamabe}
-\Delta u + c_nRu=|u|^\frac{4}{n-2}u \qquad\text{in \ } M,
\end{equation}
for any closed  Riemannian manifold $M$ of dimension $n\geq 3$. Then, $\widetilde{g}:=u^{4/(n-2)}g$ is a metric conformal to $g$ with constant scalar curvature. A comprehensive analysis of this problem was later provided by Lee and Parker in \cite{lp}.

The natural extension of the Yamabe problem to manifolds with boundary was initiated by Cherrier  \cite{che} and Escobar  \cite{e0,e1,e2,e4}, with further contributions by Han and Li \cite{hl0,hl}. The existence of positive solutions to \eqref{problem} is by now well understood; see \cite{mn} and the references therein for a detailed account.

In the present work, we are concerned with the existence of nodal (sign-changing) solutions to \eqref{problem}. While the study of such solutions for the Yamabe equation \eqref{eq:yamabe} is rather extensive, to the best of our knowledge, the existence of nodal solutions for \eqref{problem} remains widely open.

For the Yamabe problem, when $M$ is the standard sphere $\mathbb{S}^n$, W.Y. Ding established the existence of infinitely many sign-changing solutions to the  Yamabe equation \eqref{eq:yamabe} in \cite{ding}. Other types of nodal solutions on $\mathbb{S}^n$ were exhibited in \cite{dmpp1,dmpp2,c,cfs,mm}.  In \cite{fp} Fernández and Petean showed that there is a solution with precisely $\ell$ nodal domains for every $\ell\geq 2$. For closed manifolds with suitable symmetries existence of nodal solutions to \eqref{eq:yamabe} has been established in \cite{cf,cp}. Nodal solutions on products were obtained in \cite{h,p}.
In contrast, there are few results concerning the existence of sign-changing solutions to the Yamabe problem \eqref{eq:yamabe} on an arbitrary closed Riemannian manifold. In \cite{ah} Ammann and Humbert introduced a conformal invariant, called  the second Yamabe invariant, defined in terms of the second eigenvalue of the conformal Laplacian, and showed that on every closed Riemannian manifold of dimension $n\geq 11$ which is not locally conformally flat, there exists a function $u$ that realizes this invariant. This function necessarily changes sign, so the expression $\widetilde{g}:=|u|^{4/(n-2)}g$ is not a metric on $M$. In \cite{ah} it is called a \emph{generalized metric}. This result was extended in \cite[Theorem 1.2$(v)$]{cpt} to manifolds of dimension $10$ that satisfy an additional assumption. Every function that realizes the second Yamabe invariant has least energy among all sign-changing solutions. The blow-up behavior of sequences of least energy sign-changing solutions of to \eqref{eq:yamabe} was described by Premoselli and Vétois in \cite{pv}. They also showed in \cite{pv2} that in dimensions $3$ to $10$, the second Yamabe invariant is not attained for any metric on the sphere that is sufficiently close to the standard one.  \\

In this paper, we  find nodal solutions to \eqref{problem} in two particular cases:
\begin{equation}\label{scalar flat}
\begin{cases}
 -\Delta u + c_nRu=0 &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= |u|^\frac{2}{n-2}u &\text{on \ }\partial M,
\end{cases}
\end{equation}
and
\begin{equation}\label{minimal boundary}
\begin{cases}
 -\Delta u + c_nRu=|u|^\frac{4}{n-2}u &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu=0 &\text{on \ }\partial M.
\end{cases}
\end{equation}
A positive solution $u$ to \eqref{scalar flat} gives rise to a metric $\widetilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \eqref{minimal boundary}, then $\widetilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \cite{e2} and \cite{e1}.
We obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on
$M.$

Consider the quadratic form
\begin{equation}\label{eq:quadratic form}
\mathscr{Q}u:=\int_{M}(|\nabla u|^2+c_nRu^2)\,\mathrm{dv}+\int_{\partial M} d_nhu^2\,\mathrm{d} \sigma,
\end{equation}
where $\,\mathrm{dv}$ and $\,\mathrm{d} \sigma$ are the Riemannian measures on $M$ and $\partial M$ induced by the metric $g$. We assume that $M$ is \emph{positive}, this means that the first eigenvalue satisfies
\begin{align} \label{eq:M positive}
\lambda_1(M)&:=\inf_{v\in H^1(M)\smallsetminus\{0\}}\frac{\mathscr{Q}v}{\int_{M} v^2\,\mathrm{dv}}>0.
\end{align}
We prove the following results.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$ with boundary. The problem of the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary leads to a nonlinear equation of critical type for the conformal factor. More precisely, if $R=R_g$ and $h=h_g$ denote respectively the scalar curvature of $M$ and the mean curvature of its boundary associated with the metric $g$, the corresponding curvatures $\widetilde R=R_{\widetilde g}$ and $\widetilde h=h_{\widetilde g}$ for the conformal metric $\widetilde g:=u^{4\over n-2}g$ are given by a positive solution $u$ of the problem
 \begin{equation}\label{problem}
\begin{cases}
 -\Delta u + c_nRu=\widetilde Ru^{n+2\over n-2} &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= \widetilde h u^\frac{n}{n-2} &\text{on \ }\partial M,
\end{cases}
\end{equation}
where $\Delta=\Delta_g$ is the Laplacian, $\nu=\nu_g$ is the outer unit normal with respect to the metric $g$ and $c_n:=\frac{n-2}{4(n-1)}$, $d_n:=\frac{n-2}{2}$.

When $M$ is closed (that is, $\partial M=\emptyset$) the problem \eqref{problem} reduces to the well-known Yamabe problem, a higher-dimensional analogue of the uniformization theorem in complex analysis. The Yamabe problem, which seeks a metric conformal to $g$ with constant scalar curvature, was solved through the combined efforts of Yamabe \cite{yamabe}, Trudinger \cite{trudinger}, Aubin \cite{aubin} and Schoen \cite{schoen}. They proved the existence of a positive solution $u$ to the equation
\begin{equation}\label{eq:yamabe}
-\Delta u + c_nRu=|u|^\frac{4}{n-2}u \qquad\text{in \ } M,
\end{equation}
for any closed Riemannian manifold $M$ of dimension $n\geq 3$. Then, $\widetilde{g}:=u^{4/(n-2)}g$ is a metric conformal to $g$ with constant scalar curvature. A comprehensive analysis of this problem was later provided by Lee and Parker in \cite{lp}.

The natural extension of the Yamabe problem to manifolds with boundary was initiated by Cherrier \cite{che} and Escobar \cite{e0,e1,e2,e4}, with further contributions by Han and Li \cite{hl0,hl}. The existence of positive solutions to \eqref{problem} is by now well understood; see \cite{mn} and the references therein for a detailed account.

For the Yamabe problem, when $M$ is the standard sphere $\mathbb{S}^n$, W.Y. Ding established the existence of infinitely many sign-changing solutions to the Yamabe equation \eqref{eq:yamabe} in \cite{ding}. Other types of nodal solutions on $\mathbb{S}^n$ were exhibited in \cite{dmpp1,dmpp2,c,cfs,mm}. In \cite{fp} Fernández and Petean showed that there is a solution with precisely $\ell$ nodal domains for every $\ell\geq 2$. For closed manifolds with suitable symmetries existence of nodal solutions to \eqref{eq:yamabe} has been established in \cite{cf,cp}. Nodal solutions on products were obtained in \cite{h,p}.
In contrast, there are few results concerning the existence of sign-changing solutions to the Yamabe problem \eqref{eq:yamabe} on an arbitrary closed Riemannian manifold. In \cite{ah} Ammann and Humbert introduced a conformal invariant, called the second Yamabe invariant, defined in terms of the second eigenvalue of the conformal Laplacian, and showed that on every closed Riemannian manifold of dimension $n\geq 11$ which is not locally conformally flat, there exists a function $u$ that realizes this invariant. This function necessarily changes sign, so the expression $\widetilde{g}:=|u|^{4/(n-2)}g$ is not a metric on $M$. In \cite{ah} it is called a \emph{generalized metric}. This result was extended in \cite[Theorem 1.2$(v)$]{cpt} to manifolds of dimension $10$ that satisfy an additional assumption. Every function that realizes the second Yamabe invariant has least energy among all sign-changing solutions. The blow-up behavior of sequences of least energy sign-changing solutions of to \eqref{eq:yamabe} was described by Premoselli and Vétois in \cite{pv}. They also showed in \cite{pv2} that in dimensions $3$ to $10$, the second Yamabe invariant is not attained for any metric on the sphere that is sufficiently close to the standard one. \\

In this paper, we find nodal solutions to \eqref{problem} in two particular cases:
\begin{equation}\label{scalar flat}
\begin{cases}
 -\Delta u + c_nRu=0 &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= |u|^\frac{2}{n-2}u &\text{on \ }\partial M,
\end{cases}
\end{equation}
and
\begin{equation}\label{minimal boundary}
\begin{cases}
 -\Delta u + c_nRu=|u|^\frac{4}{n-2}u &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu=0 &\text{on \ }\partial M.
\end{cases}
\end{equation}
A positive solution $u$ to \eqref{scalar flat} gives rise to a metric $\widetilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \eqref{minimal boundary}, then $\widetilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \cite{e2} and \cite{e1}.
We obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on
$M.$

Consider the quadratic form
\begin{equation}\label{eq:quadratic form}
\mathscr{Q}u:=\int_{M}(|\nabla u|^2+c_nRu^2)\,\mathrm{dv}+\int_{\partial M} d_nhu^2\,\mathrm{d} \sigma,
\end{equation}
where $\,\mathrm{dv}$ and $\,\mathrm{d} \sigma$ are the Riemannian measures on $M$ and $\partial M$ induced by the metric $g$. We assume that $M$ is \emph{positive}, this means that the first eigenvalue satisfies
\begin{align} \label{eq:M positive}
\lambda_1(M)&:=\inf_{v\in H^1(M)\smallsetminus\{0\}}\frac{\mathscr{Q}v}{\int_{M} v^2\,\mathrm{dv}}>0.
\end{align}
We prove the following results.

In this paper, we find nodal solutions to \eqref{problem} in two particular cases:
\begin{equation}\label{scalar flat}
\begin{cases}
 -\Delta u + c_nRu=0 &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= |u|^\frac{2}{n-2}u &\text{on \ }\partial M,
\end{cases}
\end{equation}
and
\begin{equation}\label{minimal boundary}
\begin{cases}
 -\Delta u + c_nRu=|u|^\frac{4}{n-2}u &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu=0 &\text{on \ }\partial M.
\end{cases}
\end{equation}
A positive solution $u$ to \eqref{scalar flat} gives rise to a metric $\tilde{g}:=u^{4/(n-2)}g$ on $M$, conformally equivalent to $g$, whose scalar curvature on $M$ is zero and for which the mean curvature of the boundary is constant. On the other hand, if $u$ is a positive solution to \eqref{minimal boundary}, then $\tilde{g}:=u^{4/(n-2)}g$ is a metric on $M$, conformally equivalent to $g$, whose scalar curvature is constant and for which the mean curvature of the boundary is zero. The existence of such metrics was first studied by Escobar in \cite{e2} and \cite{e1}.
We obtain least-energy sign-changing solutions to both problems under suitable geometric assumptions on
$M.$

Consider the quadratic form
\begin{equation}\label{eq:quadratic form}
\mathscr{Q}u:=\im(|\nabla u|^2+c_nRu^2)\dv+\idm d_nhu^2\d \sigma,
\end{equation}
where $\dv$ and $\d \sigma$ are the Riemannian measures on $M$ and $\partial M$ induced by the metric $g$. We assume that $M$ is \emph{positive}, this means that the first eigenvalue satisfies
\begin{align} \label{eq:M positive}
\lambda_1(M)&:=\inf_{v\in H^1(M)\smallsetminus\{0\}}\frac{\mathscr{Q}v}{\im v^2\dv}>0.
\end{align}
We prove the following results.

\begin{theorem} \label{thm:main minimal boundary}
If $n\geq 7$ and $M$ has a nonumbilic point $\xi\in\partial M$, then the problem \eqref{minimal boundary} has a least energy sign-changing solution.
\end{theorem}

The strategy to prove Theorem \ref{thm:main scalar flat} is as follows. If $u\in H^1(M)$ is a nontrivial solution to \eqref{scalar flat}, then multiplying that problem by $u$ and integrating by parts shows that $u$ belongs to the set
$$\cN:=\Big\{u\in H^1(M): u\neq 0 \text{ and }\mathscr{Q} u=\idm |u|^{2(n-1)/(n-2)}\d \sigma\Big\}.$$
If $u$ changes sign, then multiplying \eqref{scalar flat} by the positive and negative parts $u^\pm$ of $u$ shows that $u$ belongs to
$$\cE:=\{u\in\cN:u^+\in\cN\text{ and }u^-\in\cN\}.$$
We define
$$\mu_1(M):=\inf_{u\in\cN}J(u)\qquad\text{and}\qquad\mu_2(M):=\inf_{u\in\cE}J(u),$$
where $J$ is the energy functional associated with problem \eqref{scalar flat} and defined in \eqref{eq:J}. As shown in Proposition \ref{prop:mu is invariant}, these are conformal invariants. The first is related to Escobar's invariant $Q(M,\partial M)$, defined in \cite{e1}, and is attained by a positive multiple of a minimizer of $Q(M,\partial M)$. Theorem \ref{thm:mu_2} sets a condition for $\mu_2(M)$ to be attained by a function $u\in\cE$ which is a solution to \eqref{scalar flat} that changes sign. Its Corollary \ref{cor:mu_2a} provides a friendlier condition in terms of a linear combination of two test functions. In Section \ref{sec:scalar flat} we show that a minimizer of $\mu_1(M)$ and a cut-off of a minimizer of $\mu_1(\mathbb{B}^n)$ for the unit ball $\mathbb{B}^n$ serve this purpose. The proof is based on the estimates provided by Escobar in \cite{e1}.

