A classical conjecture of Geroch asserts that the $n$-torus $\mathbb{T}^n$ admits no Riemannian metric of positive scalar curvature. The conjecture (together with its rigidity case) was settled by Schoen--Yau for $3\le n\le 7$ via minimal hypersurfaces \cite{SY79}, and in all dimensions by Gromov--Lawson using the Dirac operator \cite{GL}. At the opposite end, Gromov asked whether a Riemannian metric $g$ on $S^n$ that strictly dominates the round metric $g_{S^n}$ must have scalar curvature strictly less than $n(n-1)$ somewhere. Llarull answered this by proving that if a closed spin $n$-manifold $(M^n,g)$ satisfies $R_M\ge n(n-1)$ and admits an area non-increasing map of nonzero degree to the round sphere, then rigidity holds in the strongest sense: the map is an isometry and hence $g$ is the pullback of the round metric \cite{Llarull}. Llarull's argument proceeds via spectral estimates for a twisted Dirac operator and index theoretic input tied to the nonzero degree. Subsequent work extended Llarull's theorem by lowering the regularity of the map \cite{Bar24, CHS24, LT22}, allowing targets beyond the sphere \cite{GS02, HSS24}, and treating manifolds with boundary under mean curvature assumptions \cite{BBHW24, CZ24, HLS, Lott21}.\\

These two results-—nonexistence and rigidity on tori, and Llarull-type rigidity on spheres--mark the extremes for scalar curvature on basic topological models. Between them lies the mixed geometry of products $S^{n-m}\times\mathbb{T}^m$. The product metric $g_{S^{n-m}}+ g_{\mathbb{T}^m}$ has scalar curvature exactly $(n-m)(n-m-1)$, the Llarull threshold for the spherical factor, yet it carries $m$ macroscopic flat directions encoding torus topology. Studying rigidity at this borderline illuminates how positive scalar curvature interacts with large-scale topology: it asks how much of the sphere’s rigidity persists once one permits $m$ flat directions, and conversely how enlargeability phenomena behind the torus obstruction constrain geometry when a positively curved factor is present. In this sense, $S^{n-m}\times\mathbb{T}^m$ is a natural ``midpoint" between the sphere and the torus, and scalar-curvature rigidity for degree–nonzero maps to this product probes the precise balance between curvature and topology.\\

In this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\times\mathbb{T}^m$ for $3\leq n\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:

\begin{abstract}
We prove Llarull-type rigidity for $S^{n-m}\times\T^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\T^m$ whose spherical projection is area non-increasing, and there exists $\psi\in C^\infty(M)$ with $-\Delta_M\psi-\tfrac12|D_M\psi|^2+\tfrac12\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\textup{dist}(\partial_-M,\partial_+M)\le 2\pi\sqrt{n/((n+1)\sigma)}$ when $R_M\ge (n-m)(n-m-1)+\sigma$. The method combines stable weighted slicing with a spectral Dirac operator argument.
\end{abstract}

In this paper we establish Llarull-type rigidity for degree-nonzero maps to $S^{n-m}\times\T^m$ for $3\leq n\leq 7$, together with quantitative band-width inequalities in the incomplete setting. Our main theorem asserts:

Our approach combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$
with positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\ge 6$ was proved by different methods in \cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \cite{Tony}, which shows that when $\pr_{S^{n-m}}\circ\Phi:M^n\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\

\begin{maintheorem}\label{main;thm;2}
 Let $3\leq n+1\leq 7$ and $1\leq m\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\partial M = \partial_-M\sqcup\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\sigma > 0$ be a positive real number. Let $\Phi:(M,g)\to (S^{n-m}\times \T^m \times [-1,1], g_{S^{n-m}} + g_{\T^m} + dt^2)$ be a smooth map with the following properties:
 \begin{itemize}
 \item $\Phi$ has non-zero degree.
 \item $\Phi(\partial_{\pm}M)\subset S^{n-m}\times \T^m \times\{\pm1\}.$
 \item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
 \end{itemize}
The followings hold for the bandwidth of $M$:
\begin{enumerate}[(i)]
 \item Suppose that $R_M \geq (n-m)(n-m-1) + \sigma$. Then
 \[ \textup{dist}_g(\partial_-M, \partial_+M) \leq 2\pi\sqrt{\frac{n}{(n+1)\sigma}}.\]
 \item Suppose that $\psi$ is a smooth function on $M$ such that
 \[ -\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1) - \sigma \Big) \geq 0.\]
 Then
 \[ d(\partial_-M, \partial_+M) \leq \frac{2\pi}{\sqrt{\sigma}}.\]
\end{enumerate}
\end{maintheorem}

\begin{proposition}\label{slicing;exist;prop}
 Let $2\leq n\leq 7$ and $1\leq m\leq n-2$. Let $(M^n, g)$ be a closed, connected Riemannian manifold of dimension $n$ with a smooth map $\Phi:(M^n,g)\to (S^{n-m}\times \T^m, g_{S^{n-m}} + g_{\mathbb{T}^{m}})$ of non-zero degree such that $\pr_{S^{n-m}}\circ\Phi :(M^n,g)\to (S^{n-m}, g_{S^{n-m}})$ is 1-Lipschitz. Then there exists a stable weighted slicing
 \[ \Sigma_m\subset\Sigma_{m-1}\subset\cdots\subset\Sigma_1\subset\Sigma_0 = M^n\]
 of order $m$.

\begin{theorem}\label{spectral;Llarull;thm}
 Let $N^n$ be a closed, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $N$. Suppose that there exists a smooth function $\rho >0$ on $N$ such that
\begin{align}\label{spectral;Llarull;assumption}
 0 \leq& \int_{N}\rho|D_{N}f|^2 - \frac{1}{2}\int_{N}(n(n-1) - R_{N})\rho\, f^2 - \int_{N} \left(\Delta_{N}\log\rho + \frac{1}{2} |D_{N}\log\rho|^2
 \right) \rho\, f^2
\end{align}
for all $f\in C^{\infty}(N)$.
Suppose that $\Phi:(N,g)\to (S^n, g_{S^n})$ is a smooth map with the following properties:
 \begin{itemize}
 \item $\Phi$ has non-zero degree,
 \item $\Phi$ is 1-Lipschitz.
 \end{itemize}
 Then $\rho$ is a constant on $N$ and $\Phi$ is a Riemannian isometry.
\end{theorem}

\subsection{Proof of Theorem \ref{main;thm;2}}
In the first step, we construct a $\mu$-bubble that allows us to apply Theorem \ref{main;thm;1}. Denote the projection of $\Phi$ onto the factors by $\phi:M\to S^{n-m}\times \T^m$ and $\varphi:M\to [-1,1]$. By assumption, the map $\pr_{S^{n-m}}\circ\phi = \pr_{S^{n-m}}\circ\Phi$ is 1-Lipschitz. Let $\Theta$ be a top-form of $S^{n-m}\times\T^m$ such that $\int_{S^{n-m}\times\T^m}\Theta = 1$. Define the pull-back form $\omega = \phi^*\Theta$. By Sard's theorem, we can find a regular value $t_0\in (-1,1)$ of the map $\varphi$. Let $\hat{\Sigma} = \varphi^{-1}(t_0)$. Because $M$ is connected and $\varphi$ is continuous, and by assumption $\Phi(\partial_{-}M)\subset S^{n-m}\times \T^m \times\{-1\}$ and $\Phi(\partial_{+}M)\subset S^{n-m}\times \T^m \times\{1\}$, we see that $\varphi(M)$ is a connected subset of $[-1,1]$ containing both $-1$ and $1$. It follows that $\varphi(M) = [-1,1]$ and $ \varphi^{-1}(t)$ is non-empty for all regular values $t$. So $\hat{\Sigma}$ is a smooth, orientable and embedded hypersurface in $M$. By the coarea formula,
\begin{align}\label{bandwidth;eqn1}
 \deg(\Phi) = \int_M d\varphi\wedge\omega = \int_{-1}^1\left(\int_{\varphi^{-1}(t)}\omega\right) dt
\end{align}
On the other hand, if $t_1 < t_2$ are two regular values, the Stokes theorem gives
\begin{align}\label{bandwidth;eqn2}
 0 = \int_{\varphi^{-1}([t_1, t_2])} d\omega = \int_{\varphi^{-1}(t_2)}\omega - \int_{\varphi^{-1}(t_1)}\omega.
\end{align}
Putting (\ref{bandwidth;eqn1}) and (\ref{bandwidth;eqn2}) together, we obtain
\begin{align}\label{bandwidth;eqn3}
 \int_{\hat{\Sigma}}\omega = \deg(\Phi).
\end{align}

In conclusion, we have obtained a smooth map $\Psi:\Sigma^n\to S^{n-m}\times\T^m$ of non-zero degree such that $\pr_{S^{n-m}}\circ\Psi$ is 1-Lipschitz, and a smooth function $\log\rho$ on $\Sigma$ satisfying
\begin{align}\label{bandwidth;eqn6}
 -\Delta_{\Sigma}\log\rho -\frac{1}{2}|D_{\Sigma}\log\rho|^2 + \frac{1}{2}(R_{\Sigma} - (n-m)(n-m-1) ) > 0.
\end{align}
 Theorem \ref{main;thm;1} then implies $R_{\Sigma} = (n-m)(n-m-1)$ and $\rho$ is a constant function, contradicting to (\ref{bandwidth;eqn6}). This finishes the proof of Theorem \ref{main;thm;2}.

Our approach combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$
with positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\ge 6$ was proved by different methods in \cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \cite{Tony}, which shows that when $\pr_{S^{n-m}}\circ\Phi:M^n\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\

In the incomplete setting, we extend Gromov's torical band inequality \cite{Gro18,Gro19} to bands over sphere--torus products. Bands are among the most flexible test-objects for scalar curvature: they detect how lower bounds on scalar curvature constrain the macroscopic separation of boundary components, yielding distance-type obstructions to fill-ins, doubling, and collaring. Recent progress has deepened the connection between scalar curvature and band width; see for instance \cite{CZ24, HKKZ23, HKKZ25}. For targets of the form $S^{n-m}\times \mathbb{T}^m$, the interplay between a positively curved spherical factor and $m$ flat directions raises a natural quantitative question: how far apart can two boundary components be kept under a scalar curvature lower bound that matches the spherical threshold up to a gap $\sigma>0$? Our results below give sharp-in-scale ($\sim\sigma^{-1/2}$) upper bounds, reflecting the model behavior of constant-curvature metrics and extending the torical width control to the mixed sphere-torus regime. In particular, requiring the spherical projection to be area non-increasing  prevents macroscopic stretching in the directions where scalar curvature is most constrained, and the band-width estimates quantify this restriction. To that end, we prove:\\

\begin{maintheorem}\label{main;thm;2}
	Let $3\leq n+1\leq 7$ and $1\leq m\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\partial M = \partial_-M\sqcup\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\sigma > 0$ be a positive real number. Let  $\Phi:(M,g)\to (S^{n-m}\times \mathbb{T}^m \times [-1,1], g_{S^{n-m}} + g_{\mathbb{T}^m} + dt^2)$ be a smooth map with the following properties:
	\begin{itemize}
		\item $\Phi$ has non-zero degree.
		\item $\Phi(\partial_{\pm}M)\subset S^{n-m}\times \mathbb{T}^m \times\{\pm1\}.$
		\item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
	\end{itemize}
The followings hold for the bandwidth of $M$:
\begin{enumerate}[(i)]
    \item Suppose that $R_M \geq  (n-m)(n-m-1) + \sigma$. Then
    \[	\textup{dist}_g(\partial_-M, \partial_+M) \leq  2\pi\sqrt{\frac{n}{(n+1)\sigma}}.\]
    \item Suppose that $\psi$ is a smooth function on $M$ such that
	\[	-\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1) - \sigma \Big) \geq 0.\]
    Then
    \[	d(\partial_-M, \partial_+M) \leq  \frac{2\pi}{\sqrt{\sigma}}.\]
\end{enumerate}
\end{maintheorem}

\bigskip

\begin{remark}
For the sake of exposition we will henceforth work under the stronger hypothesis that
    \[\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})\]
is 1-Lipschitz. This assumption streamlines the notations used in the proof. All arguments, however, carry over verbatim when $n-m\geq 3$ if one merely assumes that the map is area non-increasing.
\end{remark}

To prove Theorem~\ref{main;thm;1}, the approach “combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\"ock-Lichnerowicz identity for spinors.” “Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”

To prove the main theorem, the approach “combines stable weighted slicing \cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\Sigma_m \subset \cdots \subset \Sigma_1 \subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\"ock-Lichnerowicz identity for spinors.” “Equality forces $\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”

\label{main;thm;1}
	Let $3\leq n\leq 7$ and $1\leq m\leq n-2$. Let $M^n$ be a closed, orientable, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $M$. Let $\psi$ be a smooth function on $M$ such that
	\[	-\Delta_M\psi -\frac{1}{2}|D_M\psi|^2 + \frac{1}{2}\Big(R_M - (n-m)(n-m-1)\Big) \geq 0.\]
Suppose that $\Phi:(M,g)\to (S^{n-m}\times \mathbb{T}^m, g_{S^{n-m}} + g_{\mathbb{T}^m})$ is a smooth map with the following properties:
	\begin{itemize}
		\item $\Phi$ has non-zero degree,
		\item $\pr_{S^{n-m}}\circ\Phi:(M,g)\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\geq 3$ and 1-Lipschitz when $n-m=2$.
	\end{itemize}
	Then $(M^n, g)$ is isometrically covered by $(S^{n-m}\times\mathbb{R}^m,\, g_{S^{n-m}} + g_{\mathbb{R}^m})$.,

Question:
Let $3\le n\le 7$ and $1\le m\le n-2$. Let $M^n$ be a closed, orientable, connected spin $n$-manifold with Riemannian metric $g$, and let $\psi\in C^\infty(M)$ satisfy
\[
-\Delta_M\psi-\frac12|D_M\psi|^2+\frac12\bigl(R_M-(n-m)(n-m-1)\bigr)\ge 0.
\]
Assume there is a smooth map
\[
\Phi:(M,g)\to \bigl(S^{n-m}\times \mathbb T^m,\, g_{S^{n-m}}+g_{\mathbb T^m}\bigr)
\]
of nonzero degree, where $S^{n-m}$ has its round metric and $\mathbb T^m$ its flat torus metric, such that the projection onto the sphere factor,
\[
\operatorname{pr}_{S^{n-m}}\circ \Phi:(M,g)\to (S^{n-m},g_{S^{n-m}}),
\]
is area non-increasing when $n-m\ge 3$ and is $1$-Lipschitz when $n-m=2$. Under these assumptions, which conclusion about $(M,g)$ holds?

