# Exponentially Fast Volume Limits

S. Ben Ovadia⋆, F. Rodriguez-Hertz†

###### Abstract.

Let $M$ be a $d$-dimensional closed Riemannian manifold, let
$f\in\mathrm{Diff}^{1+\beta}(M)$, and denote by $m$ the Riemannian volume form
of $M$. We prove that if $m\circ f^{-n}\xrightarrow[n\to\infty]{}\mu$
exponentially fast (see §1.2), then $\mu$ is an SRB measure.

⋆ Department of Mathematics, Eberly College of Science, Pennsylvania State
University<EMAIL_ADDRESS>
† Department of Mathematics, Eberly College of Science, Pennsylvania State
University<EMAIL_ADDRESS>
##  1\. Introduction

###  1.1. Motivation

An important object in smooth ergodic theory is SRB measures, named after
Sinai, Ruelle, and Bowen. SRB measures are invariant measures whose
conditional measures on unstable leaves are absolutely continuous w.r.t the
induced Riemannian volume on unstable leaves (see [You02] for more details and
properties of SRB measures).

Aside for potential physicality and compatibility with the Riemannian volume
in dissipative systems, SRB measures are important as possible limit points of
the Riemannian volume under the dynamics. In [Bow08] Bowen shows that for
Axiom A attractors which support an SRB measure, the volume measure of the
saturation of the attractor by stable leaves converges exponentially fast
under the dynamics to the unique SRB measure supported on the attractor (the
notion of rate of convergence relates to a fixed space of test functions).
However, in the general case it is not clear if one can expect to always
achieve an SRB measure as a limit point of the pushed Riemannian volume. In
particular, some “nice” systems do not admit an SRB measure (see [HY95]).

This gives rise to the natural question: When can we achieve an SRB measure
through pushing forwards the Riemannian volume of a smooth dynamical system?
Before we describe the results of this paper and how they relate to this
question, we wish to mention another fundamental field of studies in smooth
ergodic theory, and how it relates to this question.

The smooth realization problem posed by von Neumann is the question of what
dynamical systems $(X,T,\nu)$ (not necessarily smooth) can be realized through
a measure theoretic isomorphism as a smooth system $(M,f,m)$, where $M$ is a
closed Riemannian manifold, $f$ is a smooth diffeomorphism of $M$, and
$m=m\circ f^{-1}$ is the Riemannian volume of $M$. Notice that an immediate
restriction of the smoothly-realizable dynamical systems is having finite
metric entropy. A recent advancement in this direction is due to Dolgopyat,
Kanigowski, and Rodriguez-Hertz, where they prove that for smooth systems
which preserve volume, exponential mixing implies Bernoulli ([DKRH]).
Exponential mixing is a property of the smooth structure, as it requires
specifying a space of regular test functions on which the mixing estimates
hold; however their result nonetheless explores a restriction on the ergodic
properties of smooth systems. See also [PSS] and the corresponding discussion
in [Kat23].

A natural extension of the smooth realization problem can then be, what
dynamical systems $(X,T,\nu)$ can be realized through a measure theoretic
isomorphism as a smooth system $(M,f,\mu)$, where $M$ is a closed Riemannian
manifold, $f$ is a smooth diffeomorphism of $M$, and $\mu=\lim_{n}m\circ
f^{-n}$, where $m$ is the Riemannian volume of $M$. Similarly, $\int g\circ
f^{n}hdm\xrightarrow[]{\mathrm{exp}}\int gdm\int hdm$ when $m=m\circ f^{-1}$,
can be naturally extended to $\int g\circ
f^{n}hdm\xrightarrow[]{\mathrm{exp}}\int gd\mu\int hdm$ where $m$ is not
necessarily invariant, but $\mu$ is. Can we say that $\mu$ is Bernoulli in
that case? We believe that the answer is positive based on a consequence of
this work, as we explain in §1.2.

