# On ergodic properties of geodesic flows on uniform visibility manifolds
without conjugate points

Weisheng Wu  School of Mathematical Sciences, Xiamen University, Xiamen,
361005, P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we conduct a comprehensive study on ergodic properties of the
geodesic flow on a $C^{\infty}$ uniform visibility manifold $M$ without
conjugate points. If $M$ is a closed surface of genus $\mathfrak{g}\geq 2$
without conjugate points and with bounded asymptote, we study the geometric
properties of singular geodesics and show that if all singular geodesics are
closed, then there are at most finitely many isolated singular closed
geodesics and finitely many generalized strips. In particular, the geodesic
flow is ergodic with respect to Liouville measure under the above assumption.

Let $(M,g)$ be a closed uniform visibility manifold without conjugate points
and $X$ be its universal cover. Under the entropy gap assumption, the geodesic
flow has a unique measure of maximal entropy (MME for short) by [81, Theorem
1.2]. We develop a Patterson-Sullivan construction of this unique MME and show
that it has local product structure, is fully supported, and has the Bernoulli
property.

If we assume further that $M$ has continuous Green bundles and the geodesic
flow has a hyperbolic periodic point, using the nonuniform hyperbolic
structure on an open dense subset and the symbolic approach developed in [72],
we show that for any Hölder continuous function $\psi$ or of the form
$\psi=q\psi^{u}$ for $q\in{\mathbb{R}}$ where $\psi^{u}$ is the Jacobian of
the geodesic flow along unstable Green bundle, the equilibrium state of $\psi$
is unique under the pressure gap condition.

Under the same conditions above, we apply the mixing properties of the MME to
obtain the following two Margulis type asymptotic formulae:

$\displaystyle\\#P(t)$ $\displaystyle\sim\frac{e^{ht}}{ht}$ $\displaystyle
b_{t}(x)$ $\displaystyle\sim c(x)\frac{e^{ht}}{h}.$

where $P(t)$ is the set of free-homotopy classes containing a closed geodesic
with length at most $t$, $b_{t}(x)$ is the Riemannian volume of the ball of
radius $t>0$ around $x\in X$, $h$ denotes the topological entropy of the
geodesic flow, and the notation $f_{1}\sim f_{2}$ means that
$\lim_{t\to\infty}\frac{f_{1}(t)}{f_{2}(t)}=1$. The function
$c:X\to{\mathbb{R}}$ is continuous and called the Margulis function. We then
obtain some rigidity results on $M$ involving the Margulis function.

Finally, for a uniform visibility manifold $M=X/\Gamma$ (not necessarily
compact) without conjugate points, we show that if $\Gamma$ is non-elementary
and contains an expansive isometry, then the theorem of Hopf-Tsuji-Sullivan
dichotomy holds.

###### Key words and phrases:

###### 1991 Mathematics Subject Classification:

###### Contents

  1. 1 Introduction
  2. 2 Statement of main results
    1. 2.1 Ergodicity of Liouville measure
    2. 2.2 Measure of maximal entropy and Bowen-Margulis measures
    3. 2.3 Uniqueness of equilibrium states
    4. 2.4 Counting closed geodesics and volume asymptotics
      1. 2.4.1 Counting closed geodesics
      2. 2.4.2 Volume estimates
    5. 2.5 Margulis function and rigidity
    6. 2.6 The Hopf-Tsuji-Sullivan dichotomy
    7. 2.7 Organization of the paper
  3. 3 Geometric and ergodic preliminaries
    1. 3.1 Jacobi fields, Green bundles and Pesin set
    2. 3.2 Boundary at infinity
    3. 3.3 Busemann functions and horospheres
  4. 4 Ergodicity of Liouville measure
    1. 4.1 Singular set and Expansivity
    2. 4.2 Area of ideal triangles
    3. 4.3 Singular geodesics
  5. 5 Patterson-Sullivan measure and Bowen-Margulis measure
    1. 5.1 Patterson-Sullivan measure
    2. 5.2 Bowen-Margulis measure
    3. 5.3 Relation to the MME
    4. 5.4 Bernoulli property of Bowen-Margulis measure
      1. 5.4.1 Kolmogorov property
      2. 5.4.2 From Kolmogorov to Bernoulli
      3. 5.4.3 $\varepsilon$-regular coverings
      4. 5.4.4 Construction of the function $\theta$
  6. 6 Uniqueness of Equilibrium states
    1. 6.1 Coding of nonuniformly hyperbolic flows
    2. 6.2 Uniqueness of equilibrium states
  7. 7 Counting closed geodesics
    1. 7.1 Periodic orbits and deck transformations
    2. 7.2 Neighborhood of an expansive vector
    3. 7.3 Flow boxes and a closing lemma
    4. 7.4 Depth of intersection
    5. 7.5 Scaling and mixing calculation
    6. 7.6 Upper bound for $\\#C(t)$
    7. 7.7 Lower bound for $\\#C(t)$
    8. 7.8 Equidistribution and completion of the proof
  8. 8 Volume Asymptotics
    1. 8.1 Regular partition-cover and local uniform expansion
    2. 8.2 Using scaling and mixing
      1. 8.2.1 Intersection components
      2. 8.2.2 Depth of intersection
      3. 8.2.3 Scaling and mixing calculation
      4. 8.2.4 Integration
    3. 8.3 Summing over the regular partition-cover
  9. 9 Margulis function and rigidity
    1. 9.1 Properties of Margulis function
    2. 9.2 Rigidity in dimension two
      1. 9.2.1 Unique ergodicity of horocycle flow
      2. 9.2.2 Uniqueness of harmonic measure
      3. 9.2.3 Integral formulas for topological entropy
      4. 9.2.4 Rigidity
    3. 9.3 Flip invariance of the Patterson-Sullivan measure
  10. 10 The Hopf-Tsuji-Sullivan dichotomy
    1. 10.1 The generalized shadow lemma
    2. 10.2 Properties of the radial limit set
    3. 10.3 Dynamical properties of the geodesic flow
    4. 10.4 Ergodicity for divergent groups

##  1\. Introduction

The geodesic flow is an important topic lying in the crossing field of
dynamical systems and differential geometry. The geodesic flow on manifolds of
negative curvature and nonpositive curvature is the most natural example of
hyperbolic dynamical systems and chaotic phenomenon, and its ergodic
properties are extensively studied since the beginning of the last century. In
this article, we present several new progresses in the ergodic theory of
geodesic flows on manifolds without conjugate points, including the ergodicity
with respect to Liouville measure, uniqueness and Bernoulli properties of
measure of maximal entropy (MME for short) and equilibirum states, counting
closed geodesics and volume growth asymptotics, Margulis functions and related
rigidity phenomenon, and the Hopf-Tsuji-Sullivan dichotomy with respect to the
Bowen-Margulis measure.

Suppose that $(M,g)$ is a $C^{\infty}$ closed $n$-dimensional Riemannian
manifold, where $g$ is a Riemannian metric. The geodesic flow describes the
motion of a free particle on $M$. Suppose that a particle is at a point $p\in
M$ with unit velocity $v\in S_{p}M:=\\{v\in TM:\|v\|=1\\}$. Then the particle
will move along the geodesic that is tangent to $v$ and proceeds length $t$ by
time $t$. Let $\pi:SM\to M$ be the unit tangent bundle over $M$. For each
$v\in S_{p}M$, we always denote by $c_{v}:{\mathbb{R}}\to M$ the unique
geodesic on $M$ satisfying the initial conditions $c_{v}(0)=p$ and
$\dot{c}_{v}(0)=v$. The geodesic $c_{v}$ satisfies the geodesic equation

$\nabla_{\dot{c}_{v}(t)}\dot{c}_{v}(t)=0,$

where $\nabla$ is the Riemannian connection on $M$. The geodesic flow
$\phi=(\phi^{t})_{t\in\mathbb{R}}$ (induced by the Riemannian metric $g$) on
$SM$ is defined as:

$\phi^{t}:SM\rightarrow SM,\qquad(p,v)\mapsto(c_{v}(t),\dot{c}_{v}(t)),\ \ \ \
\forall\ t\in{\mathbb{R}}.$

A vector field $J(t)$ along a geodesic $c:{\mathbb{R}}\to M$ is called a
_Jacobi field_ if it satisfies the _Jacobi equation_ :

$J^{\prime\prime}+R(J,\dot{c})\dot{c}=0$

where $R$ is the Riemannian curvature tensor and ′ denotes the covariant
derivative along $c$.

######  Definition 1.1.

Let $c$ be a geodesic on $(M,g)$.

  1. (1)

A pair of distinct points $p=c(t_{1})$ and $q=c(t_{2})$ are called _focal_ if
there is a Jacobi field $J$ along $c$ such that $J(t_{1})=0$,
$J^{\prime}(t_{1})\neq 0$ and $\frac{d}{dt}\mid_{t=t_{2}}\|J(t)\|^{2}=0$;

  2. (2)

$p=c(t_{1})$ and $q=c(t_{2})$ are called _conjugate_ if there is a nontrivial
Jacobi field $J$ along $c$ such that $J(t_{1})=0=J(t_{2})$.

A compact Riemannian manifold $(M,g)$ is called a manifold _without focal
points/without conjugate points_ if there is no focal points/conjugate points
on any geodesic in $(M,g)$.

By definition, if a manifold has no focal points then it has no conjugate
points. All manifolds of nonpositive curvature always have no focal points. In
brief, we have

negative curvature $\displaystyle\Rightarrow\text{nonpositive
curvature}\Rightarrow\text{no focal points}$ $\displaystyle\Rightarrow\text{no
conjugate points}.$

There is a natural isomorphism between the tangent space $T_{v}(SM)$ and the
space of perpendicular Jacobi fields along the geodesic $c_{v}$ where $v\in
SM$. The stable/unstable Green bundles $G^{s}(v)/G^{u}(v),v\in SM$ are
subbundles of $TSM$ associated to the two distinguished families of Jacobi
fields constructed by Green [50], which we call stable/unstable Jacobi fields.
In general $G^{s}(v)/G^{u}(v)$ are only measurable in $v$, and they are not
necessarily transverse to each other at $v$.

The statistical properties of geodesic flows on surfaces with negative
curvature were first studied by Hadamard and Morse in the beginning of the
twentieth century. In the 1960s, Anosov [2, 3] introduced the notion of the
so-called Anosov flows by abstracting the hyperbolic behaviour of the geodesic
flows of negatively curved manifolds. Suppose that $M$ has negative curvature
everywhere $K<0$. Then the geodesic flow $\phi^{t}$ is an Anosov flow: there
exist a $\phi^{t}$-invariant $T_{v}SM=Z(v)\oplus G^{s}(v)\oplus G^{u}(v)$,
constants $0<\lambda<1$, $c>0$ such that

  1. (1)

$Z(v)$ is the vector field tangent to the orbit of the flow;

  2. (2)

$\|d\phi^{t}(\xi)\|\leq c\lambda^{t}$ for $t>0$, whenever $\xi\in G^{s}(v)$,

  3. (3)

$\|d\phi^{-t}(\eta)\|\leq c\lambda^{t}$ for $t>0$, whenever $\eta\in
G^{u}(v)$.

$T_{v}SM=Z(v)\oplus G^{s}(v)\oplus G^{u}(v)$ is called the Anosov splitting
associated with the Anosov flow. In this negative curvature case, the strong
stable/unstable bundles of Anosov spliting $G^{s}(v)/G^{u}(v)$ are given
exactly by Green bundles, which is proved to be Hölder continuous. Furthermore
it is known that they are uniquely integrable to foliations $W^{s}$ and
$W^{u}$ of $SM$, called the strong stable/unstable foliations of $\phi^{t}$.

Geodesic flows on closed manifolds with negative curvature have very rich
dynamics and broad applications. In the last century, this class of geodesic
flows have always been attracting great interests of the mathematicians in
dynamical systems and related areas. A lot of beautiful results on the
dynamics of the geodesic flows have been exhibited. Among which, the
ergodicity and mixing properties of Liouville measure, the uniqueness of
measure of maximal entropy etc., have their special importance and receive
extensive attentions.

Geodesic flows on manifolds of nonpositive curvature $K\leq 0$ have also been
extensively studied since 1970’s. The study for the geodesic flows become much
more difficult due to the existence of parallel Jacobi fields. A Jacobi field
$J(t)$ along a geodesic $c(t)$ is called _parallel_ if $J^{\prime}(t)=0$ for
all $t\in{\mathbb{R}}$. The notion of _rank_ is defined as follows.

######  Definition 1.2.

For each $v\in SM$, we define rank($v$) to be the dimension of the vector
space of parallel Jacobi fields along the geodesic $c_{v}$, and
rank($M$):=$\min\\{$rank$(v):v\in SM\\}$.

In fact by the higher rank rigidity theorem [7, 9, 10, 21, 22], any simply
connected irreducible Riemannian manifold of nonpositive curvature and of rank
greater than one is isometric to a globally symmetric space of noncompact
type. This theorem was extended by Watkins [116] to manifolds without focal
points. On the other hand, for rank one manifolds of nonpositive curvature or
more generally without focal points, the geodesic flows present certain _non-
uniformly hyperbolic_ behaviors. Ergodic properties including the ergodicity
with respect to Liouville measure and the uniqueness of MME attract many
interests and fruitful results were achieved in the last several decades (see
[66, 67, 5, 118, 17, 48, 74, 72, 27, 28, 79, 121, 119, 120] and many others).

Even weaker hyperbolicity can be enjoyed by the geodesic flows on manifolds
without conjugate points. The study of its ergodic properties faces a great
challenge and relatively few results are known so far (see for example [30,
31, 49, 81, 80, 119, 95] etc.) Our goal in this paper is to conduct a
comprehensive study on ergodic properties of geodesic flows on various classes
of manifolds without conjugate points.

##  2\. Statement of main results

For geodesic flows on manifolds without conjugate points, the very weak
hyperbolicity prevents us to use rich results from hyperbolic dynamics. On the
other hand, in contrast with manifolds with negative or nonpositive curvature,
strong geometric properties such as convexity and monotonicity, or the
powerful tools such as analysis involving the curvature are not available.
Nevertheless, Eberlein [38, 40] introduced the following uniform visibility
axiom, which provides us a rather good geometric property of the manifold.

######  Definition 2.1.

(Cf. [31, Definition 2.1]) A simply connected Riemannian manifold $X$ is a
_uniform visibility manifold_ if for every $\epsilon>0$ there exists
$L=L(\epsilon)>0$ such that whenever a geodesic segment $c:[a,b]\to X$ stays
at distance at least $L$ from some point $p\in X$, then the angle sustained by
$c$ at $p$ is less than $\epsilon$, that is,

$\angle_{p}(c):=\sup_{a\leq s,t\leq b}\angle_{p}(c(s),c(t))<\epsilon.$

If $M$ is a Riemannian manifold without conjugate points whose universal cover
$X$ is a uniform visibility manifold, then we say that $M$ is a uniform
visibility manifold.

Let $M$ be a compact manifold with nonpositive curvature $K\leq 0$. Then $M$
is a uniform visibility manifold if and only if the universal cover $X$ of $M$
contains no totally geodesic isometric imbedding of the plane
${\mathbb{R}}^{2}$, see [38].

Now we introduce various classes of manifolds without conjugate points, coming
from the Green bundles or Jacobi equations. Let $M$ be a Riemannian manifold
without conjugate points. As mentioned above, in general, $G^{s}(v)/G^{u}(v)$
are only measurable in $v$, and they are not necessarily transverse to each
other at $v$. Let us define

(1) $\mathcal{R}_{1}:=\\{v\in SM:G^{s}(v)\cap G^{u}(v)=\\{0\\}\\}.$

We say that $M$ has continuous Green bundles or continuous asymptote if
$G^{s}(v)/G^{u}(v)$ are continuous in $v$. In this case, it is clear that
$\mathcal{R}_{1}$ is an open subset of $SM$.

We say that $M$ has bounded asymptote if there exists $C>0$ such that
$\|J^{s}(t)\|\leq C\|J^{s}(0)\|$ for any $t>0$ and any stable Jacobi field
$J^{s}$. We have (cf. [64, 5.3 Satz], [49])

$\displaystyle\text{no focal points}\Rightarrow\text{bounded
asymptote}\Rightarrow\text{continuous asymptote}.$

In Section 3, we will recall the Busemann functions and the construction of
stable/unstable horospherical foliations, denoted by
$\mathcal{F}^{s}(v)/\mathcal{F}^{u}(v),v\in SM$, for manifolds without
conjugate points. Let $X$ be a simply connected uniform visibility manifold
without conjugate points. Then $\mathcal{F}^{s}(v),\mathcal{F}^{u}(v)$ are
both continuous in $v\in SX$. Denote
$\mathcal{I}(v)=\mathcal{F}^{s}(v)\cap\mathcal{F}^{u}(v)$. If
$\mathcal{I}(v)=\\{v\\}$, then $v$ is called an expansive vector. Denote the
set of expansive vectors by

$\mathcal{R}_{0}:=\\{v\in SM:\mathcal{I}(v)=\\{v\\}\\}.$

These two sets $\mathcal{R}_{0}$ and $\mathcal{R}_{1}$ are central objects in
our study of dynamical properties of the geodesic flow.

###  2.1. Ergodicity of Liouville measure

Let $\mathcal{M}_{\phi}(SM)$ be the set of $\phi^{t}$-invariant Borel
probability measures on $SM$. We say that the geodesic flow is ergodic with
respect to $\mu\in\mathcal{M}_{\phi}(SM)$ or just $\mu$ is ergodic, if
$\mu(\phi^{t}(A))=\mu(A)$ for any Borel measurable set $A\subset SM$ and any
$t\in{\mathbb{R}}$. Equivalently, by Birkhoff ergodic theorem, $\mu$ is
ergodic if and only if for $\mu\text{-a.e.}\ v\in SM$

$\lim_{t\to\pm\infty}\frac{1}{t}\int_{0}^{t}f(\phi^{s}(v))ds=\int_{SM}fd\mu,\quad\forall
f\in C(SM,\mathbb{R}).$

Here $C(SM,\mathbb{R})$ denotes the Banach space of all continuous real-valued
functions on $SM$.

The most natural invariant measure for the geodesic flow is the Liouville
measure $\nu$, which is the normalized volume measure induced by the Sasaki
metric on $SM$. Hopf (cf. [56, 57]) proved the ergodicity of the geodesic flow
with respect to the Liouville measure $\nu$ on $SM$ for compact surfaces of
variable negative curvature and for compact manifolds of constant negative
sectional curvature in any dimension. The general case for compact manifolds
of variable negative curvature was established by Anosov and Sinai (cf. [2,
3]). The ergodicity was established based on the classical Hopf argument using
stable/unstable foliations on $SM$ (cf., for example the appendix in [8]).

In 1970’s, by using his theory of nonuniform hyperbolicity, Pesin obtained a
celebrated result on the ergodicity of the geodesic flows on manifolds without
focal points satisfying the uniform visibility property (cf. Theorem 12.2.12
in [11]). Define the sets:

(2) $\displaystyle\Delta^{+}$ $\displaystyle:=\\{v\in SM:\chi(v,\xi)<0\text{\
for any\ }\xi\in G^{s}(v)\\},$ $\displaystyle\Delta^{-}$
$\displaystyle:=\\{v\in SM:\chi(v,\xi)>0\text{\ for any\ }\xi\in G^{u}(v)\\},$
$\displaystyle\Delta$ $\displaystyle:=\Delta^{+}\cap\Delta^{-},$

where $\chi$ denotes the Lyapunov exponents, and $G^{s}/G^{u}$ the
stable/unstable Green bundles. $\Delta$ is called the _regular set_ with
respect to the geodesic flow. Pesin proved the following theorem:

######  Theorem 2.2 (Pesin, cf. [11]).

For the geodesic flow on a closed surface of genus $\mathfrak{g}\geq 2$ and
without focal points, we have that $\nu(\Delta)>0$, and $\phi^{t}|_{\Delta}$
is ergodic.