In \cite{hls} Ho, Lee and Shin defined a second Yamabe invariant for the problem \eqref{scalar flat} using the second eigenvalue of the boundary conformal Laplacian, and established a condition for its realization in terms of $Q(M,\partial M)$ and $Q(\mathbb{B}^n,\partial\mathbb{B}^n)$, similar to the condition \cite[(22)]{ah} of Ammann and Humbert. Ho and Pyo did something similar for the problem \eqref{minimal boundary} in \cite{hp}. The conditions given by our Corollaries \ref{cor:mu_2a} and \ref{cor:kappa_2a} are easier to verify. In Sections \ref{sec:scalar flat} and \ref{sec:minimal boundary} we will show that they are satisfied if $n\geq 7$ and $M$ has a nonumbilic point in $\partial M$.

\begin{corollary} \label{cor:mu_2}
If there exists $u\in H^1(M)$ such that $u^+\neq 0$, $u^-\neq 0$ and
$$\left(\frac{\mathscr{Q}u^+}{\big(\idm |u^+|^q\d \sigma\big)^{2/q}}\right)^{q/(q-2)} + \left(\frac{\mathscr{Q}u^-}{\big(\idm |u^-|^q\d \sigma\big)^{2/q}}\right)^{q/(q-2)} < Q(M,\partial M)^{q/(q-2)}+Q(\mathbb{B}^n,\partial\mathbb{B}^n)^{q/(q-2)},$$
then problem \eqref{scalar flat} has a least energy sign-changing solution.
\end{corollary}

\begin{corollary}\label{cor:mu_2a}
If there exist $u,v\in H^1(M)$ such that $u,v\geq 0$, $\big((1-s)u-sv\big)|_{\partial M}$ is nontrivial for all $s\in[0,1]$ and
$$J(r(1-s)u-rsv) < \mu_1(M)+\mu_1(\mathbb{B}^n)\qquad\text{for all \ }s\in[0,1]\text{ \ and \ }r\geq 0,$$
then the problem \eqref{scalar flat} has a least energy sign-changing solution.
\end{corollary}

\begin{corollary}\label{cor:kappa_2a}
If there exist $u,v\in H^1(M)$ such that $u,v\geq 0$, \ $(1-s)u-sv$ is nontrivial for all $s\in[0,1]$ and
$$I(r(1-s)u-rsv) < \kappa_1(M)+\kappa_1(\mathbb{B}^n)\qquad\text{for all \ }s\in[0,1]\text{ \ and \ }r\geq 0,$$
then the problem \eqref{minimal boundary} has a least energy sign-changing solution.
\end{corollary}

\begin{equation}\label{eq:J}
J(u):=\frac{1}{2}\mathscr{Q}u - \frac{1}{q}\idm |u|^q\d \sigma,
\end{equation}

\begin{equation}\label{problem}
\begin{cases}
 -\Delta u + c_nRu=\tilde Ru^{n+2\over n-2} &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= \tilde h u^\frac{n}{n-2} &\text{on \ }\partial M,
\end{cases}
\end{equation}

\begin{proposition} \label{prop:mu is invariant}
The numbers $\mu_1(M)$ and $\mu_2(M)$ are conformal invariants.
\end{proposition}

\begin{theorem} \label{thm:main scalar flat}
If $n\geq 7$ and $M$ has a nonumbilic point $\xi\in\partial M$, then the problem \eqref{scalar flat} has a least energy sign-changing solution.
\end{theorem}

\begin{theorem} \label{thm:mu_2}
If
$$\mu_2(M)<\mu_1(M)+\mu_1(\mathbb{B}^n),$$
then there exists $u\in\cE$ such that $J(u)=\mu_2(M)$ and $J'(u)=0$, that is, $u$ is a least energy sign-changing solution of \eqref{scalar flat}.
\end{theorem}

\begin{theorem} \label{thm:main minimal boundary}
If $n\geq 7$ and $M$ has a nonumbilic point $\xi\in\partial M$, then the problem \eqref{minimal boundary} has a least energy sign-changing solution.
\end{theorem}

The strategy to prove Theorem \ref{thm:main scalar flat} is as follows. If $u\in H^1(M)$ is a nontrivial solution to \eqref{scalar flat}, then multiplying that problem by $u$ and integrating by parts shows that $u$ belongs to the set
$$\mathcal{N}:=\Big\{u\in H^1(M): u\neq 0 \text{ and }\mathscr{Q} u=\int_{\partial M} |u|^{2(n-1)/(n-2)}\,\mathrm{d} \sigma\Big\}.$$
If $u$ changes sign, then multiplying \eqref{scalar flat} by the positive and negative parts $u^\pm$ of $u$ shows that $u$ belongs to
$$\mathcal{E}:=\{u\in\mathcal{N}:u^+\in\mathcal{N}\text{ and }u^-\in\mathcal{N}\}.$$
We define
$$\mu_1(M):=\inf_{u\in\mathcal{N}}J(u)\qquad\text{and}\qquad\mu_2(M):=\inf_{u\in\mathcal{E}}J(u),$$
where $J$ is the energy functional associated with problem \eqref{scalar flat} and defined in \eqref{eq:J}. As shown in Proposition \ref{prop:mu is invariant}, these are conformal invariants. The first is related to Escobar's invariant $Q(M,\partial M)$, defined in \cite{e1}, and is attained by a positive multiple of a minimizer of $Q(M,\partial M)$. Theorem \ref{thm:mu_2} sets a condition for $\mu_2(M)$ to be attained by a  function $u\in\mathcal{E}$ which is a solution to \eqref{scalar flat} that changes sign. Its Corollary \ref{cor:mu_2a} provides a friendlier condition in terms of a linear combination of two test functions. In Section \ref{sec:scalar flat} we show that a minimizer of $\mu_1(M)$ and a cut-off of a minimizer of $\mu_1(\mathbb{B}^n)$ for the unit ball $\mathbb{B}^n$ serve this purpose. The proof is based on the estimates provided by Escobar in \cite{e1}.

A similar strategy is used for Theorem \ref{thm:main minimal boundary}.

In the special case when $M$ is the unit ball, $\mu_2(\mathbb{B}^n)$ is not attained. However, Almaraz and Wang showed in \cite[Corollary 1.2]{aw} that there are infinitely many sign-changing solutions to the problem \eqref{scalar flat} on the unit ball $\mathbb{B}^n$ (which is conformally equivalent to the half-space $\rn_+$) if $n\geq 4$.

We will prove a similar result for problem \eqref{minimal boundary}.

\begin{theorem}\label{thm:main ball}
If $n\geq 5$ the minimal boundary problem \eqref{minimal boundary} has infinitely many  sign-changing solutions on the unit ball $\mathbb{B}^n$.
\end{theorem}

\smallskip

In \cite{hls} Ho, Lee and Shin defined a second Yamabe invariant for the problem \eqref{scalar flat} using the second eigenvalue of the boundary conformal Laplacian, and established a condition for its realization in terms of $Q(M,\partial M)$ and $Q(\mathbb{B}^n,\partial\mathbb{B}^n)$, similar to the condition \cite[(22)]{ah} of Ammann and Humbert. Ho and Pyo did something similar for the problem \eqref{minimal boundary} in \cite{hp}. The conditions given by our Corollaries \ref{cor:mu_2a} and \ref{cor:kappa_2a} are easier to verify. In Sections \ref{sec:scalar flat} and \ref{sec:minimal boundary} we will show that they are satisfied if $n\geq 7$ and $M$ has a nonumbilic point in $\partial M$.

\begin{remark}
\emph{The argument used in our proofs of Theorems \ref{minimal boundary} and \ref{scalar flat} cannot be directly applied to the general problem \eqref{problem} due to the presence of nonlinear terms with different homogeneities: $u^{n+2\over n-2}$ on $M$ and $u^{n\over n-2}$ on its boundary. This creates technical difficulties not present in the cases analyzed here. The existence of nodal solutions for the general problem remains an interesting open problem. It is worth mentioning that Escobar studied the existence of a positive solution for problem \eqref{problem} in \cite{e4}.}
\end{remark}

The paper is organized as follows. In Section \ref{sec:setting scalar flat} we describe the variational setting for the scalar flat problem \eqref{scalar flat} and establish conditions for the existence of a least energy nodal solution to it. In Section \ref{sec:setting minimal boundary} we state the corresponding results for the minimal boundary problem \eqref{minimal boundary}. Theorem \ref{thm:main scalar flat} is proved in Section \ref{sec:scalar flat} and Theorem \ref{thm:main minimal boundary} in Section \ref{sec:minimal boundary}. Finally, Section \ref{sec:ball} is devoted to the proof of Theorem \ref{thm:main ball}.

To prove Theorem~\ref{thm:main scalar flat} the authors use a variational/minimization strategy based on Nehari-type constraints.

- If $u\in H^1(M)$ is a nontrivial solution of \eqref{scalar flat}, then “multiplying that problem by $u$ and integrating by parts” yields that $u$ lies in
\[
\mathcal{N}:=\Big\{u\in H^1(M): u\neq 0 \text{ and }\mathscr{Q} u=\int_{\partial M} |u|^{2(n-1)/(n-2)}\,\mathrm{d} \sigma\Big\}.
\]
- If $u$ changes sign, then “multiplying \eqref{scalar flat} by the positive and negative parts $u^\pm$” shows $u\in$
\[
\mathcal{E}:=\{u\in\mathcal{N}:u^+\in\mathcal{N}\text{ and }u^-\in\mathcal{N}\}.
\]
- Define the minimization levels
\[
\mu_1(M):=\inf_{u\in\mathcal{N}}J(u),\qquad \mu_2(M):=\inf_{u\in\mathcal{E}}J(u),
\]
where $J$ is the energy functional for \eqref{scalar flat}. These are “conformal invariants.” Moreover, “$\mu_1(M)$ … is attained by a positive multiple of a minimizer of $Q(M,\partial M)$.”
- Theorem~\ref{thm:mu_2} gives “a condition for $\mu_2(M)$ to be attained by a function $u\in\mathcal{E}$ which is a solution to \eqref{scalar flat} that changes sign,” and Corollary~\ref{cor:mu_2a} provides “a friendlier condition in terms of a linear combination of two test functions.”
- In Section~\ref{sec:scalar flat} they verify this condition by taking “a minimizer of $\mu_1(M)$ and a cut-off of a minimizer of $\mu_1(\mathbb{B}^n)$ for the unit ball $\mathbb{B}^n$,” with the proof “based on the estimates provided by Escobar in~\cite{e1}.”

They note that “a similar strategy is used for Theorem~\ref{thm:main minimal boundary}.”

To prove the main theorem, the authors use a variational/minimization strategy based on Nehari-type constraints.