The problem of finding a Banach space of test functions which admits certain
properties is not a trivial issue. Another instance of that same challenge is
proving the spectral gap property, which requires defining a suitable Banach
space of test functions on which the dynamics act as a linear operator with a
spectral gap. Often the space of such test functions is non-trivial in the
sense that one studies functions which are regular on stable leaves, but may
have merely measurable behavior w.r.t the topology of the ambient manifold.

Moreover, the relationship between properties such as a spectral gap (on some
“reasonable” Banach space) and exponential mixing is still an open mystery. In
what cases can one have exponential mixing without a spectral gap? These types
of questions are generally still open, while being fundamental.

Finally, an additional natural property in this family would be the
exponential convergence of the volume to an invariant measure, as in (5). This
property on its own is not enough to conclude any stronger ergodic properties
(e.g $f=\mathrm{Id}_{M}$, or even $f=A\times\mathrm{Id}_{\mathbb{S}^{1}}$
where $A$ is a volume-preserving linear Anosov map of the torus). However, we
show that it is indeed enough to conclude that the limiting measure is an SRB
measure, possibly in the degenerate sense that
$h_{\mu}(f)=\int\sum\chi^{+}d\mu=0$.

To sum up the two independent directions of study we mentioned: We wish to
understand when can limits of the pushed volume be SRB measures for
thermodynamic purposes; and also we wish to understand what properties
restrict smooth systems in terms of ergodic properties and the extended smooth
realization problem. Possible future lines of study include exploring the
relationship between different smooth ergodic properties, such as exponential
mixing and a spectral gap.

###  1.2. Main results

$M$ is a $d$-dimensional closed Riemannian manifold,
$f\in\mathrm{Diff}^{1+\beta}(M)$, and $m$ denotes the Riemannian volume form
of $M$. Let us first introduce three different notions of exponential
convergence: $\exists C>0,\alpha\in(0,1],\gamma>0$ s.t $\forall
g,h\in\mathrm{H\ddot{o}l}_{\alpha}(M)$,

$\left|\frac{1}{N}\sum_{k=n}^{n+N-1}m(g\circ f^{k})-\mu(g)\right|\leq
C\cdot\|g\|_{\alpha}\cdot e^{-\gamma\cdot\min\\{n,N\\}},$ (1)

$\Big{|}m(g\circ f^{n})-\mu(g)\Big{|}\leq C\cdot\|g\|_{\alpha}\cdot e^{-\gamma
n},$ (2)

$\left|\int g\circ f^{n}\cdot hdm-\int gd\mu\int hdm\right|\leq
C\cdot\|g\|_{\alpha}\|h\|_{\alpha}\cdot e^{-\gamma n}.$ (3)

It is clear that (3)$\Rightarrow$(2)$\Rightarrow$(1). We also say that the
volume is almost exponentially mixing (however the volume need not be
$f$-invariant) if

$\left|\int g\circ f^{n}\cdot hdm\right|\leq
C\cdot\|g\|_{\alpha}\|h\|_{\alpha}\cdot e^{-\gamma n}\text{, whenever }\int
hdm=0.$ (4)

Note that (4) is proper even when $m$ is not $f$-invariant; but when $m=m\circ
f^{-1}$ it is equivalent to $m$ being exponentially mixing. The advantage of
condition (4) is that it does not require a-priori a background $f$-invariant
measure in order to test it.

The main results of the manuscript are structured in the following way:

  1. (1)

In §2 we show that exponential convergence in the sense of (1) to an ergodic
limit point implies that the limit point is an SRB measure (not necessarily
with a positive entropy). The purpose of this section is didactic. We show in
addition that in this case where $\mu$ is ergodic, it is also a weakly
physical measure (see Definition 2.7) with a full Basin (see Theorem 2.8).

  2. (2)

In §3 we prove that exponential convergence in the sense of (1) to a limit
point (not necessarily ergodic) implies that the limit point is an SRB measure
(still not necessarily with positive entropy). Note that one cannot expect
more ergodic properties without additional assumptions.

