\begin{equation}\label{homotopy seq for unordered uni hyp curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{\H_{g,[n]/k}}, \bar x)\to \pi_1^\alg(\H_{g,[n]/k}, \bar y)\to 1,
\end{equation}

\begin{bigtheorem}\label{main result for uni curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni curve} does not split.
\end{bigtheorem}

For nonnegative integers $g$ and $n$ satisfying $2g - 2 + n > 0$ and a field $k$ of characteristic $0$, let $\M_{g,n/k}$ denote the moduli stack of proper, smooth curves of genus $g$ with $n$ ordered distinct marked points over $k$, and let
\[
\pi_{g,n/k} : \cC_{g,n/k}\to \M_{g,n/k}
\]
be the universal curve over $\M_{g,n/k}$. The symmetric group on $n$ letters, denoted $S_n$, acts on $\M_{g,n/k}$ by permuting the $n$ marked points. We denote the quotient stack $[\M_{g,n/k}/S_n]$ by $\M_{g,[n]/k}$. The $S_n$-action extends to $\cC_{g,n/k}$, and we denote the quotient stack $[\cC_{g,n/k}/S_n]$ by $\cC_{g,[n]/k}$. Thus the universal curve $\pi_{g,n/k}$ induces a family of curves of genus $g$ with $n$ unordered marked points over $\M_{g,[n]/k}$:
\[
\pi_{g,[n]/k}: \cC_{g,[n]/k}\to \M_{g,[n]/k},
\]
which we call the universal curve of genus $g$ with $n$ unordered marked points.

Denote by $\pi_{g,n}:{\mathcal C}^\an_{g,n}\to {\mathcal M}^\an_{g,n}$ and $\pi_{g,[n]}:{\mathcal C}^\an_{g,[n]}\to {\mathcal M}^\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\pi_{g,n/{\mathbb C}}$ and $\pi_{g,[n]/{\mathbb C}}$, respectively. In \cite[Thm.~1.3]{chen_uni}, Chen showed that for $n \geq 2$ and $g \geq 2$, the curve $\pi_{g,[n]}$ admits no sections. In this paper, we extend Chen's result to the algebraic fundamental groups of $\pi_{g,[n]/k}$.

Let $\bar y$ be a geometric point of $\M_{g,[n]/k}$ and $C$ the fiber of $\pi_{g,[n]/k}$ over $\bar y$. Fix a geometric point $\bar x \in C$, viewed also as a geometric point of $\cC_{g,[n]/k}$. Associated to the universal curve $\pi_{g,[n]/k}$, there is a homotopy exact sequence of algebraic fundamental groups:
\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\mathrm{alg}(C, \bar x)\to \pi_1^\mathrm{alg}(\cC_{g,[n]/k}, \bar x)\to \pi_1^\mathrm{alg}(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}

\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{g,[n]/k}, \bar x)\to \pi_1^\alg(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \(\pi_1^\orb(\H^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.
\section{Configuration Spaces of Points on a Curve}
Let $C$ be a smooth complex algebraic curve of genus $g$. For a positive integer $n$, the ordered configuration space of $n$ distinct points on $C$ is
$$
F_n(C) := \{(x_1,\dots,x_n) \in C^n \mid x_i \neq x_j \text{ for } i \neq j \}.
$$
The symmetric group $S_n$ acts freely on $F_n(C)$ by permuting the coordinates, and we define the unordered configuration space as the quotient
$$
F_{[n]}(C) := F_n(C)/S_n.
$$

