# Finite Measures of Maximal Entropy for an Open Set of Partially Hyperbolic
Diffeomorphisms

Juan Carlos Mongez  Instituto de Matemática, Universidade Federal do Rio de
Janeiro, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909,
Brazil<EMAIL_ADDRESS>and  Maria Jose Pacifico  Instituto de Matemática,
Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão,
Rio de Janeiro 21945-909, Brazil<EMAIL_ADDRESS>
###### Abstract.

We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional
central direction such that the unstable entropy exceeds the stable entropy.
Our main result proves that such maps have a finite number of ergodic measures
of maximal entropy. Moreover, any $C^{1+}$ diffeomorphism near $f$ in the
$C^{1}$ topology possesses at most the same number of ergodic measures of
maximal entropy. Our contribution is in extending the findings in [2] to
arbitrary dimensions. We believe our technique, essentially distinct from the
one in that paper, is robust and may find applications in further contexts.

MJP and JCM were partially supported by CAPES-Finance Code 001. MJP was
partially supported by CNPq-Brazil Grant No. 302565/2017-5, FAPERJ (CNE)
Grant-Brazil No. E-26/202.850/2018(239069), Pronex: E-26/010.001252/2016.

###### Contents

  1. 1 Introduction
    1. 1.1 Measures of maximal entropy
    2. 1.2 Homoclinic classes with high entropy
    3. 1.3 Dependence of the number of ergodic measures of maximal entropy on the diffeomorphism
    4. 1.4 Equilibrium states
    5. 1.5 Structure of the paper and the sketch of the proof of the Main Theorem
  2. 2 Preliminaries
    1. 2.1 Hyperbolic sets and homoclinic class
    2. 2.2 Hyperbolic measures and homoclinic classes
    3. 2.3 Partial hyperbolic systems
      1. 2.3.1  Existence of equilibrium states
      2. 2.3.2 Hyperbolic measures
    4. 2.4 Unstable entropy
      1. 2.4.1 Unstable metric entropy
      2. 2.4.2  Unstable topological entropy and the variational principle
    5. 2.5 Hausdorff distance
  3. 3 Consequences of $h^{u}(f)-h^{s}(f)>0$
  4. 4 Invariant manifolds for measures with large entropy
    1. 4.1 Invariant Manifolds with Uniform Size
    2. 4.2 The measure of $\Lambda_{\rho}$
  5. 5 Proof of Theorems A and B
  6. 6 Proof of Theorems C and D

##  1\. Introduction

###  1.1. Measures of maximal entropy

A compelling feature of dynamical systems involves the rate at which the
number of distinguishable orbits grows, a quantifiable aspect captured by the
concept of entropy. The computation of entropy can be examined through both
topological and invariant measure lenses, interconnected by the variational
principle. This principle offers a method to determine the topological entropy
of a dynamical system by exploring its invariant measures and the entropies
associated with them.

To be more precise, let $f:M\to M$ be a continuous function on a compact
metric set. Denoting by ${\mathcal{M}}_{f}$ the set of $f$-invariant
probability measures, the variational principle establishes the following
relation

(1.1) $h(f)=\sup\\{h_{\mu}(f):\mu\in{\mathcal{M}}_{f}\\},$

where $h(f)$ is the topological entropy of $f$ and $h_{\mu}(f)$ is the metric
entropy with respect to the invariant measure $\mu$.

######  Definition 1.1.

A measure of maximal entropy (m.m.e.) is a measure $\mu\in{\mathcal{M}}_{f}$
such that $h_{\mu}(f)$ achieves the supremum in equation (1.1).

Sinai, Ruelle and Bowen pioneered the study of measures of maximal entropy in
the 1970s. Sinai, a trailblazer in the field, established the existence and
finiteness of maximal entropy measures for Anosov diffeomorphisms [31].
Subsequently, Ruelle and Bowen extended and broadened this approach to
encompass uniformly hyperbolic (Axiom A) systems [3, 27, 28].

Newhouse demonstrated that $C^{\infty}$ diffeomorphisms on a compact manifold
exhibit maximum entropy measures [20]. He posed the question of whether the
count of such measures is finite, particularly for diffeomorphisms on surfaces
with positive topological entropy [21, Problem 2].

As the skew product of a linear Anosov diffeomorphism by a rational rotation
exhibits an infinite number of such measures, this question is relevant only
for surfaces. Recently, Buzzi, Crovisier and Sarig provided a positive answer
to this question in [2].

In our previous work [18], we delved into the characteristics of unstable and
stable entropy, denoted as $h^{u}(f)$ and $h^{s}(f)$ respectively, as
introduced in [33, 10]. By employing the criterion developed in [7, 25], we
established that partially hyperbolic systems possessing a unidimensional
central bundle, wherein the unstable entropy surpasses the stable entropy and
the unstable foliation is minimal, there robustly exists a unique measure of
maximal entropy.

In the present work, we withdraw the hypotheses about the minimality of the
unstable foliation and prove:

Main Theorem. Let $f:M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism
of a compact manifold $M$ with $TM=E^{u}\oplus E^{c}\oplus E^{s}$ and
${\operatorname{dim}}(E^{c})=1$. If $h^{u}(f)>h^{s}(f)$, then $f$ has a finite
number of ergodic measures of maximal entropy. Moreover, there exists a
$C^{1}$ neighborhood $U$ of $f$, such that for any $C^{1+}$ diffeomorphism $g$
within $U$, the number of ergodic measures of maximal entropy of $g$ is not
greater than the number of maximal entropy ergodic measures of $f$.

###  1.2. Homoclinic classes with high entropy

Let $f$ be a $C^{r}$ diffeomorphism on a closed manifold. Consider a
hyperbolic periodic orbit of saddle type
${{\mathcal{O}}}=\\{f^{i}(x):i=0,\ldots,p-1\\}$ such that $p\geq 1$ and
$f^{p}(x)=x$.

The stable and unstable manifolds of ${{\mathcal{O}}}$ are given by

$W^{s}({{\mathcal{O}}})=\cup_{y\in{{\mathcal{O}}}}W^{s}(y)\quad\mbox{and}\quad
W^{u}({{\mathcal{O}}})=\cup_{y\in{{\mathcal{O}}}}W^{u}(y)\quad\mbox{respectively,}$

where $W^{s}(y)=\\{x\in M:d(f^{n}(x),f^{n}(y))\to 0\mbox{ when }n\to\infty\\}$
and $W^{u}(y)=\\{x\in M:d(f^{n}(x),f^{n}(y))\to 0\mbox{ when }n\to-\infty\\}$.
These are $C^{r}$ manifolds. For details see Subsection 2.1.

Let

${{\rm Per}}_{h}(f):=\\{{{\mathcal{O}}}:{{\mathcal{O}}}\text{ is a hyperbolic
periodic orbit of saddle type}\\}.$

Given ${{\mathcal{O}}}_{1},{{\mathcal{O}}}_{2}\in{{\rm Per}}_{h}(f)$, let
$W^{s}({{\mathcal{O}}}_{1})\pitchfork W^{s}({\mathcal{O}}_{2})$ denote the
collection of transverse intersection points of $W^{s}({{\mathcal{O}}}_{1})$
and $W^{u}({{\mathcal{O}}}_{2})$. Two
${{\mathcal{O}}}_{1},{{\mathcal{O}}}_{2}\in{{\rm Per}}_{h}(f)$ are
homoclinically related if $W^{u}(O_{i})\pitchfork W^{s}(O_{j})\neq\emptyset$
for $i\neq j$, we then write ${{\mathcal{O}}}_{1}\sim{{\mathcal{O}}}_{2}$.

The homoclinic class of a hyperbolic periodic orbit of saddle type
${{\mathcal{O}}}$ is defined by

${{\rm
HC}}({{\mathcal{O}}}):=\overline{\left\\{{{\mathcal{O}}}^{\prime}\in{\rm
Per}_{h}(f):{{\mathcal{O}}}^{\prime}\sim{{\mathcal{O}}}\right\\}}\text{. }$

An ergodic invariant probability measure $\mu$ is hyperbolic of saddle type,
if it has at most one positive Lyapunov exponent, at most one negative
Lyapunov exponent, and no zero Lyapunov exponent. When $\mu$ is a hyperbolic
measure for $f$ and ${{\mathcal{O}}}\in{{\rm Per}}_{h}(f)$ we write
$\mu\sim{{\mathcal{O}}}$ when $\mu$ is homoclinically related to
${{\mathcal{O}}}$, see Subsection 2.2, definitions 2.3 and 2.6.

The next theorem establishes that for a diffeomorphism $f$ as described in the
Main Theorem, there exists a finite set of hyperbolic periodic orbits such
that every ergodic measure with high entropy is related to one of them.

######  Theorem A.

Let $f:M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact
manifold $M$ with $TM=E^{u}\oplus E^{c}\oplus E^{s}$ and
${\operatorname{dim}}(E^{c})=1$. If $h^{u}(f)>h^{s}(f)$, then for any
$h(f)\geq a>h^{s}(f)$ there exists an integer $K_{a}>0$ and $K_{a}$ hyperbolic
orbits ${{\mathcal{O}}}_{1},\cdots{{\mathcal{O}}}_{K_{a}}$ with $s-$index
${\operatorname{dim}}(E^{s})+1$ such that $h(f_{|HC({{\mathcal{O}}}_{i})})\geq
a$ for every $i=1,\cdots K_{a}$ and any $\mu\in{\mathcal{M}}_{f}^{e}$ with
$h_{\mu}(g)>a$, $\mu\sim{{\mathcal{O}}}_{i}$ for some $1\leq j\leq K_{a}$.

######  Remark 1.2.

It is important to note that the above theorem does not imply a spectral
decomposition theorem like [2, Theorem 1]. Our main difficulty arises from the
potential existence of two hyperbolic periodic orbits that are not
homoclinically related, but one is entirely contained within the other.
Consequently, it is impossible to make any statement about the entropy of the
intersection between two homoclinic classes in Theorem A.

######  Remark 1.3.

The approach used by Buzzi, Crovisier and Sarig in their notable work in [2]
establishes that $C^{\infty}$ surface diffeomorphisms with positive
topological entropy have at most a finite number of ergodic measures of
maximum entropy in general, and at most one in the case topologically
transitive, has four steps. The first step heavily uses two-dimensionality to
bind topological disks ($su$-quadrilaterals) by two stable and two unstable
segments. This allows them to obtain topological intersections between stable
and unstable manifolds of two different measures. In the second step, taking
advantage of the fact that the diffeomorphism under consideration has high
entropy and a dynamic version of a lemma by Sard, they conclude that some of
these intersections are transversal. The third step consists of proving the
uniqueness of m.m.e. between a set of hyperbolic measures that are
homoclinically related. In the fourth step, using the properties of su-
quadrilaterals in step one, they prove that two hyperbolic measures carried by
a common transitive set have intersections from their stable and unstable
foliations. Applying the results of the previous steps, they were able to
prove that any stable leaf of one measure must transversally intersect some
leaf of the unstable foliation of the other measure, leading to a homoclinic
relation between the measures. Thus, the technique used in [2] is strongly
2-dimensional and we can not repeat it in a higher dimension.

######  Remark 1.4.

In general, it does not hold that every diffeomorphism close to a
diffeomorphism with a finite number of measures of maximal entropy has also a
finite number of measures of maximal entropy. But for diffeomorphisms as in
the theorem above we prove that every $C^{1+}$ diffeomorphism sufficiently
close to $f$ also has a finite number of ergodic m.m.e., see Corollary 5.2.