However, even for surfaces of nonpositive curvature, the ergodicity of the
geodesic flows with respect to the Liouville measure $\nu$ on $SM$ is widely
open since 1980’s, due to the existence of the following “flat” geodesics.

Consider a closed surface $M$ of genus $\mathfrak{g}\geq 2$ and of nonpositive
curvature. Let

$\displaystyle\Lambda:=\\{v\in SM:K(c_{v}(t))=0,\ \forall t\in\mathbb{R}\\},$

where $K$ denotes the curvature of the point. We call $c_{v}$ a _singular or
flat geodesic_ if $v\in\Lambda$. People still do not know if $\Lambda$ is
small in measure ($\nu(\Lambda)=0$ or not), in general. However, from the
dynamical point of view, Knieper [66] showed there is an entropy gap in
arbitrary dimensions:

$h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(\phi^{1}|_{\Lambda})<h_{{\mathop{\hbox{\footnotesize\rm
top}}}}(\phi^{1}),$

where $h_{{\mathop{\hbox{\footnotesize\rm top}}}}$ denotes the topological
entropy, and $\Lambda$ denotes the irregular set of the geodesic flow which is
a counterpart of the above defined set in arbitrary dimensions. Recently,
Burns-Climenhaga-Fisher-Thompson [17] reproved the entropy gap using
specification. This means that the geodesic flow restricted on $\Lambda$ has
less complexity than the whole system. Knieper [66] (see also Burns and
Gelfert [18]) proved that on rank $1$ surfaces of nonpositive curvature, the
geodesic flow on $\Lambda$ has zero topological entropy. In higher dimensions,
it is possible to have positive entropy on $\Lambda$; an example was given by
Gromov (cf. [51]).

For geodesic flows on rank $1$ surfaces of nonpositive curvature, the orbits
inside $\Lambda$ are also believed to have simple behavior. In all the known
examples, all the orbits in $\Lambda$ are closed. In a recent survey, Burns
asked whether there exists a non-closed flat geodesic (cf. Question 6.2.1 in
[20]).

######  Conjecture 2.3 (Cf. [20, 54, 103] etc.).

Let $(M,g)$ be a smooth, connected and closed surface of genus
$\mathfrak{g}\geq 2$, which has nonpositive curvature. Then all flat geodesics
are closed and there are only finitely many homotopy classes of such
geodesics. In particular, $\nu(\Lambda)=0$, and hence the geodesic flow on
$SM$ is ergodic.

The conjecture has been verified for surfaces of genus $\mathfrak{g}\geq 2$
with nonpositive curvature [118] and without focal points [121], assuming that
the negative curvature part has finitely many connected components:

######  Theorem 2.4 (Cf. [118, 121]).

Let $(M,g)$ be a smooth, connected and closed surface of genus
$\mathfrak{g}\geq 2$, without focal points. Suppose that the set $\\{x\in
M:K(x)<0\\}$ has finitely many connected components, then there are only
finitely many homotopy classes of flat geodesics and thus $\nu(\Lambda)=0$. In
particular, the geodesic flow is ergodic.

In this paper, we study the ergodicity of the geodesic flows on surfaces of
genus $\mathfrak{g}\geq 2$ without conjugate points and with continuous Green
bundles. We first show the relation between the regular set $\Delta$ and the
set $\mathcal{R}_{1}$ of unit vectors with linearly independent stable and
unstable Green bundles, see (1). We call the geodesic $c_{v}$ singular, if
$v\in\mathcal{R}_{1}^{c}$. To achieve the goal, we explore the properties of
singular geodesics, which are also of independent interest. Our results are
summarized in the following theorem. Here, we let $\text{Per}(\phi^{t})$
denote the set of periodic points of the geodesic flow.

######  Theorem A.

Let $(M,g)$ be a smooth, connected and closed surface of genus
$\mathfrak{g}\geq 2$ without conjugate points and with continuous Green
bundles, then $\Delta$ agrees $\nu$-almost everywhere with the open dense set
$\mathcal{R}_{1}$. If furthermore $M$ has bounded asymptote, then:

  1. (1)

Let $c_{v}$ be a lift of a closed geodesic and $v\in\mathcal{R}_{1}^{c}$. Then
an ideal triangle with $c_{v}$ as an edge has infinite area;

  2. (2)

If $\mathcal{R}_{1}^{c}\subset\text{Per}(\phi^{t})$, then there is a finite
decomposition of $\mathcal{R}_{1}^{c}$:

$\mathcal{R}_{1}^{c}=\mathcal{O}_{1}\cup\mathcal{O}_{2}\cup\ldots\mathcal{O}_{k}\cup\mathcal{S}_{1}\cup\mathcal{S}_{2}\cup\ldots\cup\mathcal{S}_{l},$

where each $\mathcal{O}_{i},1\leq i\leq k$, is an isolated periodic orbit and
each $\mathcal{S}_{j},1\leq j\leq l$, consists of vectors tangent to a
generalized strip. Here $k$ or $l$ are allowed to be $0$ if there is no
isolated closed singular geodesic or no generalized strip. In particular,
$\nu(\mathcal{R}_{1}^{c})=0$, and hence the geodesic flow on $SM$ is ergodic.

We ask the following question for the future study.

######  Question 2.5.

Let $(M,g)$ be a smooth, connected and closed surface of genus
$\mathfrak{g}\geq 2$, without conjugate points, and with continuous Green
bundles. Suppose that the set $(\pi(\mathcal{R}_{1}^{c}))^{c}$ has finitely
many connected components. Does there exist a non-closed singular geodesic? If
no, we have $\mathcal{R}_{1}^{c}\subset\text{Per}(\phi^{t})$ and hence by
Theorem A, the geodesic flow is ergodic.

One essential difficulty is that the flat strip lemma fails for surfaces
without conjugate points [16], and there is lack of control on the geometric
shape of the generalized strips, see for example the proof of Lemmas 4.8 and
4.9 and Theorem 4.10. Moreover, we cannot effectively characterize the
singular geodesics using curvatures through Riccati equations.

###  2.2. Measure of maximal entropy and Bowen-Margulis measures

Metric entropy introduced by Kolmogorov [69] and Sinai, is one of the most
important invariant describing the complexity of a dynamical system. Later,
Adler-Konheim-McAndrew [1] introduced the topological entropy for a
topological dynamical system.

######  Theorem 2.6 (Variational principle for entropy).

Let $T:Y\to Y$ be a continuous self-map on a compact metric space $(Y,d)$.
Then

$\displaystyle h_{top}(T)$
$\displaystyle=\sup\\{h_{\mu}(T):\mu\in\mathcal{M}_{T}(Y)\\}$
$\displaystyle=\sup\\{h_{\mu}(T):\mu\in\mathcal{M}_{T}^{e}(Y)\\}.$

Here $\mathcal{M}_{T}(Y)$ (resp. $\mathcal{M}^{e}_{T}(Y)$) denotes the set of
all $T$-invariant (resp. ergodic) probability measures on $Y$, $h_{top}(T)$
denotes the topological entropy of $T$, and $h_{\mu}(T)$ the metric entropy of
$\mu$.

A measure $m\in\mathcal{M}_{T}(Y)$ is called a _measure of maximal entropy_
(MME for short) if $h_{m}(T)=h_{top}(T)$. The definition applies to a flow by
considering its time-one map. MME (when it exists) is another important class
of distinct invariant measures, which reflects the complexity of the whole
system.

When a closed Riemannian manifold $(M,g)$ has negative curvature, the geodesic
flow on $SM$ is a prime example of Anosov flow (cf. [2], [63, Section 17.6]).
For Anosov flows, Margulis [84] in his thesis constructed an invariant measure
under the geodesic flow. This measure is mixing, and the conditional measures
on stable/unstable manifolds contract/expand with a uniform rate under the
flow. Using different methods, Bowen constructed a MME in [12] for Axiom A
flows. Later in [13], Bowen proved that MME is unique. It turns out that the
measures constructed by Bowen and Margulis are eventually the same one, which
is called _Bowen-Margulis measure_. Bowen-Margulis measure has the Bernoulli
property [99], an ultimate mixing property for measurable dynamical systems.
The uniqueness of MME becomes a classical problem in smooth ergodic theory.
For uniformly hyperbolic systems, there are at least two typical methods, one
using Markov partitions and symbolic dynamics ([15]), the other using
topological properties such as expansiveness and specification ([14, 44]).

In 1984, A. Katok [19] conjectured that the geodesic flow on a closed rank one
manifold of nonpositive curvature admits a unique MME. In 1998, Knieper [67]
proved Katok’s conjecture. He used Patterson-Sullivan measures on the boundary
at infinity of the universal cover of $M$ to construct a measure (which is
called Knieper measure), and showed that this measure is the unique MME.

Beyond nonpositive curvature, manifolds without focal/conjugate points are
studied extensively. Following Knieper’s method, Liu, Wang and the author
construct the Knieper measure for manifolds without focal points and showed it
is the unique MME [79]. Gelfert and Ruggiero [48] also proved the uniqueness
of MME for surfaces without focal points by constructing an expansive factor
of the geodesic flow.

For general manifolds without conjugate points, uniqueness of MME is far from
being well understood. The first remarkable progress was made by Climenhaga,
Knieper and War for a class of manifolds (including all surfaces of genus at
least $2$) without conjugate points. In [30], the authors proved the
uniqueness of MME for the following manifolds in _class $\mathcal{H}$_.

######  Definition 2.7.

We say that compact manifolds without conjugate points satisfying the
following conditions are in _class $\mathcal{H}$_:

  1. (1)

There exists a Riemannian metric $g_{0}$ on $M$ for which all sectional
curvatures are negative;

  2. (2)

$M$ has uniform visibility property;

  3. (3)

The fundamental group $\pi_{1}(M)$ is residually finite: the intersection of
its finite index subgroups is trivial;

  4. (4)

There exists $h_{0}<h$ such that any ergodic invariant Borel probability
measure $\mu$ on $SM$ with entropy strictly greater than $h_{0}$ is almost
expansive (cf. [31, Definition 2.8]).

All compact surfaces of genus at least $2$ without conjugate points belong to
the class $\mathcal{H}$.

Very recently, another remarkable progress was made in a series of papers by
Gelferet-Ruggiero [49], Mamani [80] and Mamani-Ruggiero [81]:

######  Theorem 2.8.

([81, Theorem 1.2]) Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility
manifold without conjugate points. Suppose that

$\sup\\{h_{\mu}(\phi^{1}):\mu\text{\ is supported on\
}\mathcal{R}_{0}^{c}\\}<h_{{\mathop{\hbox{\footnotesize\rm top}}}}(\phi^{1}).$

Then the geodesic flow has a unique MME.

The entropy gap assumption above is equivalent to the condition (4) for
manifolds in _class $\mathcal{H}$_. It is satisfied automatically for
surfaces:

######  Theorem 2.9.

([80, Theorem 1.2]) Let $(M,g)$ be a closed $C^{\infty}$ surface without
conjugate points of genus greater than one. Then the geodesic flow has a
unique MME.

We remark that Theorem 2.8 applies to a class of manifolds larger than class
$\mathcal{H}$, since conditions (1) and (3) for class $\mathcal{H}$ are
dropped. The authors prove that there exists an expansive factor of the
geodesic flow acting on a compact metric space which is topologically mixing
and has local product structure and hence has unique MME. On the other hand,
Climenhaga, Knieper and War [30] used Morse lemma to get coarse estimates, and
successfully applied Climenhaga-Thompson’s criterion [32] using a nonuniform
version of specification and expansiveness. Climenhaga-Knieper-War [30] also
construct the unique MME via Patterson-Sullivan measures and prove that the
MME is mixing.

In this paper, for a closed uniform visibility manifold without conjugate
points, we give an explicit construction of a Bowen-Margulis measure via
Patterson-Sullivan measures on the boundary at infinity. Since the generalized
strip does not necessarily possess a submanifold structure, there is no volume
along a generalized strip. Instead, we use a measurable choice theorem to
construct a Borel measure along a generalized strip. We show that a Bowen-
Margulis measure constructed in this way has metric entropy equal to the
topological entropy, and hence coincides with the unique MME obtained in
Theorem 2.8. This shows that the unique MME has local product structure. In
fact, our Patterson-Sullivan construction of Bowen-Margulis measure also works
for noncompact uniform visibility manifold without conjugate points, which
allows us to consider the Hopf-Tsuji-Sullivan dichotomy with respect to such
Bowen-Margulis measure in the last section.

As an application of the local product structure of Bowen-Margulis measure, we
show that it has the Bernoulli property, which is the ultimate mixing property
for measurable dynamical systems. In nonpositive curvature, the Bowen-Margulis
measure is proved to be mixing in [5] and Bernoulli in [26]. For manifolds
without focal points, the Bowen-Margulis measure is known to be mixing by
[77], have the Kolmogorov property by [28] and the Bernoulli property by
[120]. We apply the method of Call and Thompson [26] to prove the Kolmogorov
property by showing that the product system of the expansive factors has a
unique MME. To lift the Kolmogorov property to the Bernoulli, we adapt the
classical argument in [88, 99] for Anosov flows. In fact, we follow closely
[29], which is for systems with nonuniformly hyperbolic structure.

######  Theorem B.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points. Suppose that

$\sup\\{h_{\mu}(\phi^{1}):\mu\text{\ is supported on\
}\mathcal{R}_{0}^{c}\\}<h_{{\mathop{\hbox{\footnotesize\rm top}}}}(\phi^{1}).$

Then the unique MME is fully supported and has the Bernoulli property.

###  2.3. Uniqueness of equilibrium states

First introduced by Ruelle [104] in 1973 for an expansive dynamical system and
later studied by Walters [114] for general systems, pressure and equilibrium
states are natural generalization of entropy and MME, provided a continuous
potential function on the phase space. They have played a fundamental role in
ergodic theory, dynamical systems and dimension theory ([12, 105, 115, 93],
etc.). The uniqueness of equilibrium states for Hölder continuous potentials
was shown for uniformly hyperbolic systems in the classical works of Bowen
[14, 15]. Moreover, the unique equilibrium state has the Bernoulli property.

Recently, Burns, Climenhaga, Fisher and Thompson [17] proved the uniqueness of
equilibrium states for certain potentials in nonpositive curvature, which
generalized Knieper’s result. Extending the approach in [17], Chen, Kao and
Park proved the uniqueness of equilibrium states for certain potentials in
[27, 28] for no focal points case. These results are obtained via Climenhaga-
Thompson approach [32] using nonuniform specification and expansiveness.
However, there are essential technical gaps in this approach to study
uniqueness of equilibrium states for no conjugate points case.

Recently, Lima-Poletti [72] reproved the uniqueness of equilibrium states in
nonpositive curvature. They developed a countable symbolic coding of the
systems and thus obtained the uniqueness of equilibrium states. Based on the
nununiform hyperbolicity of the system, we are able to apply Lima-Poletti’s
approach in no conjugate points case:

######  Theorem C.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles, and let
$\psi:SM\to{\mathbb{R}}$ be Hölder continuous or of the form $\psi=q\psi^{u}$
for $q\in{\mathbb{R}}$. Suppose that the geodesic flow has a hyperbolic
periodic point. If $P(\mathcal{R}_{1}^{c},\psi)<P(\psi)$, then $\psi$ has a
unique equilibrium state $\mu$, which is hyperbolic, fully supported and has
the Bernoulli property.

In the above theorem, $\psi^{u}$ is the geometric potential defined as

$\psi^{u}(v):=-\frac{d}{dt}\Big{|}_{t=0}\log\det(d\phi^{t}\mid_{G^{u}(v)}).$

We also have the following corollary for MME, which recovers [81, Part 2 of
Theorem 1.4].

######  Corollary C.1.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles. Suppose that the geodesic
flow has a hyperbolic periodic point. Then the geodesic flow has a unique MME,
which is hyperbolic, fully supported and has the Bernoulli property.

Compare the above corollary with Theorem B. In the setting of Theorem B, we
only know that $\mathcal{R}_{0}$ has full measure with respect to the unique
MME. However, we do not necessarily have
$\mathcal{R}_{1}\subset\mathcal{R}_{0}$. Thus we do not know whether the MME
is hyperbolic and the symbolic approach [4, Theorem 1.2] is not applicable.

###  2.4. Counting closed geodesics and volume asymptotics

####  2.4.1. Counting closed geodesics

There is correspondence between closed geodesics on $M$ and periodic orbits
under the geodesic flow, and they are important from both geometric and
dynamical points of view. Let $P(t)$ be the set of free-homotopy classes
containing a closed geodesic with length at most $t$. If $M$ has negative
curvature, each free-homotopy class contains exactly one closed geodesic, so
$\\#P(t)$ is just the number of closed geodesics with length at most $t$. If
$M$ has nonpositive curvature or without focal/conjugate points, a free-
homotopy class may have infinitely many closed geodesics.

We obtain in this paper the following Margulis-type asymptotic estimates of
$\\#P(t)$.

######  Theorem D.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles. Suppose that the geodesic
flow has a hyperbolic periodic point. Then

(3) $\displaystyle\\#P(t)\sim\frac{e^{ht}}{ht}$

where $h=h_{\text{top}}(g)$ denotes the topological entropy of the geodesic
flow on $SM$, and the notation $f_{1}\sim f_{2}$ means that
$\lim_{t\to\infty}\frac{f_{1}(t)}{f_{2}(t)}=1$.

In negative curvature, the formula (3) of asymptotic growth of the number of
closed geodesics is given by the celebrated theorem of Margulis in his 1970
thesis [83, 84]. The main tool in the proof of Margulis’ theorem is the
ergodic theory of Bowen-Margulis measure, which is the unique MME for the
Anosov flow. Margulis [84] gave an explicit construction of this measure, and
showed that it is mixing, and the conditional measures on stable/unstable
manifolds have the scaling property, (i.e., contract/expand with a uniform
rate under the geodesic flow), and are invariant under unstable/stable
holonomies. Margulis [84] then proved (3) using these ergodic properties of
the Bowen-Margulis measure.

There are other approaches to formula (3) in various settings. In the 1950’s
Huber [58] proved it for closed surfaces of constant negative curvature using
Selberg’s trace formula. Another alternative approach to (3) for Axiom A flows
is given by Parry and Pollicott [89], based on zeta functions and symbolic
dynamics. See the survey by Sharp in [84] for further progress that has been
made since Margulis’ thesis.

It took a long time to establish formula (3) for the geodesic flow on a closed
rank one manifold of nonpositive curvature, which exhibits nonuniformly
hyperbolic behavior. In 1982, Katok [62] used a closing lemma in Pesin theory
to show that for a smooth flow with topological entropy $h>0$ and no fixed
points, the exponential growth rate of the number of periodic orbits is at
least $h$. During the course of setting Katok’s conjecture on the uniqueness
of MME, Knieper constructed Patterson-Sullivan measure and used it to obtain
the following asymptotic estimates [66, 68]: there exists $C>0$ such that for
$t$ large enough

$\frac{1}{C}\frac{e^{ht}}{t}\leq\\#P(t)\leq C\frac{e^{ht}}{t}.$

Since then, efforts are made to improve the above to the Margulis-type
asymptotic estimates (3). A preprint [52], though not published, contains many
inspiring ideas to this problem. A breakthrough was made recently by Ricks
[100]: (3) is established for rank one locally CAT$(0)$ spaces, which include
rank one manifolds with nonpositve curvature.

