\label{T1}
 Let $f_{0} \in C^{1,\mu}_{0}\big(\bar{\Omega}\times\mathbb{R}^{2}\big), f_{0}\geq 0$ for some $0 <\mu< 1$, satisfying
\eqref{v2.8}.  Suppose that $h \in C^{2,\mu} (\partial\Omega)$ satisfies \eqref{v1.12} and $h > 0$. Then there exists
a unique solution $f \in C^{1;1,\lambda}_{t;(x,v)} \big((0,\infty)\times \bar{\Omega}\times \mathbb{R}^{2}\big),
\varphi\in  C^{1;3,\lambda}_{t;x}\big((0,\infty)\times \bar{\Omega}\big) $
for
some $0 < \lambda < \mu$, of the relativistic Vlasov-Poisson system \eqref{v1.5}-\eqref{v1.6} with compact support
in $x$ and $v$, where the  initial boundary conditions of $(f,\varphi)$ satisfy \eqref{v1.8}-\eqref{v1.10} and \eqref{v1.7}  respectively.

The relativistic Vlasov-Poisson system describes the collective dynamics of a collisionless plasma, where particles travel at nearly the speed of light and interact through their self-generated electric fields. The system consists of a relativistic Vlasov equation for the particle distribution function coupled with the Poisson equation for the electric potential.
In this paper, we are concerned with the global solution to the relativistic Vlasov-Poisson system (RVP) with general initial data in a convex, bounded domain $\Omega$ of two space dimensions:
 \begin{align}
&\partial_t f+\hat{v}\cdot\nabla_{x}f+\nabla_{x}\varphi\cdot\nabla_{v}f=0, \label{v1.5}\\
&\Delta\varphi=\rho, \label{v1.6}
\end{align}
where $x\in \Omega\subset \mathbb{R}^{2}$, $t>0$; $f=f(t,x,v)\geq 0$ denots  the  distribution density of particles at position $x$, time $t$, with momentum $v\in\mathbb{R}^{2}$;
$\hat{v}\in\mathbb{R}^{2}$ is the velocity that relates to the momentum $v$ according to Einstein's formula:
$$\hat{v}=\frac{v}{\sqrt{1+|v|^{2}}},$$
 $\varphi(t,x)$ is the electric potential, $\rho$  is the   charge  density   given by
 $$ \rho=\rho(t,x)= \int_{\mathbb{R}^{2}}f\, {\rm d}v,$$
 and the domain $\Omega$ is a  convex  bounded domain with $C^{5}$ boundary in $\mathbb{R}^{2}$.
 We refer readers to \cite{KF,W} for more background on the relativistic Vlasov-Poisson system.

In the non-relativistic case, one has the following Vlasov-Poisson system (VP):
\begin{align}
&\partial_t f+v\cdot\nabla_{x}f+\nabla_{x}\varphi\cdot\nabla_{v}f=0,\label{v1.1}\\
&\Delta\varphi=\rho.\label{v1.2}
\end{align}
There have been many mathematical results for the non-relativistic Vlasov-Poisson system \eqref{v1.1}-\eqref{v1.2}, and we refer to Arsen'ev \cite{A}, Batt \cite{B}, Horst \cite{H}, Bardos-Degond \cite{BD}, Pfaffelmoser \cite{P}, Lions-Perthame \cite{LP} and the references therein for the global existence of solutions and related results.
It should be emphasized that  Pfaffelmoser \cite{P} and Lions-Perthame \cite{LP} demonstrated the global existence of general initial data by differential approaches. Nonetheless, it is well known that the existence of global solutions for large data is a challenging problem for the relativistic Vlasov-Poisson system. As indicated in \cite{GS}, in general, the well-known issues that emerge in the classical VP \eqref{v1.1}-\eqref{v1.2} do not arise here since $|\frac{d}{ds}x(s)|=|\hat{v}(s)|\leq 1$  along the characteristics for RVP \eqref{v1.5}-\eqref{v1.6}, which seems better than VP. However, from the energy equality of RVP in $\mathbb{R}^{3}$,
$$  \int\int_{\mathbb{R}^{6}}\sqrt{1+|v|^{2}}f(t,x,v){\rm d}x{\rm d}v+\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla \varphi|^{2}{\rm d}x=\text{constant}, \nonumber
$$it follows that $\rho(t)\in L^{\frac{4}{3}}(\mathbb{R}^{3})$  for RVP,  while $\rho(t)\in L^{\frac{5}{3}}(\mathbb{R}^{3})$ for VP.
The primary challenge facing RVP at the moment is the loss of regularity for $\rho(t)$, which means that for general initial data, the global existence of classical  solutions  for RVP in $\mathbb{R}^{3}$  is still unknown.

For the Cauchy problem of the relativistic Vlasov-Poisson system  \eqref{v1.5}-\eqref{v1.6},  Glassey-Schaeffer\cite{GS,GS1}, and later Kiessling and Tahvildar-Zadeh \cite{KT2008} and Wang \cite{WangXC2023} established the spherically symmetric and  cylindrically symmetric solutions  in $\mathbb{R}^{3}$,
Rammaha\cite{R} proved the global existence for general initial data in $\mathbb{R}^{2}$, Had\v{z}i\'{c}-Rein \cite{H} obtained global existence and nonlinear stability.
For more results on related problems, we refer readers to \cite{GS2,GS3,GS4,GS5,GS6,GS7,GS8,K,WangXC2023} and their references.
Nevertheless, there are no mathematical studies of well-posedness for the RVP solutions in the case of domains with boundaries.
The motivation of this paper is to provide new insights into boundary-value problems in kinetic equations and to understand how boundaries influence the dynamics of RVP.

 For the distribution density, we consider the following initial and boundary conditions under which the distribution density exhibits specular reflection on the boundary:
   \begin{align}
&f(0,x,v)=f_{0}(x,v),\quad x\in \Omega,\; v\in \mathbb{R}^{2}, \label{v1.8}\\
&f(t,x,v)=f(t,x,v^{*}),\quad x\in \partial\Omega, \; v\in \mathbb{R}^{2},\; t>0, \label{v1.9}
\end{align}
 satisfying
 \begin{align}
 &f_{0}(x,v)\geq 0,  \label{v1.10}\\
 &v^{*}=v-2\big(n_{x}\cdot v\big)n_{x}, \quad (x,v)\in \partial\Omega \times \mathbb{R}^{2}, \label{v1.11}
 \end{align}
 where $n_{x}$ denotes the outer normal vector at $x\in\partial\Omega$.
Meanwhile, regarding electric potential, we analyze two different types of boundary conditions.
The first one is the Neumann boundary condition,
  \begin{align}
   &\frac{\partial \varphi}{\partial n_{x}}=h(x),\,\,\, x\in \partial \Omega,\,t>0,\label{v1.7}
   \end{align}
where the function $h$  is positive and satisfies the following compatibility condition:
\begin{align}
\int_{\Omega}f_{0}(x,v){\rm d}x{\rm d}v=\int_{\partial\Omega}h(x){\rm d}l, \label{v1.12}
\end{align}
and the second is the homogeneous Dirichlet boundary condition,
 \begin{align}
   & \varphi (t,x)=0,\,\,\, x\in \partial \Omega,\,t>0.\label{vD1.7}
 \end{align}

The well-posedness in bounded domains for VP has been extensively studied; see \cite{C,D,G1,G2,H1,HJV,HV,HV1,LM} and the references therein.
Regarding the theory of well-posedness in bounded domains, many additional issues have emerged compared to the Cauchy problem of the Vlasov-Poisson system. Tracking the evolution of the characteristic curves associated with \eqref{v1.5}-\eqref{v1.6} is one of the challenges that must be addressed, for this purpose
in \cite{G2} Guo introduced the following ``singular sets",
\begin{align}
 \Gamma=\big\{(x,v)\in\Omega\times\mathbb{R}^{3}, x\in \partial\Omega,v\in T_{x}\partial\Omega\big\},   \label{v1.13}
\end{align}
 where $T_{x}\partial\Omega\subset \mathbb{R}^{3}$ is the tangent plane to $\partial\Omega$ at the point $x$.