  3. (3)

In §4 we show that almost exponential mixing of the volume in the sense of (4)
implies the existence of an $f$-invariant $\mu$ such that $m\circ
f^{-n}\xrightarrow[n\to\infty]{}\mu$ exponentially fast (in the strong sense
of (3)), and consequently that condition (3) implies that $\mu$ must either be
the unique (and hence ergodic) SRB measure of the system, and has positive
entropy, or that $\mu$ is a Dirac mass at a fixed point which is an SRB
measure in the degenerate sense that $h_{\mu}(f)=\int\sum\chi^{+}d\mu=0$. Note
that the degenerate case cannot be ruled out, as illustrated in the remark
after Theorem 4.2.

In the case we treat in §4, when $h_{\mu}(f)>0$, we believe that the methods
of [DKRH] can be extended to show that $\mu$ is Bernoulli. This is a
consequence of the observation that the proof of [DKRH] only truly requires
the conditional measures on unstable leaves to be smooth, and Proposition 4.3
gives the right notion of exponential mixing on unstable leaves for their
methods to be extended.

In addition, notice that the assumption of (1) is formally weaker than (2).
The weaker assumption allows one to rely on some averaging in order to gain
exponential convergence, rather than just pushing forwards the volume.

Our proof relies on the following tools: We use coverings by exponential Bowen
balls of the form $B(\cdot,n,e^{-n\delta})$ which have the following three
properties:

  1. (1)

$\lim_{\delta\to
0}\limsup\frac{-1}{n}\log\mu(B(\cdot,n,e^{-n\delta}))=h_{\mu_{x}}(f)$
$\mu$-a.e, where $\mu=\int\mu_{x}d\mu(x)$ is its ergodic decomposition (see
[BORH]),

  2. (2)

If $x$ is a Pesin regular point, then for all $n$ large enough, $\forall k\leq
n$, $f^{k}[B(x,n,e^{-n\delta})]$ is contained in the Pesin chart of
$f^{k}(x)$,

  3. (3)

Subsets of Pesin blocks can be covered by exponential Bowen balls with
exponentially low multiplicity, for all $n$ large enough (see [BORH, Lemma
2.2]).

Furthermore, our proof of the results of §4 relies on the construction of fake
$cs$-foliations which are absolutely continuous in small exponential
neighborhoods of Pesin regular points. These fake foliations were constructed
by Dolgopyat, Kanigowski, and Rodriguez-Hertz in [DKRH].

The key idea of the proof of §2 and §3 is a type of shadowing argument, where
since we cannot mix on exponential Bowen balls of $n$ steps, we break down the
orbit segment of $n$ steps into $\frac{1}{\epsilon}$-many orbit segments of
$n\epsilon$-many steps. Thus we can study points which remain close to a large
measure set of “good points”, but not necessarily lie in the Bowen ball of any
“good point” for the whole $n$ steps.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Main results
  2. 2 The ergodic case
    1. 2.1 Preliminary parameter choices
    2. 2.2 Large asymptotic-volume set of points shadowed by a set of $\mu$-good points
    3. 2.3 A cover by exponential Bowen balls via concatenation
    4. 2.4 Physicality
  3. 3 The non-ergodic case
    1. 3.1 Entropy formula via entropy shadowing
  4. 4 Volume is almost exponentially mixing
    1. 4.1 Exponential decay of correlations implies exponential convergence
    2. 4.2 Positive entropy, ergodicity, and uniqueness

##  2\. The ergodic case

###  2.1. Preliminary parameter choices

For didactic purposes, we treat first the ergodic case, as the argument is
much clearer in that case. In this section (and in §3) we assume that $\exists
C,\gamma,\alpha>0$ s.t