Beyond nonpositive curvature, remarkable progress was made by Climenhaga,
Knieper and War [31] where (3) is obtained for manifolds without conjugate
points in class $\mathcal{H}$. Weaver [117] also obtained (3) for surfaces
having negative curvature outside of a collection of radially symmetric caps.
The author [120] obtained (3) for rank one manifolds without focal points.

The following is a result of equidistribution of periodic orbits with respect
to the Bowen-Margulis measure, which is crucial to prove Theorem D.

######  Theorem 2.10.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles. Suppose that the geodesic
flow has a hyperbolic periodic point, and $\epsilon\in(0,{\text{inj}}(M)/2)$
is fixed where ${\text{inj}}(M)$ is the injectivity radius of $M$. For $t>0$,
let $C(t)$ be any maximal set of pairwise non-free-homotopic closed geodesics
with lengths in $(t-\epsilon,t]$, and define the measure

$\nu_{t}:=\frac{1}{\\#C(t)}\sum_{c\in C(t)}\frac{Leb_{c}}{t}$

where $Leb_{c}$ is the Lebesgue measure along the curve $\dot{c}$ in the unit
tangent bundle $SM$.

Then the measures $\nu_{t}$ converge in the weak∗ topology to the unique MME
as $t\to\infty$.

We follow Margulis’ approach and combine ideas in [100, 31]. For example, we
construct a flow box in a geometric way which helps us to apply the local
product structure of the Bowen-Margulis measure. We use mixing properties of
the MME to count the number of self-intersections of the box under the
geodesic flow. To produce closed geodesics from such self-intersections, a
type of closing lemma is of particular importance, which is established by the
uniform visibility property. Moreover in [31], by Morse Lemma the action
induced by a deck transformation on the boundary at infinity is topologically
conjugate to the analogous one under a metric of negative curvature ([31,
Remark 2.6]). In our setting, this property is absent. We assume that the
visibility manifold $M$ has continuous Green bundles, so that we can choose
the flow box inside $\mathcal{R}_{1}$ to avoid generalized strips. See Remarks
7.5 and 7.21.

####  2.4.2. Volume estimates

A twin problem is the asymptotics of volume growth of Riemannian balls in the
universal cover. Margulis in his thesis also obtained the following result:
Consider a smooth closed manifold $M$ of negative curvature with universal
cover $X$, then

(4) $b_{t}(x)/\frac{e^{ht}}{h}\sim c(x),$

where $b_{t}(x)$ is the Riemannian volume of the ball of radius $t>0$ around
$x\in X$, $h$ the topological entropy of the geodesic flow, and
$c:X\to{\mathbb{R}}$ is a continuous function, which is called _Margulis
function_. See [83, 84].

For a closed rank one manifold of nonpositive curvature, Knieper [66] used his
measure to obtain the following asymptotic estimates: there exists $C>0$ such
that

$\frac{1}{C}\leq b_{t}(x)/e^{ht}\leq C$

for any $x\in X$. However, it is difficult to improve the above to the
Margulis-type asymptotic estimates (4), see the remark after [68, Chapter 5,
Theorem 3.1] and an unpublished preprint [53]. Recently, a breakthrough was
made by Link ([75, Theorem C]), where an asymptotic estimate for the orbit
counting function is obtained for a CAT$(0)$ space, and as a consequence (4)
is established for rank one manifolds of nonpositve curvature.

Beyond nonpositive curvature, the author [119] proved (4) for rank one
manifolds without focal points, and briefly discussed the case for manifolds
without conjugate points in class $\mathcal{H}$. Recently, Pollicott and War
[95] proved (4) for manifolds without conjugate points in class $\mathcal{H}$.

In this paper, we establish volume asymptotics (4) for uniform visibility
manifolds without conjugate points and with continuous Green bundles:

######  Theorem E.

Let $(M,g)$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles. Suppose that the geodesic
flow has a hyperbolic periodic point. Then

$b_{t}(x)/\frac{e^{ht}}{h}\sim c(x),$

where $b_{t}(x)$ is the Riemannian volume of the ball of radius $t>0$ around
$x\in X$, $h$ the topological entropy of the geodesic flow, and
$c:X\to{\mathbb{R}}$ is a continuous function.

Our proof of Theorem E also uses Margulis’ approach. We construct geometric
flow boxes and apply a type of $\pi$-convergence theorem, i.e., Lemma 7.8.
First we will establish the asymptotics formula for a pair of flow boxes using
the mixing property of the unique MME and scaling property of Patterson-
Sullivan measure. The asymptotics in (4) involves countably many pairs of flow
boxes and an issue of nonuniformity arises. To overcome this difficulty, we
will use the fact that the unique MME is supported on $\mathcal{R}_{1}^{c}$
and apply Knieper’s techniques (Lemmas 8.15 and 8.16). The assumption of
continuous Green bundles is crucial here, as well as in the proof of Lemma
8.2.

Let us compare the proofs of Theorem D and Theorem E. In the proof of Theorem
D, due to the equidistribution of closed geodesics, we only need consider one
flow box and count the number of self-intersections of the box under the
geodesic flow. But we need to deal with the axis of deck transformations, in
order to count effectively the homotopy classes of closed geodesics. On the
other hand, in the proof of Theorem E, we only need to count intersections of
flow boxes, and in fact even the uniqueness of MME is not directly used. The
technical novelties in the proof of Theorem E is that we have to deal with
countably many pairs of flow boxes and the essential difficulty caused by
nonuniform hyperbolicity. Even for one pair of flow boxes, the calculations
involved are more complicated. Moreover we also consider the intersection
components of a flow box with a face under the geodesic flow, which makes the
calculation more subtle.

###  2.5. Margulis function and rigidity

Let $M=X/\Gamma$ be a closed $C^{\infty}$ uniform visibility manifold without
conjugate points and with continuous Green bundles, and $X$ its universal
cover. Suppose that the geodesic flow has a hyperbolic periodic point. The
continuous function $c(x)$ obtained in Theorem E is called _Margulis
function_. It is $\Gamma$-invariant and hence descends to a function on $M$,
which is still denoted by $c$.

As Margulis function is closely related to Patterson-Sullivan measure, its
interactions with Bowen-Margulis measure, Lebesgue measure and harmonic
measure usually give various characterizations of locally symmetric spaces. In
this paper, we study rigidity results related to Margulis function for
manifolds without conjugate points, which extend results for manifolds of
negative curvature [125] and without focal points [119].

In negative curvature, Katok conjectured that $c(x)$ is almost always not
constant and not smooth, unless $M$ is locally symmetric. In [125] Yue
answered Katok’s conjecture, and his results can be extended to certain
manifolds without conjuagte points.

######  Theorem F.

Let $M$ be a closed $C^{\infty}$ uniform visibility manifold without conjugate
points and with continuous Green bundle, $X$ the universal cover of $M$.
Suppose that the geodesic flow has a hyperbolic periodic point. Then

  1. (1)

The Margulis function $c$ is a $C^{1}$ function.

  2. (2)

If $c(x)\equiv C$, then for any $x\in X$,

$h=\int_{\partial X}trU(x,\xi)d\tilde{\mu}_{x}(\xi)$

where $U(x,\xi)$ and $trU(x,\xi)$ are the second fundamental form and the mean
curvature of the horosphere $H_{x}(\xi)$ respectively, and $\tilde{\mu}_{x}$
is the normalized Patterson-Sullivan measure.

We note that the assumption of continuous asymptote in Theorem F guarantees
that the Busemann function is $C^{2}$. When $c(x)\equiv C$, we expect stronger
rigidity results to hold. To achieve this goal, Yue [125] studied the
uniqueness of harmonic measures associated to the strong stable foliation of
the geodesic flow in negative curvature. However, as far as we know, the
uniqueness of harmonic measures in nonpositive curvature is unknown in higher
dimension.

Nevertheless, if $\dim M=2$, then $\text{Vol}(B^{s}(x,r))=2r$ where
$B^{s}(x,r)$ is any ball of radius $r$ in a strong stable manifold. In this
case, $B^{s}(x,r)$ is just a curve. Hence the leaves of the strong stable
foliation have polynomial growth. Recently, Clotet [34] proved the unique
ergodicity of the horocycle flow on surfaces of genus at least $2$, without
conjugate points and with continuous Green bundle. Combining with this result,
we can show that there is a unique harmonic measure associated to the strong
stable foliation. Then we obtain the following rigidity result by applying
Katok’s entropy rigidity result [62, Theorem B]. In dimension two, the fact
that the geodesic flow has a hyperbolic periodic point is automatic, see the
discussion at the beginning of Section $4$.

######  Theorem G.

Let $M$ be a closed Riemannian surface of genus at least $2$ without conjugate
points and with continuous Green bundles. Then $c(x)\equiv C$ if and only if
$M$ has constant negative curvature.

Without the uniqueness of harmonic measures, we can still obtain some rigidity
results in arbitrary dimensions. The flip map $F:SM\to SM$ is defined as
$F(v):=-v$. For general uniform visibility manifolds, the Bowen-Margulis
measure $m$ is constructed by using measurable choice theorem, hence is not
necessarily flip invariant. However, for uniform visibility manifolds with
continuous Green bundles and the geodesic flow admitting a hyperbolic periodic
point, the Bowen-Margulis measure $m$ is supported on $\mathcal{R}_{1}$ and
hence it is indeed flip invariant. In this case, consider the conditional
measures $\\{\bar{\mu}_{x}\\}_{x\in M}$ of $m$ with respect to the partition
$SM=\cup_{x\in M}S_{x}M$. $\bar{\mu}_{x}$ can be identified as measures on
$\partial X$, and it would be natural to consider if $\bar{\mu}_{x}$ and the
normalized Patterson-Sullivan measures $\tilde{\mu}_{x}$ coincide. Yue [124,
123] obtained related rigidity results in negative curvature, which can be
generalized as follows.

######  Theorem H.

Let $M$ be a closed $C^{\infty}$ uniform visibility manifold without conjugate
points and with continuous Green bundle. Suppose that the geodesic flows
admits a hyperbolic periodic point. Then the conditional measures
$\\{\bar{\mu}_{x}\\}_{x\in M}$ of the Bowen-Margulis measure coincide almost
everywhere with the normalized Patterson-Sullivan measures $\tilde{\mu}_{x}$
if and only if $M$ is locally symmetric.

The assumption of continuous asymptote above again guarantees that the
Busemann function is $C^{2}$.

Both Theorem G and Theorem H give characterizations of rank one locally
symmetric spaces among certain closed manifolds without conjugagte points. In
fact, it is a well known conjecture [125, p.179] that in negative curvature
$c(x)\equiv C,x\in M$ if and only if $M$ is locally symmetric. The conjecture
is true for surfaces of negative curvature by [125, Theorem 4.3], and for a
more general class of surfaces without conjugate points by Theorem G.

###  2.6. The Hopf-Tsuji-Sullivan dichotomy

We also establish the Hopf-Tsuji-Sullivan dichotomy with respect to Bowen-
Margulis measure for the geodesic flow on certain manifolds without conjugate
points. At first, let us recall some necessary terminologies (see [72, 74]).

Let $\Omega$ be a locally compact and $\sigma$-compact Hausdorff topological
space and $\\{\phi^{t}\\}_{t\in{\mathbb{R}}}$ a flow on $\Omega$. A point
$\omega\in\Omega$ is said to be positively/negatively recurrent if there
exists a sequence $t_{n}\to\infty$ of real numbers such that $\phi^{\pm
t_{n}}\omega\to\omega.$ On the other hand, $\omega\in\Omega$ is said to be
positively/negatively divergent if for every compact set $K\subset\Omega$
there exists $T>0$ such that $\phi^{\pm t}\omega\notin K$ for all $t\geq T$.

Assume that $\mu$ is a Borel measure on $\Omega$ invariant by the flow
$\\{\phi^{t}\\}_{t\in{\mathbb{R}}}$. Then the Hopf decomposition theorem (see
for instance [55, Satz 13.1]) says that the space $\Omega$ decomposes into a
disjoint union of $\phi^{t}$-invariant Borel sets $\Omega_{C}$ and
$\Omega_{D}$ which satisfy the following properties:

  1. (1)

There does not exist a Borel subset $E\subset\Omega_{C}$ with $\mu(E)>0$ and
such that the sets $\\{\phi^{k}E\\}_{k\in{\mathbb{Z}}}$ are pairwise disjoint.

  2. (2)

There exists a Borel set $W\subset\Omega_{D}$ such that $\Omega_{D}$ is the
disjoint union of sets $\\{W_{k}\\}_{k\in{\mathbb{Z}}}$, where each $W_{k}$ is
a translate of $W$ under the flow $\phi$.

According to Poincaré’s recurrence theorem (see for example [55, Satz 13.2])
$\mu$-almost every point of $\Omega_{C}$ is positively and negatively
recurrent. On the other hand, by Hopf’s divergence theorem (see also [55, Satz
13.2]), $\mu$-almost every point of $\Omega_{D}$ is positively and negatively
divergent. This implies in particular that the sets $\Omega_{C}$ and
$\Omega_{D}$ are unique up to sets of measure zero.

The dynamical system $(\Omega,\\{\phi^{t}\\}_{t\in{\mathbb{R}}},\mu)$ is said
to be completely conservative if $\mu(\Omega_{D})=0$, and completely
dissipative if $\mu(\Omega_{C})=0$.

Finally, a dynamical system $(\Omega,\\{\phi^{t}\\}_{t\in{\mathbb{R}}},\mu)$
is called ergodic if every $\phi^{t}$-invariant Borel set $E\subset\Omega$
satisfies either $\mu(E)=0$ or $\mu(\Omega\setminus E)=0$. Hence if
$(\Omega,\\{\phi^{t}\\}_{t\in{\mathbb{R}}},\mu)$ is ergodic, then it is either
completely conservative or completely dissipative. The latter can only occur
for an infinite measure $\mu$ which is supported on a single orbit
$\\{\phi^{t}\omega:t\in{\mathbb{R}}\\}$ with $\omega\in\Omega$.

In the 1930’s, Hopf first observed that the geodesic flow on surfaces with
constant negative curvature is either completely conservative or completely
dissipative with respect to the Lebesgue measure, which in this case coincides
with the Bowen-Margulis measure. Then Tsuji [113] proved that conservativity
is equivalent to ergodicity. Later Sullivan [112] obtained these results for
manifolds of constant negative curvature in any dimension. Since then, the
study of the Hopf-Tsuji-Sullivan dichotomy has been an active topic in the
study of the ergodic theory of geodesic flows. Kaimanovich [60] proved the
dichotomy for Gromov hyperbolic metric spaces. On manifolds with pinched
negative curvature, Yue [126] showed the dichotomy for Bowen-Margulis measure,
and Paulin- Pollicott-Schapira [91] proved it for equilibrium states. Roblin
[102] also proved it for the CAT($-1$) metric spaces.

In nonpositive curvature, due to existence of flat strips, in general the Hopf
parametrization is not one-to-one which causes essential difficulties. With
very advanced technique, Link and Picaud [76] proved the dichotomy with
respect to Knieper’s measure for rank one manifolds of nonpositive curvature.
Link [74] proved the dichotomy with respect to Ricks’ measure for CAT($0$)
spaces. Recently, Liu-Liu-Wang [77] proved the dichotomy with respect to
Bowen-Margulis measure for uniform visibility manifolds satisfying Axiom $2$,
on which every pair $\xi\neq\eta\in{\partial X}$ can be connected by exactly
one geodesic.

We will establish the Hopf-Tsuji-Sullivan dichotomy with respect to a Bowen-
Margulis measure $m_{\Gamma}$ for the geodesic flow on uniform visibility
manifolds without conjugate points. Our Bowen-Margulis measure is constructed
as a product of Patterson-Sullivan measures and a measure $\lambda_{\xi,\eta}$
supported on a generalized strip $(\xi\eta)$, where a measurable choice
theorem is used. Unfortunately, for noncompact manifolds, argument of Krylov-
Bogolyubov type cannot be applied, and such Bowen-Margulis measure
$m_{\Gamma}$ is not necessarily invariant under the geodesic flow.
Nevertheless, the projection of $m_{\Gamma}$ to the quotient space is
invariant under the quotient geodesic flow. So strictly speaking, we are
talking about complete dissipativity/conservativity and ergodicity for the
quotient geodesic flow
$([SM],\\{[\phi^{t}]\\}_{t\in{\mathbb{R}}},[m_{\Gamma}])$. Since there is no
confusion, we still state our results for the geodesic flow
$(SM,\\{\phi^{t}\\}_{t\in{\mathbb{R}}},m_{\Gamma})$.

######  Theorem I.

Let $X$ be a simply connected smooth uniform visibility manifold without
conjugate points. Suppose that $\Gamma\subseteq Is(X)$ is a non-elementary
discrete group which contains an expansive isometry. Let $m_{\Gamma}$ be a
Bowen-Margulis measure on $SM=SX/\Gamma$ and $\mu_{o}$ is the Patterson-
Sullivan measure normalized so that $\mu_{o}({\partial X})=1$. Then exactly
one of the following two complementary cases holds, and the statements (i) to
(iii) are equivalent in each case. If furthermore $X$ has bounded asymptote
and $\Gamma$ contains a regular isometry, then the statements (i) to (iv) are
equivalent in each case.

  1. (1)

Case:

    1. (i)

$\sum_{\gamma\in\Gamma}e^{-\delta_{\Gamma}d(o,\gamma o)}$ converges.

    2. (ii)

$\mu_{o}(L_{\Gamma}^{\text{rad}})=0$.

    3. (iii)

$(SM,(\phi^{t})_{t\in{\mathbb{R}}},m_{\Gamma})$ is completely dissipative.

    4. (iv)

$(SM,(\phi^{t})_{t\in{\mathbb{R}}},m_{\Gamma})$ is not ergodic.

  2. (2)

Case:

    1. (i)

$\sum_{\gamma\in\Gamma}e^{-\delta_{\Gamma}d(o,\gamma o)}$ diverges.

    2. (ii)

$\mu_{o}(L_{\Gamma}^{\text{rad}})=1$.

    3. (iii)

$(SM,(\phi^{t})_{t\in{\mathbb{R}}},m_{\Gamma})$ is completely conservative.

    4. (iv)

$(SM,(\phi^{t})_{t\in{\mathbb{R}}},m_{\Gamma})$ is ergodic.

The most technical part of the proof of Theorem I is Proposition 10.10 (i.e.,
(ii)$\Rightarrow$(i) in Case (1)), particularly Lemma 10.13. Using ideas from
[76], we obtain the lower bound in Lemma 10.13 by considering a small
neighborhood of an expansive vector, which in nonpositive curvature is
Ballmann’s lemma [8, Lemma III.3.1]. The assumption of bounded asymptote is
used to get Lemma 10.15. This is a key step in applying Hopf argument to show
that conservativity implies ergodicity.

###  2.7. Organization of the paper

In Section $3$, we recall geometric properties of uniform visibility manifolds
without conjugate points, including Green bundles, boundary at infinity,
Busemann functions, generalized strips, etc.