- If $u\in H^1(M)$ is a nontrivial solution of
\begin{equation}\label{scalar flat}
\begin{cases}
 -\Delta u + c_nRu=0 &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= |u|^\frac{{2}}{n-2}u &\text{on \ }\partial M,
\end{cases}
\end{equation}
then “multiplying that problem by $u$ and integrating by parts” yields that $u$ lies in
\[
\mathcal{N}:=\Big\{u\in H^1(M): u\neq 0 \text{ and }\mathscr{Q} u=\int_{\partial M} |u|^{2(n-1)/(n-2)}\,\mathrm{d} \sigma\Big\}.
\]
- If $u$ changes sign, then “multiplying \eqref{scalar flat} by the positive and negative parts $u^\pm$” shows $u\in$
\[
\mathcal{E}:=\{u\in\mathcal{N}:u^+\in\mathcal{N}\text{ and }u^-\in\mathcal{N}\}.
\]
- Define the minimization levels
\[
\mu_1(M):=\inf_{u\in\mathcal{N}}J(u),\qquad \mu_2(M):=\inf_{u\in\mathcal{E}}J(u),
\]
where $J$ is the energy functional for \eqref{scalar flat}. These are “conformal invariants.” Moreover, “$\mu_1(M)$ … is attained by a positive multiple of a minimizer of $Q(M,\partial M)$.”

We first prove the following theorem.
\begin{theorem} \label{thm:mu_2}
If
$$\mu_2(M)<\mu_1(M)+\mu_1(\mathbb{B}^n),$$
then there exists $u\in\cE$ such that $J(u)=\mu_2(M)$ and $J'(u)=0$, that is, $u$ is a least energy sign-changing solution of \eqref{scalar flat}.
\end{theorem}
It gives “a condition for $\mu_2(M)$ to be attained by a function $u\in\mathcal{E}$ which is a solution to \eqref{scalar flat} that changes sign.”

\begin{corollary}\label{cor:mu_2a}
If there exist $u,v\in H^1(M)$ such that $u,v\geq 0$, $\big((1-s)u-sv\big)|_{\partial M}$ is nontrivial for all $s\in[0,1]$ and
$$J(r(1-s)u-rsv) < \mu_1(M)+\mu_1(\mathbb{B}^n)\qquad\text{for all \ }s\in[0,1]\text{ \ and \ }r\geq 0,$$
then the problem \eqref{scalar flat} has a least energy sign-changing solution.
\end{corollary}
This provides “a friendlier condition in terms of a linear combination of two test functions.”

Next they verify this condition by taking “a minimizer of $\mu_1(M)$ and a cut-off of a minimizer of $\mu_1(\mathbb{B}^n)$ for the unit ball $\mathbb{B}^n$,” with the proof “based on the estimates provided by Escobar in~\cite{e1}.”

They note that “a similar strategy is used for
\begin{theorem} \label{thm:main minimal boundary}
If $n\geq 7$ and $M$ has a nonumbilic point $\xi\in\partial M$, then the problem \eqref{minimal boundary} has a least energy sign-changing solution.
\end{theorem}
.”

\label{thm:main scalar flat}
If $n\geq 7$ and $M$ has a nonumbilic point $\xi\in\partial M$, then the problem \begin{equation}\label{scalar flat}
\begin{cases}
 -\Delta u + c_nRu=0 &\text{on \ } M,\\
 \frac{\partial u}{\partial\nu} + d_nhu= |u|^\frac{2}{n-2}u &\text{on \ }\partial M,
\end{cases}
\end{equation}
 has a least energy sign-changing solution.

Question:
Let $(M,g)$ be a compact Riemannian manifold with boundary, of dimension $n\ge 7$. Assume that the boundary $\partial M$ contains a nonumbilic point $\xi$ (that is, at $\xi$ the second fundamental form of $\partial M$ is not proportional to the induced metric). Let $R=R_g$ be the scalar curvature of $M$, $h=h_g$ the mean curvature of $\partial M$, $\Delta=\Delta_g$ the Laplace--Beltrami operator, $\nu=\nu_g$ the outer unit normal, and
$$c_n=\frac{n-2}{4(n-1)},\qquad d_n=\frac{n-2}{2}.$$
Consider the boundary value problem
$$\begin{cases}
-\Delta u+c_nRu=0 &\text{on } M,\\
\dfrac{\partial u}{\partial \nu}+d_nhu=|u|^{\frac{2}{n-2}}u &\text{on } \partial M.
\end{cases}$$
Which conclusion about this problem is valid under these hypotheses?

The problem admits a least energy sign-changing solution; equivalently, there exists a solution $u$ that takes both positive and negative values and has minimal associated energy among all sign-changing solutions of the problem.

Choices:
B. The problem admits a least energy sign-changing solution provided one additionally assumes the strict variational inequality $\mu_2(M)<\mu_1(M)+\mu_1(\mathbb{B}^n)$; the geometric hypothesis that $\partial M$ has a nonumbilic point by itself does not suffice to guarantee existence.
C. The problem admits a sign-changing solution, that is, there exists a nontrivial solution $u$ taking both positive and negative values on $M$.
D. The problem admits a least energy sign-changing solution whenever $n\geq 6$ and $M$ has a nonumbilic point $\xi\in\partial M$.
E. The problem admits a least energy sign-changing solution, and every such least energy sign-changing solution is obtained by minimizing the energy over the full Nehari manifold $\mathcal N$ rather than over the nodal set of sign-changing functions.

replacing the theorem's geometric sufficient condition by the intermediate strict inequality criterion

dropped the least-energy minimality conclusion while keeping existence of a nodal solution

confusing minimization over $\mathcal E$ for sign-changing states with minimization over the full Nehari manifold $\mathcal N$


\begin{algorithm}[H]
\caption{Compute an upper bound on $\lambda_F$}
\label{alg_ub}
\begin{algorithmic}[1]
\Require $0 < \delta < 1$, integers $n_{\text{iteration}} \geq 0$ and $n_K \geq 0$
\Ensure an upper bound on $\lambda_F$
\State Let $\mathcal{L}_{\text{interval}} \gets \{ ([0, 1], 0, \texttt{True}, +\infty) \}$
\cmt{$\mathcal{L}_{\text{interval}}$ is a list of \textsc{StructureOfInterval} (see Algorithm~\ref{str})}
\State Let $K \gets \textsc{FindK}(n_K)$
\cmt{See Algorithm~\ref{alg_K}}
\For{$n_{\text{iteration}}$ times}
    \State $\mathcal{L}_{\text{interval}} \gets \textsc{ComputeBounds}(\mathcal{L}_{\text{interval}}, K)$
    \cmt{\textsc{ComputeBounds} uses Equation~\eqref{in_ub} to compute bounds for new intervals}
    \State $\mathcal{L}_{\text{interval}} \gets \textsc{SplitWorstIntervals}(\mathcal{L}_{\text{interval}}, \delta)$
    \cmt{\textsc{SplitWorstIntervals} splits into two a proportion $\delta$ of intervals on which the bound is the worst}
\EndFor
\State \Return the maximum bound over all elements in $\mathcal{L}_{\text{interval}}$
\end{algorithmic}
\end{algorithm}

\begin{proof}
    To obtain the first equality for $\lambda_u$, apply the chain rule to differentiate $T^n$. The second equality for $\lambda_u$ follows directly from the semi-uniform ergodic theorem \cite[Theorem 1.9]{MR1734626}. The results for $\lambda_F$ are proven in \cite[Lemma~2.2.3, Proposition~4.2.3 and Lemma~4.2.4]{morand2024galtonwatsonprocessesdynamicalenvironments}.
\end{proof}

\subsection{Lower bound on the Lyapunov exponents}\label{s2.1}

This subsection presents a method for computing lower bounds on the Lyapunov exponents, $\lambda_F$ and $\lambda_u$.

\subsubsection{Lyapunov exponents on periodic orbits}

Proposition~\ref{prop2.1} shows that, to obtain the Lyapunov exponents $\lambda_F$ and $\lambda_u$, it is sufficient to consider the functions $F$ and $\log|T'|$ on periodic orbits. This follows from the density of ergodic measures supported on periodic orbits within the space of all ergodic measures \cite[Corollary~5]{MR718829}.

\begin{prop}\cite[Corollary~5]{MR718829}
\label{prop2.1}
    Assume (H\ref{hyp4}). For all continuous functions $f:\RZ\to \mathbb{R}$, \begin{align*}
        \sup_{\nu\in\mathcal{P}_T(\rz)}\int_{\rz} f(x) \,\mathrm{d}\nu(x)=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}f(T^ix).
    \end{align*}
In particular, \begin{align*}
    \lambda_F=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}F(T^ix) \quad \text{and} \quad \lambda_u=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}\log|T'(T^ix)|.
\end{align*}
\end{prop}

In Proposition~\ref{prop2.1}, the equalities for $\lambda_F$ and $\lambda_u$ follow from the continuity of the functions $F$ and $\log |T'|$ (Lemma~\ref{lmc}), and from Proposition~\ref{proplf}.

\subsubsection{Effective lower bound on the Lyapunov exponents}\label{eff_low}

Thanks to the periodic orbits, we can numerically provide a lower bound on the Lyapunov exponents $\lambda_F$ and $\lambda_u$, as shown in Proposition~\ref{prop2.1}. Computing a lower bound for the Lyapunov exponents requires three steps: finding periodic points, computing a lower bound for $q$, and then computing a lower bound for the Lyapunov exponents.

\paragraph{Periodic point search}~

We compute an approximation of periodic orbits of $T$ of period less than some $M\in\N$ with an error smaller than $\varepsilon>0$. To achieve this, we search for all $k\leq M$, one representative of each primitive orbit of period $k$.

We assume that $T(0)=0$ (see Remark~\ref{rem1}) and $T'>0$. Let $\widetilde{T}:\mathbb{R}\to\mathbb{R}$ be the lift of $T$ such that $\widetilde{T}(0)=0$. For $k\in\mathbb{N}^*$, the $k$-periodic points of $T$ are the points $x\in[0,1[$ such that $\widetilde{T}^k(x)-x\in\mathbb{N}$. If $d\geq 2$ is the topological degree of $T$, then there are $d^k-1$ periodic points of $T$ of period $k$ which are the unique $x_j\in[0,1[$ for $j\in\llbracket 0,d^k-2\rrbracket$ such that $\widetilde{T}^k(x_j)-x_j=j$. Using standard combinatorics, we choose one representative per primitive orbit of period $k$, and approximate it by dichotomy. We can then approximate the rest of the orbit by taking the images by $T$ of our first approximation (and use again dichotomy to obtain an error smaller than $\varepsilon$ because $T$ is expanding). For each $k$-periodic point $x$, we obtain a family of points $(y_i)_{i\in\llbracket 0,k-1\rrbracket}$ such that for all $i\in\llbracket 0,k-1\rrbracket$, $|T^i(x)-y_i|<\varepsilon$. For all $i\in\llbracket 0,k-1\rrbracket$, let $I_i=[y_i-\varepsilon,y_i+\varepsilon]$. Then, $(I_i)_{i\in\llbracket0,k-1\rrbracket}$ is a family of intervals of size $2\varepsilon$ such that for all $i\in\llbracket0,k-1\rrbracket$, $T^i(x)\in I_i$.

