# Finiteness of physical measures for diffeomorphisms with multi 1-D centers

Yongluo Cao  Department of Mathematics, Soochow University, Suzhou 215006,
Jiangsu, China.  Center for Dynamical Systems and Differential Equation,
Soochow University, Suzhou 215006, Jiangsu, China<EMAIL_ADDRESS>and  Zeya
Mi  School of Mathematics and Statistics, Nanjing University of Information
Science and Technology, Nanjing 210044, Jiangsu, China<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be a $C^{2}$ diffeomorphism on compact Riemannian manifold $M$ with
partially hyperbolic splitting

$TM=E^{u}\oplus E_{1}^{c}\oplus\cdots\oplus E_{k}^{c}\oplus E^{s},$

where $E^{u}$ is uniformly expanding, $E^{s}$ is uniformly contracting, and
${\rm dim}E_{i}^{c}=1,~{}1\leq i\leq k,~{}k\geq 1$. We prove the finiteness of
ergodic physical(SRB) measures of $f$ under the hyperbolicity of Gibbs
$u$-states.

###### Key words and phrases:

Physical measures, SRB measures, Partially hyperbolic, Lyapunov exponent

###### 2010 Mathematics Subject Classification:

37C40, 37D25, 37D30

Y. Cao was supported by National Key R&D Program of China(2022YFA1005802) and
NSFC 11790274.

Z. Mi was supported by National Key R&D Program of China(2022YFA1007800) and
NSFC 12271260.

##  1\. Introduction

Let $f$ be a $C^{2}$ diffeomorphism on a compact Riemannian manifold $M$. A
_physical measure_ of $f$ is an invariant Borel probability $\mu$ on $M$ for
which there exists a positive Lebesgue measure set of points $x\in M$ such
that for any continuous $\varphi:M\to{\mathbb{R}}$ one has

$\lim_{n\to+\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))=\int\varphi
d\mu.$

The set of points ${\mathcal{B}}(\mu,f)$ for which the above convergence holds
is called the basin of $\mu$ w.r.t. $f$. Physical measures were firstly
discovered by Sinai, Ruelle and Bowen in uniformly hyperbolic systems [18, 17,
4, 5]. In smooth ergodic theory, the physical measures are usually produced
from _Sinai-Ruelle-Bowen (SRB)_ measures. One says that an invariant measure
is an SRB measure if it admits positive Lyapunov exponents almost everywhere,
and its conditional measures along Pesin unstable manifolds are absolutely
continuous w.r.t Lebesgue measures on these manifolds. It is well known that
every ergodic SRB measure with non-zero Lyapunov exponents is a physical
measure. It remains to be a changeable problem in finding physical measures
for systems beyond hyperbolicity ( see [2, Chapter 11] and [15, 20, 22] for
related conjectures ).

In this paper, we are interested in studying physical (SRB) measures for
partially hyperbolic systems (see $\S$ 2.1 for more details) under the
hyperbolicity of _Gibbs $u$-states_. By Gibbs $u$-state, we mean an invariant
measure whose disintegration along strong unstable manifolds are absolutely
continuous w.r.t. Lebesgue measures on these manifolds. The main result of
this paper is the following:

######  Theorem A.

Let $f$ be a $C^{2}$ diffeomorphism with a partially hyperbolic splitting
$TM=E^{u}\oplus_{\succ}E_{1}^{c}\oplus_{\succ}\cdots\oplus_{\succ}E_{k}^{c}\oplus_{\succ}E^{s},$
where ${\rm dim}E_{i}^{c}=1,~{}1\leq i\leq k,~{}k\geq 1$. If every Gibbs
$u$-state of $f$ is hyperbolic111An invariant measure is hyperbolic if all its
Lyapunov exponents are non-zero., then there are finitely many ergodic
physical(SRB) measures of $f$.

According to [10], partially hyperbolic splitting as in Theorem A is abundant
among diffeomorphisms away from homoclinic tangencies. Partially hyperbolic
diffeomorphisms in Theorem A sometimes inherit some properties in uniform
hyperbolicity (see e.g. [11, 21]).

We would like to list some related results below:

  * •

In [7, Theorem C], the authors prove the existence of (ergodic) SRB measures
for partially hyperbolic diffeomorphisms with multi 1-D centers.

  * •

Under the hyperbolicity of Gibbs $u$-state, for case $k=1$, it is shown in
[12] that there exist finitely many physical measures whose basins cover a
full Lebesgue measure subset.