In Section $4$, we consider ergodicity of Liouville measure for certain
surfaces without conjugate points, mostly focusing on the properties of
singular geodesics. This proves Theorem A.

Section $5$ are devoted to the construction of Patterson-Sullivan measure and
Bowen-Margulis measure. We show that Bowen-Margulis measure is exactly the
unique MME, and it has the Bernoulli property which proves Theorem B. We prove
Theorem C, the uniqueness of equilibrium states in Section $6$.

In Section $7$, we prove Theorem D for the asymptotic formula for the number
of homotopy classes of closed geodesics. In Section $8$, we prove Theorem E
for the formula of volume asymptotics. We then prove rigidity results Theorems
F, G and H related to Margulis function in Section $9$.

In the last Section $10$, we prove the Hopf-Tsuji-Sullivan dichotomy Theorem
I.

##  3\. Geometric and ergodic preliminaries

Let $M$ be an $n$-dimensional $C^{\infty}$ uniform visibility manifold without
conjugate points. Until in last section, $M$ is assumed to be compact. We
prepare some geometric and ergodic tools which will be used in subsequent
sections.

###  3.1. Jacobi fields, Green bundles and Pesin set

In order to study the dynamics of geodesic flows, we should investigate the
geometry of the second tangent bundle $TTM$. Let $\pi:TM\rightarrow M$ be the
natural projection, i.e., $\pi(v)=p$ where $v\in T_{p}M$. The connection map
$K_{v}:T_{v}TM\rightarrow T_{\pi(v)}M$ is defined as follows. For any $\xi\in
T_{v}TM$, $K_{v}\xi:=(\nabla X)(t)|_{t=0}$, where
$X:(-\epsilon,\epsilon)\rightarrow TM$ is a smooth curve satisfying $X(0)=v$
and $X^{\prime}(0)=\xi$, and $\nabla$ is the covariant derivative along the
curve $\pi(X(t))\subset M$. Then the standard Sasaki metric on $TTM$ is given
by

$\langle\xi,\eta\rangle_{v}=\langle d\pi_{v}\xi,d\pi_{v}\eta\rangle+\langle
K_{v}\xi,K_{v}\eta\rangle,\quad\xi,\eta\in T_{v}TM.$

Recall that the _Jacobi equation_ along a geodesic $c_{v}(t)$ is

(5) $J^{\prime\prime}(t)+R(\dot{c}_{v}(t),J(t))\dot{c}_{v}(t)=0,$

where $R$ is the curvature tensor, and $J(t)$ is a Jacobi field along
$c_{v}(t)$ and perpendicular to $\dot{c}_{v}(t)$. Suppose that $J_{\xi}(t)$ is
the solution of (5) which satisfies the initial conditions

$J_{\xi}(0)=d\pi_{v}\xi,~{}~{}\frac{d}{dt}\Big{|}_{t=0}J_{\xi}(t)=K_{v}\xi.$

Then it follows that (cf. p. 386 in [11])

$J_{\xi}(t)=d\pi_{\phi^{t}v}d\phi^{t}_{v}\xi,~{}~{}\frac{d}{dt}J_{\xi}(t)=K_{\phi^{t}v}d\phi^{t}_{v}\xi.$

Using the Fermi coordinates, we can write (5) in the matrix form

(6) $\frac{d^{2}}{dt^{2}}A(t)+K(t)A(t)=0.$

We have the following classical result.

######  Proposition 3.1 (Cf. [39]).

Given $s\in\mathbb{R}$, let $A_{s}(t)$ be the unique solution of (6)
satisfying $A_{s}(0)=Id$ and $A_{s}(s)=0$. Then there exists a limit

$A^{+}=\lim_{s\rightarrow+\infty}\frac{d}{dt}\Big{|}_{t=0}A_{s}(t).$

Now we can define the _positive limit solution_ $A^{+}(t)$ as the solution of
(6) satisfying the initial conditions

$A^{+}(0)=Id,\quad\frac{d}{dt}\Big{|}_{t=0}A^{+}(t)=A^{+}.$

It is easy to see that $A^{+}(t)$ is non-degenerate for all $t\in\mathbb{R}$.
Similarly, letting $s\rightarrow-\infty$, one can define the _negative limit
solution_ $A^{-}(t)$ of (6).

For each $v\in SM$, define

$G^{s}(v):=\\{\xi\in T_{v}SM:\langle\xi,Z(v)\rangle=0\text{\ and\
}J_{\xi}(t)=A^{+}(t)d\pi_{v}\xi\\},$ $G^{u}(v):=\\{\xi\in
T_{v}SM:\langle\xi,Z(v)\rangle=0\text{\ and\
}J_{\xi}(t)=A^{-}(t)d\pi_{v}\xi\\},$

where $Z$ is the vector field generated by the geodesic flow. $G^{s}/G^{u}$
are called the stable/unstable Green bundles of the geodesic flow
respectively.

######  Proposition 3.2 (Cf. Proposition 12.1.1 in [11]).

$G^{s}(v)$ and $G^{u}(v)$ have the following properties:

  1. (1)

$G^{s}(v)$ and $G^{u}(v)$ are $(n-1)$-dimensional subspaces of $T_{v}SM$.

  2. (2)

$d\pi_{v}G^{s}(v)=d\pi_{v}G^{u}(v)=\\{w\in T_{\pi(v)}M:w\text{\ is orthogonal
to\ }v\\}$.

  3. (3)

The subspaces $G^{s}(v)$ and $G^{u}(v)$ are invariant under the geodesic flow.

  4. (4)

Let $\tau:SM\rightarrow SM$ be the involution defined by $\tau v=-v$, then

$G^{s}(-v)=d\tau G^{u}(v)\text{\ and\ }G^{u}(-v)=d\tau G^{s}(v).$

  5. (5)

If the curvature satisfies $K(p)\geq-a^{2}$ for some $a>0$, then
$\|K_{v}\xi\|\leq a\|d\pi_{v}\xi\|$ for any $\xi\in G^{s}(v)$ or $\xi\in
G^{u}(v)$.

  6. (6)

If $\xi\in G^{s}(v)$ or $\xi\in G^{u}(v)$, then $J_{\xi}(t)\neq 0$ for each
$t\in\mathbb{R}$.

  7. (7)

$\xi\in G^{s}(v)$ (respectively, $\xi\in G^{u}(v)$) if and only if

$\langle\xi,Z(v)\rangle=0\text{\ and\ }\|d\pi_{g^{t}v}dg^{t}_{v}\xi\|\leq c$

for each $t>0$ (respectively, $t<0$) and some $c>0$.

Given $\xi\in T_{v}SM$, the _Lyapunov exponent_ $\chi(v,\xi)$ is defined as

$\chi(v,\xi):=\lim_{t\to\infty}\frac{1}{t}\log\|d\phi^{t}\xi\|$

whenever the limit exists. Recall the set $\mathcal{R}_{1}$ defined in (1).

######  Theorem 3.3.

([110, Theorem 4.1]) Let $(M,g)$ be a compact manifold without conjugate
points such that Green bundles are continuous, and
$\mu\in{\mathcal{M}}_{\phi}(SM)$. If the set $\mathcal{R}_{1}$ is nonempty,
then for $\mu\text{-a.e.}\ v\in\mathcal{R}_{1}$, $\xi^{s}\in G^{s}(v)$,
$\xi^{u}\in G^{u}(v)$,

$\chi(v,\xi^{s})<0\quad\text{and}\quad\chi(v,\xi^{u})>0.$

Recall the Pesin set $\Delta$ which is defined in (2).

######  Theorem 3.4.

Let $(M,g)$ be a smooth, connected and closed visibility manifold without
conjugate points and with continuous Green bundles, and
$\mu\in{\mathcal{M}}_{\phi}(SM)$. Suppose the geodesic flow has a periodic
hyperbolic point then $\Delta$ agrees $\mu$-almost everywhere with the open
dense set $\mathcal{R}_{1}$.

###### Proof.

First note that by [81, Lemma 7.1], $\mathcal{R}_{1}$ is a nonempty, open and
dense set. By Theorem 3.3, we see that for $\mu\text{-a.e.}\
v\in\mathcal{R}_{1}$, $v\in\Delta$.

On the other hand, $\mu\text{-a.e.}\ v\in\Delta$ is Oseledets regular, more
precisely, there exist $1\leq k=k(v)\leq 2n-1$, a $\phi^{t}$-invariant
decomposition

$T_{v}SM=E^{1}(v)\oplus\cdots\oplus E^{k}(v)$

and numbers $\chi_{1}(v)<\cdots<\chi_{k}(v)$ such that
$\chi(v,\xi)=\chi_{i}(v)$ for every $\xi\in E^{i}(v)\setminus\\{0\\}$. Denote

$E^{s}(v):=\oplus_{\chi_{i}(v)<0}E^{i}(v),\
E^{u}(v):=\oplus_{\chi_{i}(v)>0}E^{i}(v),\
E^{c}(v):=\oplus_{\chi_{i}(v)=0}E^{i}(v).$

By Freire-Mañé [45],

$E^{s}(v)\subset G^{s}(v)\subset E^{s}(v)\oplus E^{c}(v),\quad E^{u}(v)\subset
G^{u}(v)\subset E^{u}(v)\oplus E^{c}(v).$

If $v\in\Delta$, then $E^{c}(v)=Z(v)$, and hence $E^{s}(v)=G^{s}(v)$ and
$E^{u}(v)=G^{u}(v)$. It follows that $G^{s}(v)\cap G^{u}(v)=\\{0\\}$, that is,
$v\in\mathcal{R}_{1}$. ∎

###  3.2. Boundary at infinity

Let $M$ be a closed Riemannian manifold without focal points, and
$\mathop{\hbox{{\rm pr}}}:X\to M$ the universal cover of $M$. Let
$\Gamma\simeq\pi_{1}(M)$ be the group of deck transformations on $X$, so that
each $\gamma\in\Gamma$ acts isometrically on $X$. Let ${\mathcal{F}}$ be a
fundamental domain with respect to $\Gamma$. Denote by $d$ both the distance
functions on $M$ and $X$ induced by Riemannian metrics. The Sasaki metrics on
$SM$ and $SX$ are also denoted by $d$ if there is no confusion.

We still denote by $\mathop{\hbox{{\rm pr}}}:SX\to SM$ and $\gamma:SX\to SX$
the map on unit tangent bundles induced by $\mathop{\hbox{{\rm pr}}}$ and
$\gamma\in\Gamma$. We use an underline to denote objects in $M$ and $SM$, e.g.
for a geodesic $c$ in $X$ and $v\in SX$, ${\underline{c}}:=\mathop{\hbox{{\rm
pr}}}c$, ${\underline{v}}:=\mathop{\hbox{{\rm pr}}}v$ denote their projections
to $M$ and $SM$ respectively. However, sometimes we omit the underline if
there is no confusion caused by notations.

We call two geodesics $c_{1}$ and $c_{2}$ on $X$ _positively asymptotic_ or
just _asymptotic_ if there is a positive number $C>0$ such that
$d(c_{1}(t),c_{2}(t))\leq C,~{}~{}\forall~{}t\geq 0.$ We say $c_{1}$ and
$c_{2}$ are _negatively asymptotic_ if the above holds for $\forall t\leq 0$.
$c_{1}$ and $c_{2}$ are said to be _bi-asymptotic_ or _parallel_ if they are
both positively and negatively asymptotic. The relation of asymptoticity is an
equivalence relation between geodesics on $X$. The class of geodesics that are
asymptotic to a given geodesic $c_{v}/c_{-v}$ is denoted by
$c_{v}(+\infty)$/$c_{v}(-\infty)$ or $v^{+}/v^{-}$ respectively. We call them
_points at infinity_. Obviously, $c_{v}(-\infty)=c_{-v}(+\infty)$. We call the
set ${\partial X}$ of all points at infinity the _boundary at infinity_. If
$v^{+}=\eta\in{\partial X}$, we say $v$ _points at $\eta.$_

######  Definition 3.5.

(Cf. [81, p. 8]) Let $(X,g)$ be a simply connected manifold without conjugate
points. We say that _geodesic rays diverge in $X$_ if for every $p\in X$ and
every $\epsilon,C>0$ there exists $T(p,\epsilon,C)$ such that for every two
geodesic rays $c_{1}\neq c_{2}$ with $c_{1}(0)=c_{2}(0)=p$ in $X$ and
$\angle_{p}(\dot{c}_{1},\dot{c}_{2})\geq\epsilon$, we have
$d(c_{1}(t),c_{2}(t))\geq C$ for every $t\geq T(p,\epsilon,C)$. The geodesics
diverge uniformly if $T(p,\epsilon,C)$ does not depend on $p$.

The uniform visibility property implies that the geodesic diverge uniformly
([106, 109]). If $X$ has no focal points, then geodesic rays diverge.

Let $\overline{X}=X\cup{\partial X}$. For each point $p\in X$ and $v\in
S_{p}X$, each points $x,y\in\overline{X}-\\{p\\}$, positive numbers $\epsilon$
and $r$, and subset $A\subset\overline{X}-\\{p\\}$, we define the following
notations:

  * •

$c_{p,x}$ is the geodesic from $p$ to $x$ and satisfies $c_{p,x}(0)=p$.

  * •

$\measuredangle_{p}(x,y)=\measuredangle(\dot{c}_{p,x}(0),\dot{c}_{p,y}(0))$.

  * •

$\measuredangle_{p}(A)=\sup\\{\measuredangle_{p}(a,b)\mid a,\ b\in A\\}$.

  * •

$\measuredangle(v,x)=\measuredangle(v,\dot{c}_{p,x}(0))$.

  * •

$C(v,\epsilon)=\\{a\in\overline{X}-\\{p\\}\mid\measuredangle(v,a)<\epsilon\\}$.

  * •

$C_{\epsilon}(v)=C(v,\epsilon)\cap{\partial X}=\\{c_{w}(+\infty)\mid w\in
S_{x}X,\angle(v,w)<\epsilon\\}$.

  * •

$TC(v,\epsilon,r)=\\{q\in\overline{X}\mid\measuredangle_{p}(c_{v}(+\infty),q)<\epsilon\\}-\\{q\in\overline{X}\mid
d(p,q)\leq r\\}$.

$TC(v,\epsilon,r)$ is called the _truncated cone_ with axis $v$ and angle
$\epsilon$. Obviously $c_{v}(+\infty)\in TC(v,\epsilon,r)$. There is a unique
topology $\tau$ on $\overline{X}$ such that for each $\xi\in{\partial X}$ the
set of truncated cones containing $\xi$ forms a local basis for $\tau$ at
$\xi$. This topology is usually called the _cone topology_.

We can also define the visual topology on ${\partial X}$. For each $p\in X$,
there is a function $f_{p}:S_{p}X\to{\partial X}$ defined by

$f_{p}(v)=v^{+},v\in S_{p}X.$

Since the geodesics diverge in $X$, $f_{p}$ is a bijection. So for each $p\in
M$, $f_{p}$ induces a topology on ${\partial X}$ from the usual topology on
$S_{p}X$. The topology on ${\partial X}$ induced by $f_{p}$ is independent of
$p\in X$, and is called the _visual topology_ on ${\partial X}$. Visual
topology on ${\partial X}$ and the manifold topology on $X$ can be extended
naturally to the cone topology on $\overline{X}$. Under the cone topology,
$\overline{X}$ is homeomorphic to the closed unit ball in $\mathbb{R}^{n}$,
and ${\partial X}$ is homeomorphic to the unit sphere $\mathbb{S}^{n-1}$. For
more details about the cone topology, see [38] and [40].

We say that $M$ is quasi-convex if there exist constants $A,B>0$ such that for
every two geodesic segments $c_{1}:[t_{1},t_{2}]\to X$ and
$c_{2}:[s_{1},s_{2}]\to X$ it holds that

$d_{H}(c_{1},c_{2})\leq
A\sup\\{d(c(t_{1}),c(s_{1})),d(c_{2}(t_{2}),c_{2}(s_{2}))\\}+B,$

where $d_{H}$ is the Hausdorff distance.

Let us collect some good geometric properties of uniform visibility manifolds,
see [77, Proposition 2.7] and [81, Teorem 2.4].

######  Proposition 3.6.

Let $M$ be a closed uniform visibility manifold with no conjugate points, and
$X$ its universal cover.

  1. (1)

([106, 109]) Geodesic rays diverge uniformly in $X$.

  2. (2)

([38]) The following map is continuous.

$\Psi:SX\times[-\infty,+\infty]\rightarrow\overline{X},\quad(v,t)\mapsto
c_{v}(t).$

  3. (3)

([40]) For any two points $\xi\neq\eta\in{\partial X}$, there is at least one
connecting geodesic of them.

  4. (4)

([38]) $X$ is quasi-convex.

  5. (5)

([38]) There exists a uniform constant $Q>0$ (called Morse constant) such that
for every two bi-asymptotic geodesic $c_{1},c_{2}$ in $X$ we have
$d_{H}(c_{1},c_{2})\leq Q.$

###  3.3. Busemann functions and horospheres

For each pair of points $(p,q)\in X\times X$ and each point at infinity
$\xi\in{\partial X}$, the _Busemann function based at $\xi$ and normalized by
$p$_ is

$b_{\xi}(q,p):=\lim_{t\rightarrow+\infty}\big{(}d(q,c_{p,\xi}(t))-t\big{)},$

where $c_{p,\xi}$ is the unique geodesic from $p$ and pointing at $\xi$. The
Busemann function $b_{\xi}(q,p)$ is well-defined since the function $t\mapsto
d(q,c_{p,\xi}(t))-t$ is bounded from above by $d(p,q)$, and decreasing in $t$
(this can be checked by using the triangle inequality).

If $v\in S_{p}X$ points at $\xi\in{\partial X}$, we also write
$b_{v}(q):=b_{\xi}(q,p).$ The level sets of the Busemann function
$b_{\xi}(q,p)$ are called the _horospheres_ centered at $\xi$. The horosphere
through $p$ based at $\xi\in{\partial X}$, is denoted by $H_{p}(\xi)$ or
$H(v)$. Define the sets

${\mathcal{F}}^{s}(v):=\\{(q,-\nabla_{q}b_{v}):q\in
H(v)\\},\quad{\mathcal{F}}^{s}(v):=\\{(q,\nabla_{q}b_{-v}):q\in H(-v)\\}.$

${\mathcal{F}}^{s}/{\mathcal{F}}^{u}$ form foliations of $SX$, called
stable/unstable horospherical foliations. For more details of the Busemann
functions and horospheres, please see [37, 108, 109]. Here we are concerned
with the continuity property of the horospheres and Busemann functions.

######  Proposition 3.7.

(Cf. for example [81, Proposition 2.1, Theorem 2.1 and Theorem 1.3]) Let $M$
be a compact manifold without conjugate points with universal cover $X$. Then
for every $v\in SX$

  1. (1)

Busemann functions $b_{v}$ are $C^{1,L}$ with $L$-Lipschitz unitary gradient
for a uniform constant $L>0$;

  2. (2)

horospheres $H(v),H(-v)\subset X$ are $C^{1,L}$ embedded $(n-1)$-dimensional
submanifolds, and ${\mathcal{F}}^{s}(v),{\mathcal{F}}^{u}(v)\subset SX$ are
Lipschitz continuous embedded $(n-1)$-dimensional submanifolds.

