# On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding
for partially hyperbolic systems

Reza Mohammadpour ID Department of Mathematics, Uppsala University, Box 480,
SE-75106, Uppsala, SWEDEN<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian
manifolds of arbitrary dimension that admit a dominated splitting
$E^{cs}\oplus E^{cu}.$ We prove that if the Lyapunov exponents along $E^{cu}$
are positive for Lebesgue almost every point, then a map $f$ is non-uniformly
expanding along $E^{cu}$ under the assumption that the cocycle
$Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support
of an ergodic Lyapunov maximizing observable measure. As a result, there
exists a physical SRB measure for a $C^{1+\alpha}$ diffeomorphism map $f$ that
admits a dominated splitting $E^{cs}\oplus E^{cu}$ under assumptions that $f$
has non-zero Lyapunov exponents for Lebesgue almost every point and that the
cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the
support of an ergodic Lyapunov maximizing observable measure.

###### Key words and phrases:

nonuniform expansion, Lyapunov exponents, SRB measures, ergodic optimization,
observable measures

###### 2010 Mathematics Subject Classification:

37D25, 37A05, 37C40, 37D30

##  1\. Introduction

Let $f:M\to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold
$M$ with a normalized Riemannian volume Lebesgue measure $\operatorname{Leb}$.
Given an $f$-invariant Borel probability measure $\mu$ in $M$, the basin of
attraction $\mu$ is the set $B(\mu)$ of the points $x\in M$ such that the
averages of Dirac measures along the orbit of $x$ converge to $\mu$ in the
weak* sense:

$\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi\left(f^{j}(x)\right)=\int\varphi
d\mu$

for any continuous $\varphi:M\rightarrow\mathbb{R}$. Then we say that $\mu$ is
a physical measure for $f$ if the basin of attraction $B(\mu)$ has positive
Lebesgue measure in $M$. We denote by Phy the set of all Physical measures of
a system.

A particular type of physical measures are called Sinai–Ruelle–Bowen (SRB)
measures (see Section 6 for the precise definition). In the 1970’s, Sinai,
Ruelle, and Bowen invented SRB measures for uniformly hyperbolic attractors
sets $\Lambda$ under the assumption that $\Lambda$ has a dense set of periodic
points, or equivalently, that $\Lambda$ is locally maximal, or that $\Lambda$
has local product structure; these three properties of uniformly hyperbolic
sets are equivalent [Shu87]. For partially hyperbolic attractors, in 1982,
Pesin and Sinai [PS80] developed a technique, called push-forward, for
constructing special measures that they called $u$-measures. These measures
share a lot of geometric characteristics with SRB measures. It is well known
that an ergodic SRB measure is physical, but all physical measures are not
SRB. The easiest counterexamples are point masses on attractive fixed points
and periodic orbits. Another interesting example is the figure-eight attractor
that has a physical measure with a positive Lyapunov exponent that is not an
SRB measure. Burns, Dolgopyat, Pesin, and Pollicott [BDPP08] showed that a
$u$-measure which has negative Lyapunov exponents for vectors in the central
direction on a set of positive measure is a unique SRB measure under some
transitivity assumptions.

Alves, Bonatti and Viana [ABV00] considered the more difficult setting of a
continuous splitting $E^{s}\oplus E^{cu}$ with uniform contraction $E^{s}$ and
non-uniform expansion $E^{cu}$. The construction of the SRB measure in this
case required a significantly more complex technique. In fact, they showed
that $f$ has finitely many ergodic physical SRB measures under the assumption
that there exists a set $H$ of positive Lebesgue measure on which $f$ is non-
uniformly expanding along $E^{cu}$: There exists an $\lambda>0$ such that for
all $x\in H$

(1.1)
$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log\left\|Df_{|E^{cu}(f^{j}(x))}^{-1}\right\|<-\lambda.$

In [ADLP17], they improved [ABV00, Theorem A] by showing that the limsup
condition (1.1) could be replaced by the liminf condition; they also use Young
towers to construct the SRB measure. Climenhaga, Dolgopyat and Pesin [CDP16]
generalized the main result on existence of SRB measures from [ABV00]. We
recommend the survey [CLP17] for further references and results.

We also mention a long-standing conjecture of Viana.

######  Conjecture 1.1.

If a smooth map $f$ has only non-zero Lyapunov exponents at Lebesgue almost
every point, then $f$ preserves an SRB measure.

Apart from the one-dimensional setting, it has been difficult to work directly
with Lyapunov exponents, therefore [ABV00, ADLP17] introduced stronger
versions of non-uniform expansion. Moreover, [ABV00] asked whether their
result is true if non-uniformly expanding condition (1.1) replaced by

(1.2)
$\limsup_{n\to\infty}\frac{1}{n}\log\left\|Df_{|E^{cu}(f^{n}(x))}^{-n}\right\|<0$

for $\operatorname{Leb}$ almost every $x$, that is positivity of all the
Lyapunov exponents in the center-unstable direction.

In this work we deal with $C^{1}$ diffeomorphisms admitting partially
hyperbolic invariant sets; the tangent bundle over the set has an invariant
dominated splitting into two subbundles (see Section 3 for the precise
definition). In this paper we show that (1.2) implies (1.1) under the
assumption that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting
with index 1 on the support of an ergodic Lyapunov maximizing observable
measure. We want to emphasize that our assumption is that the cocycle
$Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support
of a measure, which always exists (see Remark 1.4), and not on the entire
manifold. Otherwise, the result would be trivial.

###  1.1. Precise setting and the main result

Let $T:X\to X$ be a homeomorphism on a compact manifold $X$ and let
$\operatorname{Leb}$ be a Lebesgue measure normalized so that
$\operatorname{Leb}(X)=1.$ We denote by $\mathcal{M}(X)$ and
$\mathcal{M}(X,T)$ the space of all Borel probability measure on $X$, and the
space of all $T$-invariant Borel probability measures on $X$ with respect to
weak∗ topology, respectively.

We say that $\Phi:=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ is a subadditive
potential if each $\phi_{n}$ is a continuous positive-valued function on $X$
such that

$0<\phi_{n+m}(x)\leq\phi_{n}(x)\phi_{m}(T^{n}(x))\quad\forall x\in
X,m,n\in\mathbb{N}.$

Furthermore, $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ is said to be an almost
additive potential if there exists a constant $C>0$ such that for any
$m,n\in\mathbb{N}$, $x\in X$, we have

$C^{-1}\phi_{n}(x)\phi_{m}(T^{n})(x)\leq\phi_{n+m}(x)\leq
C\phi_{n}(x)\phi_{m}(T^{n}(x)).$

Let $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ be a subadditive potential. For
any $T-$invariant measure $\mu$, the Lyapunov exponent of measure $\mu$

$\chi(\mu,\Phi):=\lim_{n\rightarrow\infty}\frac{1}{n}\int\log\phi_{n}(x)d\mu(x)$

exists, by Kingman’s subadditive theorem.