  * •

Without the restriction on dimension of center direction, the existence and
finiteness of physical/SRB measures have been achieved for partially
hyperbolic diffeomorphisms for which the central Lyapunov exponents of Gibbs
$u$-states are all negative, or all positive; including diffeomorphisms with
mostly contracting center [6]; with mostly expanding center [1]; with mostly
expanding center and mostly expanding center simultaneously [14].

Observe that every SRB measure must be a Gibbs $u$-state, using the result of
[7], the hyperbolicity on Gibbs $u$-states guarantees the existence of ergodic
physical measures for diffeomorphisms in Theorem A. Hence, the main effort on
Theorem A is to show the finiteness of ergodic physical measures. Let us
remark that in the previous work [6, 1, 14], the signs of central Lyapunov
exponents w.r.t. Gibbs $u$-states are definite. Our assumption on Gibbs
$u$-states is more flexible: different Gibbs $u$-states may admit different
signs of Lyapunov exponents on the same central sub-bundle.

In [12], besides the finiteness of ergodic physical measures, the authors also
show the basin covering property of these physical measures, i.e., Lebesgue
almost every point belongs to the basin of some ergodic physical measure. It
is natural to expect that the basin covering property can hold also for the
general case $k>1$. The main difference between the special case $k=1$ and the
general case $k>1$ is that when $k=1$, the non-uniform behavior occurs only in
one fixed center.

Our strategy of the proof of Theorem A is to decompose the ergodic
physical/SRB measures into different levels by their signs of Lyapunov
exponents on centers (see Theorem 4.1), and then show the finiteness of these
measures on each level. To achieve this, we will make an effort on getting the
closed property of Gibbs $cu$-states, which is done by showing the uniformity
on central Lyapunov exponents for Gibbs $cu$-states on the same level(see
Theorem 3.4).

##  2\. Preliminary

###  2.1. Dominated splitting and partially hyperbolicity

Let $f$ be a $C^{1}$ diffeomorphism on compact Riemannian manifold $M$. Given
two $Df$-invariant subbundles $E$ and $F$, we say $E$ _dominates_ $F$ if for
every $x\in M$, $u\in E(x)$ and $v\in F(x)$ with $\|u\|=\|v\|=1$, one has that

$\|Df(x)v\|<\frac{1}{2}\|Df(x)u\|.$

We write $E\oplus_{\succ}F$ to denote this domination. A $Df$-invariant
splitting $TM=E\oplus_{\succ}F$ of the tangent bundle is called a _dominated
splitting_ if $E$ dominates $F$.

One says that $f$ is a _partially hyperbolic_ diffeomorphism if there exists a
dominated splitting
$TM=E^{u}\oplus_{\succ}E_{1}\oplus_{\succ}\cdots\oplus_{\succ}E_{k}\oplus_{\succ}E^{s}$
on the tangent bundle such that

  * •

$E^{u}$ is uniformly expanding and $E^{s}$ is uniformly contracting.

  * •

at least one of $E^{u}$ and $E^{s}$ is nontrivial.

###  2.2. Gibbs $u$-states and Gibbs $E$-states

Let $f$ be a $C^{2}$ diffeomorphism with partially hyperbolic splitting
$TM=E^{u}\oplus_{\succ}E^{cs}$. Recall that an $f$-invariant measure is a
Gibbs $u$-state if its disintegration along strong unstable manifolds is
absolutely continuous w.r.t. Lebesgue measures on these manifolds. Gibbs
$u$-states were introduced and established by Sinai-Pesin [16] for partially
hyperbolic systems.

We list following properties of Gibbs $u$-states, see [2, Subsection 11.2] for
details.

######  Proposition 2.1.

Let $f$ be a diffeomorphism with partially hyperbolic splitting
$TM=E^{u}\oplus_{\succ}F$, one has following properties:

  1. (1)

The ergodic components of any Gibbs $u$-state are Gibbs $u$-states.

  2. (2)

The set of Gibbs $u$-states is compact in weak∗-topology.

  3. (3)

For Lebesgue almost every $x\in M$, any limit measure of

$\big{\\{}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\big{\\}}_{n\in{\mathbb{N}}}$

is a Gibbs $u$-state.

Recall the following entropy formula for diffeomorphisms with dominated
splittings, see [9, Theorem F], [8, Theorem A] for the proof.