\begin{proof}
Relative completion applied to the exact sequence
\[
1 \to \pi_1^\top(C)\to\pi_1^\orb(\cC^\an_{g,n})\to \pi_1^\orb(\M^\an_{g,n})\to 1
\]
yields the exact sequence of MHS
\[
0\to\p_g\to \u_{\cC_{g,n}}\to \u_{g,n}\to 0,
\]
where $\p_g =\p_{g,1}$. This gives the exact sequence
\[
 H_1(\p_g)
 \longrightarrow H_1(\u_{\cC_{g,n}})
 \longrightarrow H_1(\u_{g,n})
 \longrightarrow 0,
\]
which implies that $H_1(\u_{\cC_{g,n}})$ is pure of weight -1.
Since \(\p_g\) is center-free, the adjoint representation
\(\adj\colon \p_g \to \Der \p_g\) is injective.
Because \(\adj\) factors through \(\u_{\cC_{g,n}}\) and the functor
\(\Gr^W_{-1}\) is exact, the composition
\[
\Gr^W_{-1}\p_g
 \longrightarrow \Gr^W_{-1}\u_{\cC_{g,n}}
 \longrightarrow \Gr^W_{-1}\Der \p_g
\]
is also injective.
Using the identifications
\(H_1(\p_g) \cong \Gr^W_{-1}\p_g\) and
\(H_1(\u_{\cC_{g,n}}) \cong \Gr^W_{-1}H_1(\u_{\cC_{g,n}})\cong\Gr^W_{-1}\u_{\cC_{g,n}}\),
we obtain the commutative diagram
\[
\xymatrix{
\Gr^W_{-1}\p_g \ar[r] \ar[d]^{\cong} &
\Gr^W_{-1}\u_{\cC_{g,n}} \ar[d]^{\cong} \\
H_1(\p_g) \ar[r] &
H_1(\u_{\cC_{g,n}}).
}
\]
Therefore, the sequence
\[
0 \longrightarrow H_1(\p_g)
 \longrightarrow H_1(\u_{\cC_{g,n}})
 \longrightarrow H_1(\u_{g,n})
 \longrightarrow 0
\]
is exact. Consequently, we obtain an \(\Sp(H)\)-decomposition
\[
H_1(\u_{\cC_{g,n}})
 \cong H_1(\p_g) \oplus H_1(\u_{g,n})
 \cong H_0 \oplus \Lambda^3_nH\cong\bigoplus_{j=0}^nH_j \oplus \Lambda^3_0H ,
\]
where \(H_0 := H_1(\p_g)\cong H\).
\end{proof}
\begin{remark}
Since $H_1(\u_{\cC_{g,n}})$ is pure of weight $-1$, the weight filtration on $\u_{\cC_{g,n}}$ coincides with its lower central series.
\end{remark}
We now turn to the hyperelliptic case.
A key fact is that \(H_1(\v_g)\) is pure of weight~\(-2\)
(see~\cite[Prop.~6.9]{wat_rk_hyp_univ}),
which leads to the following statement.
\begin{proposition}
The weight filtration on \(\v_g\) satisfies
\[
\v_g = W_{-2}\v_g,
\qquad
L^l\v_g = W_{-2l}\v_g,
\]
where \(L^l\v_g\) denotes the \(l\)-th term of the lower central series.
In particular,
\[
H_1(\v_g) \cong \Gr^W_{-2}\v_g.
\]
\end{proposition}
The following lemma describes the resulting canonical \(\Sp(H)\)-decompositions