If assume that $M$ is also a uniform visibility manifold, then

  1. (1)

the families ${\mathcal{F}}^{s}/{\mathcal{F}}^{u}$ are continuous minimal
foliations of $SM$ invariant by the geodesic flow, i.e. for every
$t\in{\mathbb{R}}$,

$\phi^{t}({\mathcal{F}}^{s/u}(v))={\mathcal{F}}^{s/u}(\phi^{t}(v));$

  2. (2)

the geodesic flow is topologically mixing;

  3. (3)

there exist $A,B>0$ such that for every $v\in SX$ and every
$w\in{\mathcal{F}}^{s}(v)$,

$d_{s}(\phi^{t}(v),\phi^{t}(w))\leq Ad_{s}(v,w)+B\quad\text{for every\ }t\geq
0$

where $d_{s}$ is the induced metric on ${\mathcal{F}}^{s}(v)$.

If assume furthermore that $M$ is a visibility manifold with continuous Green
bundles and the geodesic flow has a periodic hyperbolic point, then

  1. (1)

hyperbolic periodic points are dense on $SM$;

  2. (2)

Green bundles $G^{s}/G^{u}$ are uniquely integrable and tangent to the smooth
horospherical foliations ${\mathcal{F}}^{s}/{\mathcal{F}}^{u}$ respectively.

With assumption of continuous asymptote, we have better regularity of the
Busemann functions. That is why we have this assumption in Theorems F and H.

######  Lemma 3.8.

Let $M$ be a closed manifold without conjugate points and with continuous
Green bundle. Then the Busemann function $q\mapsto b_{v}(q)$ is $C^{2}$.
Moreover, $\triangle_{y}b_{\xi}(y,x)=trU(y,\xi)$ where $U(y,\xi)$ and
$trU(y,\xi)$ are the second fundamental form and the mean curvature of the
horosphere $H_{y}(\xi)$.

###### Proof.

By Proposition 3.7, $y\mapsto b_{\xi}(y,x)$ is $C^{1,L}$ by uniform
visibility, and $\nabla_{y}b_{\xi}(y,x)=-X$ where $X$ is the geodesic spray.
Since Green bundles $G^{s}/G^{u}$ are uniquely integrable and tangent to the
smooth horospherical foliations ${\mathcal{F}}^{s}/{\mathcal{F}}^{u}$ by
Proposition 3.7, we see that
$\triangle_{y}b_{\xi}(y,x)=\text{div}(-X)=trU(y,\xi)$ and it is continuous in
$\xi$. ∎

######  Remark 3.9.

If $X$ has bounded asymptote, then $y\mapsto b_{\xi}(y,x)$ is $C^{2}$ [41,
Theorem 1 and Proposition 5], and moreover both $\nabla_{y}b_{\xi}(y,x)=-X$
and $\triangle_{y}b_{\xi}(y,x)=trU(y,\xi)$ depends continuously on $\xi$ (see
[42, Proposition 4] and the discussion at the end of Subsection 2.3 in [71]).

The notion of continuity of $v\mapsto H(v)$ is defined as the following: let
$v_{n}$ converge to $v$, then $H(v_{n})$ converges to $H(v)$ uniformly on
compact subsets of $X$.

######  Lemma 3.10.

([110, Theorem 3.7]) Let $(M,g)$ be a compact Riemannian manifold without
conjugate points. Then geodesic rays diverge uniformly in $X$ if and only if
the collection of submanifolds $v\mapsto H(v)$, depends continuously on $v$ in
the compact open topology.

Recall that by Proposition 3.6(1), uniform visibility implies that geodesic
rays diverge uniformly. Then we have

######  Proposition 3.11.

(Cf. [92, Theorem 6.1] [108, Lemma 1.2]) Let $M$ be a closed uniform
visibility manifold without conjugate points. Then for every $p\in X$, the map
$\xi\mapsto H_{\xi}(p)$ is continuous in the following sense: if
$\xi_{n}\to\xi\in{\partial X}$ and $K\subset X$ is compact, then
$H_{\xi_{n}}(p)\cap K\to H_{\xi}(p)\cap K$ uniformly in the Hausdorff
topology.

The continuity of $v\mapsto H(v)$ is equivalent to the continuity in the
$C^{1}$ topology of the map $v\mapsto b_{v}$ uniformly on compact subsets of
$X$. So we have

######  Corollary 3.12.

Let $M$ be a closed uniform visibility manifold without conjugate points. Then
the functions $(v,q)\mapsto b_{v}(q)$ and $(\xi,p,q)\mapsto b_{\xi}(p,q)$ are
continuous on $SX\times X$ and ${\partial X}\times X\times X$ respectively.

We would like to obtain certain equicontinuity of Busemann function $v\mapsto
b_{v}(q)$. The idea is to extend the function to the compact space
$\overline{X}$. Given $x,p,q\in X$, define

$b_{x}(q,p):=d(q,x)-d(p,x).$

######  Lemma 3.13.

([30, Corollary 2.18]) Let $M$ be a closed uniform visibility manifold without
conjugate points. For each pair of points $p,q\in X$, if there is a sequence
$\\{x_{n}\\}\subset X$ with $\lim_{n\rightarrow+\infty}x_{n}=\xi\in{\partial
X}$, then

$\lim_{n\rightarrow+\infty}b_{x_{n}}(q,p)=b_{\xi}(q,p).$

######  Lemma 3.14.

Let $M$ be a closed uniform visibility manifold without conjugate points. Let
$p\in X$, $A\subset S_{p}X$ be closed, and $B\subset X$ be such that
$A^{+}:=\\{v^{+}:v\in A\\}$ and

$B^{\infty}:=\\{\lim_{n}q_{n}\in{\partial X}:q_{n}\in B\\}$

are disjoint subsets of ${\partial X}$. Then the family of functions
$A\to{\mathbb{R}}$ indexed by $B$ and given by $v\mapsto b_{v}(q)$ are
equicontinuous: for every $\epsilon>0$ there exists $\delta>0$ such that if
$\angle_{p}(v,w)<\delta$, then $|b_{v}(q)-b_{w}(q)|<\epsilon$ for every $q\in
B$.

###### Proof.

The proof is a repetition of that of [31, Lemma A.1] and thus omitted. Notice
that Lemma 3.13 is essentially used in the proof. ∎

If $X$ has uniform visibility, then it is quasi-convex and geodesic rays
diverge by Proposition 3.6. The following generalized strip lemma is a direct
corollary of [101, Lemma 3.1].

######  Lemma 3.15.

(Generalized strip lemma, [101, Lemma 3.1]) Let $M$ be a closed uniform
visibility manifold without conjugate points and $X$ be its universal
covering. Then for every $v,w\in SX$ with bi-asymptotic geodesics $c_{v}$ and
$c_{w}$, there exists a connected set $\Sigma\subset I(v)$ containing the
geodesic starting points such that any geodesic $c$ with $c(0)=q\in\Sigma$ and
$\dot{c}(0)=-\nabla b_{v}(q)$ is bi-asymptotic to both $c_{v}$ and $c_{w}$. In
particular, the set

(7) $\mathcal{S}:=\bigcup_{t\in{\mathbb{R}},q\in\Sigma}c_{\nabla b_{v}(q)}(t)$

is homeomorphic to $\Sigma\times{\mathbb{R}}$.

Recall that $\mathcal{I}(v)=\mathcal{F}^{s}(v)\cap\mathcal{F}^{u}(v)$. Denote
$I(v):=H(v)\cap H(-v)$.

######  Lemma 3.16.

([81, Corollary 2.1]) Let $M$ be a compact manifold without conjugate points
with visibility universal covering $X$ and $Q>0$ Morse constant from
Proposition 3.6. Then there exists an uniform constant $Q^{\prime}\leq Q$ such
that for every $v\in SX$ the sets $\mathcal{I}(v),I(v)$ are compact connected
sets with $\text{Diam}(\mathcal{I}(v))\leq Q$ and $\text{Diam}(I(v))\leq
Q^{\prime}$.

We can define the width of a generalized strip $\mathcal{S}$ in (7) as
$\inf_{w\in\mathcal{S}}\text{Diam}(\mathcal{I}(w)).$ Then a nontrivial
generalized strip might have width zero.

##  4\. Ergodicity of Liouville measure

In this section, we always let $M$ be a closed surface with genus
$\mathfrak{g}\geq 2$, without conjugate points and with bounded asymptote, and
$X$ its universal cover.

By Katok’s result [62, Theorem B], we have

$h^{2}\geq-2\pi E/\text{Vol}(M)$

where $h=h_{{\mathop{\hbox{\footnotesize\rm top}}}}(\phi^{1})$ is the
topological entropy of the geodesic flow and $E$ is the Euler characteristic
of $M$. The equality holds if and only if $M$ has constant negative curvature.
So in our setting, $h>0$. By the variational principle Theorem 2.6, there
exists a $\mu\in{\mathcal{M}}_{\phi}(SM)$ such that $h_{\mu}(\phi^{1})>0$.
Thus $\mu$ is a hyperbolic measure, and there exists a horseshoe on $SM$. We
know that there exists a hyperbolic periodic point under the geodesic flow
(see [62, Theorem 4.1], which is a flow version of [61, Corollary 4.4]).

So $\mathcal{R}_{1}\neq\emptyset$. By Proposition 3.7, Green bundles $G^{s/u}$
are tangent to the horospherical foliations ${\mathcal{F}}^{s/u}$ in our
setting.

###  4.1. Singular set and Expansivity

On the surface $M$, we have the Fermi coordinates $\\{e_{1}(t),e_{2}(t)\\}$
along a geodesic $c_{v}(t)$, obtained by the time $t$-parallel translations
along $c_{v}(t)$ of an orthonormal basis $\\{e_{1}(0),e_{2}(0)\\}$ where
$e_{1}(0)=\dot{c}_{v}(0)$. Thus $e_{1}(t)=\dot{c}_{v}(t)$ and
$e_{2}(t)\perp\dot{c}_{v}(t)$. Suppose that $J(t)=j(t)e_{2}(t).$ Then the
Jacobi equation (5) becomes

(8) $j^{\prime\prime}(t)+K(t)j(t)=0,$

where $K(t)=K(c_{v}(t))$ is the curvature at point $c_{v}(t)$. Let
$u(t)=j^{\prime}(t)/j(t)$. Then the Jacobi equation (8) can be written
equivalently as:

(9) $u^{\prime}(t)+u^{2}(t)+K(t)=0,$

which is called _the Riccati equation_.

For a given $\xi\in T_{v}SM$, we always let $J_{\xi}(t)$ be the unique Jacobi
field satisfying the Jacobi equation (5) under initial conditions

$J_{\xi}(0)=d\pi_{v}\xi,\ ~{}~{}\frac{d}{dt}\Big{|}_{t=0}J_{\xi}(t)=K_{v}\xi.$

Suppose that $J_{\xi}(t)$ is perpendicular to $\dot{c}_{v}(t)$, then
$J_{\xi}(t)=j_{\xi}(t)e_{2}(t)$ and $j_{\xi}(t)=\|J_{\xi}(t)\|$. Denote
$u_{\xi}(t)=j^{\prime}_{\xi}(t)/j_{\xi}(t)$. Then $u_{\xi}$ is a solution of
the Riccati equation (9).

The following proposition shows the connection between the Lyapunov exponent
$\chi(v,\xi)$ and the function $u_{\xi}$.

######  Proposition 4.1.

For any $v\in SM$ and $\xi\in G^{s}(v)$, one has

$\chi(v,\xi)=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}u_{\xi}(t)dt.$

###### Proof.

By the definition of Lyapunov exponents and Proposition 3.2(5), we have

$\displaystyle\chi(v,\xi)$
$\displaystyle=\limsup_{T\to\infty}\frac{1}{T}\log\|d\phi^{t}\xi\|=\limsup_{T\to\infty}\frac{1}{T}\log\sqrt{\|J_{\xi}(T)\|^{2}+\|J^{\prime}_{\xi}(T)\|^{2}}$
$\displaystyle=\limsup_{T\to\infty}\frac{1}{T}\log\|J_{\xi}(T)\|=\limsup_{T\to\infty}\frac{1}{T}\int_{0}^{T}(\log
j_{\xi}(t))^{\prime}dt$
$\displaystyle=\limsup_{T\to\infty}\frac{1}{T}\int_{0}^{T}\frac{j_{\xi}^{\prime}(t)}{j_{\xi}(t)}dt=\limsup_{T\to\infty}\frac{1}{T}\int_{0}^{T}u_{\xi}(t)dt.$

∎

Denote by $u(w),w\in SX$ the function satisfying Riccati equation with respect
to stable Jacobi field $J^{s}_{w}$. Then
$u(w)=\frac{(j^{s}_{w})^{\prime}}{j^{s}_{w}}$ where $j^{s}_{w}=\|J^{s}_{w}\|$.
Note that $u(w)\leq 0$ if $M$ has no focal points, but this is not necessarily
true when $M$ only has no conjugate points.

Recall that the geodesic $c_{v}$ is singular if $v\in\mathcal{R}_{1}^{c}$. The
following says that the stable (and hence unstable) Lyapunov exponent along a
singular geodesic is zero.

######  Lemma 4.2.

If $v\in\mathcal{R}_{1}^{c}$, then
$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}u(\phi^{t}(v))dt=0.$

###### Proof.

Since $v\in\mathcal{R}_{1}^{c}$, $J^{s}_{v}$ is also an unstable Jacobi field.
Since $M$ has bounded asymptote, we actually have

$\lim_{T\to\infty}\frac{1}{T}\log
j^{s}_{v}(T)=\limsup_{T\to\infty}\frac{1}{T}\log j^{s}_{v}(T).$

So the following two limits exist and coincide

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}u(\phi^{t}(v))dt=\lim_{T\to\infty}\frac{1}{T}\log
j^{s}_{v}(T).$

If $\lim_{T\to\infty}\frac{1}{T}\log j^{s}_{v}(T)>0$, by [49, Proposition
5.3], one can show that $v\in\mathcal{R}_{1}$ by explicitly constructing
another linearly independent Jacobi field, which is a contradiction.

If $\lim_{T\to\infty}\frac{1}{T}\log j^{s}_{v}(T)<0$, then
$\lim_{T\to\infty}j^{s}_{v}(T)=0$. Since $J^{s}_{v}$ is also an unstable
Jacobi field, by bounded asymptote we have for any $t\in{\mathbb{R}}$,

$j^{s}_{v}(t)\leq C\lim_{T\to\infty}j^{s}_{v}(T)=0$

which is a contradiction. So we have proved that

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}u(\phi^{t}(v))dt=0.$

∎

The proof of Theorem A(3) uses an argument based on the following
_expansivity_ property of a vector $v\in SM$ not tangent to a generalized
strip defined in (7). This argument is also used in the proof of Lemma 4.9.

######  Definition 4.3 (Cf. Definition 3.2.11 in [63]).

We say $v\in SM$ has the expansivity property if there exists a small
$\delta_{0}>0$, such that whenever $d(\phi^{t}(v),\phi^{t}(w))<\delta_{0}$,
$\forall t\in\mathbb{R}$, then $w=\phi^{t_{0}}(v)$ for some $t_{0}$ with
$|t_{0}|<\delta_{0}$.

Let $\mathcal{O}({\underline{v}})$ denote the orbit of ${\underline{v}}\in SM$
under the geodesic flow.

######  Lemma 4.4.

If ${\underline{v}}\in SM$ is not tangent to a generalized strip, then it has
the expansivity property.

###### Proof.

Assume the contrary. Then for any small $\epsilon>0$ less than the injectivity
radius of $M$, there exists a point ${\underline{w}}\in SM$ such that
${\underline{w}}\notin\mathcal{O}({\underline{v}})$ and
$d(c_{\underline{v}}(t),c_{\underline{w}}(t))<\epsilon$, $\forall
t\in\mathbb{R}$. By the choice of $\epsilon$, we can lift
$c_{\underline{v}}(t)$ and $c_{\underline{w}}(t)$ to the universal cover $X$
such that

$d(c_{v}(t),c_{w}(t))<\epsilon,\ \ \forall t\in\mathbb{R}.$

Thus by Lemma 3.15, $c_{v}(t)$ and $c_{w}(t)$ bound a generalized strip. Hence
${\underline{v}}$ is tangent to a generalized strip, a contradiction. ∎

###  4.2. Area of ideal triangles

Given distinct $x,y,z\in{\partial X}$, an ideal triangle with vertices $x,y,z$
means the region in $X$ bounded by the three geodesics joining the vertices.
In the case when at least one of $x,y,z$ is on ${\partial X}$ (the other
points can be inside $X$), we also call the region bounded by the three
geodesics an ideal triangle.

The following theorem is a version of [107, Theorem 1] for surfaces without
conjugate points.

######  Theorem 4.5.

Let $M$ be a closed surface with genus $\mathfrak{g}\geq 2$, without conjugate
points and with bounded asymptote, and $X$ its universal cover. Suppose that
$c_{v}$ is a singular geodesic in $X$, i.e., $v\in\mathcal{R}_{1}^{c}$, and
moreover it is a lift of a closed geodesic $c_{\underline{v}}$. Then every
ideal triangle having $c_{v}$ as an edge has infinite area.

###### Proof.

Suppose that $h(s)\in SX,s\in[0,a]$ is the curve in the horospherical
foliation ${\mathcal{F}}^{s}$ with $h(0)=v$. We want to show that the area of
$\pi\phi^{[0,\infty]}h[0,a]$ is infinite. Assume the contrary. Then obviously
$\liminf_{t\to\infty}l(\pi\phi^{t}h[0,a])=0$ where $l$ denotes the length of
the curve. As a consequence of bounded asymptote,

$\lim_{t\to\infty}l(\pi\phi^{t}h[0,a])=0$

Let $T_{0}$ be the least period of ${\underline{v}}$. Since
$v\in\mathcal{R}_{1}^{c}$, by Lemma 4.2

$\lim_{n\to\infty}\frac{1}{nT_{0}}\int_{0}^{nT_{0}}u(\phi^{t}(v))dt=0.$

By continuity of $u$, for any $s\in[0,a]$,

$\lim_{n\to\infty}\frac{1}{nT_{0}}\int_{0}^{nT_{0}}u(\phi^{t}(h(s)))dt=0.$

Then there exists large enough $n_{0}>0$ which depends on the unit vector
$c(s)$ such that

$-\frac{1}{n_{0}T_{0}}\int_{0}^{n_{0}T_{0}}u(\phi^{t}(h(s)))dt\leq l(\pi
h[0,s]).$

On compact manifold $M$, $\pi g^{[0,\infty]}(h[0,a])$ is contained in a small
compact neighborhood of the closed geodesic $c_{v}$. By the compactness and
continuity, we can find a uniform constant $n_{0}>0$ independent of
$s\in[0,a]$ such that

(10) $-\frac{1}{n_{0}T_{0}}\int_{0}^{n_{0}T_{0}}u(\phi^{t}(h(s)))dt\leq l(\pi
h[0,s]).$

By the argument at the beginning of this section, Green bundles $G^{s/u}$ are
tangent to the horospherical foliations ${\mathcal{F}}^{s/u}$ in our setting.
Then we have

$\displaystyle l(\pi\phi^{nn_{0}T_{0}}(h[0,a]))$
$\displaystyle=\int_{0}^{a}\|d\pi
d\phi^{nn_{0}T_{0}}h^{\prime}(s)\|ds=\int_{0}^{a}\|J^{s}_{h(s)}(nn_{0}T_{0})\|ds$
$\displaystyle=\int_{0}^{a}\|d\pi
d\phi^{(n-1)n_{0}T_{0}}h^{\prime}(s)\|e^{\int_{0}^{n_{0}T_{0}}u(\phi^{t+(n-1)n_{0}T_{0}}h(s))dt}ds$
$\displaystyle\geq\int_{0}^{a}\|d\pi
d\phi^{(n-1)n_{0}T_{0}}h^{\prime}(s)\|e^{-n_{0}T_{0}l(\pi\phi^{(n-1)n_{0}T_{0}}h([0,s]))dt}ds$
$\displaystyle\geq
l(\pi\phi^{(n-1)n_{0}T_{0}}(h[0,a]))e^{-n_{0}T_{0}l(\pi\phi^{(n-1)n_{0}T_{0}}h([0,a]))}$
$\displaystyle\geq
l(\pi\phi^{(n-1)n_{0}T_{0}}(h[0,a]))\left(1-n_{0}T_{0}l(\pi\phi^{(n-1)n_{0}T_{0}}h([0,a])\right)$

where we used (10) and $e^{x}\geq 1+x$ for $|x|$ small. Denote
$a_{n}:=l(\pi\phi^{nn_{0}T_{0}}(h[0,a]))$ and $B=n_{0}T_{0}$, we get

(11) $a_{n}\geq a_{n-1}(1-Ba_{n-1})\geq\cdots\geq
a_{0}\prod_{i=0}^{n-1}(1-Ba_{i}).$

Since we assume the contrary that the area of $\pi\phi^{[0,\infty]}h[0,a]$ is
finite, by bounded asymptote, we have $\sum_{i=0}^{\infty}a_{i}<+\infty$ and
$\lim_{n\to\infty}a_{n}=0$. Then by (11),

$\lim_{n\to\infty}\prod_{i=0}^{n-1}(1-Ba_{i})=0$

which is equivalent to $\sum_{i=0}^{\infty}a_{i}=+\infty$. We have a
contradiction and therefore the theorem follows. ∎

###  4.3. Singular geodesics

In this subsection, we discuss some important properties of the singular
geodesics. Our Theorem is a straightforward corollary of these properties. In
fact, it is closely related to the following two lemmas (Lemma 4.6 and 4.9).
The first lemma shows that if a flat geodesic converges to a closed one (no
matter singular or not), then the former geodesic must also be closed, and
coincide with the latter.