######  Proposition 2.2.

Let $f$ be a diffeomorphism with dominated splitting $TM=E\oplus_{\succ}F$.
Then for Lebesgue almost every $x\in M$, any limit measure $\mu$ of
$\\{\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\\}_{n\in{\mathbb{N}}}$
satisfies

$h_{\mu}(f)\geq\int\log|{\rm det}Df|_{E}|{\rm d}\mu.$

Recall another notion called _Gibbs $cu$-states_. Let $f$ be a diffeomorphism
with dominated splitting $TM=E\oplus_{\succ}F$. An invariant measure $\mu$
called a Gibbs $cu$-state associated to $E$ if

  * •

the Lyapunov exponents of $\mu$ along $E$ are positive,

  * •

the conditional measure of $\mu$ along Pesin unstable manifolds tangent to $E$
are absolutely continuous w.r.t. Lebesgue measure on these manifolds.

Since we need to deal with various Gibbs $cu$-states for partially hyperbolic
diffeomorphisms with several sub-bundles, for simplicity we will just call
$\mu$ a Gibbs $E$-state if it is a Gibbs $cu$-state associated to $E$. Note
that when $E$ is uniformly expanding, then Gibbs $E$-state is indeed the Gibbs
$u$-state.

Similar to property (1) of Proposition 2.1, we have the following result. See
a proof in [19, Lemma 2.4] or [7, Proposition 4.7].

######  Proposition 2.3.

Assume that $f$ is a $C^{2}$ diffeomorphism with a dominated splitting
$TM=E\oplus_{\succ}F$. If $\mu$ is a Gibbs $E$-state, then any ergodic
component of $\mu$ is also a Gibbs $E$-state.

We will use the following fact several times, which can be deduced by the
absolute continuity of Pesin stable lamination, see [22] for a proof.

######  Proposition 2.4.

Assume that $f$ is a $C^{2}$ diffeomorphism with a dominated splitting
$TM=E\oplus_{\succ}F$. If $\mu$ is an ergodic Gibbs $E$-state whose Lyapunov
exponents along $F$ are all negative, then $\mu$ is a physical measure.

The result below is abstract from [7, Theorem 4.10].

######  Proposition 2.5.

Let $f$ be a diffeomorphism with dominated splitting $TM=E\oplus_{\succ}F$. If
$\mu$ is a limit measure of the set of ergodic Gibbs $E$-states, whose
Lyapunov exponents along $E$ are uniformly positive, then $\mu$ is a Gibbs
$E$-state.

##  3\. Closed property on Gibbs $cu$-states

For the rest, we will concentrate on partially hyperbolic diffeomorphisms as
in Theorem A. For convenience, given $k\geq 1$, denote by $\mathcal{P}_{k}(M)$
the collection of $C^{2}$ diffeomorphisms in $M$ with partially hyperbolic
splitting
$TM=E^{u}\oplus_{\succ}E_{1}^{c}\oplus_{\succ}\cdots\oplus_{\succ}E_{k}^{c}\oplus_{\succ}E^{s}$
such that

  * •

${\rm dim}E_{i}=1,1\leq i\leq k$;

  * •

every Gibbs $u$-state of $f$ is hyperbolic.

We write

$F_{i}=E^{u}\oplus_{\succ}\cdots\oplus_{\succ}E_{i}^{c},\quad 0\leq i\leq k.$

For every $0\leq i\leq k$, let $G_{i}$ be the set of Gibbs $F_{i}$-states of
$f$; let $G_{i}^{erg}$ be the set of ergodic Gibbs $F_{i}$-states of $f$. For
any $f$-invariant measure $\mu$, by Oseledec’s theorem [3], for every $1\leq
i\leq k$, one can define

$\lambda^{c}_{i}(x)=\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}|_{E^{c}_{i}(x)}\|,\quad\mu\textrm{-a.e.}~{}x.$

Moreover, define

$\lambda^{c}_{i}(\mu)=\int\log\|Df|_{E^{c}_{i}}\|{\rm d}\mu.$

The main result of this section is as follows:

######  Theorem 3.1.

If $f\in\mathcal{P}_{k}(M)$, then for every $0\leq i\leq k-1$, if $\mu$ is a
limit measure of $G_{i}^{erg}$, then

  * •

$\mu$ is a Gibbs $F_{i}$-state,

  * •

and one has that

$\lambda^{c}_{i+1}(x)>0,\quad\mu\textrm{-a.e.}~{}x.$

Moreover, there are at most finitely many ergodic Gibbs $F_{k}$-states.

We start with the following observation on Gibbs $E$-states for general
diffeomorphisms with dominated splitting $TM=E\oplus_{\succ}F$.