\subsection{ A more convenient coordinate system near $\partial\Omega\times \mathbb{R}^{2}$}
 Assuming that $\partial\Omega$ is a smooth curve in $\mathbb{R}^{2}$ and its parametric equation can be expressed as $r(l)=(x_{1}(l),x_{2}(l))$, where $l$ represents the arc length parameter.
 At the point $r(l)$, we shall denote the outer normal to $\partial\Omega$  by $n(l)$.

The implicit function theorem shows that for $\delta > 0$ sufficiently small we can
parameterize uniquely the set of points  $x\in \partial\Omega + B_{\delta}(0) \subset \mathbb{R}^{2}  $ by the unique
values $(l, x_{\bot})$ satisfying the equation,
\begin{align}
x = r(l)-x_{\bot}n (l).  \label{v2.1}
\end{align}
Set
$$U(l)=\frac{dr(l)}{dl},\quad W(l)=\frac{dU(l)}{dl}; \quad \text{then}\,\;  n(l)=-\frac{W(l)}{|W(l)|}.$$
By the Frenet-Serret formulas, we can obtain a set of unit orthogonal local coordinate frames  $\big(U(l),n(l)\big)$, and then represent any vector $v\in  \mathbb{R}^{2}$ as,
\begin{align}
v = v_{\|}(l)-v_{\bot}n (l), \label{v2.2}
\end{align}
where $v_{\|}(l)=\omega U(l)\in T_{r(l)}(\partial\Omega), v_{\bot}\in \mathbb{R}.$

For the set of points  in the phase space $\Omega\times \mathbb{R}^{2}$ that are
close to the boundary $\partial\Omega\times \mathbb{R}^{2}$, we can denote $f(t,x,v)=f(t,l,x_{\bot},\omega,v_{\bot})$ by the system of coordinate $(l,x_{\bot},\omega,v_{\bot})$
and the equation of $f(t,x,v)$ will satisfy the following new form,
\begin{align}
f_{t}+\frac{1}{\sqrt{1+|v|^{2}}}\frac{\omega}{1-kx_{\bot}}\frac{\partial f}{\partial l}+\frac{v_{\bot}}{\sqrt{1+|v|^{2}}}\frac{\partial f}{\partial x_{\bot}}
+\sigma \frac{\partial f}{\partial \omega}+F\frac{\partial f}{\partial v_{\bot}}=0,\label{v2.3}
\end{align}
where $k>0$ is the curvature of boundary curves and
\begin{align}
&E=\nabla \varphi=E_{l}U(l)-E_{\bot}n(l), \quad E_{\bot}=-h, \label{v2.4}\\
&\sigma=E_{l}+\frac{v_{\bot}}{\sqrt{1+|v|^{2}}}\frac{k\omega}{1-kx_{\bot}},\quad F=E_{\bot}-\frac{1}{\sqrt{1+|v|^{2}}}\frac{k\omega^{2}}{1-kx_{\bot}}.\label{v2.5}
\end{align}
\begin{Remark}
The proof of \eqref{v2.3} is a standard change of variables  using the classical Frenet-Serret formulas; we will not provide details here.
\end{Remark}
\begin{Remark}
Note $1-kx_{\bot}>0$ for sufficiently small $x_{\bot}$. Since the domain $\Omega $ is convex, and due to $h>0$ we have
$F < 0$.
\end{Remark}

\subsection{Compatibility conditions and assumptions for the initial and boundary data}

In the process of establishing a classical solution, it is necessary that the initial data $f_{0}(x,v)$ satisfies the following compatibility conditions  at the reflection points of
$\partial\Omega\times \mathbb{R}^{2}$ (cf.  \cite{G2,H1}),
\begin{align}
&f_{0}(x,v)=f_{0}(x,v^{*}),\label{v2.6}\\
&v_{\bot}\Big[\nabla_{x}^{\bot}f_{0}(x,v^{*})+\nabla_{x}^{\bot}f_{0}(x,v)\Big]+2E_{\bot}(0,x)\nabla_{v}^{\bot}f_{0}(x,v)=0,\label{v2.7}
\end{align}
where $E_{\bot}(0, x)$ is the decomposition of the field $E (0, x)$ given by \eqref{v2.4} and
$\nabla_{x}^{\bot},\nabla_{v}^{\bot}$ are the normal components to $\partial\Omega$ of the gradients $\nabla_x,\nabla_v$ respectively.

We assume that the initial data $f_{0}(x,v)$ is constant near the singular set in order to establish a classical solution for general initial data (cf. \cite{HV}), as the characteristic curve continually   hits the boundary near the singular set, that is,
the initial data $f_{0}(x,v)$  satisfies the following flatness condition near the singular set
$\Gamma=\big\{(x,v)\in\Omega\times\mathbb{R}^{2}, x\in \partial\Omega,v\in T_{x}\partial\Omega\big\}$,
\begin{align}
  f_{0}\in C^{1,\mu},\quad f_{0}(x,v)=\text{constant,\,\,  dist}((x,v), \Gamma)\leq \delta_{0},\label{v2.8}
\end{align}
for some $\delta_{0}>0$ small.

We note that if the function $h(t,x)$ depends on time and that $\frac{\partial h}{\partial t}$ is smooth enough, the    main result below still holds.
For convenience, we assume that $\frac{\partial h}{\partial t}=0$, that is,  $h=h(x)$.

\subsection{Main results}

We define some functional spaces as follows.