$(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$; equivalently, there exists a covering map from $S^{n-m}\times \mathbb R^m$ onto $M$ that is a local isometry for the product of the round metric on $S^{n-m}$ and the Euclidean metric on $\mathbb R^m$.

Choices:
B. $(M,g)$ is isometric to $\bigl(S^{n-m}\times \mathbb T^m,\, g_{S^{n-m}}+g_{\mathbb T^m}\bigr)$ itself; in particular, the given map $\Phi$ can be taken to be a Riemannian isometry.
C. The universal cover of $(M,g)$ splits isometrically as $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$.
D. $(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times \mathbb R^m,\, g_{S^{n-m}}+g_{\mathbb R^m}\bigr)$ provided that $\operatorname{pr}_{S^{n-m}}\circ\Phi$ is $1$-Lipschitz; in fact the same conclusion holds without distinguishing the cases $n-m\ge 3$ and $n-m=2$.
E. $(M,g)$ is isometrically covered by $\bigl(S^{n-m}\times N^m,\, g_{S^{n-m}}+h\bigr)$ for some complete flat $m$-manifold $(N^m,h)$, but one cannot in general conclude that the cover is $S^{n-m}\times \mathbb R^m$ with the Euclidean metric on the second factor.

replace explicit isometric covering statement by universal-cover splitting

dimension-dependent hypothesis on spherical projection

rigid upward propagation forcing Euclidean torus directions

The stem does not explicitly reveal the conclusion, and no choice is singled out by wording alone. It presents hypotheses of a rigidity theorem and asks for the resulting geometric conclusion, so there is no direct answer leakage.

This is close to a theorem-recall item: the stem essentially states the full hypotheses and asks for the theorem's conclusion. However, it is not completely tautological because the options include stronger, weaker, and hypothesis-tampered variants that must be distinguished.

Some reasoning is needed to reject the stronger claim (B), the weaker-but-true formulation (C), the altered-hypothesis claim (D), and the plausible alternative flat-factor conclusion (E). Still, the item mainly tests precise recall/application of a known theorem rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: confusing exactness with a stronger isometry statement, settling for a weaker universal-cover splitting, overlooking the dimension-dependent hypothesis, or weakening the flat factor to an arbitrary flat manifold.

A solid advanced theorem-application MCQ with strong distractors and no answer leakage, but it leans heavily on recall of a specific rigidity theorem rather than deep generative reasoning.


\begin{thm} \label{Theorem : 1. main theorem}

Let $\psi=(\psi_i)_{i=1}^k : I \to G=\prod_{i=1}^k G_i$ be a curve such that $\mathcal{I}\circ \psi$ is a $C^l$-map for some $l >\frac{2\zeta_1} {\zeta_k}$ and $(\mathcal{I}_i\circ\psi_i)'(s) \neq 0$  for all $1\leq i\leq k$ and almost every $s\in I$. Suppose that
\begin{equation}
    \nu(\{s \in I : \psi(s) \in \Delta_H\}) = 0 \text{ for all } H\in\mathcal{H} \text{ such that } Gp_H \text{ is closed and } Gp_H \neq p_H.
\end{equation}
Then, for every $f \in C_c(L/\Gamma)$, we have
\begin{equation}
    \lim_{t\rightarrow \infty} \int_0^1 f(a(t)\psi(s)x_0)ds = \int_{L/\Gamma} f\, d\mu_L,
\end{equation}
where $x_0=e\Gamma$, $\overline{Gx_0}=L/\Gamma$, and $\mu_L$ is the unique $L$-invariant probability measure on $L/\Gamma$.
\end{thm}

\begin{prop}\label{prop: 6.  u(x) is in circle}
    Let $H\in \mathscr{H}$ such that $G p_{H}$ is closed.
    Suppose that
    \[  S_{H} = \{\mathbf{x} \in \R^{ \Sigma_{i=1}^k (n_i-1)}: u(\mathbf{x}) p_{H} \in V^{0-}_L(A) \}
    \]is nonempty.
        Then there exists $\xi_0 \in G$ and a reductive subgroup $F$ of $G$ containing $A$ such that the following conditions are satisfied.
    \begin{enumerate}
        \item $F = N_G^1\big(\xi_0H\xi_0^{-1}\big)$. In particular, if $G$ does not fix $p_H$, then $F$ is a proper reductive subgroup of $G$.
        \item $F$ is an almost direct product $S_0\cdot S_1 \cdots S_p$ for some $1 \leq p \leq k$, where $S_0$ is the largest compact normal subgroup of $F$, and for $1 \leq j \leq p$, each $S_{j}$ is of the form $\Delta_{\mathcal{J}_j}\mathrm{SO}(m_{\mathcal{J}_j},1)$ for some partition $ \mathscr{P} = \{\mathcal{J}_1, \mathcal{J}_2, \cdots, \mathcal{J}_p\}$ of $\{1, 2, \cdots, k\}$ such that for each $\mathcal{J}\in \mathscr{P}$, we have $\zeta_{j_1} = \zeta_{j_2}$ for all $j_1, j_2 \in \mathcal{J}$.
        \item \begin{equation} \label{equation: 4. in the sphere}
        S_{H} = \{\mathbf{x} \in \R^{\Sigma_{i=1}^k (n_i-1)}: \mathcal{I}\circ u(\mathbf{x}) \in \mathcal{I}(F\xi_0) \},
    \end{equation}
    which is an embedding of a product of subspheres or of subspaces of $\R^{n_i-1}$ into $\R^{ \Sigma_{i=1}^k (n_i-1)}.$
    \end{enumerate}

\end{prop}

\label{sec:intro}
\subsection{Background}

The equidistribution problem, initiated by Shah~\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a
homogeneous space under the action of a diagonal one-parameter subgroup.
The central question in this problem is to determine the precise conditions on the curve that ensure the translated
measures do not lose mass to infinity or become equidistributed in the homogeneous
space.
More precisely, the general setting of the problem can be described as follows.
Let $G$ be a semisimple Lie group, $\Gamma$ be a lattice in $G$, and $x \in G/\Gamma$.
Let $A = \{a(t): t \in \mathbb{R}\}$ be an $\mathbb{R}$-diagonalizable one-parameter subgroup of $G$, and
\begin{equation} \label{eq:horo}
U^+_G(A)=\{u\in G: a(-t)ua(t)\to e \text{ as } t\to\infty\}
\end{equation}
denote the corresponding expanding horospherical subgroup of $G$.
The problem asks: For any curve $\varphi: [0,1] \to U^+(A)$, under what conditions on $\varphi$,  do the parametric measures concentrated on $\{a(t)\varphi([0,1])x\}$ become equidistributed as $t \to \infty$ with respect to the unique $G$-invariant measure $\mu_G$ on $G/\Gamma$.

In the specific case of $G = \mathrm{SO}(n,1)$, establishing these conditions for curves yields a finer result for the equidistribution of $(n-1)$-dimensional objects in $n$-dimensional hyperbolic spaces.
Specifically, let $M$ be a hyperbolic $n$-manifold of finite Riemannian volume.
There exists a lattice $\Gamma$ in $\mathrm{SO}(n,1)$ such that $M \cong \mathbb{H}^n/\Gamma$, where $\mathbb{H}^n \cong \mathrm{SO}(n) \backslash \mathrm{SO}(n,1)$.
Let $\pi: \mathbb{H}^n \to M$ be the quotient map. In the open unit ball model of $\mathbb{H}^n$, for $0 < \alpha< 1$, we can embed the sphere $\alpha \mathbb{S}^{n-1}$ within a unit ball. As $\alpha \to 1^-$, this sphere $\alpha \mathbb{S}^{n-1}$ approaches the boundary $\partial \mathbb{H}^n = \mathbb{S}^{n-1}$.
For the rotation-invariant probability measure $\mu_{\alpha}$ concentrated on $\pi(\alpha \mathbb{S}^{n-1})$, we have:
$$\lim_{\alpha \to 1^-} \mu_{\alpha} = \mu_M$$ where $\mu_M$ is the normalized Riemannian volume measure on $M$. In other words, the measure $\mu_\alpha$ becomes equidistributed as $\alpha \to 1^-$. This is a special case of the results shown in \cite{DWR93} and \cite{EM93}. See also \cite{Randol84} for $n = 3$ case.

Shah \cite{Sha09SOn1analytic} proved that for any analytic curve $\psi: [0,1] \to \mathbb{S}^{n-1}$ such that its image is not contained in any proper subsphere, the parametric measures concentrated on $\pi(\alpha\psi)$ equidistribute to $\mu_M$ as $\alpha\to 1^-$, as compared to the measures concentrated on entire spheres $\pi(\alpha \mathbb{S}^{n-1})$. In the language of homogeneous dynamics, the condition on $\psi$ is formulated as follows: Let $A = \{a(t): t \in \mathbb{R}\}$ be a non-trivial diagonalizable one-parameter subgroup of $G=\mathrm{SO}(n,1)$. Let $P^- = \{g \in G : \lim_{t \to \infty} a(t) g a(t)^{-1} \text{ exsits in } G\}$ be the corresponding proper parabolic subgroup. The quotient space $P^-\backslash G$ can be identified with $\mathrm{SO}(n-1)\backslash \mathrm{SO}(n) \cong \mathbb{S}^{n-1}$. Suppose $\varphi:[0,1]\to U^+(A)$ be such that the projection of $\varphi(t)$ on $\mathbb{S}^{n-1}$ equals $\psi(t)$ for almost all $t$. So, the projection of $\varphi([0,1])$ on the quotient space $P^-\backslash G$ is not contained in any proper subsphere of $\mathbb{S}^{n-1}$, then the expanding translates of $\varphi([0,1])\Gamma/\Gamma$ by $\{a(t)\}_t$ become equidistributed as $t\to \infty$. Shah \cite{Sha09SOn1smooth} later generalized this result to the case where $\psi$ is a smooth function. In this setting, reflecting the differences between analytic and smooth functions, the condition required is that the projection of $\varphi$ to $P^-\backslash G$ must not map any set of positive measure into a specific countable collection of proper subspheres.

Lei Yang \cite{LYangProduct} extended the result on analytic curves to the setting of actions of $G = \big(\mathrm{SO}(n,1)\big)^k$ on finite volume homogeneous spaces $L/\Gamma$, where $G\subset L$, and described the sufficient algebraic conditions on the analytic curves for equidistribution.

Meanwhile, inspired by the work of Aka et al.\cite{Aka18}, P. Yang \cite{PYang20thesis} resolved this problem in full generality for analytic curves in a semisimple algebraic group by generalizing the concept of constraining pencils to unstable Schubert varieties.

The goal of this paper is to extend the results of Lei Yang for translates of smooth curves and provide Lie-theoretic and geometric conditions on curves to ensure equidistribution.

\subsection{Main result}
To state our main theorem, we begin with some notation.

Let $Q_n$ be a quadratic form in $n+1$ variables defined as $$Q_n(x_0, x_1, \cdots, x_n) = 2x_0x_n-(x_1^2+x_2^2+\cdots + x_{n-1}^2).$$
We identify $\mathrm{SO}(n,1)$ with $\mathrm{SO}(Q_n)=\{g\in \mathrm{SL}(n+1,\mathbb{R}): Q_n(gv)=Q_n(v)\, \forall v\in\mathbb{R}^{n+1}\}.$
For $n \geq 2$, this group has two connected components.
Throughout this paper, we let $\mathrm{SO}(n,1)$ denote its identity component $\mathrm{SO}_0(n,1)$.