######  Proposition 3.2.

Let $f$ be a diffeomorphism with dominated splitting $TM=E\oplus_{\succ}F$.
Then, we have

  * •

either there exist at most finitely many ergodic Gibbs $E$-states,

  * •

or, if $\mu$ is a limit measure of the set of ergodic Gibbs $E$-states, which
is a Gibbs $E$-state, then for $\mu$-a.e. $x\in M$, there exist non-negative
Lyapunov exponents along $F$.

###### Proof.

Suppose that there are infinitely many ergodic Gibbs $E$-states. Let
$\\{\mu_{n}\\}_{n\in{\mathbb{N}}}$ be a sequence of different ergodic Gibbs
$E$-states, which converges to $\mu$ as $n\to+\infty$, and $\mu$ is a Gibbs
$E$-state.

Arguing by contradiction, assume $A\subset M$ satisfying $\mu(A)>0$ and

$\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}|_{F(x)}\|<0,\quad\forall~{}x\in A.$

This implies that there exist ergodic components of $\mu$ whose Lyapunov
exponents along $F$ are all negative. Note that these ergodic measures are
Gibbs $E$-states by applying Proposition 2.3. By Proposition 2.4, we know that
they are also physical measures, so there are at most countably many such
measures. Consequently, one can find an ergodic component $\nu$ of $\mu$ such
that all its Lyapunov exponents along $F$ are negative, and there exists
$b\in(0,1]$ such that $\mu$ can be rewritten as

$\mu=b\nu+(1-b)\eta$

for some Gibbs $E$-state $\eta$.

By Pesin theory [3, $\S 2$ & $\S 4$ ], consider $\\{X_{k}\\}_{k\geq 1}$ as the
Pesin blocks related to $\nu$ with following properties:

  * •

$X_{k}$ is compact for every $k\geq 1$, and $\nu(X_{k})\to 1$ as
$k\to+\infty$;

  * •

there is $\delta_{k}>0$ such that every $x\in X_{k}$ admits a local Pesin
stable manifold ${\mathcal{F}}^{s}_{\delta_{k}}(x)$ and a local Pesin unstable
manifold ${\mathcal{F}}^{u}_{\delta_{k}}(x)$ with size $\delta_{k}$;

  * •

${\mathcal{F}}^{u}_{\delta_{k}}(x)$ is tangent222This means that
$T_{y}{\mathcal{F}}^{u}_{\delta_{k}}(x)=E(y)$ for every
$y\in{\mathcal{F}}^{u}_{\delta_{k}}(x)$. to $E$ and
${\mathcal{F}}^{s}_{\delta_{k}}(x)$ is tangent to $F$;

  * •

both ${\mathcal{F}}^{u}_{\delta_{k}}(x)$ and
${\mathcal{F}}^{s}_{\delta_{k}}(x)$ depend continuously on $x\in X_{k}$.

Choosing $k\in{\mathbb{N}}$ such that $\nu(X_{k})>0.$ Fixing $r\ll\delta_{k}$.
By compactness of $X_{k}$, one can take $p\in X_{k}$ such that

(1) $\nu(X_{k}\cap B(p,r))>0.$

Let

$S^{u}=\bigcup_{x\in X_{k}\cap B(p,r)}{\mathcal{F}}_{\delta_{k}}^{u}(x).$

We have $\nu(S^{u})>0$ by (1). According to the properties of
$\\{{\mathcal{F}}_{\delta_{k}}^{u}(x)\\}_{x\in X_{k}}$, one knows that

$\mathcal{S}^{u}=\\{{\mathcal{F}}_{\delta_{k}}^{u}(x):x\in X_{k}\cap
B(p,r)\\}$

is a measurable partition of the set $S^{u}$ w.r.t. $\nu$.