###  1.3. Dependence of the number of ergodic measures of maximal entropy on
the diffeomorphism

A natural question is the dependence of the simplex of m.m.e. on the
diffeomorphism. For $C^{\infty}$ surface diffeomorphisms, it is proved in [2]
that the number of ergodic m.m.e. varies semicontinuously on the
diffeomorphism. Here we achieve the same property for $C^{1+}$ partially
hyperbolic diffeomorphisms with a central bundle of dimension one and with
unstable entropy bigger than the stable entropy. To announce precisely our
result, let $N_{max}(f)$ be the number of ergodic m.m.e. of a diffeomorphism
$f$.

######  Theorem B.

Let $f:M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact
manifold $M$ with $TM=E^{u}\oplus E^{c}\oplus E^{s}$ and
$\operatorname{dim}(E^{c})=1$. If $h^{u}(f)>h^{s}(f)$. The function
$N_{max}:\operatorname{Diff}^{1+}(M)\to\mathbb{N}\cup\\{\infty\\}$ is upper
semicontinuous in the $C^{1}$ topology, that is, $\limsup_{g\to
f}N_{max}(g)\leq N_{max}(f)$.

###  1.4. Equilibrium states

Let $\phi:M\to\mathbb{R}$ be a continuous function. An $f-$invariant
probability measure $\mu$ is an equilibrium state for the potential $\phi$, if
$h(f,\mu)+\int\phi d\mu=P(f,\phi)$ where

$P(f,\phi):=\sup_{\mu\in{\mathcal{M}}_{f}}h_{\mu}(f)+\int\phi d\mu$

The next theorem gives information about equilibrium states for potentials
with a certain regularity.

######  Theorem C.

Let $f:M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact
manifold $M$ with $TM=E^{u}\oplus E^{c}\oplus E^{s}$,
${\operatorname{dim}}(E^{c})=1$ and $\phi$ a potential Hölder continuous. If
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$, then for any $P(f,\phi)\geq
a>h^{s}(f)+\sup\phi$ there exists an integer $K_{a}>0$ and $K_{a}$ hyperbolic
orbits $\mathcal{O}_{1},\cdots\mathcal{O}_{K_{a}}$ with $s-$index
${\operatorname{dim}}(E^{s})+1$ such that
$P(f_{|HC({{\mathcal{O}}}_{i})},\phi_{|HC({{\mathcal{O}}}_{i})})\geq a$ for
every $i=1,\cdots K_{a}$ and any $\mu\in{\mathcal{M}}_{f}^{e}$ with
$P_{\mu}(f,\phi)\geq a$, $\mu\sim\mathcal{O}_{i}$ for some $1\leq j\leq
K_{a}$.

As a consequence of Theorem C, we get that $C^{1+}$ diffeomorphisms in a
$C^{1}$ open neighborhood of $f$ also have a finite number of equilibrium
states, see Corollary 6.2.

Let $M_{\text{max}}(f)$ be the number of ergodic equilibrium states of a
diffeomorphism $f$. Our last result establishes the following upper
semicontinuity property for $M_{\text{max}}(f)$.

######  Theorem D.

Let $f:M\to M$ be a $C^{1+}$ partially hyperbolic diffeomorphism of a compact
manifold $M$ with $TM=E^{u}\oplus E^{c}\oplus E^{s}$ and
${\operatorname{dim}}(E^{c})=1$. If $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$, The
function $M_{max}:{{\rm Diff}}^{1+}(M)\to{\mathbb{N}}\cup\\{\infty\\}$ varies
upper semicontinuity in the $C^{1}$ topology, that is, $\limsup_{g\to
f}M_{max}(g)\leq M_{max}(f)$.

###  1.5. Structure of the paper and the sketch of the proof of the Main
Theorem

The proof establishing the finiteness of the number of ergodic measures of
maximal entropy in the Main Theorem is structured into three key steps:

  * (1)

Utilizing the assumption $h^{u}(f)>h^{s}(f)$, we established that every
ergodic measure with substantial entropy possesses a negative central Lyapunov
exponent that remains uniformly away from zero. That is, if
$h_{\mu}(f)>h^{u}(f)$ then

$\lambda_{c}(f,\mu)\leq h_{\mu}(f)-h^{u}(f),$

where $\lambda_{c}(f,\mu)$ denotes the central Lyapunov exponent with respect
to $\mu$.

  * (2)

Consider the set

$\Lambda_{\rho}(f):=\\{x\in M:\forall n\geq 1\mbox{ it holds
}\|Df^{n}(x)_{|E^{c}}\|\leq\rho^{n}\\}.$

We prove that every point in $\Lambda_{\rho}(f)$ possesses an invariant Pesin
manifold of uniform size. Particularly, for two points in $\Lambda_{\rho}(f)$
that are sufficiently close, their invariant manifolds have transversal
intersections.

  * 3)

Utilizing the classical Pliss Lemma, we establish that for every ergodic
measure with sufficiently large entropy, the set $\Lambda_{\rho}(f)$ possesses
a positive measure with respect to this measure.

Items (1), (2) and (3) imply that there is only a finite number of equivalence
classes in the set of hyperbolic measures containing measures of maximal
entropy (we are referencing the equivalence relation described in Definition
2.5). By Theorem 2.8, each of these equivalence classes possesses at most one
measure of maximal entropy. Therefore, the number of maximal entropy ergodic
measures for $f$ is finite.

To prove that diffeomorphisms close enough to $f$ have at most the same number
of ergodic m.m.e. of $f$ we proceed as follows:

  * (4)

We observe that the property $h^{u}(f)>h^{s}(f)$ is an open property in
$C^{1+}$ diffeomorphisms on $M$, that is, for any $h^{u}(f)>a>b>h^{s}(f)$,
there exists a $C^{1}$ neighborhood of $f$ such that every $C^{1+}$
diffeomorphism has unstable entropy greater than $a$ and stable entropy less
than $b$.

  * (5)

Item (4) allows us to demonstrate that the properties mentioned in Items (2)
and (3) are open. Specifically, there exists a $\beta>0$ such that for every
$C^{1+}$ diffeomorphism $g$ within a certain neighborhood of $f$, the set
$\Lambda_{\rho}(g)$ has a measure greater than $\beta$ for each ergodic
measure of maximal entropy of $g$.

  * (6)

For every diffeomorphism $g$ close enough to $f$ we establish an injective
correspondence between the ergodic measure of maximal entropy of $g$ into the
ergodic measure of maximal entropy of $f$.

The final item demonstrates that any $g$ sufficiently close to $f$ has, at
most, the same number of ergodic measures of maximal entropy as $f$.
Establishing an injective map between the ergodic measures of maximal entropy
of $g$ and those of $f$ was the most intricate part of our work. This proof
relies on two notable facts:

  * (a)

Consider a sequence of $C^{1+}$ diffeomorphisms $(g_{n})_{n\in\mathbb{N}}$
converging to $f$ in the $C^{1}$ topology. If $\mu_{n}$ represents ergodic
m.m.e’s for $g_{n}$ and $\mu_{n}$ converges to $\nu$, we prove that $\nu$ is a
measure of maximal entropy for $f$.

  * (b)

For $\mu_{n}$ and $\nu$ as in Item (a) we show that $\nu(\Lambda_{\rho}(f))>0$

Item (a) implies that $\nu$ forms a convex combination of ergodic measures of
maximal entropy for $f$. Item (b) suggests that at least one measure within
this convex combination corresponds to an ergodic measure of maximal entropy
for $f$ gives a positive measure to certain subset of $\Lambda_{\rho}(f)$.
This specific measure is the one we associate with $\mu_{n}$.

### Organization of the paper

In Section 2 we recall the main previous results that we will use. In section
3 we explore the consequences of the hypotheses $h^{u}(f)>h^{s}(f)$. The
section 4 is dedicated to proving that the set of points with Pesin manifolds
of uniform size has a uniform positive size for measures with large entropy.
In sections 5 and 6 we prove the results of our work.

##  2\. Preliminaries

###  2.1. Hyperbolic sets and homoclinic class

We will review the definition of homoclinic classes by Newhouse [19] and some
of their properties.

Let $f:M\to M$ be a diffeomorphism on a compact manifold $M$, and let
$\Lambda\subset M$ be a compact and invariant set. We say that $\Lambda$ is a
hyperbolic set for $f$ if there exists a splitting of the tangent bundle
$T\Lambda$ into two bundles $E^{s}$ and $E^{u}$ such that for all
$x\in\Lambda$, $Df_{x}E^{s}_{x}=E^{s}_{f(x)}$ and
$Df_{x}E^{u}_{x}=E^{s}_{f(x)}$ and there exist a constant $C>0$ and
$0<\lambda<1$ such that for all $x\in\Lambda$ it holds

$\left\|Df^{n}_{x}v\right\|\leq C\lambda^{n}\left\|v\right\|\mbox{ for all
}v\in E^{s}_{x}\mbox{ and }n\geq 1$

and

$\left\|Df^{-n}_{x}v\right\|\leq C\lambda^{n}\left\|v\right\|\mbox{for all
}v\in E^{u}_{x}\mbox{ and }n\geq 1$

.

It is well known that if $\Lambda$ is a hyperbolic set, then for any
$x\in\Lambda$, the following sets are $C^{1}$-immersed submanifolds [11].

$\begin{array}[]{l}W^{s}(x)=\left\\{y\in
M:\lim_{n\to\infty}d\left(f^{n}(x),f^{n}(y)\right)=0\right\\},\\\
W^{u}(x)=\left\\{y\in
M:\lim_{n\to\infty}d\left(f^{-n}(x),f^{-n}(y)\right)=0\right\\}.\end{array}$

The sets $W^{s/u}(x)$ are called the stable and unstable manifolds of $x$
respectively, and for $\varepsilon$ small,
$W^{s/u}_{\varepsilon}(x):=W^{s/u}(x)\cap B_{\varepsilon}(x)$ are called the
local stable and unstable manifolds of $x$. When a periodic point is
hyperbolic we say that its orbit is a hyperbolic periodic orbit. We denote the
set of all hyperbolic periodic orbits of $f$ by $\operatorname{Per}_{h}(f)$.
The transverse intersection of $C^{r}$-submanifolds $U,V\subset M$ is

$U\pitchfork V=\\{x\in U\cap V:T_{x}U\oplus T_{x}V=T_{x}M\\}$

Given two orbits ${{\mathcal{O}}}_{1},{{\mathcal{O}}}_{2}\in{{\rm
Per}}_{h}(f)$ we say that they are homoclinically related if

$W^{s}\left({{\mathcal{O}}}_{1}\right)\pitchfork
W^{u}\left({{\mathcal{O}}}_{2}\right)\neq\emptyset\text{ and
}W^{s}\left({{\mathcal{O}}}_{2}\right)\pitchfork
W^{u}\left({{\mathcal{O}}}_{1}\right)\neq\emptyset\text{. }$

If two orbits, ${{\mathcal{O}}}_{1}$ and ${{\mathcal{O}}}_{2}$, are
homoclinically related we write ${{\mathcal{O}}}_{1}\sim{{\mathcal{O}}}_{2}$.
The pair $({\rm Per}_{h}(f),\sim)$ defines an equivalence relation.

######  Definition 2.1.

The homoclinic class of a hyperbolic periodic orbit ${{\mathcal{O}}}$ is

${{\rm
HC}}({{\mathcal{O}}}):=\overline{\left\\{{{\mathcal{O}}}^{\prime}\in{{\rm
Per}}_{h}(f):{{\mathcal{O}}}^{\prime}\sim{{\mathcal{O}}}\right\\}}\doteq\overline{W^{s}\left({{\mathcal{O}}}\right)\pitchfork
W^{u}\left({{\mathcal{O}}}\right)}{.}$

The class ${\rm HC}({\mathcal{O}})$ is called trivial when it coincides with
${\mathcal{O}}$. See [19] for the proof of $\doteq$. The compact set ${\rm
HC}({\mathcal{O}})$ is transitive [19]. In general, different homoclinic
classes may have non-empty intersections.