Let $G = G_1\times G_2 \times \cdots \times G_k$ where each factor is $G_i = \mathrm{SO}(n_i,1)$ with $n_i \geq 2$; Unless otherwise specified, the index $i$ will always range from $1$ to $k$.

Let $\pi_i: G \to G_i$ be the projection map onto the $i$-th factor. Let $L$ be a Lie group containing $G$, and $\Gamma$ be a lattice in $L$.
Let $X = L/\Gamma$ and $x = l_0\cdot\Gamma \in L/\Gamma$ be a point whose $G$-orbit is dense in $X$. By replacing $\Gamma$ with $l_0 \Gamma l_0^{-1}$, we may assume without loss of generality that $x$ is the identity coset $x_0 = e \Gamma$.
Let $A = \{a(t) = \big( a_1(t), a_2(t), \cdots, a_k(t) \big) \}_{t \in \mathbb{R}}$ be an $\mathbb{R}$-diagonalizable one-parameter subgroup of $G$ such that each $A_i := \{a_i(t)\}_{t \in \mathbb{R}}$ is a nontrivial $\mathbb{R}$-diagonalizable subgroup of $G_i$.
By a suitable conjugation, we may write $a(t) =(a_1(t), a_2(t), \cdots, a_k(t))$ as $$\left(
\begin{pmatrix} e^{\zeta_1 t} & & \\
& I_{n_1-1} & \\
& & e^{-\zeta_1 t}
\end{pmatrix},
\begin{pmatrix} e^{\zeta_2 t} & & \\
& I_{n_2-1} & \\
& & e^{-\zeta_2 t}
\end{pmatrix},
\cdots,
\begin{pmatrix} e^{\zeta_k t} & & \\
& I_{n_k-1} & \\
& & e^{-\zeta_k t}
\end{pmatrix}\right)
$$ for some positive constants $\zeta_i>0$.
For simplicity, assume that $\zeta_i$'s are arranged in decreasing order; that is, $$\zeta_1 \geq \zeta_2 \geq \cdots \geq \zeta_k.$$

Let $K_i \cong \mathrm{SO}(n_i)$ be a maximal compact subgroup of $G_i$ and let $M_i = Z_{G_i}(A_i) \cap K_i$ and $M = M_1 \times \cdots \times M_k$.
Let $P^-_i = \{g_i \in G_i: \lim_{t \to \infty} a_i(t) g_i a_i(t)^{-1} \text{ exsits  in } G_i\}$ and $P^- = \{g \in G: \displaystyle{\lim_{t\rightarrow \infty}} a(t)ga(t)^{-1} \text{ exists in } $G$\}$.
Then
\begin{equation}
    \begin{split}
    P^-\backslash G &= (P_1^-\backslash G_1) \times (P_2^-\backslash G_2) \times \cdots \times (P_k^-\backslash G_k)\\
    &\cong (M_1\backslash K_1) \times (M_2\backslash K_2) \times \cdots \times (M_k \backslash K_k)\\
    & \cong \mathbb{S}^{n_1-1} \times \mathbb{S}^{n_2-1}\times \cdots \times \mathbb{S}^{n_k-1}.
    \end{split}
\end{equation}
Let $\mathcal{I}_i: G_i \rightarrow P_i^-\backslash G_i$ and $\mathcal{I}: G \to P^-\backslash G $ be the corresponding quotient maps. We note that the action of $G_i$ on $\mathbb{S}^{n_i-1}\cong P_i\backslash G_i$ is via Mobius transformations.

Let $\mathscr{H}$ denote the collection of proper closed and connected (Lie) subgroups $H$ of $L$ such that $H\cap \Gamma$ is a lattice in $H$ and some $\Ad_L$-unipotent one-parameter subgroup of $H$ acts ergodically on $H/(H \cap \Gamma)$ with respect to the $H$- invariant measure $\mu_H$.

Define $V_L = \bigoplus_{i =1}^{\dim L} (\bigwedge^i \mathcal{L})$ where $\mathcal{L}$ is the Lie algebra of $L$ and $L$ acts on $V_L$ via the representation  $\Ad_L$.

For any Lie subgroup $H$ of $L$, choose $p_H \in \wedge^{\dim H} \mathrm{Lie}(H)\backslash \{0\}$.
Let
\[
V_L^{0-}(A) = \{v \in V_L: \lim_{t \to \infty} a(t)v\in V_L \text{ exists}\}.
\]
We note that for $V_L^{0-}(A)$ is preserved by the action of $P^-$.

For each $H \in \mathscr{H}$, define
    \begin{align}
    \Delta_{H}&=\{g:g\in G,\, gp_H\in V^{0-}_L(A)\}
    \end{align}

\begin{defn}
    Let $\mathcal{J} \subset \{1, 2, \cdots, k\}$ be a set of indices and let $m_{\mathcal{J}} \in \mathbb{N}$ satisfying $1 \leq m_{\mathcal{J}}\leq \min_{j \in \mathcal{J}}n_j$.
    Let $\iota_j: \mathbb{S}^{m_\mathcal{J}-1} \to \mathbb{S}^{n_j-1}$ be the standard inclusion $\mathbb{S}^{m_\mathcal{J}-1} \hookrightarrow \mathbb{S}^{n_j-1}$ followed by a M\"obius transformation on $\mathbb{S}^{n_j-1}$.
    We define the diagonal M\"obius embedding $\iota_{\mathcal{J}}: \mathbb{S}^{m_\mathcal{J}-1} \to \prod_{j \in \mathcal{J}} \mathbb{S}^{n_j-1}$ by setting $\iota_{\mathcal{J}} = (\iota_j)_{j \in \mathcal{J}}$. Here, we identify $\mathbb{S}^0$ with a single point.
\end{defn}

\begin{defn}\label{def: 1. obstruction set}
    Let $\mathscr{P} = \{\mathcal{J}_1, \mathcal{J}_2, \cdots \mathcal{J}_p\}$ be a partition of $\{1, 2, \cdots, k\}$ such that for each $\mathcal{J} \in \mathscr{P}$, we have $\zeta_{j_1} = \zeta_{j_2}$ for all $j_1, j_2 \in \mathcal{J}$. For each $\mathcal{J} \in \mathscr{P}$, we choose a diagonal M\"obius embedding $\iota_\mathcal{J}$ as defined above. We then define $\iota_{\mathscr{P}} = \prod_{\mathcal{J}\in \mathscr{P}} \iota_{\mathcal{J}}: \prod_{\mathcal{J} \in \mathscr{P}}\mathbb{S}^{m_\mathcal{J}-1} \to \prod_{i=1}^k \mathbb{S}^{n_i-1}$.

    Later, in Proposition~\ref{prop: 6.  u(x) is in circle}, we will show that for each $H\in \mathcal{H}$ such that $Gp_H$ is closed, we have that $\mathcal{I}(\Delta_{H})$ equals the image of $\iota_\mathscr{P}$ defined as above, that is, $\mathcal{I}(\Delta_{H})$ is the image of a M\"obius embedding of a product of subspheres into $\prod_{i=1}^k \mathbb{S}^{n_i-1}$.

\end{defn}

Let $I = [0,1]$ be a closed interval in $\mathbb{R}$. Let $\psi_i: I \rightarrow G_i$ be a curve and $\psi = (\psi_i)_{i = 1}^k : I \to \prod_{i=1}^k G_i$.
Let $\nu$ be the Lebesgue measure on $\mathbb{R}$.

The equidistribution problem, initiated by Shah~\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a
homogeneous space under the action of a diagonal one-parameter subgroup.
The central question in this problem is to determine the precise conditions on the curve that ensure the translated
measures do not lose mass to infinity or become equidistributed in the homogeneous
space.
More precisely, the general setting of the problem can be described as follows.
Let $G$ be a semisimple Lie group, $\Gamma$ be a lattice in $G$, and $x \in G/\Gamma$.
Let $A = \{a(t): t \in \mathbb{R}\}$ be an $\mathbb{R}$-diagonalizable one-parameter subgroup of $G$, and
\begin{equation} \label{eq:horo}
U^+_G(A)=\{u\in G: a(-t)ua(t)\to e \text{ as } t\to\infty\}
\end{equation}
denote the corresponding expanding horospherical subgroup of $G$.
The problem asks: For any curve $\varphi: [0,1] \to U^+(A)$, under what conditions on $\varphi$, do the parametric measures concentrated on $\{a(t)\varphi([0,1])x\}$ become equidistributed as $t \to \infty$ with respect to the unique $G$-invariant measure $\mu_G$ on $G/\Gamma$.

Shah \cite{Sha09SOn1analytic} proved that for any analytic curve $\psi: [0,1] \to \mathbb{S}^{n-1}$ such that its image is not contained in any proper subsphere, the parametric measures concentrated on $\pi(\alpha\psi)$ equidistribute to $\mu_M$ as $\alpha\to 1^-$, as compared to the measures concentrated on entire spheres $\pi(\alpha \mathbb{S}^{n-1})$. In the language of homogeneous dynamics, the condition on $\psi$ is formulated as follows: Let $A = \{a(t): t \in \mathbb{R}\}$ be a non-trivial diagonalizable one-parameter subgroup of $G=\mathrm{SO}(n,1)$. Let $P^- = \{g \in G : \lim_{t \to \infty} a(t) g a(t)^{-1} \text{ exsits in } G\}$ be the corresponding proper parabolic subgroup. The quotient space $P^-\backslash G$ can be identified with $\mathrm{SO}(n-1)\backslash \mathrm{SO}(n) \cong \mathbb{S}^{n-1}$. Suppose $\varphi:[0,1]\to U^+(A)$ be such that the projection of $\varphi(t)$ on $\mathbb{S}^{n-1}$ equals $\psi(t)$ for almost all $t$. So, the projection of $\varphi([0,1])$ on the quotient space $P^-\backslash G$ is not contained in any proper subsphere of $\mathbb{S}^{n-1}$, then the expanding translates of $\varphi([0,1])\Gamma/\Gamma$ by $\{a(t)\}_t$ become equidistributed as $t\to \infty$. Shah \cite{Sha09SOn1smooth} later generalized this result to the case where $\psi$ is a smooth function. In this setting, reflecting the differences between analytic and smooth functions, the condition required is that the projection of $\varphi$ to $P^-\backslash G$ must not map any set of positive measure into a specific countable collection of proper subspheres.

Let $\pi_i: G \to G_i$ be the projection map onto the $i$-th factor. Let $L$ be a Lie group containing $G$, and $\Gamma$ be a lattice in $L$.
Let $X = L/\Gamma$ and $x = l_0\cdot\Gamma \in L/\Gamma$ be a point whose $G$-orbit is dense in $X$. By replacing $\Gamma$ with $l_0 \Gamma l_0^{-1}$, we may assume without loss of generality that $x$ is the identity coset $x_0 = e \Gamma$.
Let $A = \{a(t) = \big( a_1(t), a_2(t), \cdots, a_k(t) \big) \}_{t \in \mathbb{R}}$ be an $\mathbb{R}$-diagonalizable one-parameter subgroup of $G$ such that each $A_i := \{a_i(t)\}_{t \in \mathbb{R}}$ is a nontrivial $\mathbb{R}$-diagonalizable subgroup of $G_i$.
By a suitable conjugation, we may write $a(t) =(a_1(t), a_2(t), \cdots, a_k(t))$ as $$\left(
\begin{pmatrix} e^{\zeta_1 t} & & \\
& I_{n_1-1} & \\
& & e^{-\zeta_1 t}
\end{pmatrix},
\begin{pmatrix} e^{\zeta_2 t} & & \\
& I_{n_2-1} & \\
& & e^{-\zeta_2 t}
\end{pmatrix},
\cdots,
\begin{pmatrix} e^{\zeta_k t} & & \\
& I_{n_k-1} & \\
& & e^{-\zeta_k t}
\end{pmatrix}\right)
$$ for some positive constants $\zeta_i>0$.
For simplicity, assume that $\zeta_i$'s are arranged in decreasing order; that is, $$\zeta_1 \geq \zeta_2 \geq \cdots \geq \zeta_k.$$

Let $K_i \cong \mathrm{SO}(n_i)$ be a maximal compact subgroup of $G_i$ and let $M_i = Z_{G_i}(A_i) \cap K_i$ and $M = M_1 \times \cdots \times M_k$.
Let $P^-_i = \{g_i \in G_i: \lim_{t \to \infty} a_i(t) g_i a_i(t)^{-1} \text{ exsits in } G_i\}$ and $P^- = \{g \in G: \displaystyle{\lim_{t\rightarrow \infty}} a(t)ga(t)^{-1} \text{ exists in } $G$\}$.
Then
\begin{equation}
 \begin{split}
 P^-\backslash G &= (P_1^-\backslash G_1) \times (P_2^-\backslash G_2) \times \cdots \times (P_k^-\backslash G_k)\\
 &\cong (M_1\backslash K_1) \times (M_2\backslash K_2) \times \cdots \times (M_k \backslash K_k)\\
 & \cong \mathbb{S}^{n_1-1} \times \mathbb{S}^{n_2-1}\times \cdots \times \mathbb{S}^{n_k-1}.
 \end{split}
\end{equation}
Let $\mathcal{I}_i: G_i \rightarrow P_i^-\backslash G_i$ and $\mathcal{I}: G \to P^-\backslash G $ be the corresponding quotient maps. We note that the action of $G_i$ on $\mathbb{S}^{n_i-1}\cong P_i\backslash G_i$ is via Mobius transformations.