For $\mu\in(0, 1), \nabla=(\nabla_{x},\nabla_{v})$,
  \begin{align}
  &\|f\|_{C^{1,\mu}(\bar{\Omega}\times \mathbb{R}^{2})}=\sup_{(x,v),(x',v')\in\bar{\Omega}\times \mathbb{R}^{2}}\Big(\frac{|\nabla f(x,v)-\nabla f(x',v')|}{|x-x'|^{\mu}+|v-v'|^{\mu}}\Big)
 +\|f\|_{L^{\infty}(\bar{\Omega}\times \mathbb{R}^{2})},\nonumber\\
  & \|f\|_{C^{1;1,\mu}_{t;x}([0,T]\times\bar{\Omega} )}=\sup_{x,x'\in\bar{\Omega},t,t'\in[0,T]}\frac{|\nabla_{x} f(t,x)-\nabla_{x} f(t',x')|}{|x-x'|^{\mu}}
  +\|f\|_{C([0,T]\times\bar{\Omega})}   +\|f_{t}\|_{C([0,T]\times\bar{\Omega})},\nonumber\\
  & \|f\|_{C^{1;1,\mu}_{t;(x,v)}([0,T]\times\Omega\times\mathbb{R}^{2})}\nonumber\\
  &=\sup_{x,x'\in\bar{\Omega},v,v'\in\mathbb{R}^{2}, t,t'\in[0,T]}
  \frac{|\nabla_{x} f(t,x,v)-\nabla_{x} f(t',x',v')|+|\nabla_{v} f(t,x,v)-\nabla_{v} f(t',x',v')|}{|x-x'|^{\mu}+|v-v'|^{\mu}}
  \nonumber\\
  &\qquad+\|f\|_{C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})}+\|f_{t}\|_{C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})},\nonumber
  \end{align}
 where $C([0,T]\times\bar{\Omega}),C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})$ are the spaces of continous functions bounded in the uniform norm, and
\begin{align}
 C&^{1,\mu}_{0}\Big(\bar{\Omega}\times \mathbb{R}^{2}\Big)\nonumber\\
 &=\Big\{f\in C^{1,\mu}\big(\bar{\Omega}\times \mathbb{R}^{2}\big):f \,\,\text{is \,compactly \,supported,}\;  \|f\|_{C^{1,\mu}(\bar{\Omega}\times \mathbb{R}^{2})}<\infty \Big\}.\nonumber
 \end{align}

We state  our main results of this paper as follows, with respect to the two different boundary conditions of the electric potential.

The relativistic Vlasov-Poisson system describes the collective dynamics of a collisionless plasma, where particles travel at nearly the speed of light and interact through their self-generated electric fields. The system consists of a relativistic Vlasov equation for the particle distribution function coupled with the Poisson equation for the electric potential.
In this paper, we are concerned with the global solution to the relativistic Vlasov-Poisson system (RVP) with general initial data in a convex, bounded domain $\Omega$ of two space dimensions:
 \begin{align}
&\partial_t f+\hat{v}\cdot\nabla_{x}f+\nabla_{x}\varphi\cdot\nabla_{v}f=0, \label{v1.5}\\
&\Delta\varphi=\rho, \label{v1.6}
\end{align}
where $x\in \Omega\subset \mathbb{R}^{2}$, $t>0$; $f=f(t,x,v)\geq 0$ denots the distribution density of particles at position $x$, time $t$, with momentum $v\in\mathbb{R}^{2}$;
$\hat{v}\in\mathbb{R}^{2}$ is the velocity that relates to the momentum $v$ according to Einstein's formula:
$$\hat{v}=\frac{v}{\sqrt{1+|v|^{2}}},$$
 $\varphi(t,x)$ is the electric potential, $\rho$ is the charge density given by
 $$ \rho=\rho(t,x)= \int_{\mathbb{R}^{2}}f\, {\rm d}v,$$
 and the domain $\Omega$ is a convex bounded domain with $C^{5}$ boundary in $\mathbb{R}^{2}$.
 We refer readers to \cite{KF,W} for more background on the relativistic Vlasov-Poisson system.

For the distribution density, we consider the following initial and boundary conditions under which the distribution density exhibits specular reflection on the boundary:
 \begin{align}
&f(0,x,v)=f_{0}(x,v),\quad x\in \Omega,\; v\in \mathbb{R}^{2}, \label{v1.8}\\
&f(t,x,v)=f(t,x,v^{*}),\quad x\in \partial\Omega, \; v\in \mathbb{R}^{2},\; t>0, \label{v1.9}
\end{align}
 satisfying
 \begin{align}
 &f_{0}(x,v)\geq 0, \label{v1.10}\\
 &v^{*}=v-2\big(n_{x}\cdot v\big)n_{x}, \quad (x,v)\in \partial\Omega \times \mathbb{R}^{2}, \label{v1.11}
 \end{align}
 where $n_{x}$ denotes the outer normal vector at $x\in\partial\Omega$.
Meanwhile, regarding electric potential, we analyze two different types of boundary conditions.
The first one is the Neumann boundary condition,
 \begin{align}
 &\frac{\partial \varphi}{\partial n_{x}}=h(x),\,\,\, x\in \partial \Omega,\,t>0,\label{v1.7}
 \end{align}
where the function $h$ is positive and satisfies the following compatibility condition:
\begin{align}
\int_{\Omega}f_{0}(x,v){\rm d}x{\rm d}v=\int_{\partial\Omega}h(x){\rm d}l, \label{v1.12}
\end{align}
and the second is the homogeneous Dirichlet boundary condition,
 \begin{align}
 & \varphi (t,x)=0,\,\,\, x\in \partial \Omega,\,t>0.\label{vD1.7}
 \end{align}

In the process of establishing a classical solution, it is necessary that the initial data $f_{0}(x,v)$ satisfies the following compatibility conditions at the reflection points of
$\partial\Omega\times \mathbb{R}^{2}$ (cf. \cite{G2,H1}),
\begin{align}
&f_{0}(x,v)=f_{0}(x,v^{*}),\label{v2.6}\\
&v_{\bot}\Big[\nabla_{x}^{\bot}f_{0}(x,v^{*})+\nabla_{x}^{\bot}f_{0}(x,v)\Big]+2E_{\bot}(0,x)\nabla_{v}^{\bot}f_{0}(x,v)=0,\label{v2.7}
\end{align}
where $E_{\bot}(0, x)$ is the decomposition of the field $E (0, x)$ given by \eqref{v2.4} and
$\nabla_{x}^{\bot},\nabla_{v}^{\bot}$ are the normal components to $\partial\Omega$ of the gradients $\nabla_x,\nabla_v$ respectively.

We assume that the initial data $f_{0}(x,v)$ is constant near the singular set in order to establish a classical solution for general initial data (cf. \cite{HV}), as the characteristic curve continually hits the boundary near the singular set, that is,
the initial data $f_{0}(x,v)$ satisfies the following flatness condition near the singular set
$\Gamma=\big\{(x,v)\in\Omega\times\mathbb{R}^{2}, x\in \partial\Omega,v\in T_{x}\partial\Omega\big\}$,
\begin{align}
 f_{0}\in C^{1,\mu},\quad f_{0}(x,v)=\text{constant,\,\, dist}((x,v), \Gamma)\leq \delta_{0},\label{v2.8}
\end{align}
for some $\delta_{0}>0$ small.