A hyperbolic set $\Lambda$ is called locally maximal if it possesses an open
neighborhood $V$ satisfying $\Lambda=\bigcap_{n\in{\mathbb{Z}}}f^{n}(V)$. A
fundamental property of hyperbolic sets is the existence of a $\delta>0$ such
that, for any $x,y\in\Lambda$ with $d(x,y)<\delta$, the intersection
$W_{\varepsilon}^{s}(x)\pitchfork W_{\varepsilon}^{u}(x)$ is a unique point
$y\in M$. When this point $y$ belongs to $\Lambda$, we refer to $\Lambda$ as
possessing a local product structure. It is known that a hyperbolic set
$\Lambda$ is locally maximal if and only if it presents this property of local
product structure [30].

A basic set $\Lambda$ for $f$ is an $f$-invariant compact set which is
transitive, hyperbolic, and locally maximal. A totally disconnected and
infinite basic set is referred to as a horseshoe. It is known that every
homoclinic class can be approximated by horseshoes, see [13].

###  2.2. Hyperbolic measures and homoclinic classes

We will recall some of the results presented in [2]. Let $x$ be a point of
$M$. For a nonzero vector $v\in T_{x}M$, the Lyapunov exponent of $f$ at $v$
in $x$ is defined by

$\chi(x,v)=\lim_{{n\to+\infty}}\frac{1}{n}\log\left\|Df^{n}(v)\right\|$

when the limit exists.

Let ${\mathcal{M}}_{f}^{e}$ o set of the ergodic measures of $f$. If
$\mu\in{\mathcal{M}}_{f}^{e}$, the Oseledets theorem establishes that for
almost every $x\in M$ with respect to $\mu$, the Lyapunov exponents of every
nonzero vector $v\in T_{x}M$ exist [22].

######  Theorem 2.2 (Oseledets).

Let $(f,\mu)$ be as described above. Then for $\mu$ almost every point, there
are real numbers $\chi_{1}(f,\mu)<\chi_{2}(f,\mu)<\cdots<\chi_{k}(f,\mu)$, and
a splitting

$T_{x}M=E_{x,1}\oplus E_{x,2}\oplus\cdots\oplus E_{x,k}$

satisfying:

  1. (1)

$Df(E_{x,i})=E_{f(x),i}$ for $i=1,\ldots,k$.

  2. (2)

For all nonzero $v\in E_{x,i}$,

$\chi(x,v)=\lim_{{n\to\pm\infty}}\frac{1}{n}\log\left\|Df^{n}_{v}\right\|=\chi_{i}.$

The set of points for which the Oseledest Theorem holds for a certain ergodic
measure is called a Lyapunov regular set, denoted by ${{\mathcal{R}}}$.

######  Definition 2.3.

An ergodic measure $\mu$ is hyperbolic of saddle type if $|\chi_{i}(f,\mu)|>0$
for every $i$ and $\chi_{1}(f,\mu)<0<\chi_{k}(f,\mu)$. We will denote the set
of these ergodic measures by ${\mathcal{M}}_{f}^{h}$.

Note that we consider that every hyperbolic measure is an ergodic measure.

When $\mu\in{\mathcal{M}}_{f}^{h}$ we define for $\mu$-almost every $x\in M$
the splitting $T_{x}M=E^{+}_{x}(f)\oplus$ $E^{-}_{x}(f)$ where
$E^{+}_{x},E^{-}_{x}$ are linear spaces such that
$\lim_{n\rightarrow\infty}\frac{1}{n}\log\left\|Df_{x}^{n}v\right\|<0$ on
$E^{+}(x)\backslash\\{0\\}$, and
$\lim_{n\rightarrow\infty}\frac{1}{n}\log\left\|Df_{x}^{-n}v\right\|>0$ on
$E^{-}(x)\backslash\\{0\\}$.

For $f\in{\rm Diff}^{1+}(M)$ the stable Pesin set of the point $x\in M$ is

$W^{-}(x)=\left\\{y\in M:\limsup_{n\to+\infty}\frac{1}{n}\log
d\left(f^{n}(x),f^{n}(y)\right)=0\right\\}.$

Similarly one defines the unstable Pesin set as

$W^{+}(x)=\left\\{y\in M:\limsup_{n\rightarrow+\infty}\frac{1}{n}\log
d\left(f^{-n}(x),f^{-n}(y)\right)=0\right\\}.$

It is known that given $\mu\in{\mathcal{M}}_{f}^{h}$, for $\mu$ almost every
$x\in M$ the stable and unstable Pesin sets are immersed submanifolds with the
dimension ${\operatorname{dim}}(E^{-})$ and ${\operatorname{dim}}(E^{+})$
respectively and therefore are called stable and unstable Pesin manifolds
[24]. Given a hyperbolic measure $\mu$ of $f$ we define the s-index of $\mu$
as s-index$(\mu):=\operatorname{Dim}(E^{+})$. In the remaining of the section,
we consider only ergodic measures with the same s-index.

######  Definition 2.4.

For two measures $\mu_{1},\mu_{2}\in{\mathcal{M}}_{f}^{h}$, we write
$\mu_{1}\preceq\mu_{2}$ if there are two sets $\Lambda_{1},\Lambda_{2}$ such
that $\mu_{i}\left(\Lambda_{i}\right)>0$ with the property that for any point
$\left(x,y\right)\in\Lambda_{1}\times\Lambda_{2}$ we have that
$W^{+}\left(x\right)\pitchfork W^{-}\left(y\right)\neq\emptyset$.

######  Definition 2.5.

Two ergodic measures $\mu_{1},\mu_{2}\in{\mathcal{M}}_{f}^{h}$ are
homoclinically related if $\mu_{1}\preceq\mu_{2}$ and $\mu_{2}\preceq\mu_{1}$.
In this case, we write $\mu_{1}\sim\mu_{2}$. The homoclinic class of the
measure $\mu\in{\mathcal{M}}_{f}^{h}$ is the set

$HC(\mu)=\bigcup\\{\nu\in{\mathcal{M}}_{f}^{h}:\mu\sim\nu\\},$

and the topological homoclinic class is the set

$HC_{T}(\mu):=\overline{\bigcup\\{\operatorname{supp}(\nu):\nu\in{\mathcal{M}}_{f}^{h},\mu\sim\nu\\}}$

.

The relation $\left({\mathcal{M}}_{f}^{h},\sim\right)$ is an equivalence
relation [2, Proposition 2.11].

######  Definition 2.6.

Given $\mu\in{\mathcal{M}}_{f}^{h}$ and
$\mathcal{O}\in\operatorname{Per}_{h}(f)$ with $\mathcal{O}=\\{p,\cdots
f^{n-1}(p)=p\\}$. We say that $\mu$ and ${{\mathcal{O}}}$ are homoclinically
related and write ${{\mathcal{O}}}\sim\mu$ or $p\sim\mu$ if the measure
$\frac{1}{n}\sum_{j=0}^{n}\delta_{f^{j}(p)}$ is homoclinically related to the
measure $\mu$.

######  Proposition 2.7.

[2, Corollary 2.14.] For any $f\in{{\rm Diff}}^{r}(M)$, $r>1$, and any
$\mu\in{\mathcal{M}}_{f}^{h}$, the set of measures supported by periodic
orbits is dense in the set of hyperbolic ergodic measures homoclinically
related to $\mu$, endowed with the weak-$\ast$ topology. In particular, there
exists ${{\mathcal{O}}}\in{\mathcal{M}}_{f}^{h}$ such that
${{\mathcal{O}}}\sim\mu$ and

$HC_{T}(\mu)=\overline{\bigcup\left\\{\operatorname{supp}\nu:\nu\in{\mathcal{M}}_{f}^{h},\nu\sim{{\mathcal{O}}}\right\\}}=HC(\mathcal{O}).$

######  Proposition 2.8.

Let $f:M\rightarrow M$ be a $C^{r}$ diffeomorphism
$r>1,\phi:M\rightarrow{\mathbb{R}}$ be a Hölder potential, and $p$ a
hyperbolic periodic point with orbit ${{\mathcal{O}}}$. Then there is at most
one, hyperbolic, equilibrium state for $\phi$ which is homoclinically related
to ${{\mathcal{O}}}$, and its support coincides with ${{\rm
HC}}({{\mathcal{O}}})$.

###### Proof.

The proof is a consequence of [23], which is a generalization of [29], and
Theorem [2, Theorem 3.1]. ∎

######  Remark 2.9.

It is important to observe that the above proposition does not imply that for
each $\mathcal{O}\in\operatorname{Per}(f)$, $HC(\mathcal{O})$ has at most one
hyperbolic measure of maximal entropy. It can exist two orbits
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}\in\operatorname{Per}(f)$ that are not
homoclinically related to each other, each one related to a hyperbolic measure
for maximal entropy, such that $HC(\mathcal{O}_{1})\subset
HC(\mathcal{O}_{2})$. Consequently, $HC(\mathcal{O}_{2})$ could have two
ergodic measures of maximal entropy.

###  2.3. Partial hyperbolic systems

The notion of partially hyperbolic systems represents a natural extension of
the concept of uniformly hyperbolic systems.

######  Definition 2.10.

A diffeomorphism $f$ defined on a compact manifold $M$ is partially hyperbolic
if it admits a non-trivial $Df$\- invariant splitting of the tangent bundle
$TM=E^{s}\oplus E^{c}\oplus E^{u}$, such that, all unit vectors $v^{\sigma}\in
E^{\sigma}_{x}$ ($\sigma=s,c,u$) with $x\in M$ satisfy

$\|Df_{x}v^{s}\|<\|Df_{x}v^{c}\|<\|Df_{x}v^{u}\|$

for some appropriate Riemannian metric; $f$ must also satisfy that
$0<\|Df_{|E^{s}}\|<\xi<1$ and $0<\|Df^{-1}_{|E^{u}}\|<\xi<1$.

####  2.3.1.  Existence of equilibrium states

Newhouse proved that every $C^{\infty}$ diffeomorphism has m.m.e. [21], but
this is not true for diffeomorphisms with less differentiability [17]. It is
known that every Anosov diffeomorphism is expansive and therefore has
equilibrium states for every Hölder continuous potential. In general, a
partially hyperbolic system does not have equilibrium states.

Let $\phi$ be a Hölder continuous potential. Consider the function
$P_{metric}(f,\phi)$, defined as

$P_{metric}(f,\phi):{\mathcal{M}}_{f}\to\mathbb{R},\,\,\mu\mapsto
P_{\mu}(f,\phi)$

Since ${\mathcal{M}}_{f}$ is a compact set in the weak * topology, if the
metric entropy is upper semicontinuous then $P_{metric}(f,\phi)$ achieves a
maximal value and so $(f,\phi)$ has an equilibrium state.

For homeomorphisms, Bowen showed [4] that the metric entropy is upper
semicontinuous whenever $(M,f)$ is a $h$-expansive system. In particular, the
pressure $P_{\mu}(f,\phi)$ is also upper semicontinuous and then $(f,\phi)$
admits an equilibrium state for any continuous potential $\phi$. In
particular, since hyperbolic systems with one-dimensional center bundles are
$h$-expansive, [8, 9, 14], they have equilibrium states.