\paragraph{Lower bound on $q$}~

To compute a lower bound on $\lambda_F$, we must first compute a lower bound on $q \circ T$ along the periodic orbit (and thus of $q$) because $s\in[0,1]\mapsto \partial_s\varphi(x,s)$ is an increasing function for all $x\in\RZ$.

\begin{lemma}\label{lem_lb_q}
    Assume (H\ref{hyp4}) and that $T(0)=0$. Let $k\in\mathbb{N}^*$, $x\in\RZ$ be a $k$-periodic point of $T$, and $(I_i)_{i\in\llbracket0,k-1\rrbracket}$ be a family of intervals such that, for all $i\in\llbracket0,k-1\rrbracket$, $x_i\defeq T^i(x)\in I_i$. Then, for all $n\in\mathbb{N}$, \begin{align*}
    q(x)\geq \inf(\underbrace{\varphi(I_0,\varphi(I_1,\varphi(I_2,\ldots\varphi(I_{k-1},\varphi(I_0,\varphi(I_1, \ldots\varphi(I_{n-1[k]}}_{n \text{ times}},0)\ldots),
\end{align*}
where $m[k]$ is the remainder of the Euclidean division of $m$ by $k$.
Moreover, for all $i\in\llbracket0,k-1\rrbracket$, if $q_i\leq q(x_i)$, then $\inf(\varphi(I_{i-1[k]},q_i))\leq q(x_{i-1[k]})$.
\end{lemma}

\begin{proof}
   By Definition~\ref{def_q}, $q(x)\geq \varphi^{(n)}(x,0)$. The conclusion of the first inequality follows by induction, thanks to Proposition~\ref{prop5} and the $k$ periodicity of $x$. The second inequality follows from Proposition~\ref{prop1}, and because for all $x\in\RZ$, $s\mapsto \varphi(x,s)$ is increasing.
\end{proof}

\begin{algorithm}[H]
\caption{Compute a lower bound on $\lambda_F$}
\label{alg_lb}
\begin{algorithmic}[1]
\Require $\varepsilon > 0$, integer $M \geq 0$
\Ensure  a lower bound on $\lambda_F$
\State $\mathcal{L}_{\text{periodic\_orbits}} \gets \textsc{FindPeriodicOrbits}(M, \varepsilon)$
\cmt{\textsc{FindPeriodicOrbits} return all periodic orbits of $T$ of length $\leq M$, approximated with error $\leq \varepsilon$}
\State $\text{bound} \gets -\infty$

\ForAll{$\text{orbit} \in \mathcal{L}_{\text{periodic\_orbits}}$}
    \State $\text{bound\_q} \gets \textsc{LowerBoundQ}(\text{orbit}, \varepsilon)$
    \cmt{\textsc{LowerBoundQ} computes a lower bound on $q(Tx)$ for all $x$ in the orbit (see Lemma~\ref{lem_lb_q})}
    \State $\text{orbit\_bound} \gets \uo{i=1}{\text{length}(\text{orbit})}{\sum} \inf \partial_s \varphi([\text{orbit}[i]-\varepsilon,\text{orbit}[i]+\varepsilon], \text{bound\_q}[i])$
    \cmt{\eqref{in_lf}}
    \State $\text{bound} \gets \max(\text{bound}, \text{orbit\_bound})$
\EndFor
\State  \Return $\text{bound}$
\end{algorithmic}
\end{algorithm}

\begin{algorithmic}[1]
\Require $\varepsilon > 0$, integer $M \geq 0$
\Ensure  a lower bound on $\lambda_F$
\State $\mathcal{L}_{\text{periodic\_orbits}} \gets \textsc{FindPeriodicOrbits}(M, \varepsilon)$
\cmt{\textsc{FindPeriodicOrbits} return all periodic orbits of $T$ of length $\leq M$, approximated with error $\leq \varepsilon$}
\State $\text{bound} \gets -\infty$

\ForAll{$\text{orbit} \in \mathcal{L}_{\text{periodic\_orbits}}$}
    \State $\text{bound\_q} \gets \textsc{LowerBoundQ}(\text{orbit}, \varepsilon)$
    \cmt{\textsc{LowerBoundQ} computes a lower bound on $q(Tx)$ for all $x$ in the orbit (see Lemma~\ref{lem_lb_q})}
    \State $\text{orbit\_bound} \gets \uo{i=1}{\text{length}(\text{orbit})}{\sum} \inf \partial_s \varphi([\text{orbit}[i]-\varepsilon,\text{orbit}[i]+\varepsilon], \text{bound\_q}[i])$
    \cmt{\eqref{in_lf}}
    \State $\text{bound} \gets \max(\text{bound}, \text{orbit\_bound})$
\EndFor
\State  \Return $\text{bound}$
\end{algorithmic}
\end{algorithm}

Thanks to Proposition~\ref{prop2.1}, we can compute a lower bound on the Lyapunov exponent of the transformation $T$ using the same method. For all $M\in\N^*$, \begin{align*}
    \lambda_u\geq \sup_{m\leq M}\sup_{x\in \text{Per}_m(\rz)}\frac{1}{m}\sum_{i=0}^{m-1}\log|T'(T^ix)|.
\end{align*}
However, the Lyapunov exponent of the transformation $T$ does not depend on $q$, making it easier to compute.

\subsection{Upper bound on the Lyapunov exponents}\label{s2.2}

This subsection presents a useful method for computing upper bounds on the Lyapunov exponents $\lambda_F$ and $\lambda_u$. These exponents are approximated by finding upper bounds of the Birkhoff sums of the functions $F$ and $\log |T'|$ on small intervals and using the expressions given in Corollary~\ref{coro2.2}.

\subsubsection{An expression of the Lyapunov exponents for computing an upper bound}

Corollary~\ref{coro2.2} provides an alternative expression for the Lyapunov exponents $\lambda_F$ and $\lambda_u$, which is useful for computing an upper bound on these exponents.

\begin{corollary}\label{coro2.2}
 Assume (H\ref{hyp4}). For all continuous functions $f:\RZ\to \mathbb{R}$, \begin{align}\label{abc}
        \sup_{\nu\in\mathcal{P}_T(\rz)}\int_{\rz} f(x) \,\mathrm{d}\nu(x)=\sup_{x\in\rz}\inf_{k\in\mathbb{N}^*}\frac{1}{k}\sum_{i=0}^{k-1} f(T^ix).
    \end{align}
In particular,
\begin{align*}
\lambda_F=\sup_{x\in\rz}\inf_{k\in\mathbb{N}^*}\frac{1}{k}\sum_{i=0}^{k-1} F(T^ix) \quad \text{and} \quad \lambda_u=\sup_{x\in\rz}\inf_{k\in\mathbb{N}^*}\frac{1}{k}\sum_{i=0}^{k-1} \log|T'(T^ix)|.
\end{align*}
\end{corollary}

\begin{proof}Let $f:\RZ\to \mathbb{R}$ be a continuous function.
    By Proposition~\ref{prop2.1}, \begin{align*}
        \sup_{\nu\in\mathcal{P}_T(\rz)}\int_{\rz} f(x) \,\mathrm{d}\nu(x)=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}f(T^ix).
    \end{align*}
    Let $k\in\mathbb{N}^*$ and $x\in \text{Per}_k(\rz)$. By Pliss's lemma \cite[Lemma 11.8]{MR889254}, there exists $j\in\mathbb{N}$ such that \begin{align*}
        \frac{1}{k}\sum_{i=0}^{k-1}f(T^ix)= \inf_{n\in\mathbb{N}^*}\frac{1}{n}\sum_{i=0}^{n-1}f(T^{i+j}x)\leq\sup_{y\in\rz}\inf_{n\in\mathbb{N}^*}\frac{1}{n}\sum_{i=0}^{n-1} f(T^iy).
    \end{align*}
    Having established that \begin{align*}
        \sup_{\nu\in\mathcal{P}_T(\rz)}\int_{\rz} f(x) \,\mathrm{d}\nu(x)\leq\sup_{x\in\rz}\inf_{k\in\mathbb{N}^*}\frac{1}{k}\sum_{i=0}^{k-1} f(T^ix),
    \end{align*}
    it remains to prove the converse inequality to obtain Equality~\eqref{abc}. It is a direct corollary of the semi-uniform ergodic theorem \cite[Theorem 1.9]{MR1734626}. The equalities on $\lambda_F$ and $\lambda_u$ follow from the continuity of the functions $F$ and $\log |T'|$ (see Lemma~\ref{lmc}), and from Proposition~\ref{proplf}.
\end{proof}

\subsubsection{Effective upper bound on the Lyapunov exponents}\label{eff}

Computing an upper bound for the Lyapunov exponent $\lambda_F$ requires three steps, which are described in the next three paragraphs. First, we find a constant $K<1$ such that $K\geq q(x)$ for all $x\in\RZ$. Second, we compute an upper bound on $q$ using the expression $\varphi^{(n)}(x,K)$ with $n$ large. Finally, we compute an upper bound for the Lyapunov exponents using the upper bound on $q$ and the expression for the Lyapunov exponent $\lambda_F$ given in Corollary~\ref{coro2.2}.

\paragraph{Upper bound on $K$}~

In the uniformly supercritical case, there exists a constant $K<1$ such that $K\geq q(x)$ for all $x\in\RZ$ \cite[Lemma~4.1.1]{morand2024galtonwatsonprocessesdynamicalenvironments}. Lemma~\ref{lem2.1} justifies using the function $x\mapsto \varphi^{(n)}(x,K)$ with $n$ large as an upper bound on $q$.

\begin{lemma}\label{lem2.1}
Assume (H\ref{hyp4}). Let $K<1$ be such that $K\geq q(x)$ for all $x\in\RZ$. Then:\begin{itemize}
    \item For all $n\in\N$ and $x\in\RZ$, $\varphi^{(n)}(x,K)\geq q(x)$.
    \item The sequence of functions $(x\mapsto\varphi^{(n)}(x,K))_{n\in\mathbb{N}}$ converges uniformly to $q$.
\end{itemize}
\end{lemma}

\begin{proof}As $K<1$, the uniform convergence of this sequence of functions is proved by \cite[Corollary~4.1.2]{morand2024galtonwatsonprocessesdynamicalenvironments}.
Let $x\in\RZ$ and $n\in\mathbb{N}$.
As $s\mapsto \varphi^{(n)}(x,s)$ is increasing and by Proposition \ref{prop1},
\begin{align*}
    \varphi^{(n)}(x,K)&\geq\varphi^{(n)}(x,q(T^nx))=q(x).\qedhere
\end{align*}
\end{proof}

We need to find a suitable $K<1$. If there exists $N\in\N^*$ such that, for all $x\in\RZ$, $\varphi^{(N)}(x,K)\leq K$, then $K$ is suitable. However, we may choose a value of $N$ depending on $x$ by Lemma~\ref{lem_k2}.

\begin{lemma}\label{lem_k2}
Assume $(H\ref{hyp4})$. Let $K<1$ be such that for all $x\in\RZ$, there exists $N_x\in\N^*$ such that $\varphi^{(N_x)}(x,K)\leq K$. Then, $q(x)\leq K$ for all $x\in\RZ$.
\end{lemma}

\begin{proof}
Let $x\in\RZ$. We define a sequence of positive integers $(n_i)_{i\in\N}$ by the following induction relation:
\begin{align*}
\left\{\begin{array}{ll}
  & n_0=N_x, \\
  & n_{i+1}=N_{T^{n_0+\ldots+n_i}x} \text{ for all } i\in\N.
   \end{array}\right.
\end{align*}
Then for all $i\in\N$,
\begin{align*}
    \varphi^{(n_0+\ldots+n_i)}(x,K)&=\varphi^{(n_0)}(x,\varphi^{(n_1)}(T^{n_0}x,\ldots,\varphi^{(n_i)}(T^{n_0+\ldots+n_{i-1}}x,K)\ldots)\leq K,
\end{align*}
by induction, thanks to the definition of the sequence $(n_i)_{i\in\N}$. By \cite[Corollary~4.1.2]{morand2024galtonwatsonprocessesdynamicalenvironments}, \begin{align*}
    \varphi^{(n_0+\ldots+n_i)}(x,K)\underset{i\to +\infty}{\longrightarrow} q(x).
\end{align*} So, $q(x)\leq K$.
\end{proof}

\begin{remark}\label{rem2}
    In practice, it is sometimes difficult to check if a process is in the uniformly supercritical regime. However, Lemma~\ref{lem_k2} gives a sufficient condition for this to be the case.
\end{remark}

By Lemma~\ref{lem_k2}, we can certify whether a constant $0<C<1$ is an upper bound on $q$. This method is used in Algorithm~\ref{alg_K}. The algorithm returns true if it is verified that $C$ is an upper bound on $q$. It returns false if it is not verified (note that this does not necessarily imply that $C$ is not an upper bound of $q$).

\begin{algorithm}[H]
\caption{Verify whether $C$ is an upper bound on $\sup_{x\in\mathbb{R}} q(x)$}
\label{alg_K}
\begin{algorithmic}[1]
\Require $0 < C < 1$, integer $n_{\max} \geq 0$
\Ensure \texttt{True} if it is verified that $C \geq \sup_{x \in \mathbb{R}} q(x)$, \texttt{False} otherwise

\State Let $\mathcal{L}_{\text{interval}} \gets \{ [0, 1] \}$
\For{$i = 1$ to $n_{\max}$}
    \State Let $\mathcal{L}_{\text{new}} \gets$ empty list
    \cmt{$\mathcal{L}_{\text{new}}$ is the list of intervals after an iteration}
    \ForAll{$I \in \mathcal{L}_{\text{interval}}$}
        \If{for all $j = 1$ to $\left\lfloor \frac{i}{2} \right\rfloor + 1$, we have $\sup \varphi^{(j)}(I, C) > C$}
            \State Let $I_1, I_2 \gets \textsc{Split}(I)$
            \cmt{\textsc{Split} divides an interval into two intervals of the same size}
            \State Append $I_1$ and $I_2$ to $\mathcal{L}_{\text{new}}$
        \EndIf
    \EndFor
    \State $\mathcal{L}_{\text{interval}} \gets \mathcal{L}_{\text{new}}$
    \If{$\mathcal{L}_{\text{interval}}$ is empty}
        \State \Return \texttt{True}
    \EndIf
\EndFor
\State \Return \texttt{False}
\end{algorithmic}

\label{A}

In this appendix, we prove Theorem~\ref{thm2.2}, which gives the differentiability class of $q$ and the Hölder regularity of its derivatives as a function of the ratio $\frac{|\lambda_F|}{\

Understanding the regularity of invariant graphs is a central issue in the study of skew product systems \cite{MR1605989,MR1677161,MR1898800,MR3816739}. From a numerical point of view, their regularity can often be estimated by controlling Lyapunov exponents.
The numerical estimation of Lyapunov exponents has been investigated in several contexts,
notably for random matrix products \cite{MR4014663,MR4304495},
for iterated function systems contracting on average \cite{MR1751564},
and for interval transformations \cite{MR2430654,MR4538289}
(including maps with critical points, discontinuities, or unbounded derivatives).