For $\mu\in(0, 1), \nabla=(\nabla_{x},\nabla_{v})$,
 \begin{align}
 &\|f\|_{C^{1,\mu}(\bar{\Omega}\times \mathbb{R}^{2})}=\sup_{(x,v),(x',v')\in\bar{\Omega}\times \mathbb{R}^{2}}\Big(\frac{|\nabla f(x,v)-\nabla f(x',v')|}{|x-x'|^{\mu}+|v-v'|^{\mu}}\Big)
 +\|f\|_{L^{\infty}(\bar{\Omega}\times \mathbb{R}^{2})},\nonumber\\
 & \|f\|_{C^{1;1,\mu}_{t;x}([0,T]\times\bar{\Omega} )}=\sup_{x,x'\in\bar{\Omega},t,t'\in[0,T]}\frac{|\nabla_{x} f(t,x)-\nabla_{x} f(t',x')|}{|x-x'|^{\mu}}
 +\|f\|_{C([0,T]\times\bar{\Omega})} +\|f_{t}\|_{C([0,T]\times\bar{\Omega})},\nonumber\\
 & \|f\|_{C^{1;1,\mu}_{t;(x,v)}([0,T]\times\Omega\times\mathbb{R}^{2})}\nonumber\\
 &=\sup_{x,x'\in\bar{\Omega},v,v'\in\mathbb{R}^{2}, t,t'\in[0,T]}
 \frac{|\nabla_{x} f(t,x,v)-\nabla_{x} f(t',x',v')|+|\nabla_{v} f(t,x,v)-\nabla_{v} f(t',x',v')|}{|x-x'|^{\mu}+|v-v'|^{\mu}}
 \nonumber\\
 &\qquad+\|f\|_{C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})}+\|f_{t}\|_{C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})},\nonumber
 \end{align}
 where $C([0,T]\times\bar{\Omega}),C([0,T]\times\bar{\Omega}\times\mathbb{R}^{2})$ are the spaces of continous functions bounded in the uniform norm, and
\begin{align}
 C&^{1,\mu}_{0}\Big(\bar{\Omega}\times \mathbb{R}^{2}\Big)\nonumber\\
 &=\Big\{f\in C^{1,\mu}\big(\bar{\Omega}\times \mathbb{R}^{2}\big):f \,\,\text{is \,compactly \,supported,}\; \|f\|_{C^{1,\mu}(\bar{\Omega}\times \mathbb{R}^{2})}<\infty \Big\}.\nonumber
 \end{align}

We state our main results of this paper as follows, with respect to the two different boundary conditions of the electric potential.

\begin{Theorem}\label{T2}
Assume that, given $T>0$, $E\in C^{0;1,\mu}_{t;x}([0,T]\times \bar{\Omega})$ for some $\mu\in (0,1)$, and $E\cdot n=h(x)>0$ on $\partial\Omega$.
 Suppose that $f_{0}\in C^{1,\mu}_{0}(\bar{\Omega}\times \mathbb{R}^{2}), f_{0}\geq 0$
 for some $\mu > 0$.
Then there exists a unique function $f \in C^{1;1,\lambda}_{t;x,v}([0,T]\times \Omega \times \mathbb{R}^{2})$, satisfying the linear
 relativistic
Vlasov-Poisson system \eqref{vx3.1}-\eqref{vx3.3} for some $0< \lambda< \mu$. Meanwhile, the function $f$ satisfies,
\begin{align}
f(t,x,v)&\geq 0, \label{v4.5}\\
\int f(t,x,v){\rm d}x{\rm d}v&=\int f_{0}(x,v){\rm d}x{\rm d}v, \quad \forall \, t\in[0,T]. \label{v4.6}
\end{align}
\end{Theorem}

Finally, we deal with $E(t,x)$. Due to \eqref{v5.6}, and $E=\nabla \varphi$,
\begin{align}
|E(t,x)|+|\nabla E(t,x)|+[\nabla E(t,\cdot)]_{0,\lambda;x}&\leq C\Big(\|\rho\|_{C^{0,\lambda}}+\|h\|_{C^{1,\lambda}}\Big)\nonumber\\
&\leq C(T)\Big(\|f\|_{C^{1;1,\lambda}_{t;(x,v)}\big([0,T]\times\Omega\times\mathbb{R}^{2}\big)}+1\Big).\nonumber
\end{align}
Since $f(t,x,v)\in C^{1;1,\lambda}_{t;(x,v)}\big([0,T] \times\Omega\times\mathbb{R}^{2}\big)$ and $E=\nabla\varphi$, it follows that
$\nabla \rho \in C^{0,\lambda}(\Omega)$ and $E$ satisfies \eqref{v1.6}-\eqref{v1.7}. Once again applying \eqref{v5.6}, we obtain
\begin{align}
|\nabla^{2}E(t,x)|+[\nabla^{2} E(t,\cdot)]_{0,\lambda;x}&\leq C\Big(\|\nabla \rho\|_{C^{0,\lambda}}+\|\nabla h\|_{C^{1,\lambda}}\Big)\nonumber\\
&\leq C(T)\Big(\|f\|_{C^{1;1,\lambda}_{t;(x,v)}\big([0,T]\times\Omega\times\mathbb{R}^{2}\big)}+1\Big).\nonumber
\end{align}
According to the assumptions on $\varphi$, the function $\varphi_{t}$ satisfies
\begin{align}
&\Delta\varphi_{t}=\rho_{t}, \,\,x\in \Omega,t>0, \nonumber\\
&\frac{\partial \varphi_{t}}{\partial n_{x}}=0,\,\, x\in \partial \Omega,t>0,\nonumber
\end{align}
and by \eqref{v5.6},
$$|E_{t}(t,x)|\leq\|\rho_{t}\|_{C^{0,\lambda}} \leq C(T)\|f\|_{C^{1;1,\lambda}_{t;(x,v)}\big([0,T]\times\Omega\times\mathbb{R}^{2}\big)}.$$
Then the inequalities \eqref{v5.12}-\eqref{v5.14} follow.
\end{proof}
\begin{Proposition}\label{P4}
Let $ 0<\lambda<\mu$,
 suppose that $f_{0}\in C^{1,\mu}_{0}(\bar{\Omega}\times \mathbb{R}^{2}), f_{0}\geq 0$ satisfies \eqref{v2.8} and $h \in C^{1,\mu}(\partial\Omega),\,\,h(x)>0$.
Then the iterative sequence $f^{n}$ is globally defined for each $x\in \Omega,\,v\in\mathbb{R}^{2},\,0\leq t<\infty $.
 Moreover, the function $f^{n}(t,x,v)\in C^{1;1,\lambda}_{t;(x,v)}\big([0,T] \times\Omega\times\mathbb{R}^{2}\big)$ for $\forall\, T >0$ satisfies
\begin{align}
\|f^{n}\|_{L^{\infty}}&=\|f_{0}\|_{L^{\infty}} ,\label{v5.15}\\
\int \rho^{n}(t,x){\rm d}x&=\int f_{0}(x,v){\rm d}x{\rm d}v. \label{v5.16}
\end{align}
\end{Proposition}
\begin{proof}
We use induction to prove this proposition.