For any point $x\in X$, we denote by $V(x)$ the set of those Borel
probabilities on $X$ that are limits in the weak∗ topology of convergent
subsequences of the sequence

(1.3) $\gamma_{n,x}:=\frac{1}{n}\sum_{j=0}^{n-1}\delta_{T^{j}(x)},$

where $\delta_{y}$ is the Dirac delta probability measure supported at $y\in
X.$ In other words,

$V(x)=\left\\{\mu\in\mathcal{M}(X):\exists n_{j}\rightarrow+\infty\text{ such
that
}\lim_{j\rightarrow\infty}\gamma_{n_{j},x}=\mu\right\\}\subset\mathcal{M}(X,T).$

The probabilities of the sequence (1.3) are called empirical probabilities of
the orbit of $x$. Note that the empirical probabilities do not necessarily
converge.

######  Definition 1.2.

A probability measure $\mu\in\mathcal{M}(X)$ is observable or physical-like if
for any $\varepsilon>0$ the set $B_{\epsilon}(\mu)=\\{x\in
X:\operatorname{dist}(V(x),\mu)<\varepsilon\\}$ has positive Lebesgue measure.
We call $B_{\epsilon}(\mu)$ the basin of $\varepsilon$-attraction of $\mu$. We
denote by $\mathcal{O}$ the set of observable measures.

An observable measure is not necessary a physical measure; see Example 5.5. In
general, for a system, physical measures might not exist, however there always
exists an observable measure; see [CE11, Theorem 1.3].

######  Definition 1.3.

An observable measure $\mu$ is called a Lyapunov maximizing observable measure
for a subadditive potential $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ if

(1.4)
$\chi(\mu,\Phi)=\sup\left\\{\chi(\nu,\Phi)\mid\nu\in\mathcal{O}\right\\}$

Similarly, one can define a Birkhoff maximizing observable measure (see
[Zha16]).

######  Remark 1.4.

Since $\chi(.,\Phi)$ is upper semi-continuous with respect to weak* topology
and $\mathcal{O}$ is weak* compact by Theorem 2.2, the supremum is always
attained in (1.4). Therefore, a Lyapunov maximizing observable measure always
exists.

Our goal is to show that for partially hyperbolic systems, the positive
Lyapunov exponents in the center-unstable direction imply the nonuniform
expansion. It is clear that the two formulations (1.5) and (1.6) are
equivalent when the non-uniformly expanding direction is 1-dimensional, that
is, when $E^{cu}$ can be split as $E^{c}\oplus E^{u}$ with $E^{c}$ having
dimension 1 and $E^{u}$ being uniformly expanding. A distinction in our result
from similar results is that we only require the dominated spliting property
for $Df_{|E^{cu}(f)}^{-1}$ on the support of an invariant measure (ergodic
Lyapunov maximizing observable measure), which the measure always exist (see
Remark 1.4), not on the whole manifold.

The main result of this paper is the following theorem.

######  Theorem A.

Let $f:M\rightarrow M$ be a $C^{1}$ diffeomorphism on a compact Riemannian
manifold that admits a dominated splitting $TM=E^{cs}\oplus E^{cu}$. Assume
that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1
on the support of an ergodic Lyapunov maximizing observable measure for a
subadditive potential $\left\\{\log\left\|Df_{\mid
E^{cu}\left(f^{n}\right)}^{-n}\right\|\right\\}_{n=1}^{\infty}$. If Lebesgue
a.e. $x$ satisfies

(1.5)
$\limsup_{n\to\infty}\frac{1}{n}\log\left\|Df^{-n}_{|E^{cu}(f^{n}(x))}\right\|<0,$

then there are $\lambda>0$ and $K\in\mathbb{N}$ such that Lebesgue a.e. $x$
satisfies

(1.6)
$\limsup_{n\rightarrow+\infty}\frac{1}{nK}\sum_{i=1}^{n}\log\left\|Df_{|E^{cu}(f^{iK}(x))}^{-K}\right\|\leq-\lambda<0.$

Section 5 provides several examples that demonstrate observable measures and
the sub-domination of $E^{cu}$ on the support of an ergodic Lyapunov
maximizing observable measure.

######  Remark 1.5.

Theorem A can also be localized as [ABV00] asked; if (1.5) holds on a set of
positive Lebesgue measure, then (1.6) holds on some set of positive Lebesgue
measure. In order to state it, we need to define an observable measure
relative to some $C^{1}$-disk. Let $D$ be an embedded compact $C^{1}$-disc in
$M$. Then $\mu$ is called an observable measure relative to $D$, if for any
$\eta>0$, one has

$\operatorname{Leb}_{D}(\\{x\in D:\operatorname{dist}(V(x),\mu)<\eta\\})>0,$

where $\operatorname{Leb}_{D}$ is the normalized Lebesgue measure on a disc
$D$.

Crovisier, Yang and Zhang [CYZ20] generalized Theorem 2.2 to the set of all
observable measures relative to some $C^{1}$-disk $D$; denote it by
$\mathcal{O}_{D}.$ Moreover, for a set of positive $\operatorname{Leb}_{D}$
measure $B\subset D$ , we say that $\mu$ is called a observable measure
relative to $B\subset D$, if for any $\eta>0$,

$\operatorname{Leb}_{D}(\\{x\in B:\operatorname{dist}(V(x),\mu)<\eta\\})>0.$

By replacing $\mathcal{O}$ with $\mathcal{O}_{D}$ in Definition 1.3, we can
state Theorem A as follows, if (1.5) holds on a set of positive Lebesgue
measure $B\subset D$, then (1.6) holds on some set of positive Lebesgue
measure.

######  Corollary 1.6.

Let $f:M\rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a compact
Riemannian manifold that admits a dominated splitting $TM=E^{s}\oplus E^{cu}$,
where $E^{s}$ is uniformly contracting. Assume that the cocycle
$Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support
of an ergodic Lyapunov maximizing observable measure a subadditive potential
$\left\\{\log\left\|Df_{\mid
E^{cu}\left(f^{n}\right)}^{-n}\right\|\right\\}_{n=1}^{\infty}$. If Lebesgue
a.e. $x$ satisfies

(1.7)
$\limsup_{n\to\infty}\frac{1}{n}\log\left\|Df^{-n}_{|E^{cu}(f^{n}(x))}\right\|<0,$

then $f$ has finitely many ergodic physical SRB measures and the union of
their basins has full Lebesgue measure.

###### Proof.

It follows from Theorem A and [ADLP17, Theorem A]. ∎

The assumption of the uniformly contracting in Corollary 1.6 can be somewhat
relaxed, as [ABV00] explained in Section 6 of their paper. For partial
hyperbolicity systems with $E^{cu}$ non-uniformly expanding, [ABV00] proved
that the diffeomorphism admits invariant probability measures with absolutely
continuous conditional measures with respect to Lebesgue measure along the
centre-unstable direction; such measures are called Gibbs cu-states (See
[DV05] for the precise definition and the properties).