\end{defn}

Let $I = [0,1]$ be a closed interval in $\mathbb{R}$. Let $\psi_i: I \rightarrow G_i$ be a curve and $\psi = (\psi_i)_{i = 1}^k : I \to \prod_{i=1}^k G_i$.
Let $\nu$ be the Lebesgue measure on $\mathbb{R}$.

The equidistribution problem, initiated by Shah~\cite{Sha09SLnR}, concerns the limiting distribution of parameter measures on a curve segment that expands within a
homogeneous space under the action of a diagonal one-parameter subgroup.
The central question in this problem is to determine the precise conditions on the curve that ensure the translated
measures do not lose mass to infinity or become equidistributed in the homogeneous
space.
More precisely, the general setting of the problem can be described as follows.
Let $G$ be a semisimple Lie group, $\Gamma$ be a lattice in $G$, and $x \in G/\Gamma$.
Let $A = \{a(t): t \in \R\}$ be an $\R$-diagonalizable one-parameter subgroup of $G$, and
\begin{equation} \label{eq:horo}
U^+_G(A)=\{u\in G: a(-t)ua(t)\to e \text{ as } t\to\infty\}
\end{equation}
denote the corresponding expanding horospherical subgroup of $G$.
The problem asks: For any curve $\varphi: [0,1] \to U^+(A)$, under what conditions on $\varphi$, do the parametric measures concentrated on $\{a(t)\varphi([0,1])x\}$ become equidistributed as $t \to \infty$ with respect to the unique $G$-invariant measure $\mu_G$ on $G/\Gamma$.

Let $I = [0,1]$ be a closed interval in $\R$. Let $\psi_i: I \rightarrow G_i$ be a curve and $\psi = (\psi_i)_{i = 1}^k : I \to \prod_{i=1}^k G_i$.
Let $\nu$ be the Lebesgue measure on $\R$.

We will deduce Theorem \ref{Theorem : 1. main theorem} from a sharper result, Theorem \ref{Thm : 1. shrinking equidistribuiton at point}.
\begin{thm}[Equidistribution of expanding translate of shirinking pieces of a curve]\label{Thm : 1. shrinking equidistribuiton at point}
 Let the notation and conditions be as in Theorem~\ref{Theorem : 1. main theorem}.
 Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \frac{\zeta_k}{2}$ and for any $s_0 \in I\backslash E$, $f \in C_c(X)$, a sequence $\{s_n\}_{n \in \N}$ in $I$ and a sequence $\{t_n\}_{n \in\N}$ in $\R$ such that $s_n \to s_0$ and $t_n \to \infty $ as $n \to \infty$, we have
 \begin{equation}\label{equation: 1. equidist on shrinking curve}
 \lim_{n \to \infty}\int^1_0 f\big(a(t_n)\psi(s_n + e^{-mt_n}\eta)x_0\big) \, d\eta = \int_X f d\mu_L.
 \end{equation}
\end{thm}

Our approach is as follows. For each $H \in \mathscr{H}$, when the $G$ orbit of a vector $p_H$ is not closed, we employ Kempf's invariant theory \cite{Kempf78} and the technique developed by Shah and P.Yang \cite{SY24}.
Together, these enable us to replace the given representation and vector whose $G$ orbit is not closed with a different pair, provided that the new pair meets certain conditions.
We explicitly construct a new, more comuptable pair that meets the required conditions by utilizing the standard representation $\R^{n+1}$ of $\mathrm{SO}(n,1)$. Calculations with this new pair then show that the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.
Therefore, the main obstructions originate from the case where $Gp_H$ is closed. In this closed orbit case, the obstructions that $\mathcal{I}\circ\psi$ should avoid take the form of a M\"obius embedding of a product of subspheres into $\prod_{i=1}^k\mathbb{S}^{n_i-1}$. As $\mathscr{H}$ is countable, the resulting set of obstructions $\{ \Delta_H: H\in \mathcal{H} \text{ such that } Gp_H \text{ is closed} \} $ is countable. The countability of this set is essential. Unlike in the analytic case, a property holding for a smooth function on a set of positive measure does not imply it holds globally. Thus, the fact that the obstruction set is merely countable is what makes it possible for a smooth map to exist that avoids these conditions.

\begin{prop}\label{prop: 6. u(x) is in circle}
 Let $H\in \mathscr{H}$ such that $G p_{H}$ is closed.
 Suppose that
 \[ S_{H} = \{\mathbf{x} \in \R^{ \Sigma_{i=1}^k (n_i-1)}: u(\mathbf{x}) p_{H} \in V^{0-}_L(A) \}
 \]is nonempty.
 Then there exists $\xi_0 \in G$ and a reductive subgroup $F$ of $G$ containing $A$ such that the following conditions are satisfied.
 \begin{enumerate}
 \item $F = N_G^1\big(\xi_0H\xi_0^{-1}\big)$. In particular, if $G$ does not fix $p_H$, then $F$ is a proper reductive subgroup of $G$.
 \item $F$ is an almost direct product $S_0\cdot S_1 \cdots S_p$ for some $1 \leq p \leq k$, where $S_0$ is the largest compact normal subgroup of $F$, and for $1 \leq j \leq p$, each $S_{j}$ is of the form $\Delta_{\mathcal{J}_j}\mathrm{SO}(m_{\mathcal{J}_j},1)$ for some partition $ \mathscr{P} = \{\mathcal{J}_1, \mathcal{J}_2, \cdots, \mathcal{J}_p\}$ of $\{1, 2, \cdots, k\}$ such that for each $\mathcal{J}\in \mathscr{P}$, we have $\zeta_{j_1} = \zeta_{j_2}$ for all $j_1, j_2 \in \mathcal{J}$.
 \item \begin{equation} \label{equation: 4. in the sphere}
 S_{H} = \{\mathbf{x} \in \R^{\Sigma_{i=1}^k (n_i-1)}: \mathcal{I}\circ u(\mathbf{x}) \in \mathcal{I}(F\xi_0) \},
 \end{equation}
 which is an embedding of a product of subspheres or of subspaces of $\R^{n_i-1}$ into $\R^{ \Sigma_{i=1}^k (n_i-1)}.$
 \end{enumerate}

Now, since a sequence of polynomials $\{ \eta \mapsto a(t_n) u(R( e^{-mt_n} \eta))u(\varphi(s_n))\}_n$ has a bounded degree, we can let $d$ be the maximum degree of the sequence.
For $\frac{\varepsilon_0}{2}$, $d$ and $C\Gamma/\Gamma$, there exists a compact subset $\mathcal{D}$ of $\mathcal{A}$ given in Theorem \ref{Theorem: 7. linearlization for equidistribution}. Let $\Phi_1$ be a relatively compact open neighberhood of $\mathcal{D}$ in $V_L$ and $\Psi_1$ be the corresponding neighborhood of $C\Gamma/\Gamma$ in $L/\Gamma$.
Let
\begin{equation}
 I(\Psi_1,n) = \{\eta \in I: a(t_n)u(R( e^{-mt_n}\eta))u(\varphi(s_n))\Gamma/\Gamma \in \Psi_1\}.
\end{equation}
Then,
\begin{equation}\label{equation: 7. liminf of measure of open set}
 \varepsilon_0 = \mu_{s_0}(C\Gamma/\Gamma) \leq \mu_{s_0}(\Psi_1) \leq \liminf_n\mu_{s_0,t_n}(\Psi_1) = \liminf_n \nu\big(I(\Psi_1, n)\big)
\end{equation} where $\nu$ is the Lebesgue measure on $\R$.

\subsection{proof of Theorem \ref{Thm : 1. shrinking equidistribuiton at point}}
This proof follows the proof of Theorem 1.3
from \cite{ShahPYang24}.
Since we have shown that for almost every $s_0 \in I$ we have $\mu_{s_0} = \mu_L$, it remains to prove that for any sequence $t_n \to \infty$ and any $f \in C_c(X)$,
\begin{equation}
 \lim_{t_n \to \infty} \int ^1_0f\big(a(t_n)u(\varphi(s_n + e^{-mt_n}\eta)) x_0\big)d\eta = \int_X f(y) d\mu_{s_0}(y)
\end{equation} holds.
Fix $f \in C_c(X)$ and $\varepsilon>0$. Then, there exists $\delta>0$ such that for any $y$ and $z \in X$, if $y \stackrel{\delta}{\approx} z $, then $f(y) \stackrel{\varepsilon}{\approx}f(z)$.
Observe that
\begin{equation*}
\begin{split}
 a(t_n)u(\varphi(s_n + e^{-mt_n} \eta))
 & = a(t_n)u(\varphi(s_n + e^{-mt_n}\eta)-\varphi(s_n))u(\varphi(s_n))\\
 & = a(t_n)u(R(e^{-mt_n}\eta) + O(e^{-mlt_n}))u(\varphi(s_n))\\
 & = a(t_n)u(O(e^{-mlt_n}))a(t_n)^{-1}a(t_n)u(R(e^{-mt_n}\eta))u(\varphi(s_n))\\
 & = u(O(e^{(\zeta_1-ml)t_n})a(t_n)u(R(e^{-mt_n} \eta))u(\varphi(s_n)).
\end{split}
\end{equation*}

\subsection{Proof of Theorem \ref{Theorem : 1. main theorem}}
This proof follows the proof of Theorem 1.3
from \cite{ShahPYang24}.
Note that $E$ is a Lebesgue null set. By equation (\ref{3. equation: measure nondivergent}) and Theorem \ref{Thm : 1. shrinking equidistribuiton at point}, we can derive that for any bounded continuous function $f \in C_b(X)$, the equation (\ref{equation: 1. equidist on shrinking curve}) still holds. It is enough to show that for any $f \in C_b(X)$ such that $\norm{f}_{\infty} \leq 1$ and $\int_Xf \, d\mu_L =0$, and for any compact set $K$ in $I\backslash E$,
\begin{equation}\label{equation: 5. equidist on cpt set}
 \lim_{t \to \infty} \frac{1}{\nu(K)}\int_K f\big(a(t)u(\varphi(s))x_0\big)ds = 0.
\end{equation}

We will deduce Theorem \ref{Theorem : 1. main theorem} from a sharper result, Theorem \ref{Thm : 1. shrinking equidistribuiton at point}.
\begin{thm}[Equidistribution of expanding translate of shirinking  pieces of a curve]\label{Thm : 1. shrinking equidistribuiton at point}
    Let the notation and conditions be as in Theorem~\ref{Theorem : 1. main theorem}.
    Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \frac{\zeta_k}{2}$ and for any $s_0 \in I\backslash E$, $f \in C_c(X)$, a sequence $\{s_n\}_{n \in \mathbb{N}}$ in $I$ and a sequence $\{t_n\}_{n \in\mathbb{N}}$ in $\mathbb{R}$ such that $s_n \to s_0$ and $t_n \to \infty $ as $n \to \infty$, we have
    \begin{equation}\label{equation: 1. equidist on shrinking curve}
        \lim_{n \to \infty}\int^1_0 f\big(a(t_n)\psi(s_n + e^{-mt_n}\eta)x_0\big) \, d\eta = \int_X f d\mu_L.
    \end{equation}
\end{thm}

\subsection{Paper Organization and Proof Outline}
The proof of equidistribution on homogeneous spaces typically involves three main steps.
 First, one proves the non-divergence (i.e., no escape of mass) of the limit measures.
 Next, it must be shown that the limit measure is invariant under a non-trivial unipotent subgroup.
 Finally, the linearization technique is employed to demonstrate that these measures do not accumulate on lower- dimensional unipotent-invariant subvarieties immersed in the homogeneous space. Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.

In previous works (for example, \cite{Sha09SOn1smooth}, \cite{Sha09SLnR}, \cite{Sha09SOn1analytic}, and \cite{LYangProduct}), the Nondivergence Theorem by  Dani, Kleinbock, and Margulis (See \cite{DM93}, \cite{KM98}) has been a key tool to establish non-divergence of limit measures. Applying this theorem requires the given curve to satisfy a certain growth property called $(C, \alpha)$-goodness.
However, while analytic functions possess this property, smooth functions generally do not.
To address this, following the approach of Shah and P. Yang (\cite{ShahPYang24}), we approximate $\mathcal{I}\circ\psi$ at each point $s_0 \in I\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m = \frac{\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.
We first demonstrate the equidistribution of parametric measures concentrated on these polynomial curves on the shrinking intervals through a sequence of arguments presented in Sections 2, 3, and 4.