of the first homology groups of the corresponding Lie algebras
\(\v_{g,n}\) and \(\v_{\cC_{g,n}}\).
\begin{lemma}\label{H_1 decomp for rel comp of hyp mcg}
For $g\geq 2$, there are canonical $\Sp(H)$-decompositions:
$$
H_1(\v_{g,n})\cong \Gr^W_{-1}H_1(\v_{g,n})\oplus \Gr^W_{-2}H_1(\v_{g,n})\cong \bigoplus_{j=1}^nH_j\oplus H_1(\v_g)
$$
and
$$
H_1(\v_{\cC_{g,n}})\cong \Gr^W_{-1}H_1(\v_{\cC_{g,n}})\oplus \Gr^W_{-2}H_1(\v_{\cC_{g,n}})\cong \bigoplus_{j=0}^{n}H_j\oplus H_1(\v_g).
$$
\end{lemma}
\begin{proof}
The exact sequence
$$
1\to \pi_1^\top(F_n(C))\to \pi_1^\orb(\H^\an_{g,n})\to \pi_1^\orb(\H^\an_{g})\to 1
$$
induces the exact sequence of pronilpotent Lie algebras
$$
0\to \p_{g,n}\to \v_{g,n}\to \v_g\to 0,
$$
which in turn yields the exact sequence
$$
H_1(\p_{g,n})\to H_1(\v_{g,n})\to H_1(\v_g)\to 0.
$$
The adjoint action of $\v_{g,n}$ on $\p_{g,n}$ yields the Lie algebra map
$$
\v_{g,n}\to \Der\p_{g,n}.
$$
The center-freeness of $\p_{g,n}$ (see \cite{NTU}) implies that the composition
$$
\p_{g,n}\to \v_{g,n}\to \Der\p_{g,n}
$$
is an injection.
Since the functor $\Gr^W_\bullet$ is exact, the composition
$$
H_1(\p_{g,n})\cong\Gr^W_{-1}\p_{g,n}\to \Gr^W_{-1}\v_{g,n}\to \Gr^W_{-1}\Der\p_{g,n}
$$
is also injective. Since $\Gr^W_{-1}\v_{g,n}\cong \Gr^W_{-1}H_1(\v_{g,n})$ and the diagram
$$
\xymatrix{
H_1(\p_{g,n})\ar[r]\ar[dr]&H_1(\v_{g,n})\ar[d]\\
&\Gr^W_{-1}H_1(\v_{g,n})
}
$$
commutes, the sequence
\begin{equation}
0\to H_1(\p_{g,n})\to H_1(\v_{g,n})\to H_1(\v_{g})\to 0
\end{equation} is exact.
Applying the functor $\Gr^W_\bullet$, we obtain the exact sequence
\begin{equation}
0\to \Gr^W_\bullet H_1(\p_{g,n})\to \Gr^W_\bullet H_1(\v_{g,n})\to \Gr^W_\bullet H_1(\v_{g})\to 0.
\end{equation}
By \textcolor{black}{\cite[Prop.~6.9]{wat_rk_hyp_univ}},
$H_1(\v_g)$ is pure of weight $-2$, and so there is a canonical isomorphism
$$
\Gr^W_{-1}H_1(\v_{g,n})\cong \Gr^W_{-1}H_1(\p_{g,n})=H_1(\p_{g,n}).
$$
Composing with the projection $H_1(\v_{g,n})\to \Gr^W_{-1}H_1(\v_{g,n})$, we get a canonical projection $H_1(\v_{g,n})\to H_1(\p_{g,n})$. This yields a canonical splitting
$$
H_1(\v_{g,n}) \cong \Gr^W_{-1}H_1(\v_{g,n})\oplus \Gr^W_{-2}H_1(\v_{g,n}) \cong H_1(\p_{g,n})\oplus H_1(\v_g)\cong \bigoplus_{j=1}^nH_j\oplus H_1(\v_g).
$$