######  Claim.

${\rm Leb}_{\gamma}(B(\nu,f))>0$ for every $\gamma\in\mathcal{S}^{u}.$

###### Proof of the Claim.

Note that the basin ${\mathcal{B}}(\nu,f)$ has full $\nu$-measure as $\nu$ is
ergodic, and $X_{k}\cap B(p,r)\subset S^{u}$ by construction, we get

$\nu(B(p,r)\cap X_{k}\cap{\mathcal{B}}(\nu,f)\cap S^{u})>0$

Since $\nu$ is an ergodic Gibbs $E$-state, the conditional measures of
$\nu|_{S^{u}}$ along unstable disks of $\mathcal{S}^{u}$ are absolutely
continuous w.r.t. Lebesgue measures on these disks, which ensures that one can
take $\gamma_{\nu}\in\mathcal{S}^{u}$ for which there exists a subset $B$ of
$B(p,r)\cap\gamma_{\nu}\cap X_{k}\cap{\mathcal{B}}(\nu,f)$ such that

(2) ${\rm Leb}_{\gamma_{\nu}}(B)>0.$

Since $r$ is sufficiently small, we conclude that for every $x\in X_{k}\cap
B(p,r)$, ${\mathcal{F}}^{s}_{\delta_{k}}(x)$ intersects every local Pesin
unstable manifold from $\mathcal{S}^{u}$ transversely. By applying the
absolute continuity of Pesin stable lamination, (2) ensures

${\rm Leb}_{\gamma}\left(\bigcup_{z\in
B}{\mathcal{F}}_{\delta_{k}}^{s}(z)\cap\gamma\right)>0,\quad\textrm{for
every}~{}\gamma\in\mathcal{S}^{u}.$

Together with the observation that
${\mathcal{F}}_{\delta_{k}}^{s}(z)\subset{\mathcal{B}}(\nu,f)$ for every $z\in
B$, this yields the desired result. ∎

Recall that we have $\nu(S^{u})>0$, so $\mu(S^{u})\geq b\nu(S^{u})>0$. Up to
reducing the sizes of unstable disks of $\mathcal{S}^{u}$ slightly, we may
assume $\mu(\partial(S^{u}))=0,$ which implies that

$\lim_{n\to+\infty}\mu_{n}(S^{u})=\mu(S^{u}).$

In particular, there exists $n_{0}\in{\mathbb{N}}$ such that
$\mu_{n}(S^{u})>0$ for every $n\geq n_{0}$. For every $n\geq n_{0}$, since
$\mu_{n}$ is an ergodic Gibbs $E$-state, one can find
$\gamma_{n}\in\mathcal{S}^{u}$ such that ${\rm Leb}_{\gamma_{n}}$-almost every
point of $\gamma_{n}$ belongs to ${\mathcal{B}}(\mu_{n},f)$. This together
with the Claim above yields that $\mu_{n}=\nu$ for every $n\geq n_{0}$, which
is contrary to our assumption.

∎

We recall the following folklore result [7, Proposition 4.9].

######  Lemma 3.3.

Under the setting as above, we have $G_{k}\subset G_{k-1}\subset\cdots\subset
G_{0}$.

Given an invariant measure $\mu$, denote by $\mathcal{C}(\mu)$ the family of
ergodic components of $\mu$. With above preparations, we can give the next
result, which asserts a closed property on Gibbs $F_{i}$-states.

######  Theorem 3.4.

If $f\in\mathcal{P}_{k}(M)$, then for every $1\leq i\leq k$, either
$\\#G_{i}^{erg}<+\infty$ or any limit measure of $G_{i}^{erg}$ is contained in
$G_{i}$, whose Lyapunov exponents along $E_{i}^{c}$ are uniformly positive.

###### Proof.

We prove the result by induction on $1\leq i\leq k$. We start by proving the
case $i=1$. Assume that $\\#G_{1}^{erg}=+\infty$ and let $\mu$ be a limit
measure of $G_{1}^{erg}$. That is, there is a sequence of ergodic Gibbs
$F_{1}$-states $\\{\mu_{n}\\}_{n\in{\mathbb{N}}}$ such that

$\mu_{n}\xrightarrow{weak^{*}}\mu,\quad\textrm{as}~{}n\to+\infty.$

We have $\\{\mu_{n}\\}_{n\in{\mathbb{N}}}\subset G_{0}$ as $G_{1}\subset
G_{0}$ by Lemma 3.3. From item (2) of Proposition 2.1, we know $\mu\in G_{0}$.
By Proposition 3.2 and the hyperbolicity assumption on Gibbs $u$-states, we
get

$\lambda^{c}_{1}(x)>0,\quad\mu\textrm{-a.e.}~{}x.$

To complete the proof of the case $i=1$, by Proposition 2.5 it suffices to
verify that the Lyapunov exponents of $\mu$ are uniformly positive along
$F_{i}$. By contradiction, there exists a sequence of ergodic measures
$\\{\nu_{n}\\}_{n\in{\mathbb{N}}}$ in $\mathcal{C}(\mu)$ such that