######  Remark 1.7.

All results of the present paper also work for diffeomorphisms with partially
hyperbolic attractors (see Subsection 3.2).

In Section 2 we provide properties of observable measures and related
concepts. In Section 3 we study Lyapunov exponents, dominated splitting and
related concepts. In Section 4 we discuss ergodic optimization of Lyapunov
exponents and the maximal Lyapunov exponents over the set of observable
measures. In Section 5 we give some examples about observable measures. The
proof of Theorem A is given in Section 5.

Acknowledgments. The author is grateful to José Alves, David Burguet, Snir Ben
Ovadia and Kiho Park for the useful discussion. He also thanks Davi Obata for
the influential conversation. Finally, he thanks Russell Avdek for improving
the readability of the introduction of the paper.

This work was supported by the Knut and Alice Wallenberg Foundation.

##  2\. Observable measures

Let $T:X\to X$ be a homeomorphism on a compact manifold $X$ and let
$\operatorname{Leb}$ be a Lebesgue measure normalized so that
$\operatorname{Leb}(X)=1.$ We recall some basic results related with
observable measures.

######  Remark 2.1.

By the definition of observable measures, $\epsilon$-approximation in the
space $\mathcal{M}(X,T)$ can be transferred to an $\epsilon$-approximation (in
time-mean) towards an attractor in the ambient manifold $X$. In other words,
if $\mu$ is observable and $x\in B_{\epsilon}(\mu)$, then the iterates
$T^{n}(x)$ will $\epsilon$-approach the support of $\mu$ with a frequency
which is asymptotically bounded away from zero.

The set $B(\mu)=\\{x\in X:V(x)=\\{\mu\\}\\}$ is the basin of attraction of a
physical-measure $\mu$. Inspired by the latter definition, we define the basin
of attraction $\mathcal{O}$:

(2.1) $\mathcal{G}(\mathcal{O})=\\{x\in X:V(x)\subset\mathcal{O}\\}.$

For simplicity, we denote $\mathcal{G}:=\mathcal{G}(\mathcal{O})$.
$\operatorname{Leb}(\mathcal{G})=1$ by Theorem 2.2.

######  Theorem 2.2 ([CE11, Theorem 1.5]).

$\mathcal{O}$ is the smallest weak* compact containing, for Lebesgue almost
all initial states, the limits of all convergent subsequences of (1.3).

In other words, $\mathcal{O}=\bigcap_{K\in W}K$, where

$W:=\\{K\subset\mathcal{M}(X):K\text{ is compact and
}\operatorname{Leb}(\mathcal{G}(K))=1\\}.$

######  Theorem 2.3 ([CE11, Theorem 1.6]).

There are finitely many physical measures such that the union of their basins
of attraction covers $X$ Lebesgue almost everywhere if and only if
$\mathcal{O}$ is finite. In this case, $\mathcal{O}=$Phy.

######  Theorem 2.4 ([CE11, Theorem 1.7]).

If $\mathcal{O}$ is countably infinite, then there are countably infinitely
many physical measures such that their basins of attraction cover $X$ Lebesgue
almost everywhere . In this case $\mathcal{O}$ is the weak* closure of the set
of physical measures.

##  3\. Lyapunov exponents and dominated systems

We assume that $X$ is a compact metric space and $T:X\to X$ is a
homeomorphism; we denote by $\text{Hom}_{\mu}(X)$ the set of homeomorphisms
that preserve an invariant measure $\mu$. Let $\mathcal{A}:X\rightarrow
GL(d,\mathbb{R})$ be a continuous map. We define a linear cocycle
$F:X\times\mathbb{R}^{d}\rightarrow X\times\mathbb{R}^{d}$ as

$F(x,v)=(T(x),\mathcal{A}(x)v).$

We say that $F$ is generated by $T$ and $\mathcal{A}$; we will also denote it
by $(\mathcal{A},T).$ Observe $F^{n}(x,v)=(T^{n}(x),\mathcal{A}^{n}(x)v)$ for
each $n\geq 1$, where

(3.1)
$\mathcal{A}^{n}(x)=\mathcal{A}(T^{n-1}(x))\mathcal{A}(T^{n-2}(x))\cdots\mathcal{A}(x).$

$\|\mathcal{A}\|$, the Euclidean operator norm of a matrix $\mathcal{A}$, is
submultiplicative i.e.,

$0<\|\mathcal{A}^{n+m}(x)\|\leq\|\mathcal{A}^{n}(x)\|\|\mathcal{A}^{m}(T^{n}(x))\|\hskip
5.69046pt\forall x\in X,m,n\in\mathbb{N},$

therefore the potential
$\Phi_{\mathcal{A}}:=\\{\log\|\mathcal{A}^{n}\|\\}_{n=1}^{\infty}$ is
subadditive for a matrix cocycle $\mathcal{A}$. The linear cocycles generated
by the derivative of a diffeomorphism map or a smooth map $f:X\rightarrow X$
on a compact Riemannian manifold $X$ and a family of maps
$\mathcal{A}(x):=Df_{x}:T_{x}X\rightarrow T_{T(x)}X$ is called derivative
cocycles.

######  Theorem 3.1 (Oseledets).

Let $\mu$ be an $T$-invariant measure. For $\mu$ almost every $x\in X$, there
are functions $k=k(x)$,
$\lambda_{1}(x,\mathcal{A})>\lambda_{2}(x,\mathcal{A})>\ldots>\lambda_{k}(x,\mathcal{A})$
and a direct sum decompositions $\mathbb{R}^{d}=E^{1}(x)\oplus
E^{2}(x)\oplus\ldots E^{k}(x)$ such that, for every $i=1,\ldots,k:$

  * 1)

The maps $x\to k(x)$, $x\to\lambda(x,\mathcal{A})$ and $x\to E^{i}(x)$ (with
values $\mathbb{N}$, $\mathbb{R}$ and $Gr(d)$111$Gr(d)$ is the Grassmanian of
$\mathbb{R}^{d}.$) are measurable.

  * 2)

$k(T(x))=k(x)$, $\lambda_{i}(T(x),\mathcal{A})=\lambda_{i}(x,\mathcal{A})$ and
$\mathcal{A}(x)E^{i}(x)=E^{i}(T(x)).$

  * 3)

For every non-null $v\in E^{i}(x)$,

$\lim_{n\to\pm\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)v\|=\lambda_{i}(x,\mathcal{A}).$

  * 4)

For any disjoint sum of the spaces within the splitting

$\lim_{n\to\pm\infty}\frac{1}{n}\log|\sin\measuredangle(\oplus_{i\in
I}E^{i}(T^{n}(x)),\oplus_{i\in J}E^{i}(T^{n}(x))|=0$

whenever $I\cap J=\emptyset$.