To apply the linearization technique, a linear dynamical result, often called the ``basic lemma'', is required; this can be found in the aforementioned papers on equidistribution.
\cite{ShahPYang24} extended this result, originally for fixed-sized curves, to shrinking pieces of a curve for a particular case where $G = SL(n,\mathbb{R})$.
 In Section \ref{Section: Shrinking}, We prove that this linear dynamical result for shrinking pieces of a curve also holds when $G$ is a product of $\mathrm{SO}(n,1)$'s.

In Section \ref{Sec: nondiv and unip inv}, we follow standard schemes to establish the nondivergence and unipotent invariance of the limit measure for measures concentrated on shrinking pieces of polynomial curves.

In Section \ref{Sec: Equidistribution}, we identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution of the limit measure for measures concentrated on shrinking pieces of polynomial curves, thereby providing the proofs of our main results: Theorem \ref{Theorem : 1. main theorem} and Theorem \ref{Thm : 1. shrinking equidistribuiton at point}.

Our approach is as follows. For each $H \in \mathscr{H}$, when the $G$ orbit of a vector $p_H$ is not closed, we employ Kempf's invariant theory \cite{Kempf78} and the technique developed by Shah and P.Yang \cite{SY24}.
Together, these enable us to replace the given representation and vector whose $G$ orbit is not closed with a different pair, provided that the new pair meets certain conditions.
We explicitly construct a new, more comuptable pair that meets the required conditions by utilizing the standard representation $\mathbb{R}^{n+1}$ of $\mathrm{SO}(n,1)$.  Calculations with this new pair then show that the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.
Therefore, the main obstructions originate from the case where $Gp_H$ is closed. In this closed orbit case, the obstructions that $\mathcal{I}\circ\psi$ should avoid take the form of a M\"obius embedding of a product of subspheres into $\prod_{i=1}^k\mathbb{S}^{n_i-1}$. As $\mathscr{H}$ is countable, the resulting set of obstructions $\{ \Delta_H: H\in \mathcal{H} \text{ such that } Gp_H \text{ is closed} \} $ is countable. The countability of this set is essential. Unlike in the analytic case, a property holding for a smooth function on a set of positive measure does not imply it holds globally. Thus, the fact that the obstruction set is merely countable is what makes it possible for a smooth map to exist that avoids these conditions.

The post-theorem introduction says that Theorem~\ref{Theorem : 1. main theorem} is deduced from the sharper Theorem~\ref{Thm : 1. shrinking equidistribuiton at point} (equidistribution for expanding translates of \emph{shrinking} pieces of the curve).

It then outlines a standard three-step strategy for equidistribution proofs on homogeneous spaces:
(i) prove \emph{non-divergence} (no escape of mass) of limit measures;
(ii) show the limit measure is invariant under a \emph{non-trivial unipotent subgroup};
(iii) use the \emph{linearization technique} to show limit measures do not accumulate on lower-dimensional unipotent-invariant subvarieties. “Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.”

Because applying the Dani--Kleinbock--Margulis nondivergence theorem typically requires $(C,\alpha)$-goodness (automatic for analytic but not for smooth maps), the approach described is to “approximate $\mathcal{I}\circ\psi$ at each point $s_0\in I\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m=\frac{\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.” One first proves equidistribution for measures concentrated on these polynomial curves on shrinking intervals (Sections 2–4).

For linearization, a “basic lemma” is required; the paper proves the needed linear dynamical result for shrinking pieces in the product $G$ of $\mathrm{SO}(n,1)$'s (Section~\ref{Section: Shrinking}). Then “standard schemes” are followed to establish nondivergence and unipotent invariance for the limit measures attached to shrinking pieces of polynomial curves (Section~\ref{Sec: nondiv and unip inv}).

Finally (Section~\ref{Sec: Equidistribution}), the paper “identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution,” yielding Theorem~\ref{Theorem : 1. main theorem} and Theorem~\ref{Thm : 1. shrinking equidistribuiton at point}. The described obstruction analysis proceeds by: for each $H\in\mathscr{H}$ with $Gp_H$ \emph{not} closed, using Kempf’s invariant theory and Shah–P. Yang’s technique to “replace the given representation and vector” by a “new, more comuptable pair” (constructed using the standard representation $\mathbb{R}^{n+1}$ of $\mathrm{SO}(n,1)$), and then showing via calculations that “the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.” Hence “the main obstructions originate from the case where $Gp_H$ is closed,” where the avoidance conditions are “a M\"obius embedding of a product of subspheres into $\prod_{i=1}^k\mathbb{S}^{n_i-1}$.” The set of such obstructions is countable, and “the countability of this set is essential” for smooth maps, since (unlike the analytic case) positive-measure conditions do not globalize.

The post-theorem introduction says that, to prove the main theorem, it is deduced from the sharper

\begin{thm}[Equidistribution of expanding translate of shirinking  pieces of a curve]\label{Thm : 1. shrinking equidistribuiton at point}
    Let the notation and conditions be as in Theorem~\ref{Theorem : 1. main theorem}.
    Then, there exists a Lebesgue null set $E$ in $I$ such that for $m: = \frac{\zeta_k}{2}$ and for any $s_0 \in I\backslash E$, $f \in C_c(X)$, a sequence $\{s_n\}_{n \in \N}$ in $I$ and a sequence $\{t_n\}_{n \in\N}$ in $\R$ such that $s_n \to s_0$ and $t_n \to \infty $ as $n \to \infty$, we have
    \begin{equation}\label{equation: 1. equidist on shrinking curve}
        \lim_{n \to \infty}\int^1_0 f\big(a(t_n)\psi(s_n + e^{-mt_n}\eta)x_0\big) \, d\eta = \int_X f d\mu_L.
    \end{equation}
\end{thm}

(i.e. equidistribution for expanding translates of \emph{shrinking} pieces of the curve).

It then outlines a standard three-step strategy for equidistribution proofs on homogeneous spaces:
(i) prove \emph{non-divergence} (no escape of mass) of limit measures;
(ii) show the limit measure is invariant under a \emph{non-trivial unipotent subgroup};
(iii) use the \emph{linearization technique} to show limit measures do not accumulate on lower-dimensional unipotent-invariant subvarieties. “Ratner's theorem then guarantees that the limit measure is the $L$-invariant measure on its homogeneous space.”

Because applying the Dani--Kleinbock--Margulis nondivergence theorem typically requires $(C,\alpha)$-goodness (automatic for analytic but not for smooth maps), the approach described is to “approximate $\mathcal{I}\circ\psi$ at each point $s_0\in I\backslash E$ by an ($l-1$)-degree Taylor polynomial on shrinking intervals $Ie^{-mt}$ (for $m=\frac{\zeta_k}{2}$), to compensate for errors that expand due to the translation by $a(t)$.” One first proves equidistribution for measures concentrated on these polynomial curves on shrinking intervals (Sections 2–4).

For linearization, a “basic lemma” is required; the paper proves the needed linear dynamical result for shrinking pieces in the product $G$ of $\mathrm{SO}(n,1)$'s (Section~\ref{Section: Shrinking}). Then “standard schemes” are followed to establish nondivergence and unipotent invariance for the limit measures attached to shrinking pieces of polynomial curves (Section~\ref{Sec: nondiv and unip inv}).

Finally (Section~\ref{Sec: Equidistribution}), the paper “identify obstructions to equidistribution and demonstrate that avoiding them guarantees equidistribution,” yielding the main theorem and the preceding theorem. The described obstruction analysis proceeds by: for each $H\in\mathscr{H}$ with $Gp_H$ \emph{not} closed, using Kempf’s invariant theory and Shah–P. Yang’s technique to “replace the given representation and vector” by a “new, more comuptable pair” (constructed using the standard representation $\mathbb{R}^{n+1}$ of $\mathrm{SO}(n,1)$), and then showing via calculations that “the obstructions arising from this non-closed case are negligible, provided that the derivative of our curve in each component is non-zero almost everywhere.” Hence “the main obstructions originate from the case where $Gp_H$ is closed,” where the avoidance conditions are “a M\"obius embedding of a product of subspheres into $\prod_{i=1}^k\mathbb{S}^{n_i-1}$.” The set of such obstructions is countable, and “the countability of this set is essential” for smooth maps, since (unlike the analytic case) positive-measure conditions do not globalize.

Question:
Let $G=\prod_{i=1}^k G_i$ with one-parameter subgroup $A=\{a(t)\}_{t\in\mathbb R}\subset G$ given by
$$a(t)=(a_1(t),\dots,a_k(t)),\qquad a_i(t)=\begin{pmatrix} e^{\zeta_i t} & & \\ & I_{n_i-1} & \\ & & e^{-\zeta_i t}\end{pmatrix},$$
where $\zeta_1\ge \zeta_2\ge \cdots\ge \zeta_k>0$. Let $L$ be a Lie group containing $G$, let $\Gamma$ be a lattice in $L$, set $X=L/\Gamma$, and let $x_0=e\Gamma\in X$ with dense $G$-orbit $\overline{Gx_0}=X$. For each $i$, define
$$P_i^-:=\{g_i\in G_i:\ \lim_{t\to\infty} a_i(t)g_i a_i(t)^{-1}\ \text{exists in }G_i\},$$
and define
$$P^-:=\{g\in G:\ \lim_{t\to\infty} a(t)ga(t)^{-1}\ \text{exists in }G\}.$$
Let $\mathcal I_i:G_i\to P_i^-\backslash G_i\cong \mathbb S^{n_i-1}$ and $\mathcal I:G\to P^-\backslash G\cong \prod_{i=1}^k\mathbb S^{n_i-1}$ be the quotient maps. Let $I=[0,1]$ with Lebesgue measure $\nu$. Suppose a curve $\psi=(\psi_i)_{i=1}^k:I\to G$ satisfies the following:
1. $\mathcal I\circ\psi$ is a $C^l$ map for some $l>\frac{2\zeta_1}{\zeta_k}$;
2. for each $1\le i\le k$, the derivative $(\mathcal I_i\circ\psi_i)'(s)\neq 0$ for almost every $s\in I$;
3. for every subgroup $H\in\mathcal H$ such that the orbit $Gp_H$ is closed and nontrivial ($Gp_H\neq p_H$), the associated exceptional set $\Delta_H\subset G$ satisfies
$$\nu\big(\{s\in I:\ \psi(s)\in\Delta_H\}\big)=0.$$
Here $\mathcal H$, $p_H$, and $\Delta_H$ are the subgroup/vector/exceptional-set objects attached to the linearization setup. Under these hypotheses, which conclusion about the translated curve $a(t)\psi(s)x_0$ holds for every $f\in C_c(L/\Gamma)$?

For every $f\in C_c(L/\Gamma)$, one has
$$\lim_{t\to\infty}\int_0^1 f\big(a(t)\psi(s)x_0\big)\,ds=\int_{L/\Gamma} f\,d\mu_L,$$
where $\mu_L$ is the unique $L$-invariant probability measure on $L/\Gamma$.

Choices:
B.
For every $f\in C_c(L/\Gamma)$, one has
$$\lim_{t\to\infty}\int_0^1 f\big(a(t)\psi(s)x_0\big)\,ds=\int_{L/\Gamma} f\,d\mu_L,$$
provided the exceptional-set condition is imposed for all $H\in\mathcal H$ with $Gp_H\neq p_H$, without restricting to the case that $Gp_H$ is closed.
C.
For every $f\in C_c(L/\Gamma)$$ and every sequence $t_n\to\infty$, the family of averages
$$\int_0^1 f\big(a(t_n)\psi(s)x_0\big)\,ds$$
is relatively compact in $\mathbb R$, and every subsequential limit equals $\int_{L/\Gamma} f\,d\mu_L$.
D.
For every $f\in C_c(L/\Gamma)$, one has
$$\lim_{t\to\infty}\int_0^1 f\big(a(t)\psi(s)x_0\big)\,ds=\int_{L/\Gamma} f\,d\mu_L,$$
already under the weaker regularity assumption that $\mathcal I\circ\psi$ is $C^l$ for some $l>\frac{\zeta_1}{\zeta_k}$, while keeping the other hypotheses unchanged.
E.
For every $f\in C_c(L/\Gamma)$, one has
$$\lim_{t\to\infty}\int_0^1 f\big(a(t)\psi(s)x_0\big)\,ds=\int_{L/\Gamma} f\,d\mu_L,$$
assuming instead that there is at least one index $i$ for which $(\mathcal I_i\circ\psi_i)'(s)\neq 0$ for almost every $s\in I$, together with the same avoidance condition on the closed nontrivial obstructions $\Delta_H$.

closed-orbit restriction in obstruction analysis

dropped direct full-limit statement in favor of equivalent subsequential formulation

threshold $l>2\zeta_1/\zeta_k$ needed for Taylor approximation on shrinking scales

nonvanishing derivative required in each component to rule out non-closed obstructions

The stem does not explicitly reveal the correct option. It presents hypotheses and asks for the resulting conclusion, without verbal cues that single out choice A.

This is essentially a direct theorem-recall item: the hypotheses in the stem match a theorem statement, and the keyed answer is the theorem's conclusion almost verbatim.

There is some reasoning pressure because several distractors are close variants (especially the subsequential formulation in C and the weakened hypotheses in B, D, E). However, identifying A mainly requires recognizing the exact theorem statement rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: overextending the obstruction condition, weakening the regularity threshold, weakening the derivative hypothesis, and confusing the exact limit statement with a weaker subsequential version.