$\forall g\in\mathrm{H\ddot{o}l}_{\alpha}(M),\text{
}\left|\frac{1}{N}\sum_{k=n}^{n+N-1}m(g\circ f^{k})-\mu(g)\right|\leq
C\cdot\|g\|_{\alpha}\cdot e^{-\gamma\cdot\min\\{n,N\\}},$ (5)

the weakest notion of exponential convergence. Assume that $\mu$ is ergodic.
Let $\epsilon>0$, and set:

  1. (1)

Let $\mathcal{K}_{\epsilon}$ be a set s.t $\mu(\mathcal{K}_{\epsilon})\geq
e^{-\epsilon^{4}}$, $\mu(B(x,-n\epsilon,e^{-2\delta
n})),\mu(B(x,-n\epsilon,e^{-\delta
n}))=e^{-n\epsilon(h_{\mu}(f)\pm\epsilon^{2})}$ for all $n\geq n_{\epsilon}$,
for some $\delta\in(0,\epsilon^{2})$ (see [BORH]).

  2. (2)

Let $\ell=\ell(\epsilon)\in\mathbb{N}$ s.t
$\mu(\Lambda_{\ell}^{(\underline{\chi},\tau)})\geq e^{-\epsilon^{4}}$, with
$0<\tau<\min\\{\tau_{\underline{\chi}},\frac{1}{3d}\delta^{3}\\}$.

  3. (3)

Set $E_{\epsilon}:=\Lambda_{\ell}\cap\mathcal{K}_{\epsilon}$. Then
$\mu(E_{\epsilon})\geq e^{-\epsilon^{3}}$ for all sufficiently small
$\epsilon>0$. W.l.o.g assume that $\epsilon=\frac{1}{p^{2}}$, and that
$p^{2}|n$ when we choose some large $n$ s.t $e^{-\delta n}\ll\frac{1}{\ell}$,
so the ceiling values can be omitted.111That is, $n$ takes values in
$\\{m\cdot p^{2}\\}_{m\geq 1}$ for some $p\in\mathbb{N}$.

  4. (4)

Let $n\geq n_{\epsilon}$, and let $\widetilde{\mathcal{A}}^{(n)}_{\epsilon}$
be a cover of $E_{\epsilon}$ by Bowen balls $B(\cdot,-n\epsilon,e^{-2\delta
n})$ with multiplicity bounded by $e^{3d\tau n}\leq e^{\delta^{3}n}\leq
e^{\epsilon^{6}n}$, and in particular with cardinality bounded by
$\\#\mathcal{A}^{(n)}_{\epsilon}\leq
e^{n\epsilon(h_{\mu}(f)+\epsilon^{2})}e^{\epsilon^{6}n}$, as in [BORH, Lemma
2.2]. Set $\mathcal{A}^{(n)}_{\epsilon}:=\\{B(x,n\epsilon,e^{-\delta
n}):B(x,n\epsilon,e^{-2\delta
n})\in\widetilde{\mathcal{A}}^{(n)}_{\epsilon}\\}$.

###  2.2. Large asymptotic-volume set of points shadowed by a set of
$\mu$-good points

######  Lemma 2.1.

Let $0\leq i\leq(1-2\sqrt{\epsilon})\frac{1}{\epsilon}$. Then for all
$\epsilon>0$ sufficiently small, $\exists n_{\epsilon}^{\prime}\geq
n_{\epsilon}$ s.t for all $n\geq n_{\epsilon}^{\prime}$,

$\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(M\setminus\bigcup\mathcal{A}^{(n)}_{\epsilon})\leq\epsilon^{2}.$

###### Proof.

$\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(M\setminus\bigcup\mathcal{A}^{(n)}_{\epsilon})=1-\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(\bigcup\mathcal{A}^{(n)}_{\epsilon})=1-\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(\mathbb{1}_{\bigcup\mathcal{A}^{(n)}_{\epsilon}})$.