The splitting $\mathbb{R}^{d}=E^{1}(x)\oplus\ldots E^{k}(x)$ is called
Oseledets splitting.

We denote by
$\chi_{1}(x,\mathcal{A})\geq\chi_{2}(x,\mathcal{A})>\ldots\geq\chi_{d}(x,\mathcal{A})$
the Lyapunov exponents, counted with multiplicity222The multiplicity of
$\lambda_{i}(x,\mathcal{A})$ is the same as the dimension of $E^{i}(x).$, of
the cocycle $(\mathcal{A},T).$ This is the Lyapunov spectrum of the cocycle
with respect to some invariant measure $\mu$.

Let us denote $\chi(\mu,\Phi_{\mathcal{A}}):=\int\chi(.,\mathcal{A})d\mu.$ If
the measure $\mu$ is ergodic then
$\chi(x,\mathcal{A})=\chi(\mu,\Phi_{\mathcal{A}})$ for $\mu$-almost every
$x\in X.$ In this case, we denote $\chi(\mathcal{A}):=\chi(x,\mathcal{A}).$

######  Theorem 3.2.

Suppose $\\{\nu_{n}\\}_{n=1}^{\infty}$ is a sequence in $\mathcal{M}(X)$ and
$\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ is a subadditive potential over a
topological dynamical system $(X,T).$ We form the new sequence
$\\{\mu_{n}\\}_{n=1}^{\infty}$ by
$\mu_{n}=\frac{1}{n}\sum_{i=0}^{n-1}\nu_{n}oT^{i}$. Assume that $\mu_{n_{i}}$
converges to $\mu$ in $\mathcal{M}(X)$ for some subsequence $\\{n_{i}\\}$ of
natural numbers. Then $\mu$ is an $T$-invariant measure and

$\limsup_{i\rightarrow\infty}\frac{1}{n_{i}}\int\log\phi_{n_{i}}(x)d\nu_{n_{i}}(x)\leq\chi(\mu,\Phi).$

Moreover, if $\Phi$ is an almost additive potential, then

$\lim_{i\rightarrow\infty}\frac{1}{n_{i}}\int\log\phi_{n_{i}}(x)d\nu_{n_{i}}(x)=\chi(\mu,\Phi).$

###### Proof.

It follows from [FH10, Lemmas A2 and A4]. ∎

Given an almost additive potential $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$
over a topological dynamical system $(X,T)$, Feng and Huang [FH10, Lemma A.4]
proved the following lemma:

######  Lemma 3.3.

For every $\epsilon>0$, there is $\delta>0$ such that if
$\mu,\nu\in\mathcal{M}(X,T)$, and $d(\mu,\nu)<\delta$, then
$|\chi(\mu,\Phi)-\chi(\nu,\Phi)|<\epsilon.$

###  3.1. Uniform $p$-gap

Let $\mathcal{A}:X\to GL(d,\mathbb{R})$ be a linear cocycle over $(X,T)$ and
let $\mu$ be an $T-$invariant measure. For $\mu$ a.e. $x$, we denote by
$\chi_{1}(x,\mathcal{A})\geq\ldots\geq\chi_{d}(x,\mathcal{A})$ the Lyapunov
exponents of the cocycle $(\mathcal{A},T)$ (see Section 3 for the precise
definition). We say that the Lyapunov spectrum of the cocycle
$(\mathcal{A},T)$ has a uniform p-gap property if there exist a
$C^{0}$-neighborhood $\mathcal{U}_{\mathcal{A}}\subset
C^{0}(X,GL(d,\mathbb{R}))$ of $\mathcal{A}$, and $\beta>0$ such that for every
$B\in\mathcal{U}_{\mathcal{A}}$ we have

$\int(\chi_{p}(x,B)-\chi_{p+1}(x,B))d\mu(x)>\beta.$

Denote $\chi_{p}(\mathcal{A}):=\chi_{p}(x,\mathcal{A})$, when $\mu$ is
ergodic. When $\mu$ is an ergodic measure, then uniform $p-$gap property means
that there exist a $C^{0}$-neighborhood $\mathcal{U}_{\mathcal{A}}\subset
C^{0}(X,GL(d,\mathbb{R}))$ and $\beta>0$ such that for every
$B\in\mathcal{U}_{\mathcal{A}}$ we have

$\chi_{p}(B)-\chi_{p+1}(B)>\beta.$

######  Remark 3.4.

The uniform $p-$gap property is related to the well-known paper Bochi and
Viana [BV05], which can be summarized as follows: If $\mathcal{A}_{0}:X\to
GL(d,\mathbb{R})$ is a continuity point 333We say that a cocycle
$\mathcal{A}:X\to GL(d,\mathbb{R})$ is a point of continuity of the $p$-th
Lyapunov exponent $\chi_{p}$ if it is a point of continuity of the function
$\operatorname{Lyap}_{p}:C(X,GL(d,\mathbb{R}))\rightarrow\mathbb{R}$ that maps
$\mathcal{A}\mapsto\chi_{p}(\mathcal{A})$. That is, for every sequence of
cocycles $\mathcal{A}_{n}$ such that $\mathcal{A}_{n}\rightarrow\mathcal{A}$
then $\chi_{p}\left(\mathcal{A}_{n}\right)\rightarrow\chi_{p}(\mathcal{A})$.
for the Lyapunov spectrum with respect to an ergodic measure $\mu$ then for
for $\mu$-almost all points the Oseledets splitting is either dominated or
trivial. Moreover, that continuity of the Lyapunov spectrum is
$C^{0}$-generic, meaning that this dichotomy is present in a residual set in
the space of linear cocycles. Note that if a cocycle $\mathcal{A}_{0}:X\to
GL(d,\mathbb{R})$ is a point of continuity of the $p$-th and $(p+1)$-th
Lyapunov exponents and it holds that
$\chi_{p}(\mathcal{A}_{0})>\chi_{p+1}(\mathcal{A}_{0})$, then it implies that,
taking a small $\varepsilon>0$, there exists a $C^{0}$-neighbourhood of
$\mathcal{A}_{0}$ such that there is a uniform gap between the $p-$th and
$p+1-$th Lyapunov exponents.

###  3.2. Dominated splitting

Partial hyperbolicity is a relaxed form of uniform hyperbolicity that plans to
address larger families of dynamics. A main aim of their study consists in
understanding how the properties of uniformly hyperbolic systems extends.