\subsection{The sequence $Q^{n}(t)$ converge to $Q(t)$}
\begin{Proposition}\label{P7}
Let us assume that $Q(t)\leq M, Q^{n}(t)\leq M$ for $n\geq n_{0}, 0\leq t\leq T$. If $f^{n}\,\rightarrow \,f\in C^{\nu;1,\lambda}_{t;(x,v)}([0,T]\times \bar{\Omega}\times \mathbb{R}^{2})$ for any $0<\lambda<\mu, 0<\nu<1$, then we have $Q^{n}(t)\rightarrow Q(t)$ uniformly on $[0,T]$.
\end{Proposition}
\begin{proof}
Due to the Lemma \ref{L2}, we know that the characteristic curves away from the singular set in the initial state remain away from it
during their evolution. For these characteristic curves, we can estimate their difference as $n\rightarrow \infty$.
It is similar to the proof of the analogous result on Vlasov-Poisson system (cf. \cite{HV,HV1}).
In fact, since the sequence $f^{n}$ is uniformly bounded on $n$ in the space $C^{1;1,\lambda}_{t;(x,v)}([0,T]\times \bar{\Omega}\times \mathbb{R}^{2})$, similar to the proof of Theorem\ref{T2}, we see that the number of bounces is uniformly bounded with respect to $n$ in the time interval $[0,T]$.
Moreover, the time series when the characteristic curves of $f^{n}$ bounce to the boundary converge to the time when the charateristics of $f$ collide with the boundary.
Since $E^{n}\rightarrow E$, it follows that the characteristic curves $\big(X^{n}(s;0,x_{0},v_{0}),V^{n}(s;0,x_{0},v_{0})\big)$ of $f^{n}$ converge to the ones $\big(X(s;0,x_{0},v_{0}),V(s;0,x_{0},v_{0})\big)$ of $f$ between bounces. Also we have the following estimate, \begin{align}
|X^{n}(S)-X(S)|=&v_{0}\int^{s}_{0}\big(\frac{1}{\sqrt{1+|V^{n}(\tau)|^{2}}}-\frac{1}{\sqrt{1+|V(\tau)|^{2}}}\big) {\rm d}\tau \nonumber\\
&+\int^{s}_{0}\int^{\tau}_{0}\frac{E^{n}(\mu,X^{n}(\mu))-E(\mu,X^{n}(\mu))}{\sqrt{1+|V^{n}(\tau)|^{2}}}{\rm d}\mu{\rm d}\tau \nonumber\\
&+\int^{s}_{0}\int^{\tau}_{0}E(\mu,X(\mu))\big(\frac{1}{\sqrt{1+|V^{n}(\tau)|^{2}}}-\frac{1}{\sqrt{1+|V(\tau)|^{2}}}\big)\nonumber\\
\leq &C(T)\big(\int^{s}_{0}|V^{n}(s)-V(s)|{\rm d}\tau +\int^{s}_{0}|X^{n}(s)-X(s)|{\rm d}\tau\big)\nonumber\\
&+C(T)\|E^{n}-E\|_{L^{\infty}}.\nonumber
\end{align}
Similarly,
\begin{align}
|V^{n}(S)-V(S)|\leq C(T)\int^{s}_{0}|X^{n}(s)-X(s)|{\rm d}\tau +C(T)\|E^{n}-E\|_{L^{\infty}}.\nonumber
\end{align}
Let us define $Z(s)=|X^{n}(S)-X(S)|+|V^{n}(S)-V(S)|$, then
$$Z(s)\leq C(T)\int^{s}_{0}Z(\tau){\rm d}\tau+C(T)\|E^{n}-E\|_{L^{\infty}}.$$
By Gronwall's inequality, it follows that
$$Z(s)\leq C(T)\|E^{n}-E\|_{L^{\infty}}\rightarrow 0.$$
Because the number of bounces is uniformly bounded and $|V|$ remains unchanged before and after the bounce, we conclude that
the functions $|V^{n}|$ converge uniformly to $|V(s)|$ when $n\rightarrow \infty$. Meanwhile, in accordance with the definition of $Q^{n}(t)$,
we obtain that $Q^{n}(t)\rightarrow Q(t)$, and the proof is completed.
\end{proof}

Finally, we give the analogous velocity lemma for the Dirichlet problem.
\begin{Lemma}\label{L5}
For a given constant $\delta >0$, let $\Gamma_{\delta}=([\partial\Omega +B_{\delta}(0)]\cap \Omega)\times \mathbb{R}^{2},$
and $X(s;t,x,v)$, $V(s;t,x,v)$ be the characteristic curves associated with the relativistic
Vlasov-Poisson system defined previously.
Suppose that $E,\varphi (t,x)$ satisfy the assumptions of Theorem \ref{T2} and Lemma \ref{L3}, respectively. Then
the existence of solutions to the characteristic equations \eqref{v4.1}-\eqref{v4.3} can be obtained in $[0,T]$
for any $(x,v)\in \bar{\Omega}\times \mathbb{R}^{2}$.
Furthermore, the following estimate holds for any $(x, v) \in\Gamma_{\delta}$,
\begin{align}
C_{1}\Big(v^{2}_{\bot}(0)+x_{\bot}(0)\Big)\leq \Big(v^{2}_{\bot}(t)+x_{\bot}(t)\Big) \leq C_{2}\Big(v^{2}_{\bot}(0)+x_{\bot}(0)\Big), t\in[0,T],\label{v7.5}
\end{align}
for some positive constants $C_{1}, C_{2}$ depending only on
$T,f_{0},\|E\|_{L^{\infty}([0,T],C^{\frac{1}{2}}(\Omega))}, \Omega, \|j\|_{L^{\infty}}$.
\end{Lemma}
\begin{proof}
By the definition of $\alpha$ and $E=\nabla_{x}\varphi(t,x)=E_{l}\cdot U(l)-E_{\bot}n(l)$, taking the derivative of $\alpha$ with respect to $t$ along the characteristics, we get
\begin{align}
\frac{d \alpha}{d t}=&-\sqrt{1+|v|^{2}}\partial_{t}E_{\bot}(t,l,0)x_{\bot}-\frac{\partial \varphi}{\partial t}\nonumber\\
&+\frac{1}{\sqrt{1+|v|^{2}}}\cdot\frac{\omega}{1-\kappa x_{\perp}}\Big(
\big(-\sqrt{1+|v|^{2}}\partial_{l}E_{\bot}(t,l,0)+\frac{d \kappa}{d l}\omega^{2}+\kappa E_{l}\big)x_{\bot}-E_{l}\Big)\nonumber\\
&+\frac{v_{\bot}}{\sqrt{1+|v|^{2}}}\cdot\Big(-\sqrt{1+|v|^{2}}E_{\bot}(t,l,0)+\kappa \omega^{2}-E_{\bot}\Big)\nonumber\\
&+\Big(E_{l}+\frac{v_{\bot}}{\sqrt{1+|v|^{2}}}\frac{\kappa\omega}{1-\kappa x_{\bot}}\Big)\cdot\Big(-\frac{\omega}{\sqrt{1+|v|^{2}}}E_{\bot}(t,l,0)+2\kappa \omega \Big)x_{\bot}\nonumber\\
&+\Big(E_{\bot}-\frac{1}{\sqrt{1+|v|^{2}}}\cdot\frac{\kappa\omega^{2}}{1-\kappa x_{\bot}}\Big)\cdot \Big(v_{\bot}-\frac{v_{\bot}}{\sqrt{1+|v|^{2}}}E_{\bot}(t,l,0)x_{\bot}\Big).\nonumber
\end{align}
With the help of Lemmaa \ref{L3} and \ref{L4}, similarly to the discussion in Lemma \ref{L2}, we conclude that
$$\Big|\frac{d \alpha}{d t}(t,X(t),V(t))\Big|\leq C \alpha \big(1+|\log \alpha|+|\log \alpha|^{2}\big).$$
Then the lemma follows from the Gronwall inequality.
\end{proof}

\begin{Theorem}\label{T02}
 Let $f_{0} \in C^{1,\mu}_{0}\big(\bar{\Omega}\times\mathbb{R}^{2}\big), f_{0}\geq 0$ for some $0 <\mu< 1$, satisfying
\eqref{v2.8}. Then there exists
a unique solution $f \in C^{1;1,\lambda}_{t;(x,v)} \big((0,\infty)\times \bar{\Omega}\times \mathbb{R}^{2}\big),
\varphi\in  C^{1;3,\lambda}_{t;x}\big((0,\infty)\times \bar{\Omega}\big) $
for
some $0 < \lambda < \mu$, of the relativistic Vlasov-Poisson system \eqref{v1.5}-\eqref{v1.6} with compact support
in $x$ and $v$, with the  initial boundary conditions of $(f,\varphi)$ satisfying \eqref{v1.8}-\eqref{v1.10} and \eqref{vD1.7}  respectively.
\end{Theorem}
\subsection{Difficulties and strategy of the proofs}

To prove the main results, we first apply the velocity lemma to establish the well-posedness of linearized problems. Then, we construct an iterative scheme and show the convergence of the iterative sequences. The main issues to address are the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$. Finally, we use bootstrapping techniques to reach the desired conclusions.