A mathematically sophisticated but theorem-recall-heavy MCQ. Its distractors are strong, but it is largely a restatement of a known result rather than a non-tautological reasoning question.


\label{thm:main}
 Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.
  Assume that
  $$
 h_\nu(\phi)<\htop(\phi)
 $$
for all non-expansive measures
$\nu \in \MMM_\phi(SM)$.   Then the geodesic flow has a unique measure $\mu$ of maximal entropy.

 Furthermore, $\mu(\mathcal{E}) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.

\begin{proof}
   For a proof of (1), (2), (3a) see \cite[Lemma III.H 3.2, Lemma III.H 3.3]{BH99}. The assertion (3b) is a consequence
   of (3a) and Lemma \ref{lem:endpts-suffice}.
\end{proof}
Gromov introduced  in \cite{mG87} a natural topology on $\partial X$ and  on $ \bar X :=X \cup \partial X$  such that  $ \bar X $ is compact (see  also \cite[Def. III.H 3.5]{BH99}).
We will give a definition in the geometric context in subsection \ref{subsec:Noconjugatepts-Ghyp}.

\subsection{Gromov hyperbolic groups}\label{subsec:hypgroups}
\begin{definition}\label{def:Hypgroups}
Let $G$ be a finitely generated group and denote by $\CC(G, S)$ the Cayley graph of $G$ with respect to some finite generating set $S \subset G$. Then  $G$ is called Gromov hyperbolic if $\CC(G, S)$ equipped with the word metric
is a $\delta$-hyperbolic metric space for some $\delta\ge 0$
\end{definition}
\begin{remark}
\begin{itemize}
\item
If $S_1, S_2$ are finite generating sets of $G$ then the identity $\id: \CC(G, S_1) \to  \CC(G, S_2)$ is a quasi isometry.
In particular, the definition of Gromov hyperbolicity of a finitely generated group does not depend on the choice of the finite generating set.
\item Let $\Gamma$ be a group acting properly, cocompactly and isometrically on some geodesic metric space $X$.
Then $\Gamma$ is finitely generated and for any reference point $x_0 \in X$ and finite set of generators the orbit map $\Gamma \to X$ with
$\gamma \mapsto \gamma x_0$ extends to a quasi isometry between $\CC(G, S)$ and $X$ (see e.g. \cite[Prop. I.8.19]{BH99}).
In particular,  $\Gamma$ is Gromov hyperbolic if and only if $X$ is Gromov hyperbolic.
\end{itemize}
\end{remark}

Assume that $\Gamma$ is a Gromov hyperbolic group acting properly and cocompactly
by isometries on  proper metric space $X$. Consider for  $x\in X$ and $R>0$ the subset of  $\Gamma$ given by
$$
\Gamma_R(x) =  \{\gamma \in \Gamma \mid\; d(x,\gamma x) \leq R\}.
$$
 Then by a result of Coornaert \cite{mCo93} there are  constants $0 <C_1 \le C_2$ and $h>0$ such that
\begin{equation}\label{eqn:co93}
C_1 e^{hR}\le \card \ \Gamma_R(x) \le C_2 e^{hR}
\end{equation}
If $(X,g)$ is a Riemannian manifold then $h$ is the volume entropy $\hvol(g)$, i.e.
$$
h =  \lim\limits_{r \to \infty} \frac{\log \mathrm{vol} B(p,r)}{r}
$$
Define the translation length of $\gamma \in \Gamma$ by
 \begin{equation} \label{e: length}
\ell(\gamma) = \inf_{x \in X} d(x,\gamma x).
\end{equation}
Note that the infimum in
\eqref{e: length} is attained
for each $\gamma \in \Gamma$ (see
for example \cite[Prop. II.6.10]{BH99}). Furthermore, $\ell(\gamma^{-1}) = \ell(\gamma)$ and
$\ell(\alpha\gamma\alpha^{-1}) = \ell(\gamma)$ for
every isometry $\alpha$ of $X$. Therefore the length of a conjugacy class $[\gamma]$  of $\gamma$  can be defined as the translation length of a representative of  $[\gamma]$.
Using ideas developed  in \cite{gK83} and \cite{gK97} we proved in  \cite{CK02}:
\begin{theorem} \label{thm:conj-Grhyp}
Let $\Gamma$ be a group acting
properly and cocompactly by isometries on
a proper geodesic  Gromov-hyperbolic
metric space  $X$. Assume
that the Gromov boundary $\partial X$ of $X$
contains more than two points.
  For all $t \geq 0$, let
$$
P(t) = \card \{ [\gamma] \mid\; \gamma \in \Gamma \;  \; \text{is primitive and }   \; \; \ell(\gamma) \leq t \},
$$
where $ \gamma \in \Gamma$ is called primitive if it cannot be written as a proper power $\gamma = \alpha^n$ for some $\alpha \in \Gamma$ and $ n \ge 2$.

  Then there exists
constants   $A\ge 1$,  $t_0 >0$ and such that
$$
\frac{1}{A} \frac{ e^{ht} } {t} \leq P(t) \leq A e^{ht}
$$
for all $t \geq t_0$, where $h>0$  is as in \ref{eqn:co93}.
\end{theorem}
\begin{remark} \label{rem:conj-Grhyp}

This results has the following interpretation in the Riemannian setting.
Assume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group
$\pi_1(M)$. Let $\wM$ be the universal covering then $M = \wM/ \Gamma$ where  $\Gamma$   is the group of covering transformations isomorphic to $\pi_1(M)$.
As mentioned above
$h$ is the volume entropy $\hvol(g)$  of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.
By a result of Manning \cite{aM79}, the topological entropy $\htop(\phi)$ of the geodesic flow $\phi^t$ on $SM$ is larger than the volume entropy.

\end{remark}

\subsection{Closed manifolds without conjugate points and  Gromov hyperbolic fundamental groups}\label{subsec:Noconjugatepts-Ghyp}
In the following we will assume that $(M,g)$ is a closed Riemannina manifold without conjugate points and Gromov hyperbolic fundamental
group $\pi_1(M)$. As pointed out above in this case Freir\'{e}  and  Ma\~n\'e  \cite{FM82}
proved that
the volume entropy $\hvol(g)$ is equal to the topological entropy $\htop(\phi)$.

First we recall the following theorem which is foundational in the theory
of manifolds without conjugate points.
\begin{theorem}(Hadamard-Cartan)\label{thm: Hadamard-Cartan}\\
Let $(M,g)$ be a complete $n$-dimensional Riemannian manifold with
no conjugate points. Then for all $p \in M$ the exponential map
$\exp_p: T_pM \to M$ is a covering map.
In particular complete simply connected manifolds without conjugate points are diffeomorph to
$\R^n$.
\end{theorem}
\begin{remark}
\begin{enumerate}
\item[{\rm (a)}] A complete Riemannian manifold $(M,g)$ has no conjugate points iff for any pair of points on the universal cover $\wM$ there is a unique connecting geodesic geodesics with respect to the lifted metric. In particular all geodesics are minimizing.
\item[{\rm (b)}] The topology of those manifolds is to a large extend determined by the
fundamental group since the contractibility of the universal cover implies that the higher
homotopy groups are vanishing, i.e., $\pi_k(M) = 0$ for $k \ge 2$.
\item[{\rm (c)}] Manifolds of non-positive sectional curvature form an important subclass of manifolds with no conjugate points. Simply connected complete manifolds of nonpositive curvature are called Hadamard manifolds.

\end{enumerate}
\end{remark}

We assume that $(M,g)$ satisfies the divergence property, i.e.
 for any pair of geodesics  $c_1 \neq c_2$ in $( \wM, g)$ with $c_1(0) =c_2(0)$
we have
\begin{equation}\label{eqn:divergence}
\lim\limits_{t \to \infty} d(c_1(t), c_2(t)) = \infty.
\end{equation}
\begin{remark}
\begin{itemize}
\item
By a result of E. Hopf \cite{eH48} all closed non-flat surfaces without conjugate points have genus at least two. Since such surfaces
carry a metric with negative curvature their fundamental groups are Gromov hyperbolic. Furthermore,  Green \cite{wG56} showed that surfaces without conjugate have the divergence property.
\item
Until now there is no example of a closed manifold without conjugate points known where the divergence property does not hold. A sufficient condition for the divergence property is the continuity of the stable Jacobi-tensors (see \cite{ES76}). In particular, this assumption holds if $(M,g)$ has non-positive curvature, or more generally no focal points.
\end{itemize}
\end{remark}
  The following notion was introduced by Eberlein  in \cite{pEb72} and  Eberlein and  O'Neill \cite{EO73}.
\begin{definition}
A simply connected Riemannian manifold $\wM$ without conjugate points
is a \emph{(uniform) visibility manifold} if for every $\epsilon>0$ there exists $L>0$ such that whenever a geodesic $c\colon [a, b] \to\wM$ stays at distance at least $L$ from some point $p\in\wM$, then the angle sustained by $c$ at $p$ is less than $\epsilon$, that is
  \begin{equation*}
    \angle_p(c)=\sup_{a\leq s,t\leq b} \angle_p(c(s),c(t))<\epsilon.
  \end{equation*}
\end{definition}
The following Theorem is due to Ruggiero \cite{rR03},
  \begin{theorem}\label{thm:visibility}
Let $(M,g)$ be a closed manifold without conjugate points and Gromov hyperbolic fundamental group. Then $( \wM, g)$ is a visibility manifold if and only if $(M,g)$ has the divergence property.
\end{theorem}
\begin{remark}
Under the stronger assumption that $(M,g)$ is a closed manifold without conjugate points admitting a background metric of negative curvature this has been proved by Eberlein  \cite{pEb72}.
\end{remark}
Since $\wM$ is Gromov hyperbolic,  Lemma \ref{lem:GB} and the divergence property implies that the map
$f_p\colon S_p\wM \to \ideal$ defined by $f_p(v) = c_v(\infty)$ is a bijection.
The topology (sphere-topology) on $\partial \wM$ is defined such that $f_p$
becomes a homeomorphism.
Since for all $q \in \wM$ the map $f_q^{-1} f_p \colon S_p\wM \to S_q\wM$ is a homeomorphism, see \cite{pEb72},
the topology is independent on the reference point $p$.
The topologies on $\partial \wM$ and $\wM$
extend naturally
 to  $\cl (\wM): =  \wM\cup \partial \wM$
by requiring that the map
$\varphi\colon B_1(p) = \{v \in T_p \wM:   \|v\| \le 1\} \to \cl(\wM)$
defined by
\[
\varphi(v) = \begin{cases}
 \exp_p\left(\frac{v}{1-\|v\|}\right) & \|v\| < 1\\
f_p(v) & \|v\| = 1
\end{cases}
\]
is a homeomorphism. This topology, called the cone topology, was introduced by Eberlein
and O'Neill \cite{EO73} in the case of Hadamard manifolds and by Eberlein   \cite{pEb72} in the case of visibility manifolds. In
particular, $\cl( \wM) $ is homeomorphic to a closed ball in
$\mathbb{R}^n$. The relative topology on $\ideal$ coincides with the sphere topology, and the relative topology on $\wM$ coincides with the topology of $\wM$.

Note that for the case which we are considering
this compactification agrees with the compactification of  $\delta$-hyperbolic spaces do to Gromov \cite{mG87}.

For simply connected manifolds $X$ without conjugate points for $v\in SX$, the limit
\[
b_{v}(q) := \lim_{t\to\infty} \left( d(q, c_{v}(t)) - t\right)
\]
exists and is called the \emph{Busemann function} associated to $v$. In \cite{gK85} it was shown that Busemann functions are of class $C^{1,1}$ provided the sectional curvature is uniformly bounded from below.
\begin{definition}\label{def:busemann}
Let $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group.
Then according to Lemma \ref{lem:GB} and the divergence property,
for each $p\in X$ and $\xi \in \partial X$ there exists a uniquely determined vector $v\in S_p X$  such that $c_{v}(\infty) = \xi$.
We call $b_\xi(q,p) := b_{v}(q)$ the Busemann function based at $\xi$
and normalized by $b_\xi(p,p) =0$.
\end{definition}

\begin{remark}\label{rem:minmal}
The isometric action of $\Gamma=\pi_1(M)$ on $\wM$ extends to a continuous action on $\ideal$.
Since by \cite{pEb72} the geodesic flow is topologically transitive, every $\Gamma$-orbit in $\ideal$ is dense, i.e. the action on  $\ideal$ is minimal.
\end{remark}

\begin{proposition}\label{cor:busemann}
Let $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group.
Then the following holds
\begin{enumerate}
\item
For $p,q \in \wM$ and $\xi \in \ideal$ we have
$$
\lim\limits_{z \to \xi }d(q,z) -d(p,z) = b_p(q, \xi)
$$
\item
For all $p,q,z\in \wM$ we have
$$b_{q}(z,\xi) = b_p(z,\xi) - b_p(q,\xi)$$
In particular $b_{q}(z,\xi) = -  b_z(q,\xi)$
\end{enumerate}
\end{proposition}

\begin{proof}
For a proof see e.g. \cite{CKW21}
\end{proof}

\begin{MainThm}\label{thm:expansiveMME}
 Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov
 hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \delta$, where $\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow
 has a unique measure of maximal entropy $\mu$.