######  Lemma 4.6.

Suppose that $y\in\mathcal{R}_{1}^{c}\cap\text{Per}(\phi^{t})$, and
$\lim_{t\to\infty}d(\phi^{t}(y),\phi^{t}(z))=0$ where
$c_{z}(\pm\infty)=\xi_{\gamma}^{\pm}$ for some $\gamma\in\Gamma$. Then
$\mathcal{O}(y)=\mathcal{O}(z)$. In particular, $\mathcal{O}(y)$ is bi-
asymptotic to an axis of $\gamma$.

###### Proof.

$\tilde{c}_{0}$$\tilde{c}$$\phi(\tilde{c})$$\tilde{\alpha}$$\phi(\tilde{\alpha})$$\tilde{\beta}$$\phi(\tilde{\beta})$$A$$B$$C$$D$$E$$C^{\prime}$$D^{\prime}$$E^{\prime}$$A^{\prime}$$B^{\prime}$$F$
Figure 1. Proof of Lemma 4.6

We can lift geodesics $c_{z}(t),c_{y}(t)$ to the universal cover $X$, denoted
by $\tilde{c}_{1}(t)$ and $\tilde{c}(t)$ respectively, such that

$\lim_{t\to+\infty}d(\tilde{c}_{1}(t),\tilde{c}(t))=0.$

In particular, $\tilde{c}_{1}(+\infty)=\tilde{c}(+\infty)$.

Since the isometry $\gamma$ of $X$ fixes $\tilde{c}_{1}(\pm\infty)$, there
exists an axis $\tilde{c}_{0}$ of $\gamma$ such that
$\gamma(\tilde{c}_{0}(t))=\tilde{c}_{0}(t+t_{0})$ and
$\tilde{c}_{0}(\pm\infty)=\tilde{c}_{1}(\pm\infty)$. Moreover, on the boundary
of the disk ${\partial X}$, $\gamma$ fixes exactly two points
$\tilde{c}_{0}(\pm\infty)$, and for any other point $a\in{\partial
X}\setminus\\{\tilde{c}_{0}(\pm\infty)\\}$,
$\lim_{n\to+\infty}\gamma^{n}(a)=\tilde{c}_{0}(+\infty)$.

Assume $\tilde{c}(\pm\infty)$ is not fixed by $\gamma$. Then $\tilde{c}$ and
$\gamma(\tilde{c})$ do not intersect, since
$\gamma(\tilde{c})(+\infty)=\tilde{c}(+\infty)$. Replacing $\gamma$ by
$\gamma^{N}$ for a large enough $N\in\mathbb{N}$ if necessary, we know that
the position of $\gamma(\tilde{c})$ must be as shown in Figure 1. We then pick
another two geodesics $\tilde{\alpha}$ and $\tilde{\beta}$ as in Figure 1. The
image of $ABB^{\prime}A^{\prime}$ under $\gamma$ is $CEE^{\prime}C^{\prime}$.
Since $\gamma$ is an isometry, it preserves area. So the area of $ABCD$ is
equal to the region $A^{\prime}B^{\prime}DEE^{\prime}C^{\prime}$, and thus
greater than the area of the region $DEE^{\prime}D^{\prime}$. We can let
$A^{\prime}$ and $B^{\prime}$ approach $F$. In this process, the area of the
region $DEE^{\prime}D^{\prime}$ approaches the area of the ideal triangle
$DEF$. But since $c_{y}$ is a singular and closed geodesic, the area of $DEF$
is infinite by Theorem 4.5. Then $ABCD$ has infinite area which is absurd. So
$\gamma(\tilde{c}(\pm\infty))=\tilde{c}(\pm\infty)$.

Therefore $\tilde{c}(\pm\infty)=\tilde{c}_{0}(\pm\infty)$. Then either
$\tilde{c}(t)$ and $\tilde{c}_{0}(t)$ bound a strip by Lemma 3.15 or
$\tilde{c}(t)=\tilde{c}_{0}(t)$. Recall that
$\lim_{t\to+\infty}d(\tilde{c}(t),\tilde{c}_{0}(t))=0$, we must have
$\tilde{c}(t)=\tilde{c}_{0}(t)$. Hence $\mathcal{O}(y)=\mathcal{O}(z)$. ∎

######  Remark 4.7.

The above lemma holds more generally without assuming that $y,z$ are singular.
In fact, this is a rather general property of isometries on $X$. See the proof
of Lemma 7.4. The above proof is of independent interest and uses geometric
properties of singular geodesics.

To refine Lemma 4.6, we need the following lemma.

######  Lemma 4.8.

Let $M$ be a closed surface with genus $\mathfrak{g}\geq 2$, without conjugate
points and with bounded asymptote. If ${\underline{w}}$ is periodic,
${\underline{w}}^{\prime}\in{\mathcal{F}}^{s}({\underline{w}})\subset SM$ and

$\liminf_{t\to+\infty}d(c_{\underline{w}}(t),c_{{\underline{w}}^{\prime}}(t))=\delta>0,$

then $c_{\underline{w}}(t)$ and $c_{{\underline{w}}^{\prime}}(t)$ converge to
the boundaries of a nontrivial generalized strip.

###### Proof.

Suppose that
$\lim_{i\to+\infty}d(c_{\underline{w}}(s_{i}),c_{{\underline{w}}^{\prime}}(s_{i}))=\delta>0$.
By passing to a subsequence, we can assume that
$\lim_{i\to+\infty}\phi^{s_{i}}({\underline{w}})=v$ and
$\lim_{s_{i}\to+\infty}\phi^{s_{i}}({\underline{w}}^{\prime})={\underline{v}}^{\prime}$.
Then ${\underline{v}}^{\prime}\in{\mathcal{F}}^{s}({\underline{v}})$. Then by
Proposition 3.7, for any $t\in\mathbb{R}$

$d(c_{{\underline{v}}}(t),c_{{\underline{v}}^{\prime}}(t))=\lim_{s_{i}\to+\infty}d(c_{{\underline{w}}}(t+s_{i}),c_{{\underline{w}}^{\prime}}(t+s_{i}))\in[\delta,Q_{1}]$

where
$Q_{1}:=Ad_{s}(c_{{\underline{v}}}(0),c_{{\underline{v}}^{\prime}}(0))+B$.
Hence we can lift the geodesics to $X$ such that
$v^{\prime}\in{\mathcal{F}}^{s}(v)$ and

$d(c_{v}(t),c_{v^{\prime}}(t))\in[\delta,Q_{1}]\text{\ \ \ for\ }\forall
t\in\mathbb{R}.$

By Lemma 3.15, $c_{v}(t)$ and $c_{v^{\prime}}(t)$ are the boundaries of a
generalized strip of width at least $\delta$.

By assumption, ${\underline{w}}$ is periodic and thus
${\underline{v}}\in\mathcal{O}({\underline{w}})$. So there exists
${\underline{v}}_{1}\in{\mathcal{I}}({\underline{w}})$ such that
${\underline{v}}_{1}\in{\mathcal{O}}({\underline{v}}^{\prime})$.

  1. (1)

If ${\underline{w}}^{\prime}$ is on the arc of
${\mathcal{F}}^{s}({\underline{w}})$ between ${\underline{w}}$ and
${\underline{v}}_{1}$, then ${\underline{w}}^{\prime}$ is inside the
generalized strip bounded by geodesics $c_{v}(t)$ and $c_{v^{\prime}}(t)$, and
we are done.

  2. (2)

If ${\underline{v}}_{1}$ is on the arc of ${\mathcal{F}}^{s}({\underline{w}})$
between ${\underline{w}}$ and ${\underline{w}}^{\prime}$, then:

     * •

If
$\liminf_{t\to+\infty}d(c_{{\underline{v}}_{1}}(t),c_{{\underline{w}}^{\prime}}(t))=0,$
we are done since then by bounded asymptote

$\lim_{t\to+\infty}d(c_{{\underline{v}}_{1}}(t),c_{{\underline{w}}^{\prime}}(t))=0,$

and ${\underline{w}}$ and ${\underline{v}}_{1}$ are the boundaries of a
nontrivial generalized strip.

     * •

If otherwise
$\liminf_{t\to+\infty}d(c_{{\underline{v}}_{1}}(t),c_{{\underline{w}}^{\prime}}(t))>0,$
then repeating the above process produces another
${\underline{v}}_{2}\in{\mathcal{I}}({\underline{w}})$. If we are not done in
finite steps, since ${\mathcal{I}}({\underline{w}})$ has finite length by
Proposition 3.7, we will arrive at a limit point
${\underline{v}}_{\infty}\in{\mathcal{I}}({\underline{w}})$ such that
$\lim_{t\to+\infty}d(c_{{\underline{v}}_{\infty}}(t),c_{{\underline{w}}^{\prime}}(t))=0.$
Then we are done.

∎

Now Lemma 4.6 can be strengthened to the following.

######  Lemma 4.9.

Suppose that $y\in\mathcal{R}_{1}^{c}\cap\text{Per}(\phi^{t})$ and
$z\in\omega(y)$ where $c_{z}(\pm\infty)=\xi_{\gamma}^{\pm}$ for some
$\gamma\in\Gamma$. Then $\mathcal{O}(y)=\mathcal{O}(z)$. In particular,
$\mathcal{O}(y)$ is bi-asymptotic to an axis of $\gamma$.

###### Proof.

Suppose that there exist $s_{k}\to+\infty$ such that $\phi^{s_{k}}(y)\to z$.
If $\phi^{s_{k}}(y)\in{\mathcal{F}}^{s}(z)$ for some $k$ then we must have
$\lim_{t\to\infty}d(\phi^{t+s_{k}}(y),\phi^{t}(z))=0$ by Lemma 4.8 and its
proof. Then by Lemma 4.6, we have
$\mathcal{O}(\phi^{s_{k}}(y))=\mathcal{O}(z)$. So we are done.

Suppose that $\phi^{s_{k}}(y)\notin{\mathcal{F}}^{s}(z)$ for any $k$. Note
that if $y\neq z$ then $y$ and $z$ can not be tangent to a same generalized
strip. Therefore by Lemma 4.4, for any small $\epsilon_{0}>0$ and any large
$k$, there exists a $l_{k}$ with $l_{k}\to+\infty$ such that

$d(\phi^{l_{k}}(\phi^{s_{k}}(y)),\phi^{l_{k}}(z))=\epsilon_{0}\text{\ \ or\ \
}d(\phi^{l_{k}}(-\phi^{s_{k}}(y)),\phi^{l_{k}}(-z))=\epsilon_{0}$

where we take $l_{k}$ to be the smallest positive number to satisfy the above
equality. We only consider the first case. By taking a subsequence, we can
assume that

$\phi^{l_{k}}(\phi^{s_{k}}(y))\to y^{+}\text{\ \ and \ \ }\phi^{l_{k}}(z)\to
z^{+}$

as $k\to+\infty$. Then $z^{+}$ is such that
$c_{z^{+}}(\pm\infty)=\xi_{\gamma}^{\pm}$ and $d(y^{+},z^{+})=\epsilon_{0}$.

We claim that in the above process we can make sure that the footpoint of
$y^{+}$ is outside of the generalized strip bounded by $c_{\gamma}$ and
$c_{z}$, where $c_{\gamma}$ is an axis (fixed once for all) of $\gamma$.
Indeed, we can take $\phi^{s_{k}}(y)$ so that the footpoint of
$\phi^{l_{k}}(\phi^{s_{k}}(y))$ is outside of the above generalized strip. If
this is not the case, we then consider $-\phi^{s_{k}}(y)$ and $-z$ instead.
Under this change, we still get a limit point $y^{+}$ and $z^{+}$, so that
$y^{+}$ is outside of the generalized strip bounded by $c_{\gamma}$ and
$c_{z}$.

Then for any $t>0$, since $0<-t+l_{k}<l_{k}$ for large enough $k$, one has

$d(\phi^{-t}(y^{+}),\phi^{-t}(z^{+}))=\lim_{k\to+\infty}d(\phi^{-t+l_{k}+s_{k}}(y)),\phi^{-t+l_{k}}(z))\leq\epsilon_{0}.$

So $-y^{+}\in{\mathcal{F}}^{s}(-z^{+})$. Then we have the following two cases:

  1. (1)

$\liminf_{t\to\infty}d(\phi^{t}(-y^{+}),\phi^{t}(-z^{+}))=0$. Then by bounded
asymptote, $\lim_{t\to\infty}d(\phi^{t}(-y^{+}),\phi^{t}(-z^{+}))=0$. By Lemma
4.6, $c_{y^{+}}(\pm\infty)=\xi_{\gamma}^{\pm}$ and in fact $-y^{+}=-z^{+}$.
This contradicts to $d(y^{+},z^{+})=\epsilon_{0}$.

  2. (2)

$\liminf_{t\to\infty}d(\phi^{t}(-y^{+}),\phi^{t}(-z^{+}))=\delta$ for some
$\delta>0$. By Lemma 4.8, $\phi^{t}(-y^{+})$ converges to the boundary of a
generalized strip containing $-z^{+}$. Then by Lemma 4.6 $c_{y^{+}}$ and
$c_{z^{+}}$ are boundaries of a generalized strip of width at least $\delta$.
Recall that $y^{+}$ is outside of the generalized strip bounded by
$c_{\gamma}$ and $c_{z}$ which implies that we have enlarged the generalized
strip bounded by $c_{\gamma}$ and $c_{z}$ to the one bounded by $c_{\gamma}$
and $c_{y^{+}}$. Now note that $y^{+}\in\omega(y)$, so we can apply all the
arguments above to $y^{+}$ instead of $z$. Either we are arriving at a
contradiction as in case (1) and we are done, or we get an even larger
generalized strip. Since ${\mathcal{I}}(\dot{c}_{\gamma})$ has finite length
by Propostion 3.7, at last we will get a limit vector
$y_{\infty}\in{\mathcal{I}}(\dot{c}_{\gamma})$ such that
$y_{\infty}\in\omega(y)$ if we are always in case (2). Then we can apply all
the arguments above to $y_{\infty}$ and we must arrive at case (1). So we are
done.

∎

######  Theorem 4.10.

If $\mathcal{R}_{1}^{c}\subset\text{Per}(\phi^{t})$, then there is a finite
decomposition of $\mathcal{R}_{1}^{c}$:

$\mathcal{R}_{1}^{c}=\mathcal{O}_{1}\cup\mathcal{O}_{2}\cup\ldots\mathcal{O}_{k}\cup\mathcal{S}_{1}\cup\mathcal{S}_{2}\cup\ldots\cup\mathcal{S}_{l},$

where each $\mathcal{O}_{i},1\leq i\leq k$, is an isolated periodic orbit and
each $\mathcal{S}_{j},1\leq j\leq l$, consists of vectors tangent to a
generalized strip. Here $k$ or $l$ are allowed to be $0$ if there is no
isolated closed singular geodesic or no generalized strip.

###### Proof.

Assume the contrary to Theorem 4.10. Then there exists a sequence of different
vectors $x^{\prime}_{k}\in\mathcal{R}_{1}^{c}$ such that
$\lim_{k\to+\infty}x^{\prime}_{k}=x$ for some $x\in\mathcal{R}_{1}^{c}$. Here
different $x^{\prime}_{k}$ are tangent to different closed geodesics or to
different generalized strips, and $x$ is tangent to a closed geodesic or to a
generalized strip. For large enough $k$, we suppose that
$d(x^{\prime}_{k},x)<\frac{1}{k}$.

Fix a small number $\epsilon_{0}>0$. It is impossible that
$d(\phi^{t}(x^{\prime}_{k}),\phi^{t}(x))\leq\epsilon_{0}$, $\forall t>0$. For
otherwise, $c_{x^{\prime}_{k}}(t)$ and $c_{x}(t)$ are positively asymptotic
closed singular geodesics. They must be tangent to a common generalized strip
by Lemma 4.8 and Lemma 4.6. This is impossible since different
$x^{\prime}_{k}$ are tangent to different closed geodesics or to different
flat strips. Hence there exists a sequence $s_{k}\to+\infty$ such that

$d(g^{s_{k}}(x^{\prime}_{k}),g^{s_{k}}(x))=\epsilon_{0},$

and

$d(g^{s}(x^{\prime}_{k}),g^{s}(x))\leq\epsilon_{0},\ \forall 0\leq s<s_{k}.$

Let $y_{k}:=g^{s_{k}}(x)$ and $y^{\prime}_{k}:=g^{s_{k}}(x^{\prime}_{k})$.
Without loss of generality, we suppose that $y_{k}\to a$ and
$y^{\prime}_{k}\to b$. Then
$d(a,b)=\lim_{k_{i}\to+\infty}d(g^{s_{k_{i}}}(x_{k_{i}}),g^{s_{k_{i}}}(x^{\prime}_{k_{i}}))=\epsilon_{0}.$
We get that $d(a,b)=\epsilon_{0}$. For any $t<0$, since
$0<s_{k_{i}}+t<s_{k_{i}}$ for some large $k_{i}$, one has

$d(\phi^{t}(a),\phi^{t}(b))=\lim_{i\to+\infty}(d(g^{s_{k_{i}}+t}(x_{k_{i}}),g^{s_{k_{i}}+t}(x^{\prime}_{k_{i}})))\leq\epsilon_{0}.$

Hence we get that $d(\phi^{t}(a),\phi^{t}(b))\leq\epsilon_{0}$, $\forall t\leq
0$.