Now, we will discuss the major challenges. To this end, we first review the fundamental difficulties and core ideas of the initial boundary value problem from the perspective of the classic Vlasov-Poisson system.
In \cite{G1},   the global existence was proved for the case of a half-space $\mathbb{R}^{3}_{+}$, assuming that the function $f_{0}$ remains constant near the singular set. This assumption  avoids the evolution of characteristic curves that are close to the singular set.
  In regions far away from the singular set, the number of collisions within a finite time interval can be bounded uniformly by using the velocity lemma method described in
 \cite{LP}. This allows for a clear description of how the characteristic curves evolve.
  In \cite{HV}, Hwang and Vel\'{a}zquez considered the Vlasov-Poisson system in a general bounded convex domain $\Omega\subset \mathbb{R}^{3}$, they addressed  the increasing complexity of the evolution of characteristic curves near the boundary, making it challenging to provide an accurate mathematical description. In order to establish global existence, the authors initially employed geometric methods, as outlined in \cite{HV}. Their results indicate that the geometric properties of the domain have a more significant influence on the characteristic curves than their dynamics.

However, understanding the evolution of the characteristic curves associated with the relativistic Vlasov-Poisson system \eqref{v1.5}-\eqref{v1.6} becomes extremely difficult near the singular set.
First, we face the complexities that arise within the Vlasov-Poisson system, including the challenges related to the behavior of characteristic curves near the singular set and the influence of regional boundaries and other factors. In this study, we will draw on insights from previous research on the Vlasov-Poisson system. Specifically, we assume that the initial data $f_{0}$ remains constant near the singular set. Additionally, to analyze the characteristic curves close to the boundary, we will use geometric methods. This approach is especially relevant for general bounded regions. The core idea of constructing suitable velocity lemmas and applying geometric techniques to prove the existence of global solutions for the relativistic Vlasov-Poisson system in convex bounded domains of two spatial dimensions remains valid. This applies to general initial data as well.

However, unlike the Vlasov-Poisson system, additional challenges arise with the relativistic Vlasov-Poisson system.
In applying geometric methods to boundary issues, arc length plays a key role as a curve's characteristic in two dimensions. By using arc length as a parameter and considering the Frenet-Serret formulas, we can effectively describe the distribution density equation near the boundary $\partial\Omega$, thus establishing a vital connection in the geometric representation.

Moreover, identifying new coordinate variables is essential for developing appropriate velocity lemmas. These lemmas help describe scenarios where particles disperse from singular sets and face different numbers of collision barriers. In the context of the Newman boundary condition, we select coordinate variables $(\alpha,\beta)$ that satisfy the constraints previously mentioned,\begin{align}
&\alpha(t,l,x_{\bot},\omega,v_{\bot})=\frac{v_{\bot}^{2}}{2}-L(t,l,0,\omega,v_{\bot})x_{\bot},\nonumber \\
&\beta(t, l,  x_{\bot}, \omega, v_{\bot})=2\pi F(t,l,x_{\bot},\omega,v_{\bot})+\pi (1-\frac{v_{\bot}}{\sqrt{2\alpha}}),\nonumber
\end{align}
where
\begin{align}
L(t,l,0,w,x_{\bot})=\sqrt{1+|v|^{2}}, \quad F(t,l,x_{\bot},\omega,v_{\bot})=-\sqrt{1+|v|^{2}}h(x)-k\omega^{2}.\nonumber
\end{align}
This variable  $\alpha$  characterizes the distance from points on the characteristic curve to the singular set. From this, the specific significance of selecting the tangential and normal directions of the regional boundary in local coordinates can be identified. The other variable  $\beta$, by contrast, describes the number of collisions between particles and the boundary. It is observed that the number of collisions is inversely proportional to the distance to the singular set.
To ensure the regularity of the characteristic curve, the number of collisions must be bounded above uniformly within a certain time interval; it is for this purpose that assumption \eqref{v2.8} is proposed for the initial value.

For Dirichlet boundary conditions, deriving the velocity lemma is relatively more difficult.
Additionally, choosing coordinate variables $\alpha$ becomes more complex.
$$\alpha=\frac{v_{\bot}^{2}}{2}-\varphi(t,x)-L(t,l,0,\omega,v_{\bot})x_{\bot}.$$
 It requires estimating the first derivatives of the electric potential  $\varphi$. This difference in handling the first derivative of  $\varphi$ is another notable difference between the two boundary types. The elliptical nature of the electric potential $\varphi$ in general domains, combined with the lack of a precise formula for $\varphi$, makes obtaining accurate estimates challenging. The core idea involves locally flattening the boundary and incorporating it with the Green's function in a half-space $\mathbb{R}^{2}_{+}$. Additionally, using revised boundary estimates for the  equation of $\varphi$ and constructing suitable supersolutions are important strategies.

Lastly, we found that the effect of regional boundaries is similar to the behavior of characteristic curves, which differs from the Vlasov-Poisson system. It is important to note that this study mainly focuses on a two-dimensional situation.

\subsection{Organization of the paper}

The structure of the paper is as follows.
From Section 2 to Section 5, we mostly address the Newmann boundary condition case.
In Section 2 we introduce  an iterative system and then  define a sequence of functions $\{ f^{n}\}$,  the
limit function as $ n \rightarrow \infty$ is the desired global solution of the RVP system.
In Section 3 we establish the well-posedness of the linear problem.
In Section 4 we show that the convergence of the iterative sequences $ \{ f^{n}\}$  under the condition that $Q (t)$ is bounded.
In Section 5 we prove the boundedness of the function $Q (t)$.  This concludes the proof of the first theorem.
We address the Dirichlet boundary condition in Section 6 and subsequently obtain the second result.

\bigskip

To prove the main results (in particular Theorem~\ref{T1}), the introduction describes the following strategy.

1. **Linearized well-posedness via a velocity lemma:** “we first apply the velocity lemma to establish the well-posedness of linearized problems.”

2. **Iterative construction and convergence:** “Then, we construct an iterative scheme and show the convergence of the iterative sequences.” The “main issues” in this step are “the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$.”

3. **Bootstrapping to final regularity/conclusion:** “Finally, we use bootstrapping techniques to reach the desired conclusions.”