In this article, we focus on a model close to skew products, arising from the study of
Galton--Watson processes in dynamical environments in the uniformly supercritical case introduced in
\cite{morand2024galtonwatsonprocessesdynamicalenvironments}.
The invariant function of this model represents the probability of extinction.
We express the Hölder regularity of this invariant function as a ratio between two Lyapunov exponents.
The main numerical difficulty in controlling the regularity of the invariant graph lies in the fact that the Lyapunov exponent in the fibre direction depends on the invariant function itself.

\subsection*{Numerical bounds on the Lyapunov exponent in the fibre}

We consider a $\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\RZ\to\RZ$, and a continuous application $\varphi:\RZ\times[0,1]\to[0,1]$. Moreover, we assume that for all $x\in\RZ$, the function $s\mapsto\varphi(x,s)$ is analytic on $[0,1)$, increasing, convex, and that $\varphi(x,1)=1$. We denote by $\partial_s\varphi$ the derivative of $\varphi$ with respect to the second variable and by $\partial_x\varphi$ the derivative of $\varphi$ with respect to the first variable if it is well defined. For all $n\in\mathbb{N}$, we define $\varphi^{(n)}:\RZ\times[0,1]\to[0,1]$ by induction: for all $x\in\RZ$ and $s\in[0,1]$, \begin{align*}
\varphi^{(n)}(x,s)\vcentcolon=
\left\{\begin{array}{ll}
    & s \text{ if } n=0,\\
    & \varphi(x,\varphi^{(n-1)}(Tx,s)) \text{ otherwise.}
    \end{array}
    \right.
    \end{align*}
We consider the solutions $f:\RZ\to[0,1]$ of the functional equation: \begin{align}\label{invgraph}
     f(x)=\varphi(x,f(Tx)).
\end{align}
The constant function equal to $1$ is a solution of \eqref{invgraph}. We denote by $q$ the smallest solution of this functional equation. This model is closely related to models of invariant graphs of skew product systems, as presented in references \cite{MR1605989,MR1677161,MR1898800,MR3170606,MR3816739}. See \cite[Subsection 1.4]{morand2024galtonwatsonprocessesdynamicalenvironments} for details of the connections. \begin{remark}
     If we consider an invertible transformation of the circle instead of an uniformly expanding transformation, we would recover exactly a skew product system by considering the dynamics in the past. Let $\Psi$ be defined by: \fonction{\Psi}{\RZ\times [0,1]}{\RZ\times [0,1]}{(x,s)}{(T^{-1}x,\varphi(T^{-1}x,s))}
Then, for all $x\in\RZ$, $n\in\mathbb{N}$, and $s\in[0,1]$,
\begin{align*}
    \Psi^n(x,s)=(T^{-n}x,\varphi^{(n)}(T^{-n}x,s)).
\end{align*}
\end{remark}
We refer to $q$ as the invariant function of the system $(\varphi,T)$ by analogy with models of invariant graphs of skew product systems.

In this article, we study two Lyapunov exponents: \begin{itemize}
    \item the Lyapunov exponent of the invariant function $q$ in the fibre direction:
  \begin{align}\label{lf}
      \lambda_F&\vcentcolon=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log(\partial_s\varphi^{(n)}(x,q(T^n x))),
      \end{align}
    \item the Lyapunov exponent of the transformation $T$:
  \begin{align}\label{lu}
    \lambda_u\vcentcolon=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log(|(T^n)'(x)|).
  \end{align}\end{itemize}

  We obtain rigorous numerical bounds on these Lyapunov exponents. The difficulty lies in the fact that the Lyapunov exponent in the fibre depends on the invariant function, which is not explicit and may have low and unknown regularity. According to the specification property \cite[Corollary~5]{MR718829}, to obtain a lower bound on the Lyapunov exponent in the fibre, it is sufficient to have a lower bound on the invariant function only along the periodic orbits Proposition~\ref{prop2.1}. We then estimate the periodic orbits of $T$ of length at most some $M\in\mathbb{N}$. Corollary~\ref{coro2.2} allows us to obtain an upper bound on the Lyapunov exponent in the fibre. We first need to obtain an upper bound on the invariant function using arithmetic intervals and Lemma~\ref{lem2.1}.

 It is easier to control the Lyapunov exponents of the transformation and previous work has been done in more complicated cases \cite{MR2430654,MR4538289}.

  All the code has been written in the Julia programming language and is available on GitHub\footnote{\href{https://github.com/thomas-morand/holder_regularity_of_invariant_graph}{Link to code: https://github.com/thomas-morand/holder\_regularity\_of\_invariant\_graph}}.

 \subsection*{Application to the probability of extinction of Galton-Watson processes in dynamical environments in the uniformly supercritical case}

We apply these results to study the model of Galton-Watson processes in dynamical environments defined in \cite{morand2024galtonwatsonprocessesdynamicalenvironments}. These are Galton-Watson processes (stochastic branching processes in which each individual in a generation produces a random number of offspring) in which the law of reproduction varies between generations according to a dynamical system. We consider a $\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\RZ\to\RZ$ together with a continuous function $x\in\RZ\mapsto \mu_x\in\mathcal{P}(\mathbb{N})$, called the law of reproduction of the Galton-Watson process in dynamical environments (a function that associates a probability on $\mathbb{N}$ to each $x\in\RZ$). The law of reproduction of the Galton–Watson process in dynamical environments with initial environment $x\in\RZ$ at generation $n\in\mathbb{N}$ is $\mu_{T^nx}$, i.e., the law of reproduction between generations evolves under the action of $T$. For a fixed $x\in\RZ$, the process is a Galton-Watson process in varying environments, as studied in particular in \cite{MR0368197,MR0365733,MR2384553,MR4094390}. We denote by $(Z_n(x))_{n\in\mathbb{N}}$ the Galton–Watson process in dynamical environment with initial environment $x\in\RZ$ and where for all $n\in\mathbb{N}$, $Z_n(x)$ represents the size of the population at the $n$th generation.
To study this model, we use the probability generating function of the law of reproduction $\mu$, defined on $\RZ\times [0,1]$ by:
\begin{align*}
    \varphi(x,s) \vcentcolon= \sum_{k=0}^{+\infty} \mu_x(k)s^k.
\end{align*}
Moreover, for all $n\in\mathbb{N}$ and $x\in\RZ$, we consider the probability generating function of the random variable $Z_n(x)$, defined for all $s\in[0,1]$ by:
\begin{align*}
    \varphi^{(n)}(x,s) \vcentcolon=  \mathbb{E}[s^{Z_n(x)}].
\end{align*}
The probability of extinction $q:\RZ\to[0,1]$ is a function that associates to each $x\in\RZ$ the probability of extinction of the Galton-Watson process in dynamical environments with the initial environment $x$, i.e., with law of reproduction $(\mu_{T^nx})_{n\in\mathbb{N}}$. When $q(x)=1$, there is almost certain extinction of the process, and we say that $x\in\RZ$ is a bad environment.

We are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\nu$-almost all $x\in\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\ref{thm2.2} shows that the H\"older regularity and the differentiability class of $q$ are controlled by the ratio $\frac{|\lambda_F|}{\lambda_u}$.

In this article, we focus on a model close to skew products, arising from the study of
Galton--Watson processes in dynamical environments in the uniformly supercritical case introduced in
\cite{morand2024galtonwatsonprocessesdynamicalenvironments}.
The invariant function of this model represents the probability of extinction.
We express the Hölder regularity of this invariant function as a ratio between two Lyapunov exponents.
The main numerical difficulty in controlling the regularity of the invariant graph lies in the fact that the Lyapunov exponent in the fibre direction depends on the invariant function itself.

In this article, we study two Lyapunov exponents: \begin{itemize}
 \item the Lyapunov exponent of the invariant function $q$ in the fibre direction:
 \begin{align}\label{lf}
 \lambda_F&\vcentcolon=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log(\partial_s\varphi^{(n)}(x,q(T^n x))),
 \end{align}
 \item the Lyapunov exponent of the transformation $T$:
 \begin{align}\label{lu}
 \lambda_u\vcentcolon=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log(|(T^n)'(x)|).
 \end{align}\end{itemize}

We obtain rigorous numerical bounds on these Lyapunov exponents. The difficulty lies in the fact that the Lyapunov exponent in the fibre depends on the invariant function, which is not explicit and may have low and unknown regularity. According to the specification property \cite[Corollary~5]{MR718829}, to obtain a lower bound on the Lyapunov exponent in the fibre, it is sufficient to have a lower bound on the invariant function only along the periodic orbits Proposition~\ref{prop2.1}. We then estimate the periodic orbits of $T$ of length at most some $M\in\mathbb{N}$. Corollary~\ref{coro2.2} allows us to obtain an upper bound on the Lyapunov exponent in the fibre. We first need to obtain an upper bound on the invariant function using arithmetic intervals and Lemma~\ref{lem2.1}.

We apply these results to study the model of Galton-Watson processes in dynamical environments defined in \cite{morand2024galtonwatsonprocessesdynamicalenvironments}. These are Galton-Watson processes (stochastic branching processes in which each individual in a generation produces a random number of offspring) in which the law of reproduction varies between generations according to a dynamical system. We consider a $\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\RZ\to\RZ$ together with a continuous function $x\in\RZ\mapsto \mu_x\in\mathcal{P}(\mathbb{N})$, called the law of reproduction of the Galton-Watson process in dynamical environments (a function that associates a probability on $\mathbb{N}$ to each $x\in\RZ$). The law of reproduction of the Galton–Watson process in dynamical environments with initial environment $x\in\RZ$ at generation $n\in\mathbb{N}$ is $\mu_{T^nx}$, i.e., the law of reproduction between generations evolves under the action of $T$. For a fixed $x\in\RZ$, the process is a Galton-Watson process in varying environments, as studied in particular in \cite{MR0368197,MR0365733,MR2384553,MR4094390}. We denote by $(Z_n(x))_{n\in\mathbb{N}}$ the Galton–Watson process in dynamical environment with initial environment $x\in\RZ$ and where for all $n\in\mathbb{N}$, $Z_n(x)$ represents the size of the population at the $n$th generation.
To study this model, we use the probability generating function of the law of reproduction $\mu$, defined on $\RZ\times [0,1]$ by:
\begin{align*}
 \varphi(x,s) \vcentcolon= \sum_{k=0}^{+\infty} \mu_x(k)s^k.
\end{align*}
Moreover, for all $n\in\mathbb{N}$ and $x\in\RZ$, we consider the probability generating function of the random variable $Z_n(x)$, defined for all $s\in[0,1]$ by:
\begin{align*}
 \varphi^{(n)}(x,s) \vcentcolon= \mathbb{E}[s^{Z_n(x)}].
\end{align*}
The probability of extinction $q:\RZ\to[0,1]$ is a function that associates to each $x\in\RZ$ the probability of extinction of the Galton-Watson process in dynamical environments with the initial environment $x$, i.e., with law of reproduction $(\mu_{T^nx})_{n\in\mathbb{N}}$. When $q(x)=1$, there is almost certain extinction of the process, and we say that $x\in\RZ$ is a bad environment.