Let $f:M\rightarrow M$ be a $C^{1}$ diffeomorphism on a compact Riemannian
manifold. We say that $f$ is partially hyperbolic if there exists a continuous
$Df$-invariant splitting $TM=E^{cu}\oplus E^{cs}$ which satisfy: There are
$C>0$ and $0<\lambda<1$ such that for any $x\in M$

$\|Df_{|E^{cs}(x)}^{n}\|\leq C\lambda^{n}m(Df_{|E^{cu}(x)}^{n})\quad\forall
n\geq 0,$

where $m(A)=\|A^{-1}\|^{-1}.$ In particular, the angle between them is
uniformly away from zero.

We shall write $E^{s}$ and $E^{u}$ instead of $E^{cs}$ and $E^{cu}$,
respectively, if they have the uniform behavior. It is called uniformly
hyperbolic if both $E^{cs}$ and $E^{cu}$ are uniform.

More generally, we consider diffeomorphisms with partially hyperbolic
attractors, that is, compact subsets $\Lambda$ of $M$ such that

$\Lambda=\bigcap_{n>0}f^{n}(U)$

for some open neighborhood $U$ of $\Lambda$ with closure $f(U)\subset U$, and
there exists a splitting $T_{\Lambda}M=E^{cu}\oplus E^{cs}$ of the restriction
of the tangent bundle to $\Lambda$, with the same properties as before.

A splitting $TM=E^{1}\oplus E^{2}\ldots\oplus E^{k}$ is dominated if each
bundle is invariant and there exist $C>0$ and $\lambda\in(0,1)$ such that for
any $x\in M$ and any unit vectors $u\in E^{i}$, and $v\in E^{i+1},$

$\forall n\geq 0,\quad\|Df_{x}^{n}u\|\leq C\lambda^{n}\|Df_{x}^{n}v\|.$

Bochi and Gourmelon [BG09] showed that a cocycle has dominated splitting if
and only if there is a uniform exponential gap between singular values of its
iterates.

Assume that $X$ is a compact metric space. Let $\mathcal{A}:X\to
GL(d,\mathbb{R})$ be a matrix cocycle over a homeomorphism map $(X,T).$ Let
$Y\subset X$ be a non-empty $T$-invariant compact set. We say that the cocycle
$\mathcal{A}$ has dominated splitting with index $i$ on $Y$ if there are
constants $C>0$ and $0<\tau<1$ such that

$\frac{\sigma_{i+1}(\mathcal{A}^{n}(x))}{\sigma_{i}(\mathcal{A}^{n}(x))}\leq
C\tau^{n}\hskip 5.69046pt\forall x\in Y,\forall n\in\mathbb{N},$

where $\sigma_{i}$’s are singular values.

Dominated splitting can be characterized in terms of existence of invariant
cone fields [CP15]. We say that a family of convex cones
$\mathcal{C}:=(C_{x})_{x\in Y}$ is a $\ell$-dimensional invariant cone field
on $Y$ if

$\mathcal{A}(x)C_{x}\subset C_{T(x)}^{o}\hskip 5.69046pt\forall x\in Y,$

where $\ell$ is the dimension $\mathcal{C}.$

######  Theorem 3.5 ([Yem22, Main Theorem]).

Let $X$ be a compact metric space and $\mu$ be a Borel regular measure.
Suppose that $T\in\text{Hom}_{\mu}(X)$ such that $\mu$ is an ergodic measure
and $\mathcal{A}:X\to GL(d,\mathbb{R})$ is a matrix cocycle over $(X,T)$ such
that its Lyapunov spectrum has the uniform $p$-gap property. Then, there
exists a dominated splitting of index $p$ on $\text{supp}(\mu)$.

By the above theorem, the cocycle $\mathcal{A}$ is dominated with index $p$ on
the support of an ergodic measure if and only if its Lyapunov spectrum has the
uniform $p$-gap property with respect to an ergodic measure.

######  Remark 3.6.

It is clear that the uniform gap property with respect to some invariant
measure $\mu$ does not imply dominated splitting. Even, if there is a uniform
gap property on a set of full probability, then it doesn’t imply the dominated
splitting. One can find such an example in [Rui20].

In [Moh22], the author showed that if a cocycle has dominated splitting with
index 1 on some compact metric space, which can be characterized in terms of
existence of invariant cone fields (or multicones), then the potential
$\Phi_{\mathcal{A}}=\\{\log\|\mathcal{A}^{n}\|\\}_{n=1}^{\infty}$ is almost
additive.

######  Proposition 3.7 ([Moh22, Proposition 5.8]).

Let $X$ be a compact metric space, and $\mathcal{A}:X\rightarrow
GL(d,\mathbb{R})$ be a matrix cocycle over a homeomorphism map $(X,T)$. Assume
that $(C_{x})_{x\in X}$ is a 1-dimensional invariant cone field on $X$. Then,
there exists $\kappa>0$ such that for every $m,n>0$ and for every $x\in X$ we
have

$||\mathcal{A}^{m+n}(x)||\geq\kappa||\mathcal{A}^{m}(x)||\cdot||\mathcal{A}^{n}(T^{m}(x))||.$

##  4\. Ergodic optimization of Lyapunov exponents

Let $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ be a subadditive potential over a
topological dynamical system $(X,T).$ Ergodic optimization of Lyapunov
exponents concerns the supremum of the Lyapunov exponents of measures over
invariant measures:

(4.1) $\beta(\Phi):=\sup_{\mu\in\mathcal{M}(X,T)}\chi(\mu,\Phi).$

In (4.1), the supremum is always attained by an ergodic measure; such measures
will be called Lyapunov maximizing measures. This follows from the fact that
$\mathcal{M}(X,T)$ is a compact convex set whose extreme points are exactly
the ergodic measure, and $\chi(.,\Phi)$ is upper semi continuous with respect
to the weak* topology (see [Jen19, Boc18, Mor13]).

Let us stress that the maximal Lyapunov exponent can also be characterized as
follows:

(4.2) $\beta(\Phi)=\limsup_{n\to\infty}\frac{1}{n}\sup_{x\in
X}\log\phi_{n}(x)=\sup_{x\in
X}\limsup_{n\to\infty}\frac{1}{n}\log\phi_{n}(x).$

Now, assume that $T:X\to X$ is a homeomorphism on a compact manifold $X$. We
tend to investigate the maximal Lyapunov exponents over the set of observable
measures $\mathcal{O}$.

######  Theorem 4.1.

Let $\mathcal{A}:X\to GL(d,\mathbb{R})$ be a matrix cocycle over a
homeomorphism $(X,T)$. Assume that the cocycle $\mathcal{A}$ has a dominated
splitting with index 1 on the support of an ergodic Lyapunov maximizing
observable measure for the subadditive potential $\Phi_{\mathcal{A}}$. Then

(4.3)
$\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)\|=\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\\}$
(4.4) $\hskip
167.87108pt\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\operatorname{ess}\sup\log\|\mathcal{A}^{n}(x)\|,$

where the $\operatorname{ess}\sup$ denotes the essential supremum taken
against the Lebesgue measure $\operatorname{Leb}.$

######  Remark 4.2.

The inequality in the formula (4.4) could be strict; see Example 5.4.