4. **Control of boundary interactions / collisions (via adapted coordinates):** Near the singular set and boundary, the plan is to “draw on insights from previous research,” assuming “the initial data $f_{0}$ remains constant near the singular set,” and “to analyze the characteristic curves close to the boundary, we will use geometric methods.” For the (Neumann) boundary analysis, they “select coordinate variables $(\alpha,\beta)$,” where “$\alpha$ characterizes the distance from points on the characteristic curve to the singular set” and “$\beta$ … describes the number of collisions between particles and the boundary,” with the observation that “the number of collisions is inversely proportional to the distance to the singular set.” The assumption \eqref{v2.8} is “for this purpose” (to ensure the collision number is uniformly bounded).

5. **Paper roadmap matching the proof:** Sections 2–5 implement this plan: “introduce an iterative system … define a sequence of functions $\{ f^{n}\}$, the limit … is the desired global solution,” then “establish the well-posedness of the linear problem,” then “show … convergence … under the condition that $Q(t)$ is bounded,” and finally “prove the boundedness of the function $Q(t)$. This concludes the proof of the first theorem.”

To prove the main results (in particular the main theorem), the introduction describes the following strategy.

1. **Linearized well-posedness via a velocity lemma:** “we first apply the velocity lemma to establish the well-posedness of linearized problems.”

2. **Iterative construction and convergence:** “Then, we construct an iterative scheme and show the convergence of the iterative sequences.” The “main issues” in this step are “the uniform boundedness in a given function space and the prolongation of uniform estimates for the functions ${f^{n}}$.”

3. **Bootstrapping to final regularity/conclusion:** “Finally, we use bootstrapping techniques to reach the desired conclusions.”

4. **Control of boundary interactions / collisions (via adapted coordinates):** Near the singular set and boundary, the plan is to “draw on insights from previous research,” assuming “the initial data $f_{0}$ remains constant near the singular set,” and “to analyze the characteristic curves close to the boundary, we will use geometric methods.” For the (Neumann) boundary analysis, they “select coordinate variables $(\alpha,\beta)$,” where “$\alpha$ characterizes the distance from points on the characteristic curve to the singular set” and “$\beta$ … describes the number of collisions between particles and the boundary,” with the observation that “the number of collisions is inversely proportional to the distance to the singular set.” For this purpose (to ensure the collision number is uniformly bounded), they assume
\begin{align}
  f_{0}\in C^{1,\mu},\quad f_{0}(x,v)=\text{constant,\,\,  dist}((x,v), \Gamma)\leq \delta_{0},\label{v2.8}
\end{align}

5. **Paper roadmap matching the proof:** Sections 2–5 implement this plan: “introduce an iterative system … define a sequence of functions $\{ f^{n}\}$, the limit … is the desired global solution,” then “establish the well-posedness of the linear problem,” then “show … convergence … under the condition that $Q(t)$ is bounded,” and finally “prove the boundedness of the function $Q(t)$. This concludes the proof of the first theorem.”

\label{T1}
 Let $f_{0} \in C^{1,\mu}_{0}\big(\bar{\Omega}\times\mathbb{R}^{2}\big), f_{0}\geq 0$ for some $0 <\mu< 1$, satisfying
\begin{align}
  f_{0}\in C^{1,\mu},\quad f_{0}(x,v)=\text{constant,\,\,  dist}((x,v), \Gamma)\leq \delta_{0},\label{v2.8}
\end{align}
Suppose that $h \in C^{2,\mu} (\partial\Omega)$ satisfies
\begin{align}
\int_{\Omega}f_{0}(x,v){\rm d}x{\rm d}v=\int_{\partial\Omega}h(x){\rm d}l, \label{v1.12}
\end{align}
and $h > 0$. Then there exists
a unique solution $f \in C^{1;1,\lambda}_{t;(x,v)} \big((0,\infty)\times \bar{\Omega}\times \mathbb{R}^{2}\big),
\varphi\in  C^{1;3,\lambda}_{t;x}\big((0,\infty)\times \bar{\Omega}\big) $
for
some $0 < \lambda < \mu$, of the relativistic Vlasov-Poisson system
\begin{align}
&\partial_t f+\hat{v}\cdot\nabla_{x}f+\nabla_{x}\varphi\cdot\nabla_{v}f=0, \label{v1.5}\\
&\Delta\varphi=\rho, \label{v1.6}
\end{align}
with compact support
in $x$ and $v$, where the  initial boundary conditions of $(f,\varphi)$ satisfy
\begin{align}
&f(0,x,v)=f_{0}(x,v),\quad x\in \Omega,\; v\in \mathbb{R}^{2}, \label{v1.8}\\
&f(t,x,v)=f(t,x,v^{*}),\quad x\in \partial\Omega, \; v\in \mathbb{R}^{2},\; t>0, \label{v1.9}
\end{align}
\begin{align}
 &f_{0}(x,v)\geq 0,  \label{v1.10}\\
 &v^{*}=v-2\big(n_{x}\cdot v\big)n_{x}, \quad (x,v)\in \partial\Omega \times \mathbb{R}^{2}, \label{v1.11}
 \end{align}
and
\begin{align}
   &\frac{\partial \varphi}{\partial n_{x}}=h(x),\,\,\, x\in \partial \Omega,\,t>0,\label{v1.7}
   \end{align}
respectively.

Question:
Let \(\Omega\subset \mathbb{R}^2\) be a convex bounded domain with \(C^5\) boundary, let \(n_x\) denote the outer unit normal at \(x\in \partial\Omega\), and define the singular set \(\Gamma=\{(x,v)\in \Omega\times \mathbb{R}^2:\ x\in \partial\Omega,\ v\in T_x\partial\Omega\}\). For \(v\in \mathbb{R}^2\), set \(\hat v = v/\sqrt{1+|v|^2}\), and for \(x\in \partial\Omega\) define the specularly reflected velocity \(v^*=v-2(n_x\cdot v)n_x\). Let \(\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv\). Assume \(0<\mu<1\), \(f_0\in C_0^{1,\mu}(\overline\Omega\times \mathbb{R}^2)\) with \(f_0\ge 0\), and for some \(\delta_0>0\), \(f_0(x,v)\) is constant whenever \(\operatorname{dist}((x,v),\Gamma)\le \delta_0\). Assume also that \(h\in C^{2,\mu}(\partial\Omega)\), \(h>0\), and the compatibility condition \(\int_\Omega f_0(x,v)\,dx\,dv = \int_{\partial\Omega} h(x)\,dl\) holds. Which statement is true about the relativistic Vlasov-Poisson system
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\]
with initial condition \(f(0,x,v)=f_0(x,v)\), specular boundary condition \(f(t,x,v)=f(t,x,v^*)\) for \(x\in \partial\Omega\), and Neumann boundary condition \(\partial \varphi/\partial n_x = h(x)\) on \(\partial\Omega\)?