For every $B\in\mathcal{C}^{(n)}_{\epsilon}$, define $g^{(n)}_{B}$ be a
Lipschitz function s.t $g^{(n)}_{B}|_{B(x_{B},n\epsilon,e^{-2\delta n})}=1$,
$g^{(n)}_{B}|_{B(x_{B},n\epsilon,e^{-\delta n})^{c}}=0$, and
$\mathrm{Lip}(g^{(n)})\leq e^{2n\epsilon\log M_{f}}$, where
$M_{f}:=\max_{M}\\{\|d_{\cdot}f\|,\|d_{\cdot}f^{-1}\|\\}$. Notice:
$\mathbb{1}_{\bigcup\mathcal{A}^{(n)}_{\epsilon}}=\max_{B\in\mathcal{A}^{(n)}_{\epsilon}}\mathbb{1}_{B}\geq\max_{B\in\mathcal{C}_{\epsilon}^{(n)}}g^{(n)}_{B}=:g^{(n)}$.

Claim: $\mathrm{Lip}(g^{(n)})\leq e^{2n\epsilon\log M_{f}}$.

Proof: We prove that if $g_{1}$ and $g_{2}$ are $L$-Lipschitz, then $g_{1}\vee
g_{2}:=\max\\{g_{1},g_{2}\\}$ is $L$-Lipschitz. The claim for $g^{(n)}$
follows by induction. Let $x,y\in M$. If $g_{1}(x)\geq g_{2}(x)$ and
$g_{1}(y)\geq g_{2}(y)$, or if $g_{2}(x)\geq g_{1}(x)$ and $g_{2}(y)\geq
g_{1}(y)$, then

$\frac{|(g_{1}\vee g_{2})(x)-(g_{1}\vee g_{2})(y)|}{|x-y|}\leq L$

by the Lipschitz properties of $g_{1}$ and $g_{2}$.

We therefore may assume that w.l.o.g $g_{1}(x)\geq g_{2}(x)$ and $g_{1}(y)\leq
g_{2}(y)$ (otherwise switch the roles of $g_{1}$ and $g_{2}$). Then,

$g_{1}(x)\leq L\cdot|x-y|+g_{1}(y)\leq L\cdot|x-y|+g_{2}(y),$

and so

$g_{1}(x)-g_{2}(y)\leq L\cdot|x-y|.$

Similarly,

$g_{2}(y)-g_{1}(x)\leq L\cdot|x-y|.$

Therefore, $|g_{1}(y)-g_{2}(x)|\leq L\cdot|x-y|$, and so

$|(g_{1}\vee g_{2})(x)-(g_{1}\vee g_{2})(y)|=|g_{1}(x)-g_{2}(y)|\leq
L\cdot|x-y|.$

This concludes the proof of the claim.

By the exponential convergence of the volume averages given by (5),

$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(\mathbb{1}_{\bigcup\mathcal{A}^{(n)}_{\epsilon}})\geq$
$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(g^{(n)})=\mu(g^{(n)})\pm Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}$ $\displaystyle\geq$
$\displaystyle\mu(\max_{B\in\widetilde{\mathcal{A}}^{(n)}_{\epsilon}}\mathbb{1}_{B})-Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}=\mu(\bigcup\widetilde{\mathcal{A}}^{(n)}_{\epsilon})-Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}$ $\displaystyle\geq$
$\displaystyle\mu(E_{\epsilon})-Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}\geq e^{-\epsilon^{2}}-Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}.$

Then for all $\epsilon>0$ s.t $\gamma\sqrt{\epsilon}>2\log M_{f}\epsilon$ and
$\frac{\epsilon^{4}}{2}\geq 2\frac{\epsilon^{6}}{6}$, and for all $n$ large
enough so $Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon\log
M_{f}n}\leq\frac{\epsilon^{6}}{6}$, we have

$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(M\setminus\bigcup\mathcal{A}^{(n)}_{\epsilon})\leq
1-e^{-\epsilon^{2}}+Ce^{-\gamma\sqrt{\epsilon}n}e^{2\epsilon n\log
M_{f}}\leq\epsilon^{2}-\frac{\epsilon^{4}}{2}+\frac{\epsilon^{6}}{6}+\frac{\epsilon^{6}}{6}\leq\epsilon^{2}.$