####  2.3.2. Hyperbolic measures

Consider a partially hyperbolic diffeomorphism $f$ with a central dimension
one, denoted as $E^{c}$, and stable and unstable subspaces $E^{s}$ and
$E^{u}$, respectively, and $E^{u}$ with dimensions greater than zero. Let
$\mu$ be an ergodic measure associated to $f$. The measure $\mu$ is a
_hyperbolic measure of saddle type_ for $f$ if and only if for $\mu$-almost
every point $x$ in $M$, the Lyapunov exponent in the central direction is
different to zero, that is

$\lambda_{c}(\mu,f):=\lim_{n\to\infty}\frac{1}{n}\log|Df^{n}_{E^{c}}(x)|=\int\log|Df^{n}_{E^{c}}|d\mu\neq
0.$

Therefore, in the context of a partially hyperbolic system with a one-
dimensional central component, determining whether a given ergodic measure is
hyperbolic is achieved by verifying if the Lyapunov exponent in the central
direction is non-zero

###  2.4. Unstable entropy

In this section, we recall the notions of unstable $h^{u}(f)$ and stable
$h^{s}(f)$ topological entropy for a partial hyperbolic diffeomorphism $f$
introduced in [33, 10, 32].

####  2.4.1. Unstable metric entropy

Here we follow closely [32].

Consider a compact metric space $M$ with the Borel $\sigma$-algebra
${{\mathcal{B}}}$ and $\mu$ represent a Borel probability measure defined on
this space. Let ${{\mathcal{P}}}=\\{P_{i}\\}_{i\in I}$ be a partition of $M$
into measurable subsets, i.e., $M=\bigcup_{P_{i}\in\mathcal{P}}P_{i}$ such
that $P_{i}\in{{\mathcal{B}}}$, $\mu(P_{i}\cap P_{j})=0$, and
$\mu\left(\bigcup_{P_{i}\in{{\mathcal{P}}}}P_{i}\right)=\mu(M)=1$. We denote
by $\sigma({{\mathcal{P}}})$ the smallest $\sigma$-algebra containing all
$P_{i}\in{{\mathcal{P}}}$.

Given two partitions ${{\mathcal{P}}}$ and ${{\mathcal{Q}}}$, we use the
notation ${{\mathcal{P}}}\leq{{\mathcal{Q}}}$ if
$\sigma({{\mathcal{P}}})\subset\sigma({{\mathcal{Q}}})$.

######  Definition 2.11.

We say that a partition ${{\mathcal{P}}}$ is measurable (or countably
generated) with respect to $\mu$ if there exists a measurable family
$\\{A_{i}\\}_{i\in\mathbb{N}}$ and a measurable set $F$ of full measure such
that if $B\in{{\mathcal{P}}}$, then there exists a sequence $\\{B_{i}\\}$,
where $B_{i}\in\\{A_{i},A_{i}^{c}\\}$ such that $B\cap F=\bigcap_{i}(B_{i}\cap
F)$.

Consider $\pi:M\rightarrow M/P$ as the canonical projection, mapping each
point in $M$ to the corresponding element in the quotient space formed by the
atoms of the partition. This naturally gives rise to a measuring structure on
$M/P$ via the measure $\widetilde{\mu}=\pi^{*}\mu$ defining it on the
associated pushed $\sigma$-algebra. Additionally, for any point $x$ in $M$,
$P(x)$ represents the element of the partition to which $x$ belongs.

######  Definition 2.12.

Given a partition $\mathcal{P}$, a family $\\{\mu_{P}\\}_{P\in\mathcal{P}}$ is
a system of conditional measures for $\mu$ (with respect to $\mathcal{P}$) if,
for a given $\varphi\in C_{0}(M)$:

  1. (1)

$P\mapsto\int\varphi\,d\mu_{P}$ is $\tilde{\mu}$-measurable.

  2. (2)

$\mu_{P}(P)=1$ $\tilde{\mu}$-a.e.

  3. (3)

$\int_{M}\varphi\,d\mu=\int_{M/P}\left(\int_{P}\varphi\,d\mu_{P}\right)d\tilde{\mu}\,=\int_{M}\left(\int_{P(x)}\varphi\,d\mu_{P}(x)\right)\,d\mu$.

######  Theorem 2.13.

[26, Rokhlin’s Disintegration] Let $P$ be a measurable partition of a compact
metric space $M$, and $\mu$ a Borel probability measure. Then there exists a
unique disintegration by conditional probability measures for $\mu$.

Let $f:M\rightarrow M$ be a diffeomorphism of a compact manifold preserving a
probability measure $\mu$. Consider a partition
$P=\\{P_{1},P_{2},\ldots,P_{n}\\}$ into measurable subsets. Then:

$H_{\mu}(P)=\sum_{i=1}^{n}-\mu(P_{i})\log(\mu(P_{i}))=\int_{M}-\log(\mu(P(x)))\,d\mu,$

where $P(x)=P_{i}$ for $x\in P_{i}$. And

$h_{\mu}(P,f)=\lim_{n\to\infty}\frac{1}{n}H_{\mu}(P_{n})=\lim_{n\to\infty}\left(H_{\mu}(P_{n})-H_{\mu}(P_{n-1})\right),$

where $P_{n}=P\vee f^{-1}(P)\vee\ldots\vee f^{-n+1}(P)$ is a refinement of $P$
by the action of $f$.

Now let ${{\mathcal{F}}}$ be an $f$-invariant foliation, that is,
$f{{\mathcal{F}}}(x))={{\mathcal{F}}}(f(x))$, for every $x\in M$. We say that
${{\mathcal{F}}}$ is uniformly expanding if there exists $\lambda>1$ such that
$\forall x\in M$, $\|Df^{-1}_{|T_{x}(F(x))}\|<\lambda^{-1}$. A measurable
partition $\xi$ is called increasing and subordinated to ${{\mathcal{F}}}$ if
it satisfies the following properties:

  * (a)

$\xi(x)\subseteq{{\mathcal{F}}}(x)$ for $\mu$-almost every $x$.

  * (b)

$f^{-1}(\xi)\geq\xi$ (increasing property).

  * (c)

$\xi(x)$ contains an open neighborhood of $x$ in ${{\mathcal{F}}}(x)$ for
$\mu$-almost every $x$.

Finding a measurable and expanding partition that aligns with an invariant
lamination is typically a complex challenge. However, when dealing with a
uniformly expanding foliation that remains invariant under a diffeomorphism,
such a partition is always available [33]. Additionally, for expanding
flotations ${{\mathcal{F}}}$ it holds

  * (d)

$\vee^{\infty}_{n=0}f^{-n}\xi$ is the partition into points;

  * (e)

the largest $\sigma$-algebra contained in $\bigcap^{\infty}_{n=0}f^{n}(\xi)$
is ${{\mathcal{B}}{\mathcal{F}}}$ where ${{\mathcal{B}}{\mathcal{F}}}$ is the
$\sigma$-algebra of ${{\mathcal{F}}}$-saturated measurable subsets (union of
entire leaves).

Given a measure $\mu$ and two measurable partitions $\alpha$, $\beta$ of $M$,
the conditional entropy of $\alpha$ given $\beta$ is denoted by

$H_{\mu}(\alpha|\beta):=\int_{M}\log(\mu^{\beta}_{x}(\alpha(x)))d\mu(x)$

where $\mu^{\beta}_{x}$ is the conditional measure of $\mu$ with respect to
$\beta$ as Definition 2.12. For any invariant lamination $\mathcal{F}$ with a
measurable subordinated increasing partition satisfying all the properties
including (d) and (e) above, we define

$h_{\mu}(f,{{\mathcal{F}}}):=h_{\mu}(f,\xi)=H_{\mu}(f^{-1}\xi|\xi)=H_{\mu}(\xi|f\xi).$

It can be proved that the above definition is independent of $\xi$.

Let $f$ be a partially hyperbolic diffeomorphism $f$ such that
${\operatorname{dim}}(E^{u}_{f})\geq 1$ and $\mu\in{\mathcal{M}}_{f}$. We
define

$h_{\mu}^{u}(f)=h_{\mu}(f,{{\mathcal{F}}}^{u}),$

where ${{\mathcal{F}}}^{u}$ denotes the unstable foliation given by the
unstable manifolds.

Analogously, if the dimension of the stable bundle $E^{s}_{f}$ is greater than
or equal to 1, we can define the stable entropy for any
$\mu\in{\mathcal{M}}_{f}$ as

$h^{s}_{\mu}(f):=h_{\mu}(f,{{\mathcal{F}}}^{s}),$

where ${{\mathcal{F}}}^{s}$ denotes the unstable foliation given by the
unstable manifolds.

####  2.4.2.  Unstable topological entropy and the variational principle

Similar to the definition of topological entropy for continuous maps on
compact metric spaces, the concept of topological entropy along the unstable
foliation can be formulated by considering the notion of spanning (and
separated) sets within a collection of compact unstable plaques.

Let $f$ be a partially hyperbolic diffeomorphism $f$ such that
${\operatorname{dim}}(E^{u})\geq 1$. We denote by $d^{u}$ the metric induced
by the Riemannian structure on the unstable manifold and let
$d_{n}^{u}(x,y)=\max_{0\leq j\leq n-1}d^{u}\left(f^{j}(x),f^{j}(y)\right)$.
Let $W^{u}(x,\delta)$ be the open ball inside $W^{u}(x)$ centred at $x$ of
radius $\delta$ with respect to the metric $d^{u}$. Let
$N^{u}(f,\epsilon,n,x,\delta)$ be the maximal number of points in
$\overline{W^{u}(x,\delta)}$ with pairwise $d_{n}^{u}$-distances at least
$\varepsilon$. We call such a set an $(n,\varepsilon)$ u-separated set of
$\overline{W^{u}(x,\delta)}$. We can alternatively define unstable topological
entropy by employing $(n,\epsilon)$ u-spanning sets or open covers, yielding
equivalent definitions.

######  Definition 2.14.

The unstable topological entropy of $f$ on $M$ is defined by

$h_{\text{top }}^{u}(f)=\lim_{\delta\rightarrow 0}\sup_{x\in M}h_{\text{top
}}^{u}\left(f,\overline{W^{u}(x,\delta)}\right),$

where

$h_{\text{top
}}^{u}\left(f,\overline{W^{u}(x,\delta)}\right)=\lim_{\epsilon\rightarrow
0}\limsup_{n\rightarrow\infty}\frac{1}{n}\log N^{u}(f,\epsilon,n,x,\delta).$

Analogously, if the dimension of the stable bundle $E^{s}_{f}$ is greater than
or equal to 1, we can define the stable entropy for any
$\mu\in{\mathcal{M}}_{f}$ as $h^{s}(f):=h^{u}_{\mu}(f^{-1})$.

Similar to the traditional definition of entropy, we can establish a
connection between metric unstable entropy and unstable topological entropy
through a variational principle. Specifically, according to [10, Theorem D],
for a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$, the
following relation holds:

$h^{u}_{top}(f)=\sup\\{h^{u}_{\mu}(f):\mu\in{\mathcal{M}}_{f}\\}\quad\mbox{and}\quad
h^{u}_{top}(f)=\sup\\{h^{u}_{\nu}(f):\nu\in{\mathcal{M}}_{f}^{e}\\}.$

Another way to define topological unstable entropy is by considering the
unstable volume growth given by Hua, Saghin, and Xia ([12]), reminiscent of
the works by Yomdin and Newhouse ([34, 20]). In [10, Theorem C] it is shown
that the unstable topological entropy as defined here coincides with the
unstable volume growth.

###  2.5. Hausdorff distance

In this section we will define the Hausdorff distance, we follow [1, Section
7.3].