######  Lemma 4.3.

Let $\Phi=\\{\log\phi_{n}\\}_{n=1}^{\infty}$ be a subadditive potential. Then

$\displaystyle\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi_{n}(x)$
$\displaystyle\leq\sup\\{\chi(\mu,\Phi)\mid\mu\in\mathcal{O}\\}$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\operatorname{ess}\sup\log\phi_{n}(x),$

where the essential supremum taken against $\operatorname{Leb}.$

###### Proof.

Choose a point $x\in\mathcal{G}$ (see (2.1)), and then a subsequence of
integers $\\{n_{i}\\}$ such that
$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi_{n}(x)=\lim_{i\rightarrow\infty}\frac{1}{n_{i}}\log\phi_{n_{i}}(x).$
Since $x\in\mathcal{G}$, there is a subsequence $\gamma_{n_{i_{k}},x}$ of
$\gamma_{n_{i},x}$ such that converges in weak∗ topology to an observable
measure $\mu$ when $k\rightarrow\infty$ by definition. Without loss of
generality, we can suppose that $\gamma_{n_{i},x}$ converges to $\mu$ as
$i\to\infty$. Then, by Theorem 3.2

(4.5)
$\lim_{i\rightarrow\infty}\frac{1}{n_{i}}\int\log\phi_{n_{i}}(x)d\delta_{x}\leq\chi(\mu,\Phi)\leq\sup\\{\chi(\mu,\Phi)\mid\mu\in\mathcal{O}\\}.$

On the other hand,

(4.6)
$\lim_{i\rightarrow\infty}\frac{1}{n_{i}}\int\log\phi_{n_{i}}(x)d\delta_{x}=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi_{n}(x).$

Combination (4.5) and (4.6) and the arbitrariness of $x\in\mathcal{G}$ imply

$\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi_{n}(x)\leq\sup\\{\chi(\mu,\Phi)\mid\mu\in\mathcal{O}\\}.$

We consider any subset $U$ with full Lebesgue measure. We know that each
subset of full Lebesgue measure is dense, hence

$\limsup_{n\rightarrow\infty}\frac{1}{n}\sup_{x\in
U}\log\phi_{n}(x)=\limsup_{n\rightarrow\infty}\frac{1}{n}\sup_{x\in
X}\log\phi_{n}(x)=\beta(\Phi).$

By (4.2) and (4.1),

$\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi_{n}(x)\leq\sup\\{\chi(\mu,\Phi)\mid\mu\in\mathcal{O}\\}\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\operatorname{ess}\sup\log\phi_{n}(x).$

∎

Let $f:M\to M$ be a $C^{\infty}$ map on a compact smooth manifold $M$.
Kozlovski [Koz98] showed the following integral formula for the topological
entropy $C^{\infty}$ smooth systems:

$h_{\operatorname{top}}(f)=\lim_{n\rightarrow\infty}\frac{1}{n}\log\int\max_{k}\left\|\Lambda^{k}Df_{x}^{n}\right\|d\operatorname{Leb}(x),$

where $\Lambda^{k}Df_{x}^{n}$ is a mapping between $k$-exterior algebras of
the tangent spaces.

######  Corollary 4.4.

Assume that $f:M\to M$ is a $C^{\infty}$ map on a compact smooth manifold $M$.
Then

$h_{\operatorname{top}}(f)\geq\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\max_{k}\|\wedge^{k}Df_{x}^{n}\|.$

###### Proof.

We consider $\phi_{n}=\max_{k}\|\wedge^{k}Df_{x}^{n}\|$ in Lemma 4.3, then

$\displaystyle\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\max_{k}\|\wedge^{k}Df_{x}^{n}\|$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\log\operatorname{ess}\sup\max_{k}\|\wedge^{k}Df_{x}^{n}\|$
$\displaystyle\leq h_{\operatorname{top}}(f).$

∎

The above formula was demonstrated with a different method by Burguet [Bur20];
see (1.3) in [Bur20].

###### Proof of Theorem 4.1.

Lemma 4.3 implies that one side of the equality (4.3) and the inequality (4.4)
is hold. Now, we explain that how one can tackle to prove the other side of
the equality (4.3) in Theorem 4.1.

The set of observable measures $\mathcal{O}$ is weak* compact, hence there is
a Lyapunov maximizing observable measure $\mu$ for the subadditive potential
$\Phi_{\mathcal{A}}$. We assume that $\mu$ is ergodic. For $\delta>0$ in Lemma
3.3, $B_{\delta}(\mu)$, the basin of $\delta$-attraction of $\mu$, has
positive Lebesgue measure by the definition of the observable measure.

Let $\mathcal{G}\subset X$ such that $V(x)\subseteq\mathcal{O}$ for any
$x\in\mathcal{G}$. Catsigeras and Enrich showed that
$\operatorname{Leb}(\mathcal{G})=1$; see Theorem 2.2. For any (arbitrary) full
Lebesgue measure subset $U$, since we assume that the cocycle has a dominated
splitting with index 1 on the support of the ergodic Lyapunov maximizing
observable measure $\mu$, we can find a closed invariant subset $D_{U}$
contained in the following intersection:

$D_{U}\subset\mathcal{G}\cap U\cap B_{\delta}(\mu)\cap\text{supp}(\mu)$

such that the cocycle $\mathcal{A}$ has dominated splitting with index 1 on
$D_{U}.$

Dominated splitting can be characterized in terms of existence of invariant
cone fields. Therefore, there is a family of cone filed $(C_{x})_{x\in D_{U}}$
such that

$\mathcal{A}(x)(C_{x})\subset C_{T(x)}^{o}.$

By Proposition 3.7, the subadditive potential $\Phi_{\mathcal{A}}$ is almost
additive on $D_{U}$. Hence, $\mu\to\chi(\mu,\Phi_{\mathcal{A}})$ is continuous
on $\mathcal{M}(D_{U},T_{|D_{U}})$ by Lemma 3.3.

Let $y\in D_{U}\subset B_{\delta}(\mu)$. There is a subsequence of integers
$\\{n_{i}\\}$ such that $\frac{1}{n_{i}}\sum_{j=0}^{n_{i}-1}\delta_{T^{j}(y)}$
converges to a measure $\nu$ with $d(\mu,\nu)<\delta$ (distance in weak*
topology) by definition.

Now, we are going to prove the other side of the equality (4.3).