\begin{lemma}\label{H component in H_1}
For $g\geq 2$, there are isomorphisms
$$
\Hom_{\Sp}(H_1(\u_{g,[n]}), H)\cong \Q\quad \text{ and }\quad \Hom_{\Sp}(H_1(\v_{g,[n]}), H)\cong \Q.
$$
\end{lemma}
\begin{proof}
By Theorem \ref{str} and \textcolor{black}{Example} \ref{rel comp in unordered case}, we have
$$
 H^1(\pi_1^\orb(\M^\an_{g, [n]}),H)\cong \Hom_{\Sp}(H_1(\u_{g,[n]}), H).
$$
From the exact sequence
\[
\begin{tikzcd}
1 \ar[r]
 & \pi_1^\orb(\M^\an_{g,n}) \ar[d,"\cong"'] \ar[r]
 & \pi_1^\orb(\M^\an_{g,[n]}) \ar[d,"\cong"'] \ar[r]
 & S_n \ar[r] & 1 \\
 & \G_{g,n}
 & \G_{g,[n]} & &
\end{tikzcd}
\]
the \emph{Hochschild--Serre five-term exact sequence} yields
\[
1 \longrightarrow
H^1(S_n, H^{\Gamma_{g,n}})
\longrightarrow
H^1(\Gamma_{g,[n]}, H)
\longrightarrow
H^1(\Gamma_{g,n}, H)^{S_n}
\longrightarrow
H^2(S_n, H^{\Gamma_{g,n}}),
\]
where \(H\) is regarded as a \(\Gamma_{g,[n]}\)-module.
Since both $H^1(S_n,H^{\Gamma_{g,n}})$ and $H^2(S_n,H^{\Gamma_{g,n}})$ vanish,
$$H^1(\Gamma_{g,[n]},H)\cong H^1(\Gamma_{g,n},H)^{S_n}.$$
Since
\[
H^1(\pi_1^\orb(\M^\an_{g,n}), H)
 \cong H^1(\M_{g,n}^\an, \mathbb{H}_\Q)
 \cong \bigoplus_{j=1}^n \Q\,\kappa_j,
\]
the \(S_n\)-invariant part is isomorphic to \(\Q\).
Consequently,
\[
\Hom_{\Sp}\!\left(H_1(\u_{g,[n]}), H\right) \cong \Q.
\]
The same argument also works for $\Hom_{\Sp}(H_1(\v_{g,[n]}), H)\cong \Q$, using the natural isomorphism
$$
H^1(\H^\an_{g,n}, \mathbb{H}_\Q)
 \cong \bigoplus_{j=1}^n \Q\,\kappa_j^{\hyp}.
$$
\end{proof}
The natural $S_n$–action on $H_1(\u_{g,n})$ allows one to determine explicitly the decompositions of
$H_1(\u_{g,[n]})$ and $H_1(\u_{\cC_{g,[n]}})$.
\begin{lemma}\label{ab of u_{g,[n]}}
For \(g \ge 3\), there is an \(\Sp(H)\)-decomposition
\[
H_1(\u_{g,[n]})
 \cong H_1(\u_g) \oplus \!\bigg(\bigoplus_{j=1}^{n} H_j\!\bigg)_{S_n}
 = \Lambda^3_0 H \oplus H,
\]
which is compatible with the projection
\(H_1(\u_{g,n}) \to H_1(\u_{g,[n]})\)
and with the \(\Sp(H)\)-decomposition
\(H_1(\u_{g,n}) = \Lambda^3_0 H \oplus \bigoplus_{j=1}^{n} H_j\).
Furthermore, there is an \(\Sp(H)\)-decomposition
\[
H_1(\u_{\cC_{g,[n]}}) \cong H_0 \oplus \Lambda^3_0 H \oplus H.
\]