Replacing $x,x_{k}^{\prime}$ by $-x,-x_{k}^{\prime}$ respectively and applying
the same argument, we can obtain two points $a^{-},b^{-}$ such that
$d(a^{-},b^{-})=\epsilon_{0}$ and

$d(\phi^{t}(a^{-}),\phi^{t}(b^{-}))\leq\epsilon_{0},\quad\forall t\leq 0.$

Then we have the following three cases:

  1. (1)

$\liminf_{t\to\infty}d(\phi^{t}(-a),\phi^{t}(-b))=0$. By bounded asymptote,
this is equivalent to $\lim_{t\to\infty}d(\phi^{t}(-a),\phi^{t}(-b))=0$. By
Lemma 4.6, $-a=-b$. This contradicts to $d(a,b)=\epsilon_{0}$.

  2. (2)

$\liminf_{t\to\infty}d(\phi^{t}(-a^{-}),\phi^{t}(-b^{-}))=0$. Again by Lemma
4.6, $-a^{-}=-b^{-}$. This contradicts to $d(a^{-},b^{-})=\epsilon_{0}$.

  3. (3)

$\liminf_{t\to\infty}d(\phi^{t}(-a),\phi^{t}(-b))>0$ and
$\liminf_{t\to\infty}d(\phi^{t}(-a^{-}),\phi^{t}(-b^{-}))>0$.

For case (3), by Lemmas 4.8 and 4.9, $c_{a}$ and $c_{x}$ coincide. And
moreover, $c_{x}$ and $c_{b}$ are boundaries of a generalized strip of width
$\delta_{1}>0$. Similarly, $c_{x}$ and $c_{b^{-}}$ are boundaries of a
generalized strip of width $\delta_{2}>0$. We claim that these two generalized
strips lie on the different sides of $c_{x}$. Indeed, we choose $\epsilon_{0}$
small enough and consider the $\epsilon_{0}$-neighborhood of the closed
geodesic $c_{x}$ which contains two regions lying on the different sides of
$c_{x}$. By the definition of $b$ and $b^{-}$, they must lie in different
regions as above. This implies the claim.

In this way we get a generalized strip of width $\delta_{1}+\delta_{2}$ and
$x$ is tangent to the interior of this generalized strip. Since
$g^{s_{k}}(x^{\prime}_{k})\to b$, we can repeat all the arguments above to
$b,g^{s_{k}}(x^{\prime}_{k})$ instead of $x,x_{k}^{\prime}$. Then either we
are arriving at a contradiction as in case (1) or case (2) and we are done, or
we get a generalized strip of width greater than $\delta_{1}+\delta_{2}$ and
$b$ is tangent to the interior of the generalized strip. Proceeding in this
way, either we arrive at case (1) or case (2) in finite steps, or at last we
will get a limit generalized strip which cannot be enlarged by Proposition
3.16. Applying the above argument to this limit generalized strip, we have to
arrive at case (1) or case (2) and thus obtain a contradiction. So we are done
with the proof. ∎

###### Proof of Theorem A.

By the argument at the beginning of this section, the geodesic flow has a
periodic hyperbolic point. Then by Theorem 3.4, $\Delta$ agrees $\nu$-almost
everywhere with the open dense set $\mathcal{R}_{1}$. This proves the first
statement in Theorem A. Then Theorem A(1) follows from Theorem 4.5, and
Theorem A(2) follows from Theorem 4.10. ∎

##  5\. Patterson-Sullivan measure and Bowen-Margulis measure

###  5.1. Patterson-Sullivan measure

In this subsection we assume that $X$ is a simply connected uniform visibility
manifold without conjugate points. We will recall the construction of the
Patterson-Sullivan measure presented in [78] and obtain some properties,
especially the shadowing lemma.

######  Definition 5.1.

Let $X$ be a simply connected uniform visibility manifold without conjugate
points and $\Gamma$ a discrete subgroup of $\text{Iso}(X)$, the group of
isometries of $X$. For a given constant $r>0$, a family of finite Borel
measures $\\{\mu_{p}\\}_{p\in X}$ on ${\partial X}$ is called an
$r$-dimensional _Busemann density_ if

  1. (1)

for any $p,\ q\in X$ and $\mu_{p}$-a.e. $\xi\in{\partial X}$,

$\frac{d\mu_{q}}{d\mu_{p}}(\xi)=e^{-r\cdot b_{\xi}(q,p)}$

where $b_{\xi}(q,p)$ is the Busemann function;

  2. (2)

$\\{\mu_{p}\\}_{p\in X}$ is $\Gamma$-equivariant, i.e., for all Borel sets
$A\subset{\partial X}$ and for any $\gamma\in\Gamma$, we have

$\mu_{\gamma p}(\gamma A)=\mu_{p}(A).$

In [78], the authors constructed Busemann density via Patterson-Sullivan
construction. For each pair of points $(p,p_{0})\in X\times X$ and
$s\in\mathbb{R}$, _Poincaré series_ is defined as

$P(s,p,p_{0}):=\sum_{\alpha\in\Gamma}e^{-s\cdot d(p,\alpha p_{0})}.$

This series may diverge for some $s>0$. The number

$\delta_{\Gamma}:=\inf\\{s\in\mathbb{R}\mid P(s,p,p_{0})<+\infty\\}$

is called the _critical exponent of $\Gamma$_. By the triangle inequality, it
is easy to check that $\delta_{\Gamma}$ is independent of $p$ and $p_{0}$. We
say $\Gamma$ is of _divergent type_ if the Poincaré series diverges when
$s=\delta_{\Gamma}$. Whether $\Gamma$ is of divergent type or not is
independent of the choices of $p$ and $p_{0}$.

Fix a point $p_{0}\in X$. For each $s>\delta_{\Gamma}$ and $p\in X$, consider
the measure:

$\mu_{p,s}:=\frac{\sum_{\alpha\in\Gamma}e^{-s\cdot d(p,\alpha
p_{0})}\delta_{\alpha p_{0}}}{\sum_{\alpha\in\Gamma}e^{-s\cdot d(p_{0},\alpha
p_{0})}},$

where $\delta_{\alpha p_{0}}$ is the Dirac measure at point $\alpha p_{0}$.
The following properties of $\mu_{p,s}$ can be easily checked by using the
triangle inequality:

  1. (1)

$e^{-s\cdot d(p,p_{0})}\leq\mu_{p,s}(\overline{X})\leq e^{s\cdot d(p,p_{0})}$.

  2. (2)

$\Gamma p_{0}\subset\text{supp}\mu_{p,s}\subset\overline{\Gamma p_{0}}$, where
$\Gamma p_{0}$ is the orbit of $p_{0}\in X$ under the action of $\Gamma$.

Here in constructing a Busemann density, we consider the case that the group
$\Gamma$ is of divergent type first. When $\Gamma$ is of convergent type,
Patterson showed in [90] that one can weight the Poincaré series with a
positive monotone increasing function to make the refined Poincaré series
diverge at $s=\delta_{\Gamma}$. More precisely, we consider a positive
monotonely increasing function $g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$
with $\lim_{t\rightarrow+\infty}\frac{g(x+t)}{g(t)}=1$, $\forall
x\in\mathbb{R}^{+}$, such that the weighted Poincaré series

$\widetilde{P}(s,p,p_{0}):=\sum_{\alpha\in\Gamma}g(d(p,\alpha
p_{0}))e^{-s\cdot d(p,\alpha p_{0})}$

diverges at $s=\delta_{\Gamma}$. For the construction of the function $g$, see
[90, Lemma 3.1]. Then for the weighted Poincaré series
$\widetilde{P}(s,p,p_{0})$, we can always construct a Busemann density by
using the same method as follows.

Now for $p\in X$, consider a weak⋆ limit

$\lim_{s_{k}\downarrow h}\mu_{p,s_{k}}=\mu_{p}.$

Since $P(s,p_{0},p_{0})$ is divergent for $s=\delta_{\Gamma}$ and $\Gamma$ is
discrete, it is obvious that $\text{supp}(\mu_{p})\subset\overline{\Gamma
p_{0}}\cap{\partial X}$.

######  Proposition 5.2.

There is a sequence $s_{k}\to\delta_{\Gamma}$ as $k\to\infty$ such that for
every $p\in X$ the weak∗ limit $\lim_{k\to\infty}\mu_{p,s_{k}}=:\mu_{p}$
exists. Moreover, $\\{\mu_{p}\\}_{p\in X}$ is a $\delta_{\Gamma}$-dimensional
Buseman density.

###### Proof.

The proof is the same as that of [30, Proposition 5.1]. Note that we do not
need to assume that $X$ admits a compact quotient. ∎

######  Remark 5.3.

The uniform visibility condition is essential in the above construction. For
example, under that condition, $\partial X$ with cone topology is well defined
and the continuity of Busemann functions is guaranteed.

###  5.2. Bowen-Margulis measure

Let $P:SX\to{\partial X}\times{\partial X}$ be the projection given by
$P(v)=(v^{-},v^{+})$. Denote by $\mathcal{I}^{P}:=P(SX)=\\{P(v):v\in SX\\}$
the subset of pairs in ${\partial X}\times{\partial X}$ which can be connected
by a geodesic. Note that the connecting geodesic may not be unique. Since $X$
has uniform visibility, every pair $\xi\neq\eta$ in ${\partial X}$ can be
connected by a geodesic, i.e.,

$\partial^{2}X=({\partial X}\times{\partial
X})\setminus\\{(\xi,\xi):\xi\in{\partial X}\\}.$

Fix a point $p\in X$, we can define a $\Gamma$-invariant measure
$\overline{\mu}$ on $\partial^{2}X$ by the following formula:

(12)
$d\overline{\mu}(\xi,\eta):=e^{\delta_{\Gamma}\cdot\beta_{p}(\xi,\eta)}d\mu_{p}(\xi)d\mu_{p}(\eta),$

where $\beta_{p}(\xi,\eta):=-\\{b_{\xi}(q,p)+b_{\eta}(q,p)\\}$ is the Gromov
product, and $q$ is any point on a geodesic $c$ connecting $\xi$ and $\eta$.
It is easy to see that the function $\beta_{p}(\xi,\eta)$ does not depend on
the choice of $c$ and $q$. In geometric language, the Gromov product
$\beta_{p}(\xi,\eta)$ is the length of the part of a geodesic $c$ between the
horospheres $H_{\xi}(p)$ and $H_{\eta}(p)$.

Let $M=X/\Gamma$. Define an equivalence relation on $SM$ by writing $v\sim w$
iff $w\in{\mathcal{I}}(v)$. Write $[v]={\mathcal{I}}(v)$ for the equivalence
class of $v$. Let $\tilde{\chi}:SX\to SX/\sim$ and $\chi:SM\to SM/\sim$ be the
quotient maps. These are continuous maps when we equip the quotient spaces
with the metric

$d([v],[w])=\min\\{d(v^{\prime},w^{\prime}):v^{\prime}\in[v],w^{\prime}\in[w]\\}.$

The geodesic flow $\phi^{t}$ takes equivalence classes to equivalence classes,
$\phi^{t}[v]=[\phi^{t}v]$, hence it induces a flow $\psi^{t}:SM/\sim\to
SM/\sim$.

######  Theorem 5.4.

([81, Theorem 1.1]) Let $M$ be a compact $C^{\infty}$ $n$-dimensional manifold
without conjugate points and with uniform visibility universal cover $X$. Then
the geodesic flow $\phi^{t}$ is time-preserving semi-conjugate to the
continuous flow $\psi^{t}$ acting on a compact metric space $SM/\sim$ of
topological dimension at least $n-1$. Moreover, $\psi^{t}$ is expansive,
topologically mixing and has a local product structure.

We do not need compactness of $M$ in the following construction. Since
${\mathcal{I}}(v)$ is compact and $SM$ is a Polish space, the measurable
selection theorem of Kuratowski and Ryll-Nardzewski [111, Section 5.2]
guarantees existence of a Borel measurable map $V:SM/\sim\to SM$ such that
$\chi\circ V$ is the identity. Recall that $\mathop{\hbox{{\rm pr}}}:SX\to SM$
is the natural projection. Then for every $v\in SX$, $\mathop{\hbox{{\rm
pr}}}^{-1}(V(\mathop{\hbox{{\rm pr}}}[v]))$ intersects $[v]$ in a single
point, which we denote by $\tilde{V}([v])$. We conclude that
$\tilde{V}:SX/\sim\to SX$ is a measurable map such that
$\tilde{\chi}\circ\tilde{V}$ is the identity, and moreover
$\tilde{V}(\gamma_{*}[v])=\gamma_{*}\tilde{V}([v])$ for any $\gamma\in\Gamma$.
Now define a measure $\lambda_{\xi,\eta}$ on each $P^{-1}(\xi,\eta)$ by fixing
any $v\in P^{-1}(\xi,\eta)$ and putting for a Borel measurable set $A\subset
SX$

$\lambda_{\xi,\eta}(A):=\text{Leb}\\{t\in{\mathbb{R}}:\tilde{V}([\phi^{t}v])\in
A\\}.$

Note that this is independent of the choice of $v$. Use this to define a
measure $m$ on $SX$ by

$m(A)=\int_{\partial^{2}X}\lambda_{\xi,\eta}(A)d\overline{\mu}(\xi,\eta).$

Observe that $m$ is $\Gamma$-invariant, and it descends to a Borel measure
$m_{\Gamma}$ on $SM$.

The measure $m_{\Gamma}$ is not necessarily invariant under the geodesic flow,
since along a flow orbit inside a generalized strip, the choice for the value
of measurable function $V$ may vary. However, the measure
${\underline{m}}_{\Gamma}=\chi_{*}m_{\Gamma}$ is a flow invariant measure on
$SM/\sim$ because $\chi_{*}\lambda_{\xi,\eta}$ is flow invariant on each
$P^{-1}(\xi,\eta)/\sim$.

Now let us suppose that $M$ is compact and then clearly $m_{\Gamma}$ is
finite. Without loss of generality we scale the measure so that
$m_{\Gamma}(SM)=1$. Then the set $\chi_{*}^{-1}{\underline{m}}_{\Gamma}$ of
Borel probability measures is weak∗ compact and closed under $(\phi^{t})_{*}$
for all $t\in{\mathbb{R}}$. So the usual argument from the Krylov-Bogolyubov
theorem for producing an invariant probability measure shows that there is a
flow invariant Borel probability measure $\mu$ on $SM$ with
$\chi_{*}\mu={\underline{m}}_{\Gamma}$. This lifts to a $\Gamma$-invariant and
flow invariant Borel measure $\tilde{\mu}$ on $SX$ by

$\tilde{\mu}(A)=\int_{SM}\\#(\text{pr}^{-1}({\underline{v}})\cap
A)d\mu({\underline{v}}).$

We will call both $m_{\Gamma}$ and $\mu$ Bowen-Margulis measures of the
geodesic flow on $SM$.

######  Remark 5.5.

The above construction of $\mu$ for a compact manifold $M$ without conjugate
points which satisfies the first two conditions in Definition 2.7 of class
$\mathcal{H}$ has appeared in [30]. In the last section, we will prove the
Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on a uniform visibility
(not necessarily compact) manifold without conjugate points with respect to
the Bowen-Margulis measure $m_{\Gamma}$.

###  5.3. Relation to the MME

In this subsection, we always assume that $M=X/\Gamma$ is a compact uniform
visibility manifold with universal cover $X$. We shall show that the Bowen-
Margulis measure $\mu$ coincides with the unique MME under the setting of
Theorem 2.8.

By Freire and Mañé’s Theorem (cf. [45]), the topological entropy of the
geodesic flow coincides with the volume entropy, i.e., for any $x\in X$

$h_{\text{top}}(\phi^{1})=\lim_{r\to\infty}\frac{\log\text{Vol}B(x,r)}{r}$

where $B(x,r)$ is the open ball of radius $r$ centered at $x$ in $X$. Denote
$h=h_{\text{top}}(\phi^{1})$. By the proof of [79, Lemma 3.3], we have
$\delta_{\Gamma}=h$ where $\delta_{\Gamma}$ is the critical exponent of
$\Gamma=\pi_{1}(M)$.

######  Lemma 5.6.

For any $p\in X$, $\emph{\text{supp}}(\mu_{p})={\partial X}$.

###### Proof.

Suppose $\text{supp}(\mu_{p})\neq{\partial X}$, then there exists
$\xi\in{\partial X}$ which is not in $\text{supp}(\mu_{p})$. Then we can find
an open neighborhood $U$ of $\xi$ in ${\partial X}$ such that $\mu_{p}(U)=0$.
Since $M$ is compact, the geodesic flow is topologically transitive and hence
the action of $\Gamma$ on ${\partial X}$ is minimal (cf. [38, Proposition 3.6
and the proof of Proposition 2.8]). Then $\bigcup_{\alpha\in\Gamma}\alpha
U={\partial X}$. So for any $\eta\in{\partial X}$, there is an element
$\alpha\in\Gamma$ such that $\eta\in\alpha U$. By the $\Gamma$-invariance
$\mu_{\alpha p}(\alpha U)=\mu_{p}(U)=0$. Since $\mu_{p}$ is equivalent to
$\mu_{\alpha p}$, we have $\mu_{p}(\alpha U)=0$. Therefore
$\text{supp}(\mu_{p})=\emptyset$, a contradiction. ∎

For each $\xi\in{\partial X}$ and $p\in X$, define the projections

  * •

$pr_{\xi}:X\rightarrow{\partial X},~{}~{}q\mapsto c_{\xi,q}(+\infty)$,

  * •

$pr_{p}:X\backslash\\{p\\}\rightarrow{\partial X},~{}~{}q\mapsto
c_{p,q}(+\infty)$,

where $c_{\xi,q}$ is the geodesic satisfying $c_{\xi,q}(-\infty)=\xi$ and
$c_{\xi,q}(0)=q$. For $A\subset X$ and $p\in X$, $pr_{p}(A)=\\{pr_{p}(q)\mid
q\in A\backslash\\{p\\}\\}.$

######  Lemma 5.7.

There exists $R>0$ such that for all $x\in\overline{X}$ and $p\in X$, the
shadow $pr_{x}B(p,R)$ contains an open set in ${\partial X}$, where $B(p,R)$
is the open ball of radius $R$ centered at $p$.

###### Proof.

For $x\in\overline{X}$ and $p\in X$, let $v=-\dot{c}_{p,x}(0)$. For small
$\rho>0$, the set $A(v,\rho):=\\{c_{w}(\infty):\angle_{p}(v,w)<\rho\\}$ is
open in ${\partial X}$. Since $X$ is a uniform visibility manifold, for any
$\eta\in A(v,\rho)$, there exists at least one geodesic $c_{x,\eta}$
connecting $x$ and $\eta$. For any $0<\epsilon<\pi-\rho$, let $R=R(\epsilon)$
be the constant in Definition 2.1 of uniform visibility manifolds. Assume that
the geodesic $c_{x,\eta}$ does not intersect the ball $B(p,R)$, then
$\angle_{p}(-v,\dot{c}_{p,\eta}(0))<\epsilon$. Therefore,

$\angle_{p}(v,-v)\leq\angle_{p}(v,\dot{c}_{p,\eta}(0))+\angle_{p}(-v,\dot{c}_{p,\eta}(0))<\rho+\epsilon<\pi$

which is a contradiction. Thus $A(v,\rho)\subset pr_{x}B(p,R)$ and the lemma
follows. ∎

######  Proposition 5.8.

Let $X$ be a simply connected uniform visibility manifold which has a compact
quotient, and $\\{\mu_{p}\\}_{p\in X}$ be an $\delta_{\Gamma}$-dimensional
Busemann density on ${\partial X}$. Fix $r>R$ where $R$ is from Lemma 5.7.