######  Definition 2.15.

Let $A$ and $B$ be subsets of the metric space $(M,d)$. The Hausdorff distance
between $A$ and $B$, denoted by $d_{H}(A,B)$ is defined by

$d_{H}(A,B)=\max\\{\sup_{a\in A}d(a,B),\sup_{b\in B}d(b,A)\\}.$

Let ${{\mathcal{M}}}(M)$ be the set of all the closed subsets of $M$. It is
known that $({{\mathcal{M}}}(M),d_{H})$ is a metric space and when $M$ is
compact, the metric space $({{\mathcal{M}}}(M),d_{H})$ is compact.

######  Remark 2.16.

Let $(K_{n}){n\in{\mathbb{N}}}$ be a convergent sequence of compact subsets of
the compact metric space $(M,d)$ converging to $K$ in the Hausdorff Metric. If
$x_{n}\in K_{n}$ forms a convergent sequence in $(M,d)$ converging to $x$,
then $x\in K$. Moreover, if $(\mu_{k})_{k\in{\mathbb{N}}}$ are probability
measures converging to $\mu$ in the weak * topology, then
$\mu(K)\geq\limsup(\mu_{n}(K_{n}))$.

##  3\. Consequences of $h^{u}(f)-h^{s}(f)>0$

Let ${\rm Diff}^{1+}(M)$ be the set of $C^{1+}$ diffeomorphisms defined in
$M$. From now on, $f$ will be a $C^{1+}$ partially hyperbolic diffeomorphism
with $1$-dimensional central bundle and non trivial stable and unstable
bundles, that is, ${\operatorname{dim}}(E^{\sigma})>0(\sigma=s,u)$.

For $f\in{\rm Diff}^{1+}(M)$ and $\mu$ ergodic by the Ledrappier and Young
formula, [15, 16] and [5], we get that

(3.1) $h_{\mu}(f)\leq
h_{\mu}^{u}+\sum_{\lambda_{c}(f,\mu)>0}\lambda_{c}(f,\mu)$

and if the central Lyapunov exponent of $\mu$ is non-positive then, by the
definition of unstable metric entropy we get

(3.2) $h_{\mu}(f)=h_{\mu}^{u}(f).$

######  Lemma 3.1.

Consider $f:M\to M$ such that $h^{u}(f)>h^{s}(f)$. If $\mu$ is an ergodic
measure such that $h_{\mu}(f)>h^{s}(f)$, then
$\lambda_{c}(f,\mu)<h^{s}(f)-h_{\mu}(f)<0$.

###### Proof.

Let $\mu$ be an ergodic measure for $f$ with $h_{\mu}(f)>h^{s}(f)$.

######  Claim 1.

The central Lyapunov exponent of $\mu$ is negative, i.e.,
$\lambda_{c}(f,\mu)<0$.

###### Proof.

Suppose that the central Lyapunov exponent of $\mu$ is non-negative, i.e.,
$\lambda^{c}(\mu,f)\geq 0$. Applying (3.2) to the stable metric entropy, we
have

$h_{\mu}(f)=h^{s}_{\mu}(f)$

and by hypothesis, $h_{\mu}(f)>h^{s}(f)$, therefore $h_{\mu}^{s}(f)>h^{s}(f)$,
which contradicts the variational principle for stable entropy. ∎

By $h_{\mu}(f)>h^{s}(f)$ and Claim 1, we have that $\lambda_{c}(f,\mu)<0$, and
therefore by 3.1,

$h_{\mu}(f)\leq h^{s}(f)-\lambda_{c}(f,\mu).$

Hence, the proof is concluded.

∎

######  Remark 3.2.

It is worth noting that under the hypotheses of Lemma 3.1, every ergodic
measure with large entropy is hyperbolic and the metric entropy coincides with
the unstable metric entropy. Furthermore, the central Lyapunov exponent is
negative, so its $s$-index is $\operatorname{dim}(E^{s})+1$. This leads us to
inquire whether two measures with large entropy exhibit homoclinic relation as
defined in the definition 2.5.

Now, we will introduce an analogous result for the pressure, but before doing
so, let us establish some notations. For every $\mu\in{\mathcal{M}}_{f}$ and
$\phi:M\to{\mathbb{R}}$, a continuous potential, we define:

$P_{\mu}(f,\phi):=h_{\mu}(f)+\int\phi d\mu$

and

$P_{\mu}^{u/s}(f,\phi):=h_{\mu}^{u/s}(f)+\int\phi d\mu$

Furthermore,

$P^{u/s}(f,\phi):=\sup_{\mu\in{\mathcal{M}}_{f}}\left\\{P_{\mu}^{u/s}(f,\phi)\right\\}.$

Observe that the condition $P^{u}(f,\phi)>P^{s}(f,\phi)$ is not sufficient to
guarantee positive entropy for the equilibrium states of $(f,\phi)$. When
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$, every measure of maximal entropy $\mu$
satisfies $P_{\mu}(f,\phi)>h^{s}(f)+\sup\phi$. So,

######  Corollary 3.3.

Let $\phi:M\to{\mathbb{R}}$ be a continuous potential such that
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. If $\mu$ is an ergodic measure such
that $P_{\mu}(f,\phi)>h^{s}(f)+\sup\phi$ then
$\lambda_{c}(f,\mu)<h_{\mu}^{s}(f)-h^{s}(f)<0$.

###### Proof.

Let us consider $\mu\in{\mathcal{M}}_{f}^{e}$ such that
$P_{\mu}(f,\phi)>h^{s}(f)+\sup\phi$.

Since $\mu$ is a probability measure, we have $\sup\phi\geq\int\phi d\mu$.
Therefore, given that $h_{\mu}(f)+\int\phi d\mu>h^{s}(f)+\sup\phi$, we can
conclude that:

$h_{\mu}(f)>h^{s}(f)$

Now, since $h^{u}(f)>h^{s}(f)$, by Lemma 3.1, we obtain:

$\lambda_{c}(f,\phi)\leq h^{s}(f)-h_{\mu}(f)<0$

Thus, we have completed the proof. ∎

Consider the map $h:{{\rm Diff}}^{1+}(M)\to{\mathbb{R}}$ such that $h(f)$ is
the topological entropy of $f$. The next Lemma shows that the property
$h^{u}(f)>h^{s}(f)$ is open:

######  Lemma 3.4.

If $h^{u}(f)>a_{1}>a_{2}>h^{s}(f)$, then there exists a $C^{1}$ neighborhood
${{\mathcal{U}}}$ of $f$ such that for every $g\in{{\mathcal{U}}}\cap{{\rm
Diff}}^{1+}(M)$, $h^{u}(g)>a_{1}>a_{2}>h^{s}(f)$, the function $h:{{\rm
Diff}}^{1+}(M)\cap\mathcal{U}\to\mathbb{R}$ is continuous.

###### Proof.

The lemma is a consequence of [18, Corollary 3.6] and [18, Corollary 3.7]. ∎

##  4\. Invariant manifolds for measures with large entropy

In this section, we will investigate the measurement of the set of points with
invariant manifolds of uniform size from the perspective of measures with high
entropy.

###  4.1. Invariant Manifolds with Uniform Size

We see that ergodic measures with large entropy are hyperbolic measures, see
Lemma 3.1. Almost every point of a hyperbolic measure possesses stable and
unstable invariant manifolds, see Section 2.3.2. However, a crucial point to
note is the absence of uniform control over the size of these invariant
manifolds. Consequently, there may exist points that are arbitrarily close to
each other such that their corresponding unstable and stable manifolds may not
intersect. This suggests the potential existence of an infinite number of
maximal entropy measures that are not homoclinically related. To avoid this
kind of problem, in this section, we construct, for partially hyperbolic
diffeomorphisms with central dimension one a set where the Pesin invariant
manifolds have uniform size.

Recall that $f:M\to M$ denotes a $C^{1+}$ partially hyperbolic diffeomorphism
with $1$-dimensional central direction with non trivial stable and unstable
bundles, that is, ${\operatorname{dim}}(E^{\sigma})>0(\sigma=s,u)$. Given
$1>\rho\geq\xi>0$ where $\xi$ is as in 2.10, define

(4.1) $\Lambda_{\rho}(f)=\Lambda_{\rho}:=\\{x\in M:\forall n\geq 1\mbox{ it
holds }\|Df^{n}(x)_{|E^{c}}\|\leq\rho^{n}\\}$

######  Remark 4.1.

Note that $\Lambda_{\rho}$ is a compact set. Moreover $f$ is a partially
hyperbolic diffeomorphism with central bundle one and such that for every
$x\in\Lambda_{\rho}$ we have that

$\|Df^{n}_{x}(v)\|<\rho^{n}\mbox{ for every }v\in E^{c}\mbox{ and }n\geq 1$

and the splitting $E^{cs}(x)\oplus E^{u}(x)$ varies continuously. Therefore,
the classical result presented in [11] guarantees the existence of invariant
Pesin manifolds of uniform size. More precisely, there exists
$\delta_{r}=\delta_{r}(\rho)$ and $\delta_{l}=\delta_{l}(\rho)$ such that
every $x\in\Lambda_{\rho}(f)$ unstable and stable manifold have a size larger
than $\delta_{r}$ and if $x,y\in\Lambda_{\rho}(f)$ and $d(x,y)<\delta_{l}$
then $W^{u/s}_{\delta_{r}}(x)\pitchfork W^{s/u}_{\delta_{r}}(y)$ is non empty.
Furthermore, the invariant manifolds exhibit continuous variations concerning
both the diffeomorphisms involved and the chosen points. In other words, given
$f$ and $g$ are $C^{1+}$ close enough in the $C^{1}$ topology, for any $x$ in
$\Lambda_{\rho}(f)$ and $y$ in $\Lambda_{\rho}(g)$ where $x$ and $y$ are
close, the invariant manifolds $W^{s/u}_{\delta_{r}}(x)$ are also close to
$W^{s/u}_{\delta_{r}}(y)$ within the $C^{1}$ topology.

###  4.2. The measure of $\Lambda_{\rho}$

Now, we know that points in $\Lambda_{\rho}(f)$ have invariant manifolds of
uniform size. We will use this set to prove that there exists a finite number
of homoclinic classes containing measures with large entropy. For this, it is
crucial that this set is relevant for measures with large entropy. Therefore,
we aim to demonstrate that the set $\Lambda_{\rho}(f)$ possesses a positive
measure for measures with significant entropy. To achieve our goals, we
require the following version of the Pliss Lemma.

######  Lemma 4.2.

[6, Lemma 3.1] For any $\varepsilon>0$, $\alpha_{1}<\alpha_{2}$ and any
sequence $(a_{i})\in(\alpha_{1},+\infty)^{{\mathbb{N}}}$ satisfying

$\limsup_{n\to+\infty}{\frac{a_{0}+\cdots a_{n-1}}{n}}\leq\alpha_{2},$

there exists a sequence of integers $0<n_{1}<n_{2}<\cdots$ such that

  * (a)

for any $l\geq 1$ and $n>n_{l}$, one has
$\frac{a_{n_{l}}+\cdots+a_{n-1}}{n-n_{l}}\leq\alpha_{2}+\varepsilon$

  * (b)

the upper density $\limsup\frac{l}{n_{l}}$ is larger than
$\frac{\varepsilon}{\alpha_{2}+\varepsilon-\alpha_{1}}$.

We define $m(f):=\min_{x\in M}\\{\log\|Df(x)_{|E^{c}_{x}}\|\\}$. Note that
$m(f)$ varies continuously on $f$ in the $C^{1}$ topology.