We are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\nu$-almost all $x\in\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\ref{thm2.2} shows that the H\"older regularity and the differentiability class of $q$ are controlled by the ratio $\frac{|\lambda_F|}{\lambda_u}$.

We consider a $\mathcal{C}^1$ uniformly expanding transformation of the circle $T:\RZ\to\RZ$, and a continuous application $\varphi:\RZ\times[0,1]\to[0,1]$. Moreover, we assume that for all $x\in\RZ$, the function $s\mapsto\varphi(x,s)$ is analytic on $[0,1)$, increasing, convex, and that $\varphi(x,1)=1$. We denote by $\partial_s\varphi$ the derivative of $\varphi$ with respect to the second variable and by $\partial_x\varphi$ the derivative of $\varphi$ with respect to the first variable if it is well defined. For all $n\in\N$, we define $\varphi^{(n)}:\RZ\times[0,1]\to[0,1]$ by induction: for all $x\in\RZ$ and $s\in[0,1]$, \begin{align*}
\varphi^{(n)}(x,s)\defeq
\left\{\begin{array}{ll}
 & s \text{ if } n=0,\\
 & \varphi(x,\varphi^{(n-1)}(Tx,s)) \text{ otherwise.}
 \end{array}
 \right.
 \end{align*}
We consider the solutions $f:\RZ\to[0,1]$ of the functional equation: \begin{align}\label{invgraph}
 f(x)=\varphi(x,f(Tx)).
\end{align}
The constant function equal to $1$ is a solution of \eqref{invgraph}. We denote by $q$ the smallest solution of this functional equation. This model is closely related to models of invariant graphs of skew product systems, as presented in references \cite{MR1605989,MR1677161,MR1898800,MR3170606,MR3816739}. See \cite[Subsection 1.4]{morand2024galtonwatsonprocessesdynamicalenvironments} for details of the connections. \begin{remark}
 If we consider an invertible transformation of the circle instead of an uniformly expanding transformation, we would recover exactly a skew product system by considering the dynamics in the past. Let $\Psi$ be defined by: \fonction{\Psi}{\RZ\times [0,1]}{\RZ\times [0,1]}{(x,s)}{(T^{-1}x,\varphi(T^{-1}x,s))}
Then, for all $x\in\RZ$, $n\in\mathbb{N}$, and $s\in[0,1]$,
\begin{align*}
 \Psi^n(x,s)=(T^{-n}x,\varphi^{(n)}(T^{-n}x,s)).
\end{align*}
\end{remark}
We refer to $q$ as the invariant function of the system $(\varphi,T)$ by analogy with models of invariant graphs of skew product systems.

We are therefore in the context of the model described in the first part of the introduction, and the probability of extinction $q$ is the smallest solution of the functional equation~\eqref{invgraph}, it is thus the invariant function of our model. If, for all ergodic probability measures, the survival probability is positive for $\nu$-almost all $x\in\RZ$, then we are in the uniformly supercritical case. In this setting, Theorem~\ref{thm2.2} shows that the H\"older regularity and the differentiability class of $q$ are controlled by the ratio $\frac{|\lambda_F|}{\lambda_u}$.

In Theorem~\ref{thm2.2}, the case $k=0$ is given by \cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}. We then proceed by induction on $k\in\mathbb{N}$. We know that the family of functions $\big(x\in\mathbb{\RZ}\mapsto\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ converges uniformly to $q$ and is $(k+1)$-times differentiable. We then show that the sequence $\big(x\in\RZ\mapsto\partial_x^k\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ converges uniformly to a function (whose Hölder regularity is controlled) which is therefore the $k$th derivative of $q$. To do this, we express \begin{align*}
 x\in\mathbb{\RZ}\mapsto\partial_x^k\varphi^{(n)}(x,0)
\end{align*} as a series of differentiable functions for which we control the uniform norm and the Hölder semi-norms as functions of the Lyapunov exponents.

\encad{For $k\in\mathbb{N}$, let:
\begin{hyp}[H\ref{hyp4}(k)]\label{hyp4}~
\begin{enumerate}[label=\alph*)]
 \item $T$ is a $\mathcal{C}^{k+1}$ uniformly expanding transformation of $\RZ$.
 \item For all $\nu\in\mathcal{E}_T(\RZ)$, $\mathbb{E}_\nu[\log m(\cdot)]>0$ (uniformly supercritical case).\label{US}
 \item $\varphi$ is $\mathcal{C}^{k+1}$ on $\RZ\times[0,1)$.
 \item $q(x)>0$ for all $x\in\RZ$.
\end{enumerate}
\end{hyp}}\medskip

\begin{theorem}\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}\label{thm2.6}
 Let $\alpha\in(0,1]$. Assume (H\ref{hyp4}), and that $\alpha<\frac{|\lambda_F|}{\lambda_u}$. Then $q$ is $\alpha$-Hölder continuous.
\end{theorem}

\begin{prop}\cite[Corollary~5]{MR718829}
\label{prop2.1}
 Assume (H\ref{hyp4}). For all continuous functions $f:\RZ\to \mathbb{R}$, \begin{align*}
 \sup_{\nu\in\mathcal{P}_T(\rz)}\int_{\rz} f(x) \,\mathrm{d}\nu(x)=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}f(T^ix).
 \end{align*}
In particular, \begin{align*}
 \lambda_F=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}F(T^ix) \quad \text{and} \quad \lambda_u=\sup_{k\in\mathbb{N}^*}\sup_{x\in \text{Per}_k(\rz)}\frac{1}{k}\sum_{i=0}^{k-1}\log|T'(T^ix)|.
\end{align*}
\end{prop}

Let $N\geq 2$, $\omega\in\RZ$, and $\lambda,\varepsilon\in\mathbb{R}$ be such that $|\varepsilon|<\frac{N-1}{2\pi}$. To determine if we are in the uniformly supercritical case, we need to check if \begin{align}\label{cond}
 \uo{\nu\in\mathcal{P}_{T_{N,\varepsilon}}(\rz)}{}{\inf} \mathbb{E}_\nu[\log m_{\lambda,\omega}(\cdot)]>0.
 \end{align}
 For all $x\in\RZ$, \begin{align*}
 \log m_{\lambda,\omega}(x)=\lambda+\cos(2\pi(x+\omega)).
 \end{align*}
 Thus, Inequality~\eqref{cond} can be expressed as:\begin{align}\label{thie}
 \lambda> -\uo{\nu\in\mathcal{P}_{T_{N,\varepsilon}}(\rz)}{}{\inf} \mathbb{E}_\nu[\cos(2\pi(\cdot+\omega))].
 \end{align}
 In particular, \begin{itemize}
 \item if $\lambda>1$, we are in the uniformly supercritical case, and thus, Hypothesis~$H\ref{hyp4}(k)$ is satisfied for all $k\in\mathbb{N}$.
 \item if $N=2$ and $\varepsilon=0$,
 Inequality~\eqref{thie} was studied in \cite{MR1785392}. The maximum of $\mathbb{E}_\nu[\cos 2\pi (\cdot+\omega)]$ is reached on a Sturm measure and is periodic for all $\omega\in\RZ$ outside a certain set of Hausdorff dimension zero. \cite[Annexe]{MR1785392} gives the corresponding maximising Sturm measure for some values of $\omega\in\RZ$.
 \end{itemize}

Let $k\geq1$. We suppose that the result holds for all $k'\leq k-1$. Let $\alpha\in(0,1]$. Assume $(H\ref{hyp4}(k))$, and that $k+\alpha <\frac{|\lambda_F|}{\lambda_u}$. By Proposition~\ref{prop_dq}, for all $x\in\RZ$,
 \begin{align*}
 q^{(k)}(x)=\sum_{\ell=0}^{+\infty}f_{\ell}(x),
 \end{align*}
 with\begin{align*}
 f_{\ell}(x)=(T^\ell)'(x)^k\dsphn{ x}{q(T^{\ell+1}x)}{\ell}H_{k}(T^\ell x).
 \end{align*}
Moreover, for all $x\in\RZ$, \begin{align*}
 H_k(x)= P_k[ (T^{(z)}(x))_{ z\leq k}, (\dxsph{x}{q(Tx)}{z_1}{z_2})_{z_1+z_2\leq k}, (q^{(z)}(Tx))_{z<k}].
\end{align*}
 As $T$ is $\mathcal{C}^{k+1}$, $\varphi$ is $\mathcal{C}^{k+1}$ on $\RZ\times[0,K]$, and by the induction hypothesis, \begin{itemize}
 \item $T^{(z)}$ is Lipschitz for $z\leq k$,
 \item $x\mapsto \dxsph{x}{q(Tx)}{z_1}{z_2}$ is Lipschitz for $z_1+z_2\leq k$,
 \item $q^{(z)}\circ T$ is Lipschitz for $z< k$.
 \end{itemize}
 Thus, $H_k$ is Lipschitz, so $\alpha$-Hölder continuous. Therefore, for all $\ell\in\N$, $f_\ell$ is $\alpha$-Hölder continuous.

\begin{align}\label{invgraph}
     f(x)=\varphi(x,f(Tx)).
\end{align}

\begin{theorem}\label{thm2.2}
     Let $k\in\N$ and $\alpha\in(0,1]$. Assume:
     \begin{enumerate}[label=\alph*)]
     \item $k+\alpha <\frac{|\lambda_F|}{\lambda_u}$.
    \item $T$ is a $\mathcal{C}^{k+1}$ uniformly expanding transformation of $\RZ$.
    \item For all $\nu\in\mathcal{E}_T(\RZ)$, $\nu(q^{-1}(\{1\}))=0$ (uniformly supercritical case).
    \item $\varphi$ is $\mathcal{C}^{k+1}$ on $\RZ\times[0,1)$.
    \item $q(x)>0$ for all $x\in\RZ$.
\end{enumerate} Then $q$ is $\mathcal{C}^k$, and $q^{(k)}$ is $\alpha$-Hölder continuous.
\end{theorem}

In Theorem~\ref{thm2.2}, the case $k=0$ is given by \cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}. We then proceed by induction on $k\in\mathbb{N}$. We know that the family of functions $\big(x\in\mathbb{\RZ}\mapsto\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ converges uniformly to $q$ and is $(k+1)$-times differentiable. We then show that the sequence $\big(x\in\RZ\mapsto\partial_x^k\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ converges uniformly to a function (whose Hölder regularity is controlled) which is therefore the $k$th derivative of $q$. To do this, we express \begin{align*}
    x\in\mathbb{\RZ}\mapsto\partial_x^k\varphi^{(n)}(x,0)
\end{align*} as a series of differentiable functions for which we control the uniform norm and the Hölder semi-norms as functions of the Lyapunov exponents.

  The Hölder regularity of invariant graphs in skew product systems has been studied in \cite{MR1605989,MR1677161} for invertible transformations. In \cite{MR1898800}, the authors investigate the $\mathcal{C}^k$ regularity for a hyperbolic dynamical system. In \cite{MR3816739}, the authors consider the case where the Lyapunov exponent in the fibre direction is zero on a set of periodic orbits.

\subsection*{Outline}

In Section~\ref{section1}, we present the model of Galton-Watson processes in dynamical environments. We also describe the main objects studied in this model (and give some of their properties): the probability generating function $\varphi$ and the probability of extinction $q$.

In Section~\ref{sect_effective}, we present techniques and algorithms for computing rigorous bounds on the invariant function $q$ and on the Lyapunov exponents $\lambda_F$ and $\lambda_u$. Subsection~\ref{s2.1} presents the lower bound on the Lyapunov exponents (Algorithm~\ref{alg_lb}), whereas Subsection~\ref{s2.2} presents the upper bound (Algorithm~\ref{alg_ub}).

In Section~\ref{section3}, we provide an example of Galton-Watson process in dynamical environments to which we can apply the previous results. We analyze the numerical results and discuss the limitations of these algorithms as well as possible alternative methods.

Finally, Appendix~\ref{A} contains the proof of Theorem~\ref{thm2.2}.