$\displaystyle\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(y)\|$
$\displaystyle\geq\limsup_{i\rightarrow\infty}\frac{1}{n_{i}}\log\|\mathcal{A}^{n_{i}}(y)\|$
$\displaystyle=\underbrace{\limsup_{i\rightarrow\infty}\frac{1}{n_{i}}\int\log\|\mathcal{A}^{n_{i}}(y)\|d\delta_{y}}_{\text{
\leavevmode\hbox to11.75pt{\vbox
to11.75pt{\pgfpicture\makeatletter\hbox{\hskip 5.87407pt\lower-5.87407pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{-2.25555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys<EMAIL_ADDRESS>
Since $\frac{1}{n_{i}}\sum_{j=0}^{n_{i}-1}\delta_{T^{j}(y)}\rightarrow\nu$ (in
weak*topology) and $\Phi_{\mathcal{A}}$ is almost addtive on $D_{U}$, by
Theorem 3.2,

$\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\underbrace{\chi(\nu,\Phi_{\mathcal{A}})}_{\text{
\leavevmode\hbox to11.75pt{\vbox
to11.75pt{\pgfpicture\makeatletter\hbox{\hskip 5.87407pt\lower-5.87407pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.67407pt}{0.0pt}\pgfsys@curveto{5.67407pt}{3.13374pt}{3.13374pt}{5.67407pt}{0.0pt}{5.67407pt}\pgfsys@curveto{-3.13374pt}{5.67407pt}{-5.67407pt}{3.13374pt}{-5.67407pt}{0.0pt}\pgfsys@curveto{-5.67407pt}{-3.13374pt}{-3.13374pt}{-5.67407pt}{0.0pt}{-5.67407pt}\pgfsys@curveto{3.13374pt}{-5.67407pt}{5.67407pt}{-3.13374pt}{5.67407pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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By continuity (Lemma 3.3), for given $\epsilon>0,$

$\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\geq\chi(\mu,\Phi_{\mathcal{A}})-\epsilon.$

That implies

$\sup_{x\in
U}\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)\|\geq\chi(\mu,\Phi_{\mathcal{A}})-\epsilon.$

As we consider arbitrary $U$ and $\epsilon,$

(4.7)
$\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)\|\geq\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\\}.$

Combination (4.7) and Lemma 4.3 complete the proof. ∎

######  Corollary 4.5.

Under conditions of Theorem 4.1, we assume that the set of observable measures
$\mathcal{O}$ is at most countable, then we have

$\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)\|=\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\\}=\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\textbf{Phy}\\}.$

###### Proof.

It is enough to prove that

$\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\\}=\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\textbf{Phy}\\},$

by Theorem 4.1.

If the set of observable measures $\mathcal{O}$ is finite, then
$\mathcal{O}=\textbf{Phy}$ by Theorem 2.3, and if $\mathcal{O}$ is countable
infinite, then $\mathcal{O}$ is the weak*closure of the set of Phy measures by
Theorem 2.4. In both cases, the equality is clear. ∎

######  Remark 4.6.

The physical measures may not exist, if the set $\mathcal{O}$ is uncountable.
For instance, identity map on a manifold has observable measures but has no
physical measures. Hence, the above result fails in general when the set
$\mathcal{O}$ is uncountable.

##  5\. Examples

In the following section, we will present a set of simple examples that
highlight the concept of sub-domination of $E^{cu}$ and observable measures.

######  Example 5.1.

Let $f:M\to M$ be a $C^{1+\alpha}$ diffeomorphism on a smooth compact
Riemannian manifold $M$ that admits a dominated splitting $TM=E^{cs}\oplus
E^{cu}$. Assume that any Lyapunov maximal observable measures $\mu$ for a
subadditive potential $\left\\{\log\left\|Df_{\mid
E^{cu}\left(f^{n}\right)}^{-n}\right\|\right\\}_{n=1}^{\infty}$ is hyperbolic
ergodic $f$-invariant Borel probability measure with positive entropy
$h_{\mu}(f)>0.$ Then, one can use [Gel16, Theorem 1] to find a sub-domination
of $E^{cu}$.

######  Example 5.2.

We consider derivative cocycles as follows. Let $f:M\to M$ be a $C^{1}$
diffeomorphism on a smooth compact Riemannian manifold $M$ that admits a
dominated splitting $TM=E^{cs}\oplus E^{cu}$. Assume that the cocycle
$Df_{\mid E^{cu}\left(f\right)}^{-1}$ satisfies the uniform 1-gap property
with respect to an ergodic Lyapunov maximal observable measures $\mu$ for a
subadditive potential $\left\\{\log\left\|Df_{\mid
E^{cu}\left(f^{n}\right)}^{-n}\right\|\right\\}_{n=1}^{\infty}$444Note that in
the preceding definition, the base $f$ and the measure $\mu$ remain unaffected
by the choice of $B\in\mathcal{U}_{Df_{\mid E^{cu}\left(f\right)}^{-1}}$.,
i.e., there are a $C^{0}$-neighborhood $\mathcal{U}_{Df_{\mid
E^{cu}\left(f\right)}^{-1}}$ of $Df_{\mid E^{cu}\left(f\right)}^{-1}$, and
$\beta>0$ such that for every $B\in\mathcal{U}_{Df_{\mid
E^{cu}\left(f\right)}^{-1}}$ we have

$\int\left(\chi_{1}(x,B)-\chi_{2}(x,B)\right)d\mu(x)>\beta.$

Then, by Theorem 3.5, the cocycle $Df_{\mid E^{cu}\left(f\right)}^{-1}$ has a
dominated splitting with index 1 on the support of the ergodic Lyapunov
maximal observable measures $\mu$.

######  Example 5.3.

By Theorem 2.3, the unique SRB measure $\mu$ is the unique observable measure
for any transitive $C^{1+\alpha}$ Anosov diffeomorphism.

######  Example 5.4.

We are going to show that the inequality in the formula (4.3) could be strict.
Consider the above example, there are infinitely many other ergodic measures,
which are not observable (e.g. those supported on the periodic orbits). Hence,
by [Jen06, Theorem 1], one can find a continuous function whose integral with
respect to some ergodic measure is larger than its integral with respect to
any observable measure $\mu$.

######  Example 5.5.

Let $g:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be a transitive $C^{2}$ Anosov
diffeomorphism. By Sinai’s theorem, there exists a $g$-ergodic physical
measure $\mu$ on the two-torus, which is a SRB measure for $g$. Denote by
$f:\mathbb{T}^{3}\to\mathbb{T}^{3}$ the $C^{2}$ map on the three- dimensional
torus $\mathbb{T}^{3}$, defined by $f(x,y,z)=(x,g(y,z)).$ For Lebesgue almost
all initial states $(x,y,z)\in\mathbb{T}^{3}$, the sequence (1.3) of the
empirical probabilities converges to a measure $\mu_{x}=\delta_{x}\times\mu$,
which is supported on a 1-dimensional unstable manifold injectively immersed
in the two-torus $\\{x\\}\times\mathbb{T}^{2}$. Measures $\mu_{x}$ are
mutually singular for different values of $x\in S^{1}$ as they are supported
on disjoint compact two-tori embedded on $\mathbb{T}^{2}$. The basin of
attraction $B(\mu_{x})$ has zero Lebesgue measure in the ambient manifold,
therefore, none of the probabilities $\mu_{x}$ is physical for $f$. Moreover,
the set of all those measures $\mu_{x}$, which is weak* compact, coincides
with the set $\mathcal{O}$ of observable measures for $f$ by Theorem 2.2.