In \cite[Thm.~2.6]{chen_uni}, Chen also proved an analogous result for the hyperelliptic case. Let $\H_{g,n/k}$ be the closed substack of $\M_{g,n/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ ordered marked points. Denote the quotient stack $[\H_{g,n/k}/S_n]$ by $\H_{g,[n]/k}$; this is a closed substack of $\M_{g,[n]/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ unordered marked points. Restricting $\pi_{g,n/k}$ to $\H_{g,n/k}$, we obtain the universal hyperelliptic curve
\[
\pi^\hyp_{g,n/k}: \cC_{\H_{g,n/k}}\to \H_{g,n/k}.
\]
Extending the $S_n$-action to $\cC_{\H_{g,n/k}}$ yields a family over $\H_{g,[n]/k}$:
\[
\pi^\hyp_{g,[n]/k}:\cC_{\H_{g,[n]/k}}\to \H_{g,[n]/k},
\]
which we call the universal hyperelliptic curve of genus $g$ with $n$ unordered marked points.

Denote by $\pi^{\mathrm{hyp}}_{g,n}:{\mathcal C}^\an_{\H_{g,n}}\to {\mathcal H}^\an_{g,n}$ and $\pi^{\mathrm{hyp}}_{g,[n]}:{\mathcal C}^\an_{\H_{g,[n]}}\to {\mathcal H}^\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\pi^\hyp_{g,n/{\mathbb C}}$ and $\pi^\hyp_{g,[n]/{\mathbb C}}$, respectively. In \cite[Thm.~2.6]{chen_uni}, Chen showed that for $n \geq 2$ and $g \geq 2$, the hyperelliptic universal curve $\pi^{\mathrm{hyp}}_{g,[n]}$ admits no sections.

We extend Chen's result to the algebraic fundamental groups of $\pi^\hyp_{g,[n]/k}$. Associated to this universal hyperelliptic curve, there is a homotopy exact sequence of algebraic fundamental groups:
\begin{equation}\label{homotopy seq for unordered uni hyp curve}
1\to \pi_1^\mathrm{alg}(C, \bar x)\to \pi_1^\mathrm{alg}(\cC_{\H_{g,[n]/k}}, \bar x)\to \pi_1^\mathrm{alg}(\H_{g,[n]/k}, \bar y)\to 1,
\end{equation}
where $\bar y$ is a geometric point of $\H_{g,[n]/k}$, $C$ is the fiber of $\pi^\hyp_{g,[n]/k}$, and $\bar x$ is a geometric point of $C$.

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

A key tool used in this paper is the relative completion of a discrete group, or, in the case of a profinite group, its continuous relative completion. A detailed account of their constructions and basic properties can be found in \textcolor{black}{\cite{hain_infpretor, hain_hodge_rel, Hain15}}. In particular, for the algebraic fundamental groups of the stacks considered here, their continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \({\mathbb Q}\) to \(\Q_\ell\).

For example, the relative completion of the orbifold fundamental group
\(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is a proalgebraic group over \({\mathbb Q}\), defined as an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group. Its Lie algebra carries a natural mixed Hodge structure (MHS), and hence admits a weight filtration defined over \({\mathbb Q}\). Each graded piece of the associated graded Lie algebra of the completion is an \(\Sp(H)\)-representation. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between the pronilpotent Lie algebras of the corresponding completions.

The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.

A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \(\Q\) to \(\Q_\ell\). For example, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \(\Q\), whose graded pieces are \(\Sp(H)\)-representations. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}; in the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.

A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \(\Q\) to \(\Q_\ell\). For example, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \(\Q\), whose graded pieces are \(\Sp(H)\)-representations. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in establishing the main theorem; in the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of the following theorem.

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

\label{main result for uni curve}
If $g\geq 3$, then the homotopy exact sequence
\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{g,[n]/k}, \bar x)\to \pi_1^\alg(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}
does not split.

Question:
Let $g,n$ be nonnegative integers with $2g-2+n>0$, and let $k$ be a field of characteristic $0$. Let $\M_{g,n/k}$ be the moduli stack of proper smooth genus-$g$ curves with $n$ ordered distinct marked points, let $S_n$ act by permuting the marked points, and write $\M_{g,[n]/k}=[\M_{g,n/k}/S_n]$ for the moduli stack of genus-$g$ curves with $n$ unordered marked points. Let
\[
\pi_{g,[n]/k}: \mathcal C_{g,[n]/k}\to \mathcal M_{g,[n]/k}
\]
be the induced universal curve. For a geometric point $\bar y\in \mathcal M_{g,[n]/k}$, let $C$ be the fiber of $\pi_{g,[n]/k}$ over $\bar y$, and fix a geometric point $\bar x\in C$, viewed also as a geometric point of $\mathcal C_{g,[n]/k}$. Then one has the homotopy exact sequence of algebraic fundamental groups
\[
1\to \pi_1^{\mathrm{alg}}(C,\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to 1.
\]
If $g\ge 3$, which statement about this exact sequence holds?