  1. (1)

There exists $l=l(r)$ such that $\mu_{p}(pr_{x}B(p,r))\geq l$, for all $p\in
X$ and $x\in\overline{X}$;

  2. (2)

there is a positive constant $a=a(r)$ such that for any $x\in
X\setminus\\{p\\}$ and $\xi=c_{p,x}(-\infty)$, we have

$\frac{1}{a}e^{-hd(p,x)}\leq\mu_{p}(pr_{\xi}B(x,r))\leq ae^{-hd(p,x)};$

  3. (3)

there is a positive constant $b=b(r)$ such that for any $p,x\in X$,

$\frac{1}{b}e^{-hd(p,x)}\leq\mu_{p}(pr_{p}B(x,r))\leq be^{-hd(p,x)}.$

###### Proof.

The proof is the same as that of [30, Proposition 5.4], for which Lemmas 5.6
and 5.7 are crucial. We have established these two lemmas based on the uniform
visibility and the compactness of $M$. ∎

Let $\mu$ be the Bowen-Margulis measure of the geodesic flow on $SM$, and
$d_{1}$ be the Knieper metric on $SM$ defined as

$d_{1}({\underline{v}},{\underline{w}}):=\max_{t\in[0,1]}d(c_{\underline{v}}(t),c_{\underline{w}}(t)).$

Consider a finite measurable partition $\mathcal{A}=\\{A_{1},...,A_{l}\\}$ of
$SM$ satisfying $\mu(\partial A_{i})=0,i=1,\cdots l$ and that the radius of
every $A_{i}$ under $d_{1}$ is less than $\epsilon<\min\\{\text{Inj}(M),R\\}$,
where $R>0$ is the constant in Lemma 5.7. Let
$\mathcal{A}^{(n)}_{\phi}:=\vee^{n-1}_{i=0}\phi^{-i}\mathcal{A}$,
$n=1,2,\cdots$. For each $\alpha\in\mathcal{A}^{(n)}_{\phi}$, we know that if
${\underline{v}}\in\alpha$, then

$\alpha\subset\bigcap^{n-1}_{k=0}\phi^{-k}B_{d_{1}}(\phi^{k}{\underline{v}},\epsilon),$

where $B_{d_{1}}(\phi^{k}{\underline{v}},\epsilon)$ denotes the
$\epsilon$-ball of $\phi^{k}{\underline{v}}\in SM$ under the Knieper metric
$d_{1}$. If ${\underline{w}}\in\alpha$, then
$d(c_{\underline{v}}(t),c_{\underline{w}}(t))\leq\epsilon$ for all $0\leq
t\leq n$.

######  Proposition 5.9.

There exists a constant $a>0$ such that for any
$\alpha\in\mathcal{A}^{n}_{\phi}$, we have $\mu(\alpha)\leq e^{-hn}a$. As a
consequence, $\mu$ is a MME.

###### Proof.

Step 1. Let $p\in X$ be the reference point used in the definition of
$\bar{\mu}$ in (12), and let $v$ be a lift of ${\underline{v}}$ such that
$d(p,\pi{\underline{v}})\leq\text{diam}M=:r_{0}$. Since
$\text{diam}\alpha<\text{Inj}(M)$, we can lift $\alpha$ to $\tilde{\alpha}$ in
$X$ such that for any $v,w\in\tilde{\alpha}$, then
$d(c_{v}(t),c_{w}(t))\leq\epsilon$ for all $0\leq t\leq n$. Let $c_{v}(n)=x$,
then there exists constant $r_{1}>0$,

$P(\tilde{\alpha})\subset\bigcup_{\eta\in pr_{x}(B(p,r_{1}))}\\{\eta\\}\times
pr_{\eta}(B(x,\epsilon)).$

Indeed, let $\eta=c_{w}(-\infty)$ and $c_{\eta,x}$ the geodesic connecting
$\eta$ and $x$ such that $c_{\eta,x}(n)=x$. Since $c_{\eta,x}$ and $c_{w}$ are
negatively asymptotic, by Proposition 3.7, $d(c_{\eta,x}(t),c_{w}(t))\leq
Ad_{s}(c_{\eta,x}(n),c_{w}(n))+B\leq 2A\epsilon+B=:R_{1}$ for any $0\leq t\leq
n$. Then $d(c_{\eta,x}(0),p)\leq d(c_{\eta,x}(0),\pi w)+d(\pi w,p)\leq
R_{1}+r_{0}+\epsilon=:r_{1}.$ This implies that $\eta\in pr_{x}(B(p,r_{1}))$.

Step 2. $\tilde{\mu}(\tilde{\alpha})\leq e^{-hn}a$ for some constant $a>0$.

Indeed, for any $\eta\in pr_{x}(B(p,r_{1}))$, choose a point $q\in B(p,r_{1})$
that lies on $c_{\eta,x}$. First notice that

$d(q,x)\geq d(x,\pi v)-d(\pi v,p)-d(p,q)\geq n-r_{0}-r_{1}.$

Then by Proposition 5.8(2),

(13) $\mu_{p}(pr_{\eta}(B(x,\epsilon)))\leq
e^{hd(p,q)}\mu_{q}(pr_{\eta}(B(x,\epsilon)))\leq
e^{hr_{1}}ae^{-hd(q,x)}\leq\tilde{b}e^{-hn}.$

From definition we have
$\lambda_{\xi,\eta}(\tilde{\alpha})\leq\text{diam}\tilde{\alpha}$. Recall that
$\text{diam}\tilde{\alpha}\leq\epsilon$. We have

$m_{\Gamma}(\tilde{\alpha})\leq\bar{\mu}(P(\tilde{\alpha}))\text{diam}\tilde{\alpha}$

and the same holds for $(\phi^{t})_{*}m_{\Gamma}$. Since $\tilde{\mu}$ is a
limit of a convex combination of such measures, we also have

(14)
$\tilde{\mu}(\tilde{\alpha})\leq\bar{\mu}(P(\tilde{\alpha}))\text{diam}\tilde{\alpha}.$

At last, notice that for any $(\xi,\eta)\in P(\tilde{\alpha})$,
$\beta_{p}(\xi,\eta)\leq 2r_{1}$ since any geodesic connecting $\xi$ and
$\eta$ intersects $B(p,r_{1})$. Combining this with (13), (14) and the fact
that $\mu_{p}(pr_{x}((B(p,r_{1})))\leq\mu_{p}({\partial X})$, we get
$\tilde{\mu}(\tilde{\alpha})\leq e^{-hn}a$ for some constant $a>0$.

Step 3. We have

$H_{\mu}(\mathcal{A}^{(n)}_{\phi})=-\sum_{\alpha\in\mathcal{A}^{(n)}_{\phi}}\mu(\alpha)\log\mu(\alpha)\geq
n\cdot h-\log a,$

which implies that

$h_{\mu}(\phi^{1})\geq
h_{\mu}(\phi^{1},\mathcal{A})=\lim_{n\rightarrow\infty}\frac{1}{n}H_{\mu}(\mathcal{A}^{(n)}_{\phi})\geq
h.$

Then we have $h_{\mu}(\phi^{1})=h$. ∎

######  Proposition 5.10.

Let $X$ be a simply connected uniform visibility manifold which has a compact
quotient $M$. Suppose that the geodesic flow on $SM$ has a unique MME giving
full weight to $\mathcal{R}_{0}$. Then for any $A\subset SX$,

(15)
$\tilde{\mu}(A)=\int_{\partial^{2}X}\lambda_{\xi,\eta}(A)d\bar{\mu}(\xi,\eta)$

where $\lambda_{\xi,\eta}$ is the Lebesgue measure on $P^{-1}(\xi,\eta)$ which
is a single geodesic for $\bar{\mu}\text{-a.e.}\ (\xi,\eta)$.

###### Proof.

We claim that $P^{-1}(\xi,\eta)$ is a single geodesic for
$\bar{\mu}\text{-a.e.}\ (\xi,\eta)$. Indeed, if otherwise we must have
$\tilde{\mu}(\mathcal{R}_{0}^{c})>0$, a contradiction. Then the right side of
(15) immediately gives an invariant measure and indeed it coincides with
$\tilde{\mu}$. ∎

######  Proposition 5.11.

Let $X$ be a simply connected uniform visibility manifold which has a compact
quotient $M$. Suppose that the geodesic flow on $SM$ has a unique MME giving
full weight to $\mathcal{R}_{0}$. Then the MME is fully supported on $SM$.

###### Proof.

It is enough to prove that $\tilde{\mu}(U)>0$ for any open subset $U\subset
SX$. Recall that the Patterson-Sullivan measures have full support on
${\partial X}$ and thus $\bar{\mu}(P(U))>0$. Then by (15), $\tilde{\mu}(U)>0$.
The proposition follows. ∎

The above proposition proves the first statement of Theorem B.

######  Remark 5.12.

Under the assumption of Proposition 5.10, we see that $\mu$ and $m_{\Gamma}$
defined before Remark 5.5 represents the same measure. In the following, we
use $m_{\Gamma}$ or just $m$ to denote this measure, i.e., the Bowen-Margulis
measure and the unique MME, since the notation $\mu$ usually denotes an
abstract measure.

###  5.4. Bernoulli property of Bowen-Margulis measure

We prove Theorem B in this subsection. To start, we recall some necessary
definitions.

######  Definition 5.13 (Kolmogorov property).

Let $T:Y\to Y$ be an invertible measure-preserving transformation in a
Lebesgue space $(Y,\mathcal{B},\mu)$. We say that $T$ has the _Kolmogorov
property_ , or that $T$ is _Kolmogorov_ , if there is a sub-$\sigma$-algebra
$\mathcal{K}$ of $\mathcal{B}$ which satisfies

  1. (1)

$\mathcal{K}\subset T\mathcal{K}$,

  2. (2)

$\bigvee_{i=0}^{\infty}T^{i}\mathcal{K}=\mathcal{B}$,

  3. (3)

$\bigcap_{i=0}^{\infty}T^{-i}\mathcal{K}=\\{\emptyset,Y\\}$.

Consider $k\in{\mathbb{N}}$ and a probability vector
$p=(p_{0},\cdots,p_{k-1})$, i.e., $p_{i}\geq 0,i=0,\cdots,k-1$ and
$\sum_{i=0}^{k-1}p_{i}=1$. A _Bernoulli shift_ is a transformation $\sigma$ on
the space $\Sigma_{k}:=\\{0,\cdots,k-1\\}^{{\mathbb{Z}}}$ given by
$\sigma((x_{n})_{n\in{\mathbb{Z}}})=(x_{n+1})_{n\in{\mathbb{Z}}}$ which
preserves the Bernoulli measure $p^{{\mathbb{Z}}}$ on $\Sigma_{k}$.

######  Definition 5.14 (Bernoulli property).

Let $T:Y\to Y$ be an invertible measure-preserving transformation in a
Lebesgue space $(Y,\mathcal{B},\mu)$. We say that $T$ has the _Bernoulli
property_ or is called _Bernoulli_ if it is measurably isomorphic to a
Bernoulli shift.

Obviously, the Bernoulli property implies the Kolmogorov property.

######  Definition 5.15.

A measure-preserving flow $(Y,(f^{t})_{t\in{\mathbb{R}}},\mu)$ is said to have
the Kolmogorov (resp. Bernoulli) property if for every $t\neq 0$, the
invertible measure-preserving transformation $(Y,f^{t},\mu)$ has the
Kolmogorov (resp. Bernoulli) property.

For the the geodesic flow on manifolds of nonpositive curvature, Call and
Thompson [26] showed that the unique equilibrium state for certain potential
has the Kolmogorov property and the unique MME has the Bernoulli property,
based on the earlier work of [70]. These two results are extended to manifolds
without focal points in [28, Theorem B] and [120, Theorem C] respectively, via
methods developed in [26]. Recently, Araujo-Lima-Poletti [4] shows that the
unique equilibrium state for certain potential has the Bernoulli property by
symbolic approach.

####  5.4.1. Kolmogorov property

For compact uniform visibility manifolds without conjugate points, we first
show the following.

######  Proposition 5.16.

Let $(M,g)$ be a closed $C^{\infty}$ visibility manifold without conjugate
points. Suppose that

$\sup\\{h_{\mu}(\phi^{1}):\mu\text{\ is supported on\
}\mathcal{R}_{0}^{c}\\}<h_{{\mathop{\hbox{\footnotesize\rm top}}}}(\phi^{1}).$

Then $m$ (i.e., the unique MME) is Kolmogorov.

We first recall Ledrappier’s criterion for Kolmogorov property of equilibrium
states [70].

######  Definition 5.17.

Let $f:Y\to Y$ be a continuous self-map on a compact metric space $(Y,d)$. For
$\varepsilon>0$ and $x\in Y$, denote

$\Gamma_{\varepsilon}(x):=\\{y\in Y:d(f^{n}(x),f^{n}(y))<\varepsilon,\forall
n\in\mathbb{Z}\\},$

and

$h_{\text{loc}}(f,\varepsilon):=\sup\\{h_{\mathop{\hbox{\footnotesize\rm
top}}}(f,\Gamma_{\varepsilon}(x))):\ x\in Y\\}.$

We say that $(Y,f)$ is entropy expansive if there is $\varepsilon_{0}>0$ such
that

$h_{\text{loc}}(f,\varepsilon_{0})=0,$

and $(Y,f)$ is asymptotically entropy expansive if

$\lim_{\varepsilon\to 0}h_{\text{loc}}(f,\varepsilon)=0$.

We say that $\nu\in{\mathcal{M}}_{f}(Y)$ is almost entropy expansive at scale
$\varepsilon_{0}>0$ if

$h_{\mathop{\hbox{\footnotesize\rm top}}}(f,\Gamma_{\varepsilon}(x)))=0$

for $\nu\text{-a.e.}\ x\in Y$.

######  Theorem 5.18 (Ledrappier [70], Call-Thompson [26]).

Let $(Y,f^{t})$ be a continuous flow on a compact metric space such that
$f^{t}$ is asymptotically entropy expansive for all $t\neq 0$, and let
$\psi\in C(Y,{\mathbb{R}}$). Consider $(Y\times Y,f^{t}\times f^{t})$ which is
the product of two copies of $(Y,f^{t})$, and
$\Psi(x_{1},x_{2}):=\psi(x_{1})+\psi(x_{2})$. If $\Psi$ has a unique
equilibrium state in $\mathcal{M}_{f^{t}\times f^{t}}(Y\times Y)$, then the
unique equilibrium state for $\psi$ in $\mathcal{M}_{f^{t}}(Y)$ has the
Kolmogorov property.

We state again Theorem 5.4 here, since we will consider the Kolmogorov
property of the expansive factor of the geodesic flow.

######  Theorem 5.19.

(Mamani-Ruggiero [81, Theorem 1.1]) Let $M$ be a compact $C^{\infty}$ manifold
of dimension $n$ without conjugate points and with uniform visibility
universal cover $X$. Then the geodesic flow is time-preserving semi-conjugate
to a continuous expansive flow $\psi^{t}$ acting on a compact metric space
$Y:=SM/\sim$ of topological dimension at least $n-1$. Moreover, $\psi^{t}$ is
topologically mixing and has a local product structure.

Denote $P(t,\epsilon)$ the set of periodic orbits ${\mathcal{O}}$ of
$(Y,\psi^{t})$ with period in $[T-\epsilon,T+\epsilon]$. Define the measure

$\nu_{t}:=\frac{1}{\\#P(t,\epsilon)}\sum_{{\mathcal{O}}\in
P(t,\epsilon)}\delta_{\mathcal{O}}$

where $\delta_{\mathcal{O}}$ is an invariant probability measure supported on
the orbit ${\mathcal{O}}$. If $\nu=\lim_{t\to\infty}\nu_{t}$, we say that
$\nu$ is the limit of the invariant measures supported by periodic orbits of
$\psi^{t}$.

######  Lemma 5.20.

Let $\nu$ be the unique MME of $(Y,\psi^{t})$. Then $\nu\times\nu$ is the
unique MME of $(Y\times Y,\psi^{t}\times\psi^{t})$. Moreover, $\nu\times\nu$
is the limit of the invariant measures supported by periodic orbits of
$\psi^{t}\times\psi^{t}$.

###### Proof.

Since $(Y,\psi^{t})$ has the specification property, by [26, Lemma 3.1],
$(Y\times Y,\psi^{t}\times\psi^{t})$ has the specification property as well.
However, $(Y\times Y,\psi^{t}\times\psi^{t})$ is not necessarily expansive
even if $(Y,\psi^{t})$ is expansive. We overcome this difficulty following
ideas of [26].

Denote $\Gamma_{\epsilon}(x):=\\{y\in
Y:d(\psi^{t}x,\psi^{t}y)<\epsilon,\forall t\in{\mathbb{R}}\\}$ and

$\Gamma_{\epsilon}(x,y):=\\{(x^{\prime},y^{\prime})\in Y\times
Y:d(\psi^{t}x,\psi^{t}y),d(\psi^{t}x^{\prime},\psi^{t}y^{\prime})<\epsilon,\forall
t\in{\mathbb{R}}\\}.$

Note that
$\Gamma_{\epsilon}(x,y)=\Gamma_{\epsilon}(x)\times\Gamma_{\epsilon}(y)$. The
set of non-expansive points at scale $\epsilon>0$ is defined as

$\text{NE}(\epsilon):=\\{x\in
Y:\Gamma_{\epsilon}(x)\nsubseteq\psi_{[-s,s]}(x),\forall s>0\\}.$

According to [26, Definition 4.2], the set of product non-expansive points at
scale $\epsilon>0$ is denoted by

$\text{NE}^{\times}(\epsilon):=\\{(x,y)\in Y\times
Y:\Gamma_{\epsilon}(x,y)\nsubseteq\psi_{[-s,s]}(x)\times\psi_{[-s,s]}(y),\forall
s>0\\}.$

By [26, Lemma 4.4],

$\text{NE}^{\times}(\epsilon)\subset(Y\times\text{NE}(\epsilon))\cup(\text{NE}(\epsilon)\times
Y).$

Since $\psi^{t}$ is expansive, $\text{NE}(\epsilon)=\emptyset$ and hence
$\text{NE}^{\times}(\epsilon)=\emptyset$. Thus any
$\mu\in\mathcal{M}_{\psi^{t}\times\psi^{t}}(Y\times Y)$ is product expansive
at any scale $\epsilon$, in the sense that
$\mu(\text{NE}^{\times}(\epsilon))=0$. It follows from [26, Proposition 4.7]
that every subset $Q\subset Y\times Y$ invariant under the
${\mathbb{R}}^{2}$-action (i.e.,
$\\{\psi^{s}\times\psi^{t}):s,t\in{\mathbb{R}}\\}$) can be approximated by
elements of adapted partitions for $(t,\epsilon)$-separated sets.

Now we recall some results obtained from Climenhaga-Thompson’s proof [32] of
uniqueness of MME for $\psi^{t}\times\psi^{t}$. In fact, even Bowen’s original
proof (cf. [14] or [63, Theorem 20.1.3]) is applicable to our setting. We can
obtain similarly as [26, Theorem 6.3 and Lemma 6.4]:

  * •

$\nu$ is weakly mixing, which is equivalent to that $\nu\times\nu$ is ergodic;

  * •