##  6\. Proof of Theorem A

Tian [Tia18] showed that the negative Lyapunov exponents of all observable
measures imply non-uniform expansion.

######  Theorem 6.1 ([Tia18, Proposition A]).

Let $T:X\to X$ be a topological dynamical system on a compact metric space
$X$. Assume that $a_{n}:X\to\mathbb{R}$, $b_{n}:X\to\mathbb{R}$, $n\geq 0$,
are two sequences of continuous functions such that for every $x\in
X$,$n,k\geq 0$,

(6.1) $a_{n+k}(x)\leq a_{n}(T^{k}(x))+a_{k}(x),\quad b_{n+k}(x)\leq
b_{n}(T^{k}(x))+b_{k}(x);$

and

(6.2) $a_{n}(x)\leq a_{n+k}(x)+b_{k}(T^{n}(x)).$

If

$\lim_{n\to\infty}\frac{1}{n}\int a_{n}(x)d\mu(x)<0$

for every $\mu\in\mathcal{O}$, then there are $\lambda>0$ and $K\in\mathbb{N}$
such that Lebesgue almost every $x$ satisfies

$\limsup_{n\to\infty}\frac{1}{nK}\sum_{i=0}^{n-1}a_{K}(T^{iK}(x))\leq-\lambda<0.$

###### Proof of Theorem A.

Let $\mathcal{A}=Df_{|E^{cu}(f)}^{-1}$. Note that condition (1.5) of Theorem A
is equivalent to

(6.3)
$\operatorname{ess}\sup\limsup_{n\to\infty}\frac{1}{n}\log\|Df^{-n}_{|E^{cu}(f^{n}(x))}\|<0.$

On the other hand,

$\operatorname{ess}\sup\limsup_{n\to\infty}\frac{1}{n}\log\|Df^{-n}_{|E^{cu}(f^{n}(x))}\|=\sup\\{\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\\},$

by Theorem 4.1. Therefore, by (6.3),

(6.4) $\chi(\mu,\Phi_{\mathcal{A}})<0$

for any $\mu\in\mathcal{O}.$

We denote $a_{n}(x):=\log\|Df_{|_{E^{cu}(f^{n}(x))}}^{-n}\|$ and
$b_{n}(x)=\log\|Df_{|E^{cu}(x)}^{n}\|$. They satisfy conditions (6.1) and
(6.2) of Theorem 6.1. Moreover, Lyapunov exponents of all observable measures
are negative (see (6.4)). Then there are $\lambda>0$ and $K\in\mathbb{N}$ such
that Lebesgue a.e. $x$ satisfies

$\limsup_{n\rightarrow+\infty}\frac{1}{nK}\sum_{i=1}^{n}\log\left\|Df_{|E^{cu}(f^{iK}(x))}^{-K}\right\|\leq-\lambda<0.$

by Theorem 6.1.

∎

Let $f:M\to M$ be a $C^{1}$ diffeomorphism on a compact Riemanian manifold
$M$. Let $\mathcal{A}:=Df.$ We say that a Borel invariant measure $\mu$ is
hyperbolic if for $\mu$ a.e. $x$, $\chi_{i}(x,\mathcal{A})\neq 0$ and
$\chi_{i}(x,\mathcal{A})<0<\chi_{k(x)}(x,\mathcal{A})$, which $k(x)\leq\dim
M,$ that is

$\chi_{k(x)}(x,\mathcal{A})\leq\ldots\leq\chi_{t+1}(\mathcal{A})<0<\chi_{t}(x,\mathcal{A})\leq\ldots\leq\chi_{1}(x,\mathcal{A}).$

We say that a Borel invariant measure $\mu$ is SRB (Sinai–Ruelle–Bowen)
measure if $\mu$ is hyperbolic and admitting a system of conditional measures
such that the conditional measures on unstable manifolds are absolutely
continuous with respect to the Lebesgue measures $\operatorname{Leb}$ on these
manifolds induced by the restriction of the Riemannian structure [DV05, Pes77,
You02]; we denote by SRB the set of SRB measures. Recently, [MYZ22] proved
that a partially hyperbolic attractor for a $C^{1}$ vector field with two
dimensional center supports an SRB measure.

######  Corollary 6.2.

Under conditions of Corollary 1.6, let $\mathcal{A}=Df.$ Then

$\displaystyle\operatorname{ess}\sup\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\mathcal{A}^{n}(x)\|$
$\displaystyle=\sup\bigg{\\{}\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\mathcal{O}\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\textbf{Phy}\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}\chi(\mu,\Phi_{\mathcal{A}})\mid\mu\in\textbf{SRB}\bigg{\\}}.$

###### Proof.

First, we are going to show that $\mathcal{O}=\textbf{Phy}=\textbf{SRB}.$ By
Corollary 1.6, there exist finite ergodic SRB measures and the union of their
basins forms a set with Lebesgue full measure; denote them by
$\mu_{1},\ldots,\mu_{k}$. We define define
$\nu_{i}(B)=\frac{1}{K}(\mu_{i}(B)+\mu_{i}(f^{-1}(B))+\ldots+\mu_{i}(f^{-K-1}(B))$
555$K$ appears in Theorem A. for any Borel measurable set $B$, then $\nu_{i}$
is ergodic, and SRB for $f$ that form a compact subset of $\mathcal{O}$ and
the union of the basins of $\nu_{i}$ forms a set with Lebesgue full measure.
$\mathcal{O}$ coincides with these finite $\nu_{i}$, by Theorem 2.2. Then the
result follows from Corollary 4.5. ∎

We consider the largest Lyapunov exponent in the $E^{cs}$-direction

$\lambda^{cs}(x):=\limsup_{n\to\infty}\frac{1}{n}\log\|Df_{|E^{cs}(x)}^{n}(x)\|.$

We say that $E^{cs}$ is mostly contracting if for any local unstable manifold
$\gamma^{u}$, then we have $\lambda^{cs}(x)<0$ for a positive
$\operatorname{Leb}_{\gamma^{u}}$ measure set of points $x\in\gamma^{u}.$

Let $f:M\rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a compact
Riemannian manifold that admits a dominated splitting $TM=E^{cs}\oplus E^{u}$
with $E^{cs}$ mostly contracting. Bonatti and Viana [BV00] showed that $f$ has
finitely many ergodic physical SRB measures and the union of their basins has
full Lebesgue measure. Therefore, one can use their result to prove Corollary
6.2 for the above case.

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