The exact sequence does not split; equivalently, there is no group homomorphism $\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)$ whose composition with $\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)$ is the identity.

Choices:
B.
The exact sequence splits after passing to the maximal pro-$\ell$ quotient for every prime $\ell$; equivalently, for each $\ell$ there is a continuous homomorphism
\[\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)^{(\ell)}\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)^{(\ell)}\]
whose composition with the induced projection to \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)^{(\ell)}\) is the identity.
C.
The exact sequence is not canonically split; in particular, there is no distinguished or natural group homomorphism
\[\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\]
whose composition with \(\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\) is the identity.
D. The exact sequence splits whenever one works over a field of characteristic \(0\); equivalently, there exists a group homomorphism \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\) after extending scalars to some algebraic closure of \(k\), and hence the sequence is split exact.
E. If \(g\ge 2\), then the exact sequence does not split; equivalently, for every such \(g\) there is no group homomorphism \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\) whose composition with \(\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\) is the identity.

non-splitting survives relative completion and all pro-\ell realizations

absolute nonexistence of any splitting replaced by nonexistence of a canonical/natural splitting

descent/base-change from characteristic-0 field to algebraic closure preserves a hypothetical splitting

genus threshold g\ge 3 weakened to g\ge 2

The stem does not state or paraphrase the non-splitting conclusion; it only sets up the universal curve and the exact sequence. No explicit answer leakage is present.

The item is essentially a direct recall of a theorem: for g≥3, one asks which statement about the exact sequence holds, and the correct choice restates the theorem almost verbatim.

There is some reasoning pressure because the distractors vary by strength, genus range, and passage to pro-ℓ quotients, but the core task is still mainly theorem recognition rather than genuine derivation or synthesis.

The distractors are mathematically plausible and distinct: they test common confusions about weaker statements, boundary cases (g≥2 vs g≥3), and behavior under pro-ℓ or base change. They are well aligned with realistic failure modes.

A mathematically sharp MCQ with strong distractors and no answer leakage, but it mostly tests recall of a stated theorem rather than generative reasoning.


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 $\mathcal N$

The stem does not explicitly reveal the correct option, and it does not contain a direct verbal cue that uniquely points to choice A over the nearby alternatives. A solver still has to distinguish the exact theorem-level conclusion from weaker or altered variants.

This is essentially a theorem-recall item: the stem gives the precise hypotheses of a known existence result and asks for the corresponding conclusion. The correct answer is basically the theorem statement restated.

There is some pressure to compare closely related conclusions: a weaker true statement (existence of a sign-changing solution), a modified dimension threshold, and a stronger-looking but false variational claim. However, the item mainly tests recognition/recall rather than genuine derivation or multi-step reasoning.

The distractors are plausible and mathematically targeted. They reflect common failure modes: confusing sufficient conditions with sharper geometric hypotheses, choosing a weaker true conclusion, misremembering the dimension threshold, or misstating the variational characterization.

A solid theorem-recall MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than generative mathematical thinking.


\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

The stem does not explicitly reveal the correct option, and no phrase directly states the exact regularity conclusion. It presents hypotheses and asks for the resulting conclusion.

This is essentially a theorem-recall item: the hypotheses are listed in full and the correct choice is the theorem's conclusion almost verbatim. It tests recognition of the stated result more than nontrivial inference.

There is some reasoning required to distinguish the exact conclusion from stronger or weaker nearby statements, especially among options A, C, D, and E. However, the item mostly rewards recall of the precise theorem rather than substantial generative reasoning.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, others are natural overstrengthenings involving higher differentiability, borderline Hölder exponents, or Lipschitz regularity. These align with common failure modes in regularity questions.

A mathematically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, but it is largely tautological and only moderately tests genuine reasoning.