$\lim_{n\to+\infty}\lambda^{c}_{1}(\nu_{n})=0.$

Up to considering subsequences, we assume that $\nu_{n}$ converges to an
$f$-invariant measure $\nu$ as $n\to+\infty$. Consequently, we have

(3) $\lambda^{c}_{1}(\nu)=\lim_{n\to+\infty}\lambda^{c}_{1}(\nu_{n})=0.$

Since we have shown $\mu\in G_{0}$, and $\\{\nu_{n}\\}_{n\in{\mathbb{N}}}$ are
ergodic components of $\mu$, the first item (1) of Proposition 2.1 gives that
$\nu_{n}\in G_{0}$ for every $n\in{\mathbb{N}}$. So, $\nu$ is a limit measure
of the set of ergodic Gibbs $u$-states. Using Proposition 3.2 again and the
hyperbolicity of Gibbs $u$-states, we deduce that $\lambda^{c}_{1}(x)>0$ for
$\nu\textrm{-a.e.}~{}x$. It follows from Birkhoff’s ergodic theorem that

$\lambda^{c}_{1}(\nu)=\int\lambda^{c}_{1}(x){\rm d}\nu>0,$

where we use the fact ${\rm dim}E_{1}^{c}=1$. This gives the contradiction to
(3). Thus, the case $i=1$ is verified.

Inductively, we assume that the theorem holds for $i=\ell$($\ell<k$), and we
then show that it is true for $i=\ell+1$. Assume that
$\\#G_{\ell+1}^{erg}=+\infty$. Let $\\{\mu_{n}\\}_{n\in{\mathbb{N}}}$ be a
sequence of ergodic measures in $G_{\ell+1}$, which converges to $\mu$ as
$n\to+\infty$. In view of Proposition 2.5, we need only to show

(4)
$\inf_{\nu\in\mathcal{C}(\mu)}\left\\{\lambda^{c}_{\ell+1}(\nu)\right\\}>0.$

As Lemma 3.3 tells us that $G_{\ell+1}\subset G_{\ell}\subset G_{0}$ and we
assume that the theorem is true for $i=\ell$, we get $\mu\in G_{\ell}$. By
Theorem 3.2, together with the assumption of Gibbs $u$-states we know that
$\mu$ admits only positive Lyapunov exponents along $E_{\ell+1}^{c}$.

Now we show (4) is true. By contradiction we assume that there exists a
sequence of ergodic measures
$\\{\nu_{n}\\}_{n\in{\mathbb{N}}}\subset\mathcal{C}(\mu)$, which converges to
an $f$-invariant measure $\nu$ and satisfies

(5)
$\lim_{n\to+\infty}\lambda^{c}_{\ell+1}(\nu_{n})=\lambda^{c}_{\ell+1}(\nu)=0.$

Since $\mu\in G_{\ell}$, by Proposition 2.3 we know that
$\\{\nu_{n}\\}_{n\in{\mathbb{N}}}$ is a sequence of ergodic measures in
$G_{\ell}\subset G_{0}$.

As we have assumed that this theorem is true for $i=\ell$, the limit measure
$\nu$ of $\\{\nu_{n}\\}_{n\in{\mathbb{N}}}$ is also a Gibbs $F_{\ell}$-state.
By Proposition 3.2, we have

$\lambda^{c}_{\ell+1}(x)\geq 0,\quad\nu\textrm{-a.e.}~{}x.$

Noting that $\nu\in G_{0}$ guaranteed by item (2) of Proposition 2.1, the
assumption on Gibbs $u$-states then gives

$\lambda^{c}_{\ell+1}(x)>0,\quad\nu\textrm{-a.e.}~{}x.$

Therefore, using the Birkhoff’s ergodic theorem again, we get

$\lambda^{c}_{\ell+1}(\nu)=\int\lambda^{c}_{\ell+1}(x){\rm d}\nu>0,$

contradicting to the convergence (5). Therefore, (4) is true and we complete
the proof of Theorem 3.4.

∎

We now ready to prove Theorem 3.1.