######  Lemma 4.3.

Suppose that $h^{u}(f)>h^{s}(f)$. If $\mu\in{\mathcal{M}}_{f}^{e}$ with
$h_{\mu}(f)>h^{s}(f)$ then for every $e^{\lambda_{c}(f,\mu)}<\rho<1$ we have
that
$\mu(\Lambda_{\rho})\geq\frac{\log(\rho)-\lambda_{c}(f,\mu)}{\log(\rho)-m(f)}$

###### Proof.

Let $\mu\in{\mathcal{M}}_{f}$ be an ergodic measure with
$h_{\mu}(f)>h^{s}(f)$. Define $\Lambda_{\mu}$ as the set of regular points of
$\mu$, which constitutes a full $\mu$-measure set and are defined as the
points which satisfy the property that for any $x\in\Lambda_{\mu}$, as $n$
tends to $\pm\infty$, we have

$\frac{1}{|n|}\sum_{j=0}^{n-1}\delta_{f^{-j}(x)}\to\mu,$

where $\delta_{x}$ is the Dirac measure at point $x$.

Since $h^{u}(f)>h^{s}(f)$ and $h_{\mu}(f)>h^{s}(f)$, Lemma 3.1 implies that
$\lambda_{c}(f,\mu)<h^{s}(f)-h_{\mu}(f)<0$. Now, let
$0<e^{\lambda_{c}(f,\mu)}<\rho<1$ and define the set

$\Lambda(f)=\Lambda_{\rho}\cap\Lambda_{\mu}\cap\mathcal{R}.$

Note that $\mu(\Lambda(f))=\mu(\Lambda_{\rho})$.

For every $x\in\Lambda(f)$, we have that

$\displaystyle\lim_{n\to\infty}\frac{1}{n}\log(\|Df^{n}(x)_{|E^{c}_{x}}\|)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log(\|Df(f^{j}(x))_{|E^{c}(f^{j}(x))}\|)$
$\displaystyle=\lambda_{c}(f,\mu)$

We will now apply Lemma 4.2. Take $\varepsilon=\log(\rho)-\lambda_{c}(f,\mu)$,
$\alpha_{1}=m(f)$, $\alpha_{2}=\lambda_{c}(f,\mu)$, and consider the sequence

$(\log(\|Df(f^{j}(x))_{|E^{c}(f^{j}(x))}\|))_{j\in{\mathbb{N}}}.$

This results in a sequence of integers $(n_{l})_{l\in{\mathbb{N}}}$ such that
for each $l\in{\mathbb{N}}$ and $n>n_{l}$, we find that

$\frac{1}{n-n_{l}}\sum_{j=n_{l}}^{n-1}\log\|Df({f^{j}(x))_{|E^{c}_{f^{j}(x)}}}\|<\log(\rho).$

Consequently, we obtain
$\|Df^{n}(f^{n_{l}}(x))_{|E^{c}(f^{n_{l}}(x))}\|<\rho^{n}$ for every
$n\in{\mathbb{N}}$. Thus, for each $l$, $f^{n_{l}}(x)\in\Lambda_{\mu}$.
Applying Birkhoff’s theorem and the Pliss Lemma, we arrive at the following
estimate:

(4.2)
$\mu(\Lambda_{f})\geq\limsup_{l\to\infty}\frac{l}{n_{l}}\geq\frac{\log(\rho)-\lambda_{c}(f,\mu)}{\log(\rho)-m(f)}>0.$

∎

######  Corollary 4.4.

Let $\phi:M\to{\mathbb{R}}$ be a potential continuous such that
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. If $\mu$ is an ergodic measure such
that $h_{\mu}(f)+\int\phi d\mu>h^{s}(f)+\sup\phi$ then for every
$e^{\lambda_{c}(f,\mu)}<\rho<1$ we have that

$\mu(\Lambda_{\rho})\geq\frac{\log(\rho)-\lambda_{c}(f,\mu)}{\log(\rho)-m(f)}.$

###### Proof.

As in the proof of Corollary 3.3, the inequality $h_{\mu}(f)+\int\phi
d\mu>h^{s}(f)+\sup\phi$ implies

$h_{\mu}(f)>h^{s}(f)$

so, by Lemma 4.3 we have that

$\mu(\Lambda_{\rho})\geq\frac{\log(\rho)-\lambda_{c}(f,\mu)}{\log(\rho)-m(f)}.$

∎

######  Corollary 4.5.

Suppose that $h^{u}(f)>h^{s}(f)$. For every $h^{u}(f)>a>h^{s}(f)$ and
$\frac{h^{s}(f)-a}{2}<\log(\rho)<0$ there exist a $C^{1}$ neighborhood
${\mathcal{V}}$ of $f$ and a real number $\beta>0$ such that for every
$g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$ holds.

  1. (a)

$h^{u}(g)>\frac{h^{u}(f)+a}{2}$ and $h^{s}(g)<\frac{h^{s}(f)+a}{2}$,

  2. (b)

If $\mu\in{\mathcal{M}}_{g}^{e}$ with $h_{\mu}(g)\geq a$, then
$\mu(\Lambda_{\rho}(g))\geq\beta$.

###### Proof.

For the item (a), it is enough to consider $a_{1}=\frac{h^{u}(f)+a}{2}$ and
$a_{2}=\frac{h^{s}(f)+a}{2}$ in Lemma 3.4.

For the item (b), let $\mu\in{\mathcal{M}}_{g}^{e}$ such that $h_{\mu}(g)\geq
a$. By Lemma 3.1, we have:

(4.3) $\lambda_{c}(g,\mu)\leq h^{s}(g)-h_{\mu}(g).$

Using the first item, we also have:

(4.4) $h^{s}(g)-h_{\mu}(g)<\frac{h^{s}(f)+a}{2}-h_{\mu}(g).$

Since 4.3, 4.4 and $h_{\mu}(g)\geq a$, it follows:

$\lambda_{c}(g,\mu)<\frac{h^{s}(f)-a}{2}.$

Consider $\log(\rho)>\frac{h^{s}(f)-a}{2}$, so,
$e^{\lambda_{c}(g,\mu)}<\rho<0$. Therefore, by Lemma 4.3, we have:

$\mu(\Lambda_{\rho}(g))\geq\frac{\log(\rho)-\lambda_{c}(g,\mu)}{\log(\rho)-m(g)}\geq\frac{\log(\rho)-\frac{h^{s}(f)-a}{2}}{\log(\rho)-m(g)}>0.$

Note that since $m(g)$ varies continuously in the $C^{1}$ topology, we can
choose $\beta>0$ such that:

$\frac{\log(\rho)-\frac{h^{s}(f)-a}{2}}{\log(\rho)-m(g)}>\beta$

for every $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$, where ${{\mathcal{V}}}$
is an open neighborhood of $f$ as in item (a) and with small size such that
$m(g)$ is close to $m(f)$. Thus, we conclude the proof. ∎

######  Corollary 4.6.

Let $\phi:M\to{\mathbb{R}}$ be a potential continuous such that
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. For every $a$ such that
$h^{u}(f)+\inf\phi>a>h^{s}(f)+\sup\phi$ and
$\frac{h^{s}(f)-a+\inf\phi}{2}<\log(\rho)<1$ it holds

  1. (a)

For every $g\in{\mathcal{V}}$,

$h^{u}(g)+\inf\phi>\frac{h^{u}(f)+\inf\phi+a}{2}\quad\mbox{and}\quad
h^{s}(g)+\sup\phi<\frac{h^{s}(f)+\sup\phi+a}{2}.$

  2. (b)

If $\mu\in{\mathcal{M}}_{g}^{e}$ with $P_{\mu}(g)\geq a$, then
$\mu(\Lambda_{\rho}(g))\geq\beta$.

###### Proof.

For item (a), it is enough to consider $a_{1}=\frac{h^{u}(f)+\inf\phi+a}{2}$
and $a_{2}=\frac{h^{s}(f)+\sup\phi+a}{2}$ in Lemma 3.4.

For item (b), let $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$ and
$\mu\in{\mathcal{M}}_{g}$ such that $P_{\mu}(f)\geq a$, where
${{\mathcal{V}}}$ is the neighborhood of $f$ given by item $(a)$. By Corollary
3.3, we have:

(4.5) $\lambda_{c}(g,\mu)\leq h^{u}(g)-h_{\mu}(g).$

Now, on one side, since $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$, we have
$h^{s}(g)+\sup\phi<\frac{h^{s}(f)+a+\sup\phi}{2}$ and therefore:

(4.6) $h^{s}(f)<\frac{h^{s}(f)+a-\sup\phi}{2}.$

On the other side, as $h_{\mu}(f)+\int\phi d\mu\geq a$ and
$\sup\phi\geq\int\phi d\mu$, we have:

(4.7) $h_{\mu}(f)\geq a-\sup\phi.$

Applying 4.9 and 4.10 in 4.8 we get:

$\lambda_{c}(g,\mu)<\frac{h^{s}(f)+\sup\phi-a}{2}.$

Now, consider $\log(\rho)>\frac{h^{s}(f)+\sup\phi-a}{2}$. So,
$e^{\lambda_{c}(g,\mu)}<\rho<1$, and therefore by Corollary 4.4:

$\mu(\Lambda_{\rho}(g))\geq\frac{\log(\rho)-\lambda_{c}(g,\mu)}{\log(\rho)-m(g)}\geq\frac{\log(\rho)-\frac{h^{s}(f)+\sup\phi-a}{2}}{\log(\rho)-m(g)}.$

Note that since $m(g)$ varies continuously in the $C^{1}$ topology, we can
choose $\beta>0$ such that:

$\frac{\log(\rho)-\frac{h^{s}(f)+\sup\phi-a}{2}}{\log(\rho)-m(g)}>\beta$

for every $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$, reducing the size of
the neighborhood ${\mathcal{V}}$ when necessary. Thus, we conclude the proof.
∎

######  Corollary 4.7.

Let $\phi:M\to\mathbb{R}$ be a potential continuous such that
$h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. For every $a$ such that
$h^{u}(f)+\inf\phi>a>h^{s}(f)+\sup\phi$ and
$\frac{h^{s}(f)-a+\inf\phi}{2}<\log(\rho)<1$ it holds

  1. (a)

For every $g\in{\mathcal{V}}$,

$h^{u}(g)+\inf\phi>\frac{h^{u}(f)+\inf\phi+a}{2}\quad\mbox{and}\quad
h^{s}(g)+\sup\phi<\frac{h^{s}(f)+\sup\phi+a}{2}.$

  2. (b)

If $\mu\in{\mathcal{M}}_{g}^{e}$ with $P_{\mu}(g)\geq a$, then
$\mu(\Lambda_{\rho}(g))\geq\beta$.

###### Proof.

For item (a), it is enough to consider $a_{1}=\frac{h^{u}(f)+\inf\phi+a}{2}$
and $a_{2}=\frac{h^{s}(f)+\sup\phi+a}{2}$ in Lemma 3.4.