There exists a unique solution pair \((f,\varphi)\) such that for some \(0<\lambda<\mu\),
\[
f\in C_{t;(x,v)}^{1;1,\lambda}\big((0,\infty)\times \overline\Omega\times \mathbb{R}^2\big),
\qquad
\varphi\in C_{t;x}^{1;3,\lambda}\big((0,\infty)\times \overline\Omega\big),
\]
with \(f\) compactly supported in \((x,v)\), and \((f,\varphi)\) satisfies
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\qquad
\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv,
\]
for \(t>0\), \(x\in \Omega\), \(v\in \mathbb{R}^2\), together with
\[
f(0,x,v)=f_0(x,v),
\qquad
f(t,x,v)=f(t,x,v^*) \text{ for } x\in \partial\Omega,
\qquad
\frac{\partial\varphi}{\partial n_x}=h(x) \text{ on } \partial\Omega.
\]

Choices:
B.
There exists a unique solution pair \((f,\varphi)\) such that for some \(0<\lambda<\mu\),
\[
f\in C_{t;(x,v)}^{1;1,\lambda}\big((0,\infty)\times \overline\Omega\times \mathbb{R}^2\big),
\qquad
\varphi\in C_{t;x}^{1;3,\lambda}\big((0,\infty)\times \overline\Omega\big),
\]
with \(f\) compactly supported in \((x,v)\), and \((f,\varphi)\) satisfies
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\qquad
\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv,
\]
for \(t>0\), \(x\in \Omega\), \(v\in \mathbb{R}^2\), together with
\[
f(0,x,v)=f_0(x,v),
\qquad
f(t,x,v)=f(t,x,v^*) \text{ for } x\in \partial\Omega,
\qquad
\frac{\partial\varphi}{\partial n_x}=h(x) \text{ on } \partial\Omega,
\]
and one may take \(\lambda=\mu\).
C.
There exists at least one solution pair \((f,\varphi)\) such that for some \(0<\lambda<\mu\),
\[
f\in C_{t;(x,v)}^{1;1,\lambda}\big((0,\infty)\times \overline\Omega\times \mathbb{R}^2\big),
\qquad
\varphi\in C_{t;x}^{1;3,\lambda}\big((0,\infty)\times \overline\Omega\big),
\]
with \(f\) compactly supported in \((x,v)\), and \((f,\varphi)\) satisfies
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\qquad
\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv,
\]
for \(t>0\), \(x\in \Omega\), \(v\in \mathbb{R}^2\), together with
\[
f(0,x,v)=f_0(x,v),
\qquad
f(t,x,v)=f(t,x,v^*) \text{ for } x\in \partial\Omega,
\qquad
\frac{\partial\varphi}{\partial n_x}=h(x) \text{ on } \partial\Omega.
\]
D.
There exists a unique solution pair \((f,\varphi)\) such that for some \(0<\lambda<\mu\),
\[
f\in C_{t;(x,v)}^{1;1,\lambda}\big((0,\infty)\times \overline\Omega\times \mathbb{R}^2\big),
\qquad
\varphi\in C_{t;x}^{1;3,\lambda}\big((0,\infty)\times \overline\Omega\big),
\]
with \(f\) compactly supported in \((x,v)\), and \((f,\varphi)\) satisfies
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\qquad
\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv,
\]
for \(t>0\), \(x\in \Omega\), \(v\in \mathbb{R}^2\), together with
\[
f(0,x,v)=f_0(x,v),
\qquad
f(t,x,v)=f(t,x,v^*) \text{ for } x\in \partial\Omega,
\qquad
\frac{\partial\varphi}{\partial n_x}=h(x) \text{ on } \partial\Omega,
\]
for every nonnegative \(f_0\in C_0^{1,\mu}(\overline\Omega\times\mathbb{R}^2)\) satisfying the mass compatibility condition, without requiring that \(f_0\) be constant on \(\{\operatorname{dist}((x,v),\Gamma)\le \delta_0\}\).
E.
There exists a unique solution pair \((f,\varphi)\) such that for some fixed \(\lambda\in(0,1)\) depending only on \(\mu\) and \(\Omega\),
\[
f\in C_{t;(x,v)}^{1;1,\lambda}\big((0,\infty)\times \overline\Omega\times \mathbb{R}^2\big),
\qquad
\varphi\in C_{t;x}^{1;3,\lambda}\big((0,\infty)\times \overline\Omega\big),
\]
with \(f\) compactly supported in \((x,v)\), and \((f,\varphi)\) satisfies
\[
\partial_t f+\hat v\cdot \nabla_x f+\nabla_x\varphi\cdot \nabla_v f=0,
\qquad
\Delta \varphi = \rho,
\qquad
\rho(t,x)=\int_{\mathbb{R}^2} f(t,x,v)\,dv,
\]
for \(t>0\), \(x\in \Omega\), \(v\in \mathbb{R}^2\), together with
\[
f(0,x,v)=f_0(x,v),
\qquad
f(t,x,v)=f(t,x,v^*) \text{ for } x\in \partial\Omega,
\qquad
\frac{\partial\varphi}{\partial n_x}=h(x) \text{ on } \partial\Omega.
\]


\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, assume geodesic rays in the universal cover have the divergence property, and assume \(\pi_1(M)\) is Gromov hyperbolic and residually finite. Let \(\phi^t:SM\to SM\) be the geodesic flow on the unit tangent bundle, let \(\phi=\phi^1\), and let \(\mathcal M_\phi(SM)\) be the set of \(\phi^t\)-invariant Borel probability measures on \(SM\). Let \(\mathcal E\subset SM\) be the expansive set, meaning the set of vectors \(v\in SM\) for which there exists \(\varepsilon>0\) such that whenever \(w\in SM\) satisfies \(d(c_v(t),c_w(\mathbb R))<\varepsilon\) for all \(t\in\mathbb R\), the geodesics \(c_v\) and \(c_w\) agree up to a time shift. Call \(\nu\in\mathcal M_\phi(SM)\) non-expansive if \(\nu(\mathcal E)=0\). Assume that \(h_\nu(\phi)<h_{\mathrm{top}}(\phi)\) for every non-expansive measure \(\nu\in\mathcal M_\phi(SM)\), where a measure of maximal entropy is an invariant measure \(\mu\) with \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\). Which statement holds?

There exists exactly one \(\phi^t\)-invariant Borel probability measure \(\mu\) on \(SM\) such that \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\). Moreover, \(\mu(\mathcal E)=1\), \(\mu\) is mixing, and \(\operatorname{supp}(\mu)=SM\).

Choices:
B. There exists exactly one \(\phi^t\)-invariant Borel probability measure \(\mu\) on \(SM\) such that \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\). Moreover, every measure of maximal entropy is non-expansive, i.e. \(\mu(\mathcal E)=0\), while \(\mu\) is mixing and \(\operatorname{supp}(\mu)=SM\).
C. There exists a unique \(\phi^t\)-invariant Borel probability measure \(\mu\) on \(SM\) such that \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\).
D. For every non-expansive measure \(\nu\in\mathcal M_\phi(SM)\), one has \(h_\nu(\phi)\le h_{\mathrm{top}}(\phi)\); consequently there exists exactly one \(\phi^t\)-invariant Borel probability measure \(\mu\) on \(SM\) such that \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\), and moreover \(\mu(\mathcal E)=1\), \(\mu\) is mixing, and \(\operatorname{supp}(\mu)=SM\).
E. There exists exactly one ergodic \(\phi^t\)-invariant Borel probability measure \(\mu\) on \(SM\) such that \(h_\mu(\phi)=h_{\mathrm{top}}(\phi)\). Moreover, \(\mu(\mathcal E)=1\), and every \(\phi^t\)-invariant Borel probability measure with full support on \(SM\) is mixing.