###### Proof of Theorem 3.1.

For $0\leq i\leq k-1$, let $\mu$ be a limit measure of $G_{i}^{erg}$. We know
from Proposition 2.1 and Theorem 3.4 that $\mu$ is a Gibbs $F_{i}$-state,
which is also a Gibbs $u$-state by Lemma 3.3. So, $\mu$ is hyperbolic by
definition of $\mathcal{P}_{k}(M)$. This together with Proposition 3.2 ensures
that for $\mu$-a.e. $x\in M$, there exist positive Lyapunov exponents along
$F:=E_{i+1}^{c}\oplus_{\succ}\cdots\oplus_{\succ}E^{s}$. As $E_{i+1}^{c}$ has
dimension one, $\lambda^{c}_{i+1}(x)$ is just the largest Lyapunov exponent
along $F$. Consequently, we obtain

$\lambda^{c}_{i+1}(x)>0,\quad\mu\textrm{-a.e.}~{}x.$

To show the result on Gibbs $F_{k}$-states, assume by contradiction that there
are infinitely many ergodic Gibbs $F_{k}$-states. This implies that there
exists a sequence of different ergodic Gibbs $F_{k}$-states, which converges
to an invariant measure $\mu$. By Theorem 3.4, $\mu$ must be a Gibbs
$F_{k}$-state as well. However, from Proposition 3.2 we get that $\mu$ have
positive Lyapunov exponents along $E^{s}$, which is a contradiction. Thus, the
proof is complete. ∎

As a byproduct of Theorem 3.4, we also have the following interesting result,
which asserts the uniformity on Lyapunov exponents of Gibbs $F_{i}$-states
along $F_{i}$, $1\leq i\leq k$.

######  Corollary 3.5.

Let $f$ be a $C^{2}$ diffeomorphism as in Theorem 3.4, then there exists
$\alpha>0$ such that for every Gibbs $F_{i}$-state $\mu$, one has that

$\lambda^{c}_{i}(x)>\alpha,\quad\mu\textrm{-a.e.}~{}x.$

###### Proof.

Assuming contrary, there exists $1\leq i\leq k$ and a sequence of Gibbs
$F_{i}$-states $\\{\mu_{n}\\}_{n\in{\mathbb{N}}}$ for which there exists an
ergodic component $\nu_{n}$ of $\mu_{n}$ for every $n\in{\mathbb{N}}$ so that

$\lim_{n\to+\infty}\lambda^{c}_{i}(\nu_{n})=0.$

Without loss of generality, we suppose that $\\{\nu_{n}\\}_{n\in{\mathbb{N}}}$
converges to some $f$-invariant measure $\nu$ when $n\to+\infty$. Note that
$x\mapsto\log\|Df|_{E_{i}^{c}(x)}\|$ is continuous. By the weak∗ convergence
we have

(6) $\lambda^{c}_{i}(\nu)=\lim_{n\to+\infty}\lambda^{c}_{i}(\nu_{n})=0.$

By Proposition 2.3, we know each $\nu_{n}$ is an ergodic Gibbs $F_{i}$-state
for every $n\in{\mathbb{N}}$. Hence, $\nu$ is a limit measure of the set of
ergodic Gibbs $F_{i}$-states. By Theorem 3.4, there exists $a>0$ such that

$\lambda^{c}_{i}(x)>a,\quad\nu\textrm{-a.e.}~{}x.$

Noting ${\rm dim}E_{i}^{c}=1$, by applying Birkhoff’s ergodic theorem (see
e.g. [23, Theorem 1.14]) one obtains

$\lambda_{i}^{c}(\nu)=\int\lambda_{i}^{c}(x){\rm d}\nu>a,$

which is in contradiction with convergence (6). ∎

##  4\. Proof of Theorem A

Now we begin to prove Theorem A. Assume that $f\in\mathcal{P}_{k}(M)$. For
every $0\leq i\leq k$, put

${\mathcal{G}}_{i}=\left\\{\mu:\mu\in G_{i}~{}\textrm{is ergodic
and}~{}\lambda^{c}_{i+1}(\mu)<0\right\\}.$

Recall that $G_{i}$ is the space of Gibbs $F_{i}$-states for every $0\leq
i\leq k$, respectively; and

$\lambda^{c}_{i+1}(\mu)=\int\log\|Df|_{E_{i+1}^{c}}\|{\rm d}\mu.$

Note that here we set $E_{k+1}^{c}=E^{s}$, and $\lambda^{c}_{k+1}(\mu)$
denotes the maximal Lyapunov exponent of $\mu$ along $E^{s}$, which is
negative automatically as the uniform contraction on $E^{s}$.

To give the proof of Theorem A, we need the following result.

######  Theorem 4.1.

Under the setting of Theorem A, the set of ergodic physical(SRB) measures
coincides with $\bigcup_{0\leq i\leq k}{\mathcal{G}}_{i}$.