For item (b), let $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$ and
$\mu\in{\mathcal{M}}_{g}$ such that $P_{\mu}(f)\geq a$, where
${{\mathcal{V}}}$ is the neighborhood of $f$ given by item $(a)$. By Corollary
3.3, we have:

(4.8) $\lambda_{c}(g,\mu)\leq h^{u}(g)-h_{\mu}(g).$

Now, on one side, since $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$, we have
$h^{s}(g)+\sup\phi<\frac{h^{s}(f)+a+\sup\phi}{2}$ and therefore:

(4.9) $h^{s}(f)<\frac{h^{s}(f)+a-\sup\phi}{2}.$

On the other side, as $h_{\mu}(f)+\int\phi d\mu\geq a$ and
$\sup\phi\geq\int\phi d\mu$, we have:

(4.10) $h_{\mu}(f)\geq a-\sup\phi.$

Applying 4.9 and 4.10 in 4.8 we get:

$\lambda_{c}(g,\mu)<\frac{h^{s}(f)+\sup\phi-a}{2}.$

Now, consider $\log(\rho)>\frac{h^{s}(f)+\sup\phi-a}{2}$. So,
$e^{\lambda_{c}(g,\mu)}<\rho<1$, and therefore by Corollary 4.4:

$\mu(\Lambda_{\rho}(g))\geq\frac{\log(\rho)-\lambda_{c}(g,\mu)}{\log(\rho)-m(g)}\geq\frac{\log(\rho)-\frac{h^{s}(f)+\sup\phi-a}{2}}{\log(\rho)-m(g)}.$

Note that since $m(g)$ varies continuously in the $C^{1}$ topology, we can
choose $\beta>0$ such that:

$\frac{\log(\rho)-\frac{h^{s}(f)+\sup\phi-a}{2}}{\log(\rho)-m(g)}>\beta$

for every $g\in{\mathcal{V}}\cap{{\rm Diff}}^{1+}(M)$, reducing the size of
the neighborhood ${\mathcal{V}}$ when necessary. Thus, we conclude the proof.
∎

##  5\. Proof of Theorems A and B

In this section, we prove theorems A and B. Recall that $f:M\to M$ denotes a
$C^{1+}$ partially hyperbolic diffeomorphism of a compact manifold $M$ with
$TM=E^{u}\oplus E^{c}\oplus E^{s}$ and ${\operatorname{dim}}(E^{c})=1$.

###### Proof of Theorem A.

Let $a>0$ be a real number such that $h^{u}(f)\geq a>h^{s}(f)$. By Lemma 3.1,
we have that for every ergodic measure $\mu$ with $h_{\mu}(f)\geq a$, it
holds:

$\lambda_{c}(f,\,u)\leq h^{s}-a<0.$

Fix $e^{h^{s}(f)-a}<\rho<1$ and consider $\Lambda_{\rho}\subset M$, and
$\delta_{l}>0$ as in 2.10 and Remark 4.1 respectively.

######  Claim 2.

There exist $K_{a}\in\mathbb{N}$ and many finite $\mu_{1},\cdots\mu_{K_{a}}$
ergodic measures with $h_{\mu_{i}}(f)\geq a$ such that if
$\mu\in{\mathcal{M}}_{f}^{e}$ with $h_{\mu}(f)\geq a$, then $\mu\in
HC(\mu_{i})$ for some $i\leq K_{a}$.

###### Proof.

By Lemma 4.3, for every ergodic $\mu\in{\mathcal{M}}_{f}^{e}$ with
$h_{\mu}(f)\geq a$, we have:

$\mu(\Lambda_{\rho})>0.$

Thus, for every ergodic measure $\mu$ of $f$ with $h_{\mu}(f)\geq a$, there
exists $x_{\mu}\in\Lambda_{\rho}$ such that:

(5.1) $\mu(B_{\frac{\delta_{l}}{3}}(x_{\mu})\cap\Lambda_{\rho})>0.$

Now, let $\\{\mu_{\gamma}\\}$ be a sequence of different ergodic measures of
$f$ such that $h_{\mu_{\gamma}}(f)\geq a$ and choose $x_{\mu_{\gamma}}$ holds
5.1. Due to the compactness of $M$, there are two measures $\mu_{\gamma_{1}}$
and $\mu_{\gamma_{2}}$ such that:

$d(x_{\mu_{\gamma_{1}}},x_{\mu_{\gamma_{2}}})<\frac{\delta_{l}}{3}.$

By Remark 4.1, for $\mu_{\gamma_{1}}$-almost every $x_{1}\in
B_{\frac{\delta_{l}}{3}}(x_{\mu_{\gamma_{1}}})\cap\Lambda_{\rho}$ and
$\mu_{\gamma_{2}}$-almost every $x_{2}\in
B_{\frac{\delta_{l}}{3}}(x_{\mu_{\gamma_{2}}})\cap\Lambda_{\rho}$, we obtain
that $W^{u/s}(x_{1})\pitchfork W^{s/u}(x_{2})$ are both not empty, and
therefore, $\mu_{\gamma_{1}}\sim\mu_{\gamma_{2}}$.

Therefore, there is only a finite number of homoclinic classes in the set of
hyperbolic measures containing measures with metric entropy greater than or
equal to $a$. Finally, since every ergodic measure with metric entropy respect
$f$ greater or equal than $a$ is a hyperbolic measure, we can choose
$\mu_{1},\ldots,\mu_{K_{a}}$ as representatives of each distinct homoclinic
class, each having entropy greater than or equal to $a$. Thus, we conclude the
proof of the claim. ∎

To finish the proof, we simply choose $p_{1},\ldots,p_{K_{a}}$ with orbits
${{\mathcal{O}}}_{1},\ldots,{{\mathcal{O}}}_{K_{a}}$ such that
${{\mathcal{O}}}_{i}\sim\mu_{i}$ for every $1\leq i\leq K_{a}$ and
$HC({\mathcal{O}}_{i})=HC_{T}(\mu_{i})$. This is possible by Proposition 2.7.
∎

######  Corollary 5.1.

If $h^{u}(f)>h^{s}(f)$ then $f$ has a finite number of measures of maximal
entropy.

###### Proof.

Since $f$ is a partially hyperbolic system with a central bundle of dimension
one, it possesses measures of maximal entropy. By Lemma 3.1, every ergodic
measure of maximal entropy is hyperbolic, so according to Proposition 2.7, for
every ergodic m.m.e. $\mu$ we can select an orbit
$\mathcal{O}\in\operatorname{Per}_{h}(f)$ such that $\mu\sim{{\mathcal{O}}}$
and by Proposition 2.8 there is at most one hyperbolic measure of maximal
entropy associated with ${{\mathcal{O}}}$. Consequently, $\mu$ is the unique
measure of maximal entropy homoclinically related to ${{\mathcal{O}}}$.
Therefore every homoclinic class in ${\mathcal{M}}_{f}^{h}$ has at most one
measure of maximal entropy.

Finally, by Theorem A, the number of homoclinic classes in
${\mathcal{M}}_{f}^{h}$ containing a measure of maximal entropy is finite.
Therefore $f$ has finite ergodic measures of maximal entropy.

∎

######  Corollary 5.2.

If $h^{u}(f)>h^{s}(f)$, then there exists a $C^{1}$ neighborhood
${\mathcal{U}}$ of $f$ such that every $C^{1+}$ diffeomorphism
$g\in{\mathcal{U}}$ has a finite number of measures of maximum entropy.

###### Proof.

By Lemma 3.4 there exists a neighborhood ${\mathcal{U}}$ of $f$ such that if
$g\in{\mathcal{U}}\cap\operatorname{Diff}^{1+}(M)$ so
$h^{u}(g)>\frac{h^{u}(f)+h^{s}(f)}{2}>h^{s}(g)$. Finally, by Corollary 5.1
every $g\in{\mathcal{U}}\cap{{\rm Diff}}^{1+}(M)$ has many finite ergodic
measures of maximal entropy.

∎

###### Proof of Theorem B.

Let ${\mathcal{U}}$ be a $C^{1}$ neighborhood of $f$ as in Corollary 5.2 and
Lemma 3.4, which means that for every $g\in{\mathcal{U}}\cap{{\rm
Diff}}^{1+}(M)$, $h^{u}(g)>h^{s}(g)$, $g$ has many finite measures of maximal
entropy, and the entropy varies continuously in ${\mathcal{U}}\cap{{\rm
Diff}}^{1+}(M)$.

Let $k>0$ be the number of measures of maximal entropy of $f$ given by
Corollary 5.1. Let $\nu_{1},\ldots,\nu_{k}$ be these ergodic measures of
maximal entropy of $f$, and $p_{1}(f),\ldots,p_{k}(f)$ be periodic points of
$f$ with orbit ${{\mathcal{O}}}_{1},\ldots,{{\mathcal{O}}}_{k}$, given by
Proposition 2.7 such that ${{\mathcal{O}}}_{i}\sim\nu_{i}$,
$HC({{\mathcal{O}}}_{i})=HC_{T}(\nu_{i})$ and

$h(f_{|HC({{\mathcal{O}}}_{i})})=h(f).$

Let us consider a sequence of $C^{1+}$ diffeomorphisms, denoted as $g_{n}$,
where each $g_{n}$ belongs to $\c{U}$ and the sequence $(g_{n})_{n}$ converges
to $f$ in the $C^{1}$ topology. Let $p_{i}(g_{n})$ be the unique hyperbolic
periodic point of $g_{n}$ derived from the structural stability of $p_{i}(f)$.

######  Lemma 5.3.

If $\mu_{n}\in{\mathcal{M}}_{g_{n}}^{e}$ is a m.m.e entropy for $g_{n}$, then
there exists an index $j$ and a subsequence $(\mu_{n_{l}})_{l\geq 0}$ such
that $\mu_{n_{l}}\sim p_{j}(g_{n})$ for every $l\geq 0$.

###### Proof.

Let $\rho$ be a real number such that $h^{s}(g_{n})-h^{u}(g_{n})<\log(\rho)<0$
for every $n$ and let $\Lambda_{\rho}(g_{n})$ be as defined in 4.1. By
Corollary 4.5, there exists $\beta>0$ such that $\Lambda_{\rho}(g_{n})\geq
2\beta$ and therefore we can choose a compact subset
$\tilde{\Lambda}_{\rho}(g_{n})\subset\Lambda_{\rho}(g_{n})$ of regular points
with $\mu(\tilde{\Lambda}_{\rho}(g_{n}))\geq\beta$. Let $n_{l}$ be an
increased sequence of natural numbers such that $\mu_{n_{l}}$ converges to
$\nu$ in the weak * topology and $\Lambda_{\rho}(g_{n_{l}})$ converges to a
compact set $K$ in the Hausdorff metric. Observe that
$K\subset\Lambda_{\rho}(f)$; so, by Remark 2.16, there is a compact subset
$\tilde{\Lambda}_{\rho}(f)\subset K$ of regular points for $f$ with
$\nu(\tilde{\Lambda}_{\rho}(f))\geq\frac{\beta}{2}$.

Observe that $\nu$ is not necessarily an ergodic measure however it is a
maximal entropy measure of $f$. In effect, since
$\mu_{n}\in{\mathcal{M}}_{g_{n}}^{e}$ and $\mu_{n}$ is m.m.e. for $g_{n}$, by
Lemma 3.1, we have that (3.2) holds, that is

$h^{u}_{\mu_{n}}(g_{n})=h_{\mu_{n}}(g_{n})$

Furthermore, as shown in [33, Theorem A], we have:

$\limsup_{n}h^{u}_{\mu_{n_{l}}}(g_{n})\leq h_{\nu}^{u}(f).$

Since each $\mu_{n}$ is a measure of maximal entropy for $g_{n}$, the
topological entropy is a continuous function on ${{\mathcal{U}}}\cap{{\rm
Diff}}^{r+1}(M)$ and $h^{u}_{\mu}(f)\leq h_{\mu}(f)$, it follows that $\nu$ is
also a measure of maximal entropy for $f$, therefore, the measure $\nu$ is a
convex combination of $\nu_{1},\cdots,\nu_{k}$, that is,
$\nu=\sum_{i}a_{i}\nu_{i}$ where $0\leq a_{i}\leq 1$ so, there is $j$ such
that $\nu_{j}(\tilde{\Lambda}_{\rho}(f))\geq\frac{\beta}{2}$.