Furthermore, $\mu(\EE) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.
\end{MainThm}

\begin{MainThm}\label{thm:main}
 Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.
  Assume that
  $$
 h_\nu(\phi)<\htop(\phi)
 $$
for all non-expansive measures
$\nu \in \MMM_\phi(SM)$.   Then the geodesic flow has a unique measure $\mu$ of maximal entropy.

 Furthermore, $\mu(\EE) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.

\end{MainThm}

\label{sec:intro}
If $f: X \to X$ is a homeomorphism of a compact metric space $X$,
the topological entropy $h_{\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an
exponential scale. On the other hand, measure-theoretic entropy $h_{\mu}(f)$ is an invariant of
measurable dynamics reflecting the average complexity of the system relative
to a measure $\mu$ contained in the space of $f$-invariant Borel probability measure $\MMM_f(X)$ on $X$.
The relation between these notions is provided by the variational principle which implies (see e.g. \cite{pW82} for details)
$$
h_{\mathrm{top}} (f) = \sup \{h_\mu (f) \;|\; \mu \in \MMM_f(X)\}.
$$
A measure $\mu \in \MMM_f(X)$ such that $h_{\mu}(f) =h_{\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).
For continuous flows $\phi^t: X \to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\phi := \phi^1$.

In the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.
If $(M,g)$ is a closed Riemannian manifold without conjugate points
(all geodesics are globally minimizing if lifted to the universal cover $\widetilde{M}$) then by a theorem of Freir\'{e} and Ma\~n\'e \cite{FM82} the topological entropy $\htop (\phi)$ of the geodesic
flow $\phi^t: SM \to SM$ agrees with the volume entropy $\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \subset \widetilde{M}$ in the universal covering $\widetilde{M}$ of $M$ and is given by
$$
\hvol(g) = \lim\limits_{r \to \infty} \frac{\log \mathrm{vol} B(p,r)}{r}
$$
For manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points,
an easier proof of the result of Freir\'{e} and Ma\~n\'e were previously given by Manning \cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points
is not a local condition. Properties like the convexity of distance functions which
 hold for manifolds of non-positive sectional curvature and even no focal points
are not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds
of non-positive sectional curvature, fail under the assumption of no conjugate points \cite{kB92}.

In this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.
Hyperbolicity implies the existence of a non-trivial expansive set $\mathcal{E} \subset SM$ consisting of flow lines
which diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \in SM$ is contained in $\mathcal{E}$ if there is some
 $\epsilon >0$ such that for all $w \in SM $ with $d(c_v(t), c_w(\mathbb{R})) < \epsilon$ for all $t \in \mathbb{R}$,
the geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set
is invariant under the geodesic flow.

Geodesic flows are called expansive if $SM =\mathcal{E}$.
In case that $(M,g)$ has no conjugate points it was proved by Ruggiero \cite{rR94} that the expansiveness of the geodesic
flow
implies that the fundamental group $\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \infty) \to \widetilde{M} $ on the universal cover we have
$\sup_{t \ge 0} d(c_1(t),c_2(t)) = \infty$.
While negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.

In the following we call an invariant Borel probability measure $\nu \in \MMM_\phi(SM)$ of the geodesic flow $\phi^t: SM \to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\nu(\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.

In the following we call an invariant Borel probability measure $\nu \in \MMM_\phi(SM)$ of the geodesic flow $\phi^t: SM \to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\nu(\EE) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.

In this section we prove first that $\mu$ is the unique MME under the following conditions.
\begin{theorem}\label{thm:uniqueMME-inj}
Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and with Gromov hyperbolic fundamental group. Let $\delta $ the Gromov hyperbolicity constant of the universal cover $\wM$ and assume that the injectivity radius $\inj(M)$ of $M$ is larger than $16 \delta$. Assume that at least one of the following two
conditions is fulfilled:
\begin{enumerate}
\item
The entropy of non-expansive measures is strictly smaller than the topological entropy, i.e.
$$ h_\nu(\phi)< \htop(\phi) \; \; \text{for all} \; \; \nu \in \MMM_\phi(SM) \; \; with \; \;\nu(\EE)=0$$
\item
The expansive set $\EE \subset SM$ has non-empty interior and the geodesic flow $\phi^t: SM \to SM$ is entropy expansive
at some scale larger than $8 \delta$.
\end{enumerate}
Then the measure $\mu$ is the unique measure of maximal entropy of the geodesic flow.
\end{theorem}

Now we can prove Theorem \ref{thm:main} and Theorem \ref{thm:expansiveMME} using Theorem \ref{thm:uniqueMME-inj}
as follows.
\begin{proof}
Let $(M,g)$ be a closed manifold Riemannian manifold without conjugate points and divergence property.
 Furthermore, assume that the fundamental group of $M$ is residually finite and Gromov hyperbolic fundamental group with Gromov hyperbolicity.
 By Proposition \ref{prop:large-inj} there is a smooth Riemannian manifold $N$ and is for some
$k \in \N$ a locally isometric $k$ to $1$ covering map $p\colon N\to M$ such that the injectivity radius of $N$ is larger
than $8 \delta$. In particular $N$ has no conjugate, the divergence property
and the geodesic $\phi^t_{SM}: SM \to SM$ is $k$ to $1$ factor of the
geodesic flow $\phi^t_{SN} : SN \to SN$ as well. If $\mu$ is a invariant Borel probability measure on $SM$
we denote by $\tmu$ the canonical lift defined by
$$
\tmu(A) = \int_{SM} \frac{1} {k}\card \{dp^{-1} (v) \cap A \}d\mu(v),
$$
where $dp :SN \to SM$ is the differential of $p$.
Obviously is $\tmu$ is $\phi^t_{SN}$ invariant Borel probability measure with $dp_*\tmu =\mu$ and
$h_{\tmu}(\phi_{SN}) = h_{\mu}(\phi_{SM})$.
Since the expansive set of $SN$ is the lift of the expansive set in $SM$
and the expansivity constants of $\phi^t_{SM}: SM \to SM$ and $\phi^t_{SN} : SN \to SN$ agree
the assumptions of Theorem \ref{thm:main} or Theorem \ref{thm:expansiveMME} imply that
$\phi^t_{SN} : SN \to SN$ fullfils the assumptions
of Theorem \ref{thm:uniqueMME-inj}. Hence $\phi^t_{SN} : SN \to SN$ has a unique measure of maximal entropy.
 If $\mu_1$ and $\mu_2$ are measures of maximal entropy for $\phi^t_{SM}$
then their lifts $\tmu_1$ and $\tmu_2$ are measures of maximal entropy for $\phi^t_{SN}$ as well.
In particular $\tmu_1 = \tmu_2$ therefore $\mu_1 = \mu_2 = \mu$.

For the construction of Green bundles we need to define stable and unstable Jacobi tensors. Since $(M.g)$ has no conjugate points
for each $r > 0$, there exists orthogonal Jacobi tensors along $c_v$ with
\begin{align*}
S_{v,r}(0) = \Id_{v^\bot}, &\quad S_{v,r}(r) = 0, \\
U_{v,r}(0) = \Id_{v^\bot}, &\quad U_{v,r}(-r) = 0.
\end{align*}
Note that we have $U_{v,r}(t) = S_{-v,r}(-t)$.
The stable and unstable Jacobi tensor along $c_v$ are defined via the following initial conditions
$$ S_v(0) = \Id_{v^\bot}, \quad S_v'(0) = \lim_{r \to \infty} S_{v,r}'(0), $$
and similarly
$$ U_v(0) = \Id_{v^\bot}, \quad U_v'(0) = \lim_{r \to \infty} U_{v,r}'(0). $$
The existence of $\lim_{r \to \infty }S_{v,r}'(0)$ follows from the monotonicity $S'_{v,r}(0) <S'_{v,s}(0)$ for $r < s$ and the upper bound
$S'_{v,r}(0) < U'_{v,l}(0)$ for all $r, l > 0$. Moreover, we introduce the symmetric endomorphisms
$$ S(v) = S'_v(0) \quad \text{and} \quad U(v) = U'_v(0). $$
In particular $U(v) -S(v) \ge 0$. Since $U_{v,r}(t) = S_{-v,r}(-t)$ for all $t \in \R$ and $r>0$ we obtain $U_{v}(t) = S_{-v}(-t)$ and therefore
$$
U(v) = U'_v(0) = - S'_{-v}(0) = -S(-v).
$$
Furthermore,
\begin{equation}\label{eqn:invGbundles}
U(\phi^tv) = U'_v(t) U^{-1}_v(t) \; \text{and} \;S(\phi^tv) = S'_v(t) S^{-1}_v(t)
\end{equation}
 \begin{definition}\label{def:Green bundles}
 Let $(M,g)$ be a manifold without conjugate points. Then the subbundles $E^s$ and $E^$ of $TSM$ defined by
 $$
 E^s(v) =\{(x, U(v) x) \mid x \in T_{\pi (v)}M, x\perp v \}
 $$
 and
 $$
 E^u(v) =\{(x, U(v) x) \mid x \in T_{\pi (v)}M, x\perp v \}
 $$
 are called the stable and unstable Green bundles
 \end{definition}
 \begin{remark}
 From the equations \ref{eqn:invGbundles} follows the invariance of the Green bundles under the geodesic flow $\phi^t$, i.e.
 $$
 D\phi^t(v)E^s(v) = E^s(\phi^t(v)) \; \text{and} \; D\phi^t(v)E^u(v) = E^u(\phi^t(v))
 $$
 for all $t \in \R$.
 \end{remark}
 For the metric
entropy of the geodesic flow,
Freir\'{e} and Ma\~n\'e \cite{FM82}
 obtained, based on Ruelle's inequality \cite{dR78} and Pesin's \cite{jPe77} formula the following estimate.
 \begin{theorem}\label{thm: FM}
Let $(M,g)$ be a compact manifold without conjugate points and $\mu \in \MMM_\phi (SM) $ be a
$\phi^t$-invariant
 probability measure for the geodesic flow.
Then
$$
h_\mu(\phi^t) \le \int\limits_{SM} \tr U d \mu.
$$
Furthermore equality holds if $\mu$
is given by the Liouville measure.
\end{theorem}
\subsection{Positivity of the Liouville entropy}
In the \cite{gK85} we showed that for manifolds without conjugate points and continuous asymptote positivity the metric entropy
of the Liouville measure provided the topological entropy is positive as well.
Since the paper is not easily accessible we provide the proof in this paper.
\begin{theorem}\label{Kn-thesis}
Let $(M,g)$ be a compact manifold without conjugate points and continuous Green bundles.
If the the geodesic flow $\phi^t$ has positive topological entropy $\htop(\phi) $ then the metric entropy $h_\lambda(\phi)$ of the geodesic flow with respect to the Liouville measure $\lambda$.
\end{theorem}
\begin{proof}
Assume that $\htop(\phi) >0$. Then by the variational principle there exists a measure $\mu \in \MMM_\phi (SM) $ such that
$$
h_\mu(\phi) \ge \frac{1}{2} \htop(\phi) >0
$$
Let $F: SM \to SM$ be the flip map given by $F(v) = -v$. Then the push forward measure $ \mu_F$ given
by $ \mu_F(A) = \mu (F(A))$ for all measurable sets $A \subset SM$ is flow-invaraint as well since
$$
 \mu_F(\phi^t A) = \mu ( F(\phi^t(A)) = \mu ( \phi^{-t} F(A)) = \mu(F(A)) = \mu_F(A)
 $$
 Consider the measure
 $\nu = \frac{1}{2}( \mu + \mu_F) $. The affine property of the metric entropy
 yields
 $$
 h_\nu (\phi) = \frac{1}{2}h_\mu (\phi) + \frac{1}{2}h_{\mu_F} (\phi) >0
 $$
 From theorem \ref{thm: FM} we obtain $h_\nu (\phi_t) \le \int\limits_{SM} \mathrm{tr} U(v) d \nu $.
 Since $\tr U(v) = -\tr S(-v)$ the flip invariance of $\nu$ implies
 $$
 \int\limits_{SM} \tr U(v) d \nu = \int\limits_{SM} -\tr S(v) d \nu
$$
Hence
$$
0 < h_\nu (\phi_t) \le \frac{1}{2}\int\limits_{SM} \tr(U(v) - S(v)) d \nu
$$
Since $\tr(U(v) - S(v)) \ge 0$ and by the continuity of the Green Bundles $v \mapsto U(v)-S(v)$ is continuous the set
$$
Q:= \{v \in SM \mid \mathrm{tr} (U(v) - S(v) )> 0 \}
$$
is open and non-empty by $ 0 < h_\nu (\phi_t)$.
In particular $\lambda(Q) >0$ . Using that
$ h_\lambda (\phi_t) = \int\limits_{SM} \mathrm{tr} U(v) d \lambda$ holds
the flip invariance of the Liouville measure $ \lambda$ yields
$$
h_\lambda (\phi_t) = \frac{1}{2}\int\limits_{SM} \mathrm{tr} (U(v) - S(v) )d \lambda >0.
$$