The proof of Theorem~\ref{thm2.2} is described as follows. The case $k=0$ is already known (\cite[Theorem 1.3.8]{morand2024galtonwatsonprocessesdynamicalenvironments}). For general $k\in\mathbb{N}$, the authors \emph{proceed by induction on $k$}. They use that the family $\big(x\mapsto\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ \emph{converges uniformly to $q$} and is \emph{$(k+1)$-times differentiable}. They then \emph{show that} the sequence $\big(x\mapsto \partial_x^k\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ \emph{converges uniformly to a function} “whose Hölder regularity is controlled”, and conclude that this limit is \emph{the $k$th derivative of $q$}. To prove the uniform convergence and Hölder control, they \emph{express} $x\mapsto\partial_x^k\varphi^{(n)}(x,0)$ \emph{as a series of differentiable functions} and \emph{control the uniform norm and Hölder semi-norms as functions of the Lyapunov exponents}.

To prove the main theorem, the authors proceed as follows. The case $k=0$ is already known (Morand, \emph{Galton-Watson processes in dynamical environments}, 2024, Theorem 1.3.8). For general $k\in\mathbb{N}$, the authors \emph{proceed by induction on $k$}. They use that the family $\big(x\mapsto\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ \emph{converges uniformly to $q$} and is \emph{$(k+1)$-times differentiable}. They then \emph{show that} the sequence $\big(x\mapsto \partial_x^k\varphi^{(n)}(x,0)\big)_{n\in\mathbb{N}}$ \emph{converges uniformly to a function} “whose Hölder regularity is controlled”, and conclude that this limit is \emph{the $k$th derivative of $q$}. To prove the uniform convergence and Hölder control, they \emph{express} $x\mapsto\partial_x^k\varphi^{(n)}(x,0)$ \emph{as a series of differentiable functions} and \emph{control the uniform norm and Hölder semi-norms as functions of the Lyapunov exponents}.

\label{thm2.2}
     Let $k\in\mathbb{N}$ and $\alpha\in(0,1]$. Assume:
     \begin{enumerate}[label=\alph*)]
     \item $k+\alpha <\frac{|\lambda_F|}{\lambda_u}$.
    \item $T$ is a $\mathcal{C}^{k+1}$ uniformly expanding transformation of $\RZ$.
    \item For all $\nu\in\mathcal{E}_T(\RZ)$, $\nu(q^{-1}(\{1\}))=0$ (uniformly supercritical case).
    \item $\varphi$ is $\mathcal{C}^{k+1}$ on $\RZ\times[0,1)$.
    \item $q(x)>0$ for all $x\in\RZ$.
\end{enumerate} Then $q$ is $\mathcal{C}^k$, and $q^{(k)}$ is $\alpha$-Hölder continuous.,

Question:
Let \(T:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}\) be a map of the circle, let \(\varphi:(\mathbb{R}/\mathbb{Z})\times[0,1]\to[0,1]\), and let \(q:\mathbb{R}/\mathbb{Z}\to[0,1]\) be the invariant function satisfying \(q(x)=\varphi(x,q(Tx))\). Define the Lyapunov exponents
\[
\lambda_F=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log\big(\partial_s\varphi^{(n)}(x,q(T^n x))\big),
\qquad
\lambda_u=\lim_{n\to\infty}\frac{1}{n}\sup_{x\in\mathbb{R}/\mathbb{Z}}\log\big(|(T^n)'(x)|\big),
\]
where \(\varphi^{(n)}\) denotes the \(n\)-step fibre iterate associated with \(\varphi\). Assume that \(k\in\mathbb{N}\), \(\alpha\in(0,1]\), \(k+\alpha<|\lambda_F|/\lambda_u\), \(T\) is a \(\mathcal{C}^{k+1}\) uniformly expanding transformation of \(\mathbb{R}/\mathbb{Z}\), for every ergodic \(T\)-invariant probability measure \(\nu\in\mathcal{E}_T(\mathbb{R}/\mathbb{Z})\) one has \(\nu(q^{-1}(\{1\}))=0\), \(\varphi\) is \(\mathcal{C}^{k+1}\) on \((\mathbb{R}/\mathbb{Z})\times[0,1)\), and \(q(x)>0\) for all \(x\in\mathbb{R}/\mathbb{Z}\). Which conclusion about \(q\) holds under these hypotheses?

\(q\) is of class \(\mathcal{C}^k\), and its \(k\)-th derivative \(q^{(k)}\) is \(\alpha\)-H\"older continuous.

Choices:
B. \(q\) is of class \(\mathcal{C}^{k+1}\), and its \((k+1)\)-st derivative \(q^{(k+1)}\) is \(\alpha\)-H\"older continuous.
C. \(q\) is of class \(\mathcal{C}^k\).
D. \(q\) is of class \(\mathcal{C}^k\), and its \(k\)-th derivative \(q^{(k)}\) is \(\beta\)-H\"older continuous for every \(\beta\in(0,1]\) such that \(k+\beta\leq |\lambda_F|/\lambda_u\).
E. \(q\) is of class \(\mathcal{C}^k\), and its \(k\)-th derivative \(q^{(k)}\) is Lipschitz continuous.

available differentiability order from the series/induction argument

dropped the \(\alpha\)-H\"older conclusion for \(q^{(k)}\)

strict inequality and fixed exponent \(\alpha\) replaced by uniform all-\(\beta\) borderline claim

H\"older control of the limit upgraded to Lipschitz regularity


\begin{thm}\label{thm: explicit ribbon expansion in full generality}
  Let $R^{(1)}, \ldots, R^{(k)}$ be ribbons, where each
  \( R^{(i)} = \lambda^{(i)}/\mu^{(i)} \) satisfies
  \( \ell(\lambda^{(i)})\le n \). Then for any $I \subseteq [n-1]$, we
  have
\begin{align*}
    &\imm_{\T{I}} \left( \tJT_{R^{(1)}}( \vec{x}^{(1)})
    *\cdots *
    \tJT_{R^{(k)}}(\vec{x}^{(k)}) \right) \\
    &= \sum_{\nu_1 \vdash m_1, \, \ldots, \, \nu_k \vdash m_k}
    \left(
    \sum_{
    \substack{
    I_i \subseteq [a_i,b_i-1] \\
    I_1 \cap \cdots \cap I_k = I
    }} \,
    \prod_{i=1}^k
    f^{\nu_i} \left( \NDes(R^{(i)}) \cup d(I_i, R^{(i)}) \right)
    \right)
    s_{\nu_1}(\vec{x}^{(1)})
    \cdots
    s_{\nu_k}(\vec{x}^{(k)}),
\end{align*}
where $m_i$ is the size of $R^{(i)}$ and $[a_i, b_i-1] = \{j:\lambda^{(i)}_j>\mu^{(i)}_j\}$.
\end{thm}

\begin{thm}\label{thm: representation theoretic model for Phi(s_R)}
  Let \( R \) be a ribbon of size \(n\), and consider the
  \(\mathfrak{S}_n\times \mathfrak{S}_n\)-module
\[
    \tilde{H}_{|\Des(R')|-1}((\tilde{B}_n)_{\Des(R')})
\]
as described above. Then we have
\[
\Phi_{\vec x, \vec y} (s_R)
=\omega\left(\Frob_{\vec x, \vec y}
\left(\tilde{H}_{|\Des(R')|-1}((\tilde{B}_n)_{\Des(R')})\right)\right).
\]
\end{thm}

\begin{prop}\label{prop: counterexample}
  Let $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$
  be a collection of connected skew shapes
  that is not essentially $(3 \times 2)$-avoiding.
  Then there exists a multinetwork $\vec{H}$ for this collection satisfying $ f_1(\beta(\vec{H})) = \sgn(\beta(\vec{H})) < 0$.
\end{prop}

A matrix is said to be \emph{totally positive} if all of its minors are nonnegative. Total positivity arises in various areas of mathematics and has wide-ranging applications. Originally studied by Gantmacher and Krein \cite{Gantmacher1935,Gantmakher1937} in the context of classical analysis and numerical interpolation, total positivity has since become a central topic in representation theory, geometry, mathematical physics, and algebraic combinatorics (see the survey by Fomin \cite{Fomin2010} and references therein).

By the Cauchy--Binet theorem, it is immediate that the product
\( MN \) of two totally positive matrices \( M \) and \( N \) is also
totally positive. However, their Hadamard (entrywise) product
\( M * N \) need not be totally positive in general.
Nevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product.
A \textit{Toeplitz matrix} with \textit{Toeplitz sequence $(a_0,a_1,\ldots)$} is an infinite, upper-triangular matrix
of the form $(a_{j-i})_{i,j=0}^\infty$.
The \emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of
univariate polynomials with nonnegative coefficients and real roots.
Mal\'{o} \cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.

For an entire function \( p(t) = \prod_{i\ge0}(1 + \alpha_it) \) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \( \alpha_i \ge 0 \) by variables \( x_i \), then the coefficient of $t^k$ in \( p(t) \) becomes the \emph{elementary symmetric function} \( e_k(\vec{x}) \) in the variables \( \vec x = (x_1, x_2 ,\dots) \). In this way, the Toeplitz matrix whose entries are given by the coefficients of \( p(t) \) can be regarded as a special case of the following matrix:
\[
   M(\vec{x}) = \bigl(e_{j-i}(\vec{x})\bigr)_{i,j \ge 0}.
\]
The minors of this matrix are determinants of the \emph{(dual) Jacobi--Trudi matrices}\footnote{Our convention differs from the standard one by a conjugation of the skew shape \( \lambda/\mu \); see \Cref{rmk: nonstandard convention}.}
\[
   \tJT_{\lambda/\mu}(\vec{x}) := \bigl(e_{\lambda_i - \mu_j - i + j}(\vec{x})\bigr)_{i,j=1}^{\ell(\lambda)}.
\]
Here \(\lambda/\mu\) is a skew shape, and \( \ell(\lambda) \) denotes the number of parts of the partition \( \lambda \). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \( M(\vec{x}) \) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers.
In connection with Mal\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \( M(\vec{x}) \) and \( M(\vec{y}) \), when \( x_i, y_j \ge 0 \), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \( M(\vec{x}) \) and \( M(\vec{y}) \) for distinct sequences of variables \( \vec{x} = (x_1, x_2, \dots) \) and \( \vec{y} = (y_1, y_2, \dots) \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

In the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\vec{x})$ with an arbitrary minor of $M(\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \cite{Rhoades2005}.

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

By the Cauchy--Binet theorem, it is immediate that the product
\( MN \) of two totally positive matrices \( M \) and \( N \) is also
totally positive. However, their Hadamard (entrywise) product
\( M * N \) need not be totally positive in general.
Nevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product.
A \textit{Toeplitz matrix} with \textit{Toeplitz sequence $(a_0,a_1,\ldots)$} is an infinite, upper-triangular matrix
of the form $(a_{j-i})_{i,j=0}^\infty$.
The \emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of
univariate polynomials with nonnegative coefficients and real roots.
Mal\'{o} \cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.

For an entire function \( p(t) = \prod_{i\ge0}(1 + \alpha_it) \) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \( \alpha_i \ge 0 \) by variables \( x_i \), then the coefficient of $t^k$ in \( p(t) \) becomes the \emph{elementary symmetric function} \( e_k(\vec{x}) \) in the variables \( \vec x = (x_1, x_2 ,\dots) \). In this way, the Toeplitz matrix whose entries are given by the coefficients of \( p(t) \) can be regarded as a special case of the following matrix:
\[
 M(\vec{x}) = \bigl(e_{j-i}(\vec{x})\bigr)_{i,j \ge 0}.
\]
The minors of this matrix are determinants of the \emph{(dual) Jacobi--Trudi matrices}\footnote{Our convention differs from the standard one by a conjugation of the skew shape \( \lambda/\mu \); see \Cref{rmk: nonstandard convention}.}
\[
 \tJT_{\lambda/\mu}(\vec{x}) := \bigl(e_{\lambda_i - \mu_j - i + j}(\vec{x})\bigr)_{i,j=1}^{\ell(\lambda)}.
\]
Here \(\lambda/\mu\) is a skew shape, and \( \ell(\lambda) \) denotes the number of parts of the partition \( \lambda \). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \( M(\vec{x}) \) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers.
In connection with Mal\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \( M(\vec{x}) \) and \( M(\vec{y}) \), when \( x_i, y_j \ge 0 \), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \( M(\vec{x}) \) and \( M(\vec{y}) \) for distinct sequences of variables \( \vec{x} = (x_1, x_2, \dots) \) and \( \vec{y} = (y_1, y_2, \dots) \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
 The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
 monomial positive. In other words, every minor
\[
 \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

In the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\vec{x})$ with an arbitrary minor of $M(\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \cite{Rhoades2005}.

\begin{conj}\label{conj: stronger sokal's conjecture}
 Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
 skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
 multi-symmetric function
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
 is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

\begin{remark}\label{rmk: nonstandard convention}
Our definition of dual Jacobi--Trudi matrices
differs from the standard convention by a conjugation of $\lambda / \mu$.
When using the standard convention, the reader should take this change into account.
For example, results stated in terms of $(3 \times 2)$-avoiding skew shapes
would instead concern $(2 \times 3)$-avoiding skew shapes.
\end{remark}

\begin{conj}\label{conj: stronger sokal's conjecture}
 Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
 skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
 multi-symmetric function
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
 is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

We focus on this restricted class of skew shapes for three key reasons. First, \Cref{conj: Sokal's conjecture} and \Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \times 2$ block. For example, the expression $\det(\tJT_{(2,2,2)}(\vec{x}) * \tJT_{(2,2,2)}(\vec{y}))$ is not Schur positive.