By Proposition 2.1 we can choose $x\in W^{s}(p_{j}(f))\cap W^{u}(p_{j}(f))$
such that

$\nu_{j}(B_{\frac{\delta_{1}}{4}}(x)\cap\tilde{\Lambda}_{\rho}(f))>0,$

where $0<\delta_{1}$ is small enough such that for every $z\in
B_{\frac{\delta_{1}}{2}}(x)\cap\tilde{\Lambda}_{\rho}(f)$,

$W^{s/u}_{\delta_{1}}(z)\pitchfork W^{s,u}_{\delta_{1}}(x)\neq\emptyset.$

Since the invariant manifolds of the points $z\in\Lambda_{\rho}(g_{n_{l}})$
vary continuously in $g_{n_{l}}$ and $z$, for every ${n_{l}}$ large enough,
there exists $y_{n_{l}}\in\tilde{\Lambda}_{\rho}(g_{n})$ such that
$d(y_{n_{l}},x)<\delta/4$ and such that

$\mu_{{n_{l}}}(B_{\frac{\delta_{1}}{4}}(y_{n_{l}})\cap\tilde{\Lambda}_{\rho}(g_{n_{l}}))>0,$

and for every $z\in
B_{\frac{\delta_{1}}{4}}(y_{n_{l}})\cap\tilde{\Lambda}_{\rho}(g_{n_{l}})$ we
have

$W^{s/u}_{\delta_{1}}(z)\pitchfork W^{s,u}_{\delta_{1}}(x)\neq\emptyset$

Since the invariant manifolds $W^{s/u}(p_{j}(g_{n_{l}}))$ converge to
$W^{s/u}(p_{j}(f))$ in the $C^{1}$ topology, there exists a sequence
$x_{n_{l}}\in W^{s}(p_{j}(g_{n_{l}}))\cap W^{u}(p_{j}(g_{n}))$ such that
$(x_{n_{l}})_{{\mathbb{N}}}$ converges to $x$ and
$W^{s/u}_{\delta_{1}}(x_{n_{l}})$ converges to $W^{s/u}_{\delta_{1}}(x)$ in
the $C^{1}$ topology. Therefore, for $n$ large enough,

$W^{s/u}_{\delta_{1}}(z)\pitchfork
W^{s,u}_{\delta_{1}}(x_{n_{l}})\neq\emptyset$

for every $z\in
B_{\frac{\delta_{1}}{2}}(y_{n_{l}})\cap\tilde{\Lambda}_{\rho}(g_{n_{l}})$.
Thus, we have that $p_{j}(g_{n_{l}})\sim\mu_{{n_{l}}}$, for every ${n_{l}}$
big and the proof is concluded.

∎

Now, observe that $g_{n}$ cannot have two ergodic measures of maximal entropy
that are homonically related to the same $p_{i}(g_{n})$ because they are not
related to each other. Therefore, for sufficiently large $n$, every $g_{n}$
has at most $k$ ergodic measures of maximal entropy, otherwise, according to
Lemma 6.3, there exists a subsequence $g_{n_{l}}$ such that every $g_{n_{l}}$
has at most two different ergodic measures of maximal entropy that are
homoclinically related to the same $p_{i}(g_{n})$. Thus, the proof is
concluded. ∎

##  6\. Proof of Theorems C and D

In this section, we prove theorems C and D. As before, $f:M\to M$ denotes a
$C^{1+}$ partially hyperbolic diffeomorphism of a compact manifold $M$ with
$TM=E^{u}\oplus E^{c}\oplus E^{s}$ and ${\operatorname{dim}}(E^{c})=1$ and
$\phi:M\to\mathbb{R}$ denote a potential Hölder continuous.

###### Proof of Theorem C.

Suppose that $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. Let $b>0$ be a real number
such that $P(f,\phi)\geq b>h^{s}(f)+\sup\phi$. Note that each ergodic measure
$\mu$ of $f$ with $P_{\mu}(f,\phi)\geq b$ holds that

$h_{\mu}(f)\geq b-\sup\phi>h^{s}(f).$

Consider $a=b-\sup\phi$ in Theorem A so there exist $K_{a}\in\mathbb{N}$,
hyperbolic periodic orbits ${{\mathcal{O}}}_{1}\cdots{{\mathcal{O}}}_{K_{a}}$
such that if $\mu\in{\mathcal{M}}_{f}^{e}$ with $h_{\mu}(f)\geq b-\sup\phi$,
$\mu\sim{{\mathcal{O}}}_{i}$ for some $1\leq i\leq K_{a}$.

Finally, we choose the $K_{b}$ hyperbolic periodic orbits between
{$\mathcal{O}_{1},\cdots,\mathcal{O}_{K_{b}}$} which are homoclinically
related to an ergodic measure that has metric pressure associated greater or
equal to $b$. This concludes the proof. ∎

######  Corollary 6.1.

If $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$ then $f$ has a finite number of
measures of maximal entropy.

###### Proof.

As proven in Corollary 5.1. Subsection 2.3.1, Theorem 2.8, Proposition 2.7 and
Lemma 3.3 imply that equilibrium states exist for $(f,\phi)$, each ergodic
equilibrium state $\mu$ is hyperbolic and there exists an orbit
${{\mathcal{O}}}\in{{\rm Per}}_{h}(f)$ such that $\mu$ is the unique
equilibrium state homoclinically related to it. Consequently, every homoclinic
class in ${\mathcal{M}}_{f}^{h}$ can have, at most, one measure of maximal
entropy.

By Theorem C the number of homoclinic classes in ${\mathcal{M}}_{f}^{h}$ with
equilibrium states is finite. Therefore $(f,\phi)$ has a finite number of
equilibrium states.

∎

######  Corollary 6.2.

If $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$, then there exists a $C^{1}$
neighborhood ${\mathcal{U}}$ of $f$ such that every $C^{1+}$ diffeomorphism
$g\in{\mathcal{U}}$, $(g,\phi)$ has a finite number of equilibrium states.

###### Proof.

Taking $a_{1}=\sup\phi$ and $a_{2}=\inf\phi$ in Lemma 3.4 there exists a
$C^{1}$ neighborhood ${\mathcal{U}}$ of $f$ such that for every $g\in{{\rm
Diff}}^{1+}(M)\cap{\mathcal{U}}$ holds $h^{u}(g)>\sup\phi>\inf\phi>h^{s}(g)$.
So, applying Corollary 6.1 we get that each $g\in{{\rm
Diff}}^{1+}(M)\cap{\mathcal{U}}$ is such that $(g,\phi)$ has a finite number
of ergodic equilibrium states. ∎

###### Proof of Theorem D.

Suppose that $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$. Consider $a_{1}=\sup\phi$
and $a_{2}=\inf\phi$, by Lemma 3.4 and Corollary 6.2 there exists a $C^{1}$
neighborhood ${\mathcal{U}}$ of $f$ such that each $g\in{{\rm
Diff}}^{1+}(M)\cap{\mathcal{U}}$ holds $h^{u}(f)>\sup\phi>\inf\phi>h^{s}(f)$
and $(g,\phi)$ has many finite ergodic equilibrium states.

By the Corollary 6.1 the number of equilibrium estates for $(f,\phi)$ is
$k>0$. Let $\nu_{1},\cdots\nu_{k}$ be these ergodic equilibrium states and
$p_{1}(f)\cdots p_{k}(f)$ the periodic hyperbolic points with orbits
${{\mathcal{O}}}_{1}\cdots{\O}_{k}$ respectively of $f$ such that
${{\mathcal{O}}}_{i}\sim\mu_{i}$ and

$P(f_{|HC(\mathcal{O}_{i})},\phi_{|HC(\mathcal{O}_{i})})=P(f,\phi)$

Let $(g_{n})_{n\in\mathbb{N}}$ be a sequence of $C^{1+}$ diffeomorphisms in
${\mathcal{U}}$ converging to $f$ in the $C^{1}$ topology. For every $g_{n}$
and $p_{i}(f)$ consider $p_{i}(g_{n})$ the hyperbolic periodic point given by
the structural stability of $p_{i}(f)$.

######  Lemma 6.3.

If $\mu_{n}\in{\mathcal{M}}_{g_{n}}^{e}$ is an equilibrium state for $g_{n}$,
then there exists an index $j$ and a subsequence $(\mu_{n_{l}})_{l\geq 0}$
such that $\mu_{n_{l}}\sim p_{j}(g_{n})$ for every $l\geq 0$.

###### Proof.

We will proceed as in Lemma 6.3. Let $\rho$ be a real number as in Corollary
4.7 and for every $n$ consider $\Lambda_{\rho}(g_{n})$ as in 4.1. By Corollary
4.7, there exists $\beta>0$ such that $\Lambda_{\rho}(g_{n})\geq 2\beta$ and
therefore we can choose a compact subset
$\tilde{\Lambda}_{\rho}(g_{n})\subset\Lambda_{\rho}(g_{n})$ of regular points
with $\mu(\tilde{\Lambda}_{\rho}(g_{n}))\geq\beta$. Let $n_{l}$ be an
increased sequence of natural numbers such that $\mu_{n_{l}}$ converges to
$\nu$ in the weak * topology and $\Lambda_{\rho}(g_{n_{l}})$ converges to a
compact set $K$ in the Hausdorff metric. Observe that
$K\subset\Lambda_{\rho}(f)$; so, by Remark 2.16, there is a subset
$\tilde{\Lambda}_{\rho}(f)\subset K$ as a compact set of regular points for
$f$ with $\nu(\tilde{\Lambda}_{\rho}(f))\geq\frac{\beta}{2}$.

The measure $\nu$ is an equilibrium state of $(f,\phi)$. In effect, since
$\mu_{n}\in{\mathcal{M}}_{g_{n}}^{e}$ and $\mu_{n}$ is an equilibrium state of
$g_{n}$, by Lemma 3.3, we have that

$h^{u}_{\mu_{n}}(g_{n})=h_{\mu_{n}}(g_{n})$

Furthermore, as shown in [33, Theorem A], we have:

$\limsup_{n_{l}}h^{u}_{\mu_{n_{l}}}(g_{n})\leq h_{\nu}^{u}(f)$

and as $\mu_{n_{l}}\to\nu$ when $n\to\infty$,

$\lim_{n\to\infty}\int\phi d\mu_{n_{l}}=\int\phi d\nu.$

Therefore, since each $\mu_{n}$ is an equilibrium state of $g_{n}$, the
topological pressure is a continuous function on ${{\mathcal{U}}}\cap{{\rm
Diff}}^{r+1}(M)$ and $h^{u}_{\mu}(f)\leq h_{\mu}(f)$, we conclude that $\nu$
is also an equilibrium state of $(f,\phi)$.

Thus, the measure $\nu$ is a convex combination of $\nu_{1},\cdots,\nu_{k}$,
and there is $j$ such that
$\nu_{j}(\tilde{\Lambda}_{\rho}(f))\geq\frac{\beta}{2}$. Consider $p_{j}(f)$
such that $p_{j}\sim\mu_{j}$ and repeating the argument in Lemma 6.3, by the
continuity of the invariant manifolds, we conclude that for every $l$ large
enough $p_{i}(g_{n_{l}})\sim\mu_{n_{l}}$. Thus, the proof is concluded. ∎

Finally, with the same argument of the proof of Theorem B, we conclude $g_{n}$
has at most $k$ ergodic equilibrium states when $n$ is big enough.

∎

Acknowledgements. We are thankful to Jiagang Yang for helpful conversations on
this work.

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