\begin{remark}
\begin{itemize}
\item Closed Riemannian manifolds without conjugate points and Gromov hyperbolic fundamental group have positive topological
entropy (see \cite{CK02} and Remark  \ref{rem:conj-Grhyp}).
\item It follows from the work of Newhouse \cite{sN89} that for smooth geodesic flows $\phi^t$
there exists a measure $\nu \in \MMM_\phi(SM)$ of maximal entropy, i.e. $h_\nu(\phi)= \htop(\phi)$.
Hence, the assumption of our theorem forces  $\nu(\mathcal{E})$ to be positive. In particular, the expansive set is not empty.
\item If $(M,g)$ is a closed and non-flat  surface without conjugate points, then all assumptions of Theorem \ref{thm:main}
hold (see \cite{CKW21}). Therefore such surfaces have a unique MME.
\item If $(M,g)$ is a closed smooth Riemannian manifold without conjugate points and expansive geodesic flow (i.e $\mathcal{E} =SM$)
 the uniqueness of the MME has been obtained by Bosch\'e in \cite{aB18}.
As we mentioned above a closed manifold without conjugate points and expansive geodesic flow has the divergence
property and Gromov hyperbolic fundamental group. Since for expansive geodesic flows non-expansive measures obviously do not exist,   the result of Bosch\'e follows from our theorem as a special case.
\item If $(M,g)$ is a closed rank 1 manifold of non-positive curvature the rank 1 (regular)  set consists of orbits of the geodesic flow which do not have non-trivial parallel Jacobi-fields orthogonal to the geodesic (see e.g. \cite{gK97} or \cite{gK98}).
It is a consequence of the flat strip theorem that the regular set is contained in the expansive set.
In  \cite{gK98} we showed that for closed rank 1 manifolds the geodesic flow has a unique MME. Furthermore, the measure has full weight
on the regular, and hence, on the expansive set. A different proof of the uniqueness of the MME was later given in  \cite{BCFT}.
Since there are examples of closed rank 1 manifolds whose fundamental group has $(\mathbb{Z}^2,+)$ as a subgroup, the
fundamental group is generally not Gromov hyperbolic (see \cite{gK98}).
\end{itemize}
\end{remark}
In \cite{CKW21} Climenhaga, War and the author proved Theorem \ref{thm:main} under the more special condition of a background
metric of negative curvature and
the slightly stronger entropy gap assumption
$$
 \sup\{h_\nu(\phi) : \nu\in \MMM_\phi(SM), \nu(\mathcal{E})=0\} < \htop(\phi).
 $$
 for non-expansive measures.
 The entropy gap would follow from our condition that non-expansive measure do not have maximal
 entropy provided the expansive set is open.  Namely, if $\mathcal{E}$ is open and the entropy gap would not hold there would exist a sequence of measures
 $\nu_n \in \MMM_\phi(SM)$   with $\nu_n(\mathcal{E})=0 $ converging weakly to $\nu$ and $\lim_{n \to \infty}h_{\nu_n}(\phi) = \htop(\phi)$. Since  $\mathcal{E}$ is open, $\nu(\mathcal{E}) \le \liminf_{n \to \infty}\nu_n(\mathcal{E})= 0$ and by upper semi-continuity of entropy this would yield $h_{\nu}(\phi) = \htop(\phi)$. But this contradicts our assumption that  $h_{\nu}(\phi) <\htop(\phi)$ for non-expansive measures.
 However, to our knowledge openness of the expansive set is not known in our setting.

The assumption of a background
metric of negative curvature implies that the fundamental group is Gromov hyperbolic (see subsection \ref{subsec:hypgroups}).
Due to the uniformization theorem for surfaces and the proof of the geometrization conjecture in dimension three
Gromov hyperbolicity of the fundamental group implies the existence of a metric with even constant negative curvature.
However,  for closed manifolds of dimension bigger than three,  Gromov hyperbolicity of the fundamental group and existence of a metric without conjugate points might not be enough to yield a metric of negative curvature. In any case, to provide a solution to this question is a very difficult problem.

In \cite{CKW21} the proof of the uniqueness of the measure of maximal entropy used the background metric  of negative curvature
to establish with the help of the Morse Lemma a coarse specification property
for the geodesic flow.  Applying the work of Climenhaga and Thompson \cite{CT16}, the specification property was used
to prove the uniqueness of the MME.

 However, the proof of the above theorem does not require specification but relied on methods derived in a paper
 of the author  \cite{gK98} on the uniqueness of the measure of maximal entropy for geodesic flows on non-positively curved rank 1 manifolds.\\

 There is a interesting and quite flexible notion due to Bowen \cite{rBo72}, called entropy expansiveness  which hold for many dynamical systems for which expansiveness fails (see section \ref{sec:e-expansiveness}).

 Given a closed Riemannian manifold $(M,g)$ without conjugate points and   $(\widetilde{M},g)$
 be the universal cover with the lifted Riemannian metric denoted again by $g$. For $v \in S\widetilde{M}$ and $\rho >0$ we define
 the set
 $$
 Z_{\rho} (v) = \{w \in S\widetilde{M} \mid d(c_v(t), c_w(t)) \le \rho, t \in \mathbb{R} \}.
$$
 The geodesic flow $\phi^t: SM \to SM$  is called entropy expansive at scale $\rho >0$
if $\htop(\tilde \phi, Z_\rho(v))  = 0$, where $\tilde \phi^t$ is the geodesic flow lifted to $S \widetilde{M}$.
Due to the flat strip theorem geodesic flows on manifolds of non-positive curvature or more generally no focal points are entropy expansive at any scale (see \cite[proposition 3.3]{gK98}).
This also holds for non-flat surfaces without conjugate points \cite[Lemma 4.5]{GKOS14} even so the flat strip theorem fails in this case \cite{kB92}. As far as we know, there is no example of a closed Riemannian manifold with metric without conjugate known for which the
geodesic flow is not entropy expansive for all or even some $\rho >0$. Alternatively to Theorem \ref{thm:main}, we obtain
the uniqueness of the MME under the following conditions.
\begin{MainThm}\label{thm:expansiveMME}
 Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov
 hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \delta$, where $\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow
 has a unique measure of maximal entropy $\mu$.

Furthermore, $\mu(\mathcal{E}) =1$, $\mu$  is mixing and fully supported on the unit tangent bundle.
\end{MainThm}

 Using the results of Climenhaga, War and the author proved in \cite[Theorem 1.2]{CKW21}
 together with Theorem \ref{thm:conj-Grhyp} and Remark \ref{rem:conj-Grhyp},
 we can conclude that the measure of maximal entropy is given by the limiting distribution of closed orbits.

Furthermore, as in \cite[Theorem 1.2]{CKW22}, an estimate on the growth of pairwise non-free-homotopic closed geodesics,
obtained by Margulis \cite{gM69} in the case of negative curvature, follows. More precisely:

 \begin{MainThm}\label{thm:closed geodesics}
	Let $(M,g)$ be a closed Riemannian manifold such that the assumption in Theorem \ref{thm:main}
	or Theorem \ref{thm:expansiveMME} hold. Denote for $T>0$  by $\mathcal{P}(T)$ be any maximal set of pairwise non-free-homotopic closed geodesics of minimal length in the free homotopy classes and $P(T) =\mathrm{card} \mathcal{P}(T) $ its cardinality. Consider the measures
\begin{equation*}\label{eqn:nut}
\mu_T= \frac {1}{P(T)} \sum_{c\in \mathcal{P}(T)} \frac{\Leb_c}{T},
\end{equation*}
where $\Leb_c$ is Lebesgue measure (length) along the curve $\dot{c}$ in the unit tangent bundle $SM$. Then
\begin{enumerate}
\item \label{eqn:distr}

The measures $\mu_T$ converge  in the weak* topology  as $T\to \infty$ to the measure of maximal entropy.

\item
Furthermore,
\begin{equation*}\label{eqn:margulis}
P(T) \sim \frac{e^{hT}}{hT},
\end{equation*}
which means that the ratio of $P(T)$ and  $\frac{e^{hT} }{hT} $ converges $1$ as $T \to \infty$.
\end{enumerate}

\end{MainThm}

In \cite{CKW21} the uniqueness of the MME in Theorem~\ref{thm:main} was proved (under a background metric of negative curvature) by using the Morse Lemma to establish a \emph{coarse specification property} for the geodesic flow, and then applying the work of Climenhaga--Thompson \cite{CT16} to deduce uniqueness of the MME from specification.

The text also explains why their (slightly stronger) \emph{entropy gap} assumption
\[
\sup\{h_\nu(\phi): \nu\in\MMM_\phi(SM),\,\nu(\mathcal E)=0\}<h_{\mathrm{top}}(\phi)
\]
would follow from the present hypothesis “$h_\nu(\phi)<h_{\mathrm{top}}(\phi)$ for all non-expansive measures” \emph{if} the expansive set $\mathcal E$ were open: otherwise one could take measures $\nu_n$ with $\nu_n(\mathcal E)=0$ and $h_{\nu_n}(\phi)\to h_{\mathrm{top}}(\phi)$; by openness, the weak* limit $\nu$ still has $\nu(\mathcal E)=0$, and by upper semi-continuity of entropy one gets $h_\nu(\phi)=h_{\mathrm{top}}(\phi)$, contradicting the assumption.

For the present theorem, the text says that its proof “does not require specification” and instead “relied on methods derived in” \cite{gK98} (developed for uniqueness of the MME for geodesic flows on non-positively curved rank 1 manifolds).

Question:
Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points, and let $\phi^t:SM\to SM$ be its geodesic flow on the unit tangent bundle. Write $\phi=\phi^1$, so that $h_\nu(\phi)$ denotes the metric entropy of the time-$1$ map and $h_{\mathrm{top}}(\phi)$ its topological entropy. Assume that geodesic rays on the universal cover $\widetilde M$ have the divergence property: for every pair of geodesic rays $c_1,c_2:[0,\infty)\to\widetilde M$, one has $\sup_{t\ge 0} d(c_1(t),c_2(t))=\infty$. Assume also that the fundamental group $\pi_1(M)$ is Gromov hyperbolic and residually finite. Define the expansive set $\mathcal E\subset SM$ by declaring that $v\in\mathcal E$ if there exists $\varepsilon>0$ such that for every $w\in SM$ with $d(c_v(t),c_w(\mathbb R))<\varepsilon$ for all $t\in\mathbb R$, the geodesics $c_v$ and $c_w$ agree up to a time shift. Call a $\phi^t$-invariant Borel probability measure $\nu$ non-expansive if $\nu(\mathcal E)=0$. Suppose that every non-expansive measure $\nu\in\mathcal M_\phi(SM)$ satisfies
$$
h_\nu(\phi)<h_{\mathrm{top}}(\phi).
$$
Which conclusion about the geodesic flow holds?

There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu(\mathcal E)=1$, $\mu$ is mixing for the geodesic flow, and $\operatorname{supp}(\mu)=SM$.

Choices:
B. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, every $\phi^t$-invariant Borel probability measure gives full weight to the expansive set, $\nu(\mathcal E)=1$ for all $\nu\in\mathcal M_\phi(SM)$, and $\mu$ is mixing with $\operatorname{supp}(\mu)=SM$.
C. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu(\mathcal E)=1$.
D. There exists a measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$ with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$, and every measure of maximal entropy is mixing for the geodesic flow and fully supported on $SM$; however, the hypotheses do not force uniqueness of the measure of maximal entropy or the conclusion $\mu(\mathcal E)=1$.
E. There exists a unique measure of maximal entropy $\mu\in\mathcal M_\phi(SM)$, meaning a unique $\phi^t$-invariant Borel probability measure with $h_\mu(\phi)=h_{\mathrm{top}}(\phi)$. Moreover, $\mu$ is ergodic and supported on the expansive set $\mathcal E$, but one cannot conclude that $\mu$ is mixing or that $\operatorname{supp}(\mu)=SM$.

dropped mixing and full-support conclusions

uniqueness of MME and full-weight expansivity of the MME

mixing and full support strengthened down to mere ergodicity/support on expansive set

The stem states hypotheses and definitions but does not reveal the conclusion. The correct answer is not explicitly or trivially encoded in the wording.

This is essentially a theorem-conclusion identification question: the stem lists a detailed hypothesis package and asks for the resulting conclusion. However, the choices do force discrimination among stronger, weaker, and overgeneralized claims rather than merely repeating one sentence verbatim.

The item requires some precision in selecting the strongest justified conclusion, especially distinguishing uniqueness, mixing, full support, and the scope of expansivity. Still, it mostly tests recall/recognition of a theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and well-targeted: one is an overgeneralization to all invariant measures, one is a weaker true statement, and others confuse uniqueness, mixing, ergodicity, and support. These reflect realistic failure modes.

A solid but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it leans more toward precise recognition of a known result than toward genuinely generative reasoning.