In \Cref{sec:step1 m expansion}, we prove \Cref{thm: TL pos for 3x2 avoiding}, i.e., that the Temperley--Lieb immanant
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
is $s$-positive for any $\tau \in \K_n$, provided that each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape that does not contain a $3 \times 2$ block of cells. We say that such shapes are $(3 \times 2)$-\emph{avoiding}.

\begin{thm}\label{thm: TL multinetwork expansion}
 Let \( \tau\in \K_n \) and suppose
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ are
 $(3 \times 2)$-avoiding skew shapes, with
 \( \ell(\lambda^{(i)})\le n \) for all \( i \). If $\tau=\T{I}$ for
 some $I \subseteq [n-1]$, we have
 \[
 \imm_{\tau} \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) * \cdots * \tJT_{\lambda^{(k)} /
 \mu^{(k)}}(\vec{x}^{(k)}) \right) = \sum_{\vec{H}}
 2^{\epsilon(\vec{H})} \wt_{\underline{\vec x}}(\vec{H}),
 \]
 where the sum is over all $(\underline{\lambda}, \underline{\mu})$-multinetworks
 $\vec{H}$ with $\beta(\vec{H}) = 2^{\epsilon(\vec{H})} \B{I}$.
 If $\tau$ is not of this form, the immanant is $0$.

\begin{cor}\label{cor:all TL imms s positive}
 For any $\tau \in \K_n$ and any $(3 \times 2)$-avoiding skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ with
 \( \ell(\lambda^{(i)})\le n \) for all \( i \), the immanant
 \[ \imm_\tau \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) *\cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}}(\vec{x}^{(k)})
 \right)
\] is Schur positive.
\end{cor}

\begin{proof}
 By \Cref{thm: TL multinetwork expansion}, we may assume that \( \tau=\tau(I) \)
 for some \( I\subseteq [n-1] \).
 Recall that a
 \((\underline{\lambda}, \underline{\mu})\)-multinetwork \(\vec{H}\)
 is a tuple \((H^{(1)}, \ldots, H^{(k)})\) in which each \(H^{(i)}\)
 is a \((\lambda^{(i)} ,\mu^{(i)})\)-subnetwork. By \Cref{lem: beta
 for multinetworks}, the condition
 \(\beta(\vec{H}) = 2^{\epsilon(\vec{H})}\B{I}\) is equivalent to
 \(I_1 \cap \cdots \cap I_k = I\), where
 \(\beta(H^{(i)})=2^{\epsilon(H^{(i)})}\B{I_i}\) for each \( i \).
 Therefore, the equation in \Cref{thm: TL multinetwork expansion} can be
 rewritten as
 \begin{multline}
 \imm_{\tau} \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) * \cdots * \tJT_{\lambda^{(k)} /
 \mu^{(k)}}(\vec{x}^{(k)}) \right) \\
 = \sum_{\substack{I_1,\dots,I_k \subseteq [n-1] \\ I_1 \cap \cdots \cap I_k = I}} \,
 \prod_{i=1}^k
 \left( \sum_{\substack{H^{(i)} \\ \beta(H^{(i)}) = 2^{\epsilon(H^{(i)})}\B{I_i}}}
 2^{\epsilon(H^{(i)})} \wt_{\vec{x}^{(i)}}(H^{(i)}) \right) .
 \label{eq:network expansion verbose}
 \end{multline}
 By \Cref{thm: TL multinetwork expansion}, for each $i \in [n]$, we
 have
 \[
 \sum_{\substack{H^{(i)} \\ \beta(H^{(i)}) = 2^{\epsilon(H^{(i)})}\B{I_i}}}
 2^{\epsilon(H^{(i)})} \wt_{\vec{x}^{(i)}}(H^{(i)})
 = \imm_{\T{I_i}} \left( \tJT_{\lambda^{(i)} / \mu^{(i)}}(\vec{x}^{(i)}) \right).\]
By \cite[Proposition 3]{RHOADES2006793} and \cite[Proposition 5 and the paragraph above it]{RHOADES2006793}, each Temperley--Lieb immanant of a single Jacobi--Trudi matrix is Schur positive. Thus the expression in \eqref{eq:network expansion verbose} is Schur positive.
\end{proof}

The proof of \Cref{thm: TL multinetwork expansion} had two primary steps.
First, we expanded each Temperley--Lieb immanant as a sum over subnetworks, i.e., wrote
\[\imm_{\tau} \left( \tJT_{\lambda^{(1)} / \mu^{(1)}}(\vec{x}^{(1)}) *\cdots * \tJT_{\lambda^{(k)} / \mu^{(k)}}(\vec{x}^{(k)}) \right) = \sum_{\vec{H}} f_\tau(\beta(\vec{H})) \wt_{\underline{\vec x}}(\vec{H}).\]
Then, we argued that $f_\tau(\beta(\vec{H})) \geq 0$ whenever $\vec{H}$ is a multinetwork for a collection of $(3 \times 2)$-avoiding skew shapes.
The result below demonstrates that the proof method for \Cref{thm: TL multinetwork expansion}
will not work for collections of skew shapes that are not essentially $3 \times 2$ avoiding.
Note that if \( \tau=1 \), then \( f_\tau=f_1 \) is the sign character \( \sgn \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

\begin{thm}\label{thm: TL pos for 3x2 avoiding}
Suppose $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ is a collection of skew shapes each not containing a $3 \times 2$ block of cells. Then for any Temperley--Lieb immanant $\imm_{\tau}$, the multi-symmetric function
\[\imm_{\tau} \left(\tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
* \cdots *
\tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)}) \right)\]
is Schur positive.
\end{thm}

We focus on this restricted class of skew shapes for three key reasons. First, \Cref{conj: Sokal's conjecture} and \Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \times 2$ block. For example, the expression $\det(\tJT_{(2,2,2)}(\vec{x}) * \tJT_{(2,2,2)}(\vec{y}))$ is not Schur positive.

Second, the proof of \Cref{thm: TL pos for 3x2 avoiding} involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005}. Said techniques can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma.
However, we demonstrate that a similar sign-reversing involution argument cannot work to prove \Cref{conj: Sokal's conjecture} or \Cref{conj: stronger sokal's conjecture} when the collection of skew shapes is not \textit{essentially $(3 \times 2)$-avoiding}, a mild generalization of the condition in \Cref{thm: TL pos for 3x2 avoiding}. See \Cref{prop: counterexample} and the ensuing discussion.

Third, skew shapes avoiding a $3 \times 2$ block generalize the well-known \emph{ribbons}, which are skew shapes avoiding a $2 \times 2$ block. In the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a ribbon, we can write the Schur expansion from \Cref{thm: TL pos for 3x2 avoiding} explicitly in terms of standard Young tableaux with restricted descent sets.
See \Cref{thm: explicit ribbon expansion in full generality}.

In particular, when taking the Hadamard product of a ribbon Jacobi--Trudi matrix with
itself, we obtain the following the following explicit formula.

\begin{thm}\label{thm: Schur expansion of JT_R*JT_R}
Let \( R \) be a ribbon of size \( m \). Then we have a manifestly positive Schur expansion
\[
    \det\left( \tJT_R(\vec x) * \tJT_R(\vec y) \right) = \sum_{\lambda, \mu \vdash m} \sum_{\substack{I, J \subseteq [m-1] \\ I \cap J = \Des(R')}} f^{\lambda}(I) f^{\mu}(J) \, s_\lambda(\vec x) s_\mu(\vec y),
  \]
  where \( s_\lambda \) is the Schur function, \( f^{\lambda}(I) \)
  is the number of standard Young tableaux of shape \( \lambda \)
  with descent set \( I \), and \( \Des(R) \) is the descent set of
  \( R \).
\end{thm}
We explore the latter result further by constructing an
\( \mathfrak{S}_n \times \mathfrak{S}_n \)-module whose Frobenius
image equals the determinant in \Cref{thm: Schur expansion of
  JT_R*JT_R}, thereby establishing Schur positivity from the
representation-theoretic perspective. See \Cref{thm: representation theoretic model for Phi(s_R)}.

The rest of this paper is organized as follows. In Section~\ref{Sec:
  Preliminaries}, we review background material on multi-symmetric
functions, lattice paths, immanants, and the representation theory of products of symmetric groups.
Section~\ref{Sec: main proofs for 3x2 avoiders} is devoted to the proofs of Theorems~\ref{thm: TL pos for 3x2 avoiding}~and~\ref{thm: Schur expansion of JT_R*JT_R}.
Section \ref{Sec: rep theory construction} concerns the construction of the \( \mathfrak{S}_n \times \mathfrak{S}_n \)-module that realizes \Cref{thm: Schur expansion of JT_R*JT_R}.
Finally, Section~\ref{Sec: Concluding remarks} discusses some open problems and
directions for future research.

The post-theorem introduction indicates that the proof of \Cref{thm: TL pos for 3x2 avoiding} "involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005}," which "can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma." No further step-by-step outline for proving \Cref{thm: TL pos for 3x2 avoiding} is given beyond this description (and the note that Section~\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).

The post-theorem introduction indicates that the proof of the main theorem “involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005},” which “can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma.” No further step-by-step outline for proving the main theorem is given beyond this description (and the note that Section~\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).

\label{thm: TL pos for 3x2 avoiding}
Suppose $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ is a collection of skew shapes each not containing a $3 \times 2$ block of cells. Then for any Temperley--Lieb immanant $\imm_{\tau}$, the multi-symmetric function
\[\imm_{\tau} \left(\tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
* \cdots *
\tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)}) \right)\]
is Schur positive.,

Question:
Let \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\) be skew shapes, each of which does not contain a \(3\times 2\) block of cells. For each \(i\), let \(\vec{x}^{(i)}\) be a set of variables, let \(e_r(\vec{x}^{(i)})\) denote the elementary symmetric function of degree \(r\), and define the (dual) Jacobi--Trudi matrix by
\[
\operatorname{tJT}_{\lambda^{(i)}/\mu^{(i)}}(\vec{x}^{(i)})
:= \bigl(e_{\lambda^{(i)}_a-\mu^{(i)}_b-a+b}(\vec{x}^{(i)})\bigr)_{a,b=1}^{\ell(\lambda^{(i)})}.
\]
If \(*\) denotes the Hadamard (entrywise) product of matrices, and \(\operatorname{imm}_{\tau}\) is any Temperley--Lieb immanant (an immanant-type matrix functional generalizing the determinant), which of the following conclusions about
\[
\operatorname{imm}_{\tau}\!
\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is valid? Here, Schur positive means that when the resulting multi-symmetric function is expanded in the product Schur basis in the variable sets \(\vec{x}^{(1)},\ldots,\vec{x}^{(k)}\), all coefficients are nonnegative.

For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!
\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive.

Choices:
B.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive, provided that the collection of skew shapes does not contain a \(3\times 2\) block of cells in the union of all shapes.
C.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is monomial positive.
D.
There exists a Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), depending only on \(k\), such that for every collection of skew shapes \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\) each not containing a \(3\times 2\) block of cells, the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive.
E.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive for arbitrary skew shapes \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\), without any restriction excluding a \(3\times 2\) block of cells.