###### Proof.

To begin with, we show the existence of ergodic physical measures. From the
result [7, Theorem C], there exists an ergodic SRB measure $\mu$ for $f$,
which is a Gibbs $u$-state by definition. We know also that $\mu$ is
hyperbolic by assumption. Consequently, $\mu$ is an ergodic physical measure
by using the absolute continuity of Pesin stable lamination.

By definition, for any $0\leq i\leq k$, any invariant measure in
${\mathcal{G}}_{i}$ is a hyperbolic ergodic SRB measure, which is also a
physical measure by Proposition 2.4. Thus, it suffices to show that every
ergodic physical measure is contained in ${\mathcal{G}}_{i}$ for some $0\leq
i\leq k$.

By item (3) of Proposition 2.1 and Proposition 2.2, there is a subset $\Delta$
of $M$ with full Lebesgue measure such that for any $x\in\Delta$, any limit
measure $\eta$ of
$\\{\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\\}_{n\in{\mathbb{N}}}$ is a
Gibbs $u$-state and satisfies

(7) $h_{\eta}(f)\geq\int\log|{\rm det}Df|_{F_{i}}|{\rm
d}\eta,\quad\forall~{}1\leq i\leq k.$

Assume that $\mu$ is an ergodic physical measure, thus ${\mathcal{B}}(\mu,f)$
admits positive Lebesgue measure. Thus ${\mathcal{B}}(\mu,f)\cap\Delta$
exhibits positive Lebesgue measure. By taking
$x\in{\mathcal{B}}(\mu,f)\cap\Delta$, one gets form definition of
${\mathcal{B}}(\mu,f)$ that

$\lim_{n\to+\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}(x)}=\mu.$

So, $\mu$ is a Gibbs $u$-state and satisfies (7) form the definition of
$\Delta$.

Since any Gibbs $u$-state is hyperbolic by assumption, $\mu$ is an ergodic
hyperbolic Gibbs $u$-state. Consequently, one can take $i\in\\{0,\cdots,k\\}$
such that all the Lyapunov exponents of $\mu$ along $F_{i}$ are positive and
$\lambda^{c}_{i+1}(\mu)<0$. By Ruelle’s inequality, one has

$h_{\mu}(f)\leq\int\log|{\rm det}Df|_{F_{i}}|{\rm d}\mu.$

On the other hand, inequality (7) says that

$h_{\mu}(f)\geq\int\log|{\rm det}Df|_{F_{i}}|{\rm d}\mu.$

Therefore,

$h_{\mu}(f)=\int\log|{\rm det}Df|_{F_{i}}|{\rm d}\mu.$

By the result of Ledrappier-Young [13, Theorem A], $\mu$ is an SRB measure,
which is contained in ${\mathcal{G}}_{i}$ by its sign of Lyapunov exponents. ∎

Now, we can give the proof of Theorem A.

###### Proof of Theorem A.

In view of Theorem 4.1, it suffices to show that for each $0\leq i\leq k$,
${\mathcal{G}}_{i}$ is a finite set. We know from Theorem 3.1 that
${\mathcal{G}}_{k}$ is finite. Now we assume $0\leq i\leq k-1$.

By contradiction, there exists a sequence of measures
$\\{\mu_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{G}}_{i}$ and some invariant
measure $\mu$ such that

$\mu_{n}\xrightarrow{weak^{*}}\mu,\quad\textrm{as}~{}n\to+\infty.$

By Theorem 3.1, $\mu$ is a Gibbs $F_{i}$-state, which satisfies

(8) $\lambda^{c}_{i+1}(x)>0,\quad\mu\textrm{-a.e.}~{}x\in M.$

On the other hand, since
$\\{\mu_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{G}}_{i}$, we know by
definition that $\lambda^{c}_{i+1}(\mu_{n})<0$ for every $n\in{\mathbb{N}}$.
The convergence $\mu_{n}\xrightarrow{weak^{*}}\mu$ yields

(9) $\lambda^{c}_{i+1}(\mu)=\lim_{n\to+\infty}\lambda^{c}_{i+1}(\mu_{n})\leq
0.$

By Birkhoff’s ergodic theorem, we also have

$\lambda^{c}_{i+1}(\mu)=\int\lambda^{c}_{i+1}(x){\rm d}\mu.$

This together with (9) yields that there exists a subset $A\subset M$ such
that $\mu(A)>0$ and

$\lambda^{c}_{i+1}(x)<0,\quad\forall~{}x\in A,$

which contradicts (8) and we complete the proof.

∎

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