∎

######  Definition 2.2.

$\mathcal{S}_{n}:=\\{x\in\bigcup\mathcal{A}^{(n)}_{\epsilon}:\text{for all
}0\leq
i\leq(1-2\sqrt{\epsilon})\frac{1}{\epsilon},f^{-in\epsilon}(x)\in\bigcup\mathcal{A}^{(n)}_{\epsilon}\\}$

is the set of points in $\bigcup\mathcal{A}^{(n)}_{\epsilon}$ which are
shadowed by $E_{\epsilon}$ for at least $(1-2\sqrt{\epsilon})n$-many steps
backwards.

######  Theorem 2.3.

Let $n^{\prime}_{\epsilon}\geq 0$ as in Lemma 2.1, then for all $\epsilon>0$
sufficiently small and for all $n\geq n_{\epsilon}^{\prime}$,

$\frac{1}{\sqrt{\epsilon}n}\sum_{k=n(1-\sqrt{\epsilon})}^{n-1}m\circ
f^{-k}(\mathcal{S}_{n})\geq e^{-\epsilon^{\frac{3}{4}}}.$

###### Proof.

Let $n\geq n_{\epsilon}^{\prime}$. Let $B\in\mathcal{C}^{(n)}_{\epsilon}$. We
break down the pull-back of $B$ as follows:
$f^{-n}[B]=f^{-n\epsilon}\circ\cdots\circ f^{-n\epsilon}[B]$, where the
composition chain has $\frac{1}{\epsilon}$ many steps.

$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=n(1-\sqrt{\epsilon})}^{n-1}m\circ
f^{-k}(\mathcal{S}_{n})=$
$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=n(1-\sqrt{\epsilon})}^{n-1}m\circ
f^{-k}(\\{x\in\bigcup\mathcal{A}^{(n)}_{\epsilon}:\text{for all }0\leq
i\leq(1-2\sqrt{\epsilon})\frac{1}{\epsilon},f^{-in\epsilon}(x)\in\bigcup\mathcal{A}^{(n)}_{\epsilon}\\})$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=n(1-\sqrt{\epsilon})}^{n-1}m\circ
f^{-k}\left(\bigcap_{i=0}^{(1-2\sqrt{\epsilon})\frac{1}{\epsilon}}f^{in\epsilon}[\bigcup\mathcal{A}^{(n)}_{\epsilon}]\right)$
$\displaystyle\geq$ $\displaystyle
1-\sum_{i=0}^{(1-2\sqrt{\epsilon})\frac{1}{\epsilon}}\frac{1}{\sqrt{\epsilon}n}\sum_{k=n(1-\sqrt{\epsilon})}^{n-1}m\circ
f^{-k}(M\setminus f^{in\epsilon}[\bigcup\mathcal{A}^{(n)}_{\epsilon}])$
$\displaystyle=$ $\displaystyle
1-\sum_{i=0}^{(1-2\sqrt{\epsilon})\frac{1}{\epsilon}}\frac{1}{\sqrt{\epsilon}n}\sum_{k=-i\epsilon
n+n(1-\sqrt{\epsilon})}^{-i\epsilon n+n-1}m\circ
f^{-k}(M\setminus\bigcup\mathcal{A}^{(n)}_{\epsilon})$ $\displaystyle\geq$
$\displaystyle
1-(\frac{1-2\sqrt{\epsilon}}{\epsilon}+1)\cdot\epsilon^{2}\text{
}(\because\text{Lemma }\ref{firstLemma}).$

For all $\epsilon>0$ small enough so $1-\epsilon-\epsilon^{2}\geq
e^{-\epsilon^{\frac{3}{4}}}$, the theorem follows. ∎

###  2.3. A cover by exponential Bowen balls via concatenation

######  Definition 2.4.

Let
$\mathcal{C}_{\epsilon}^{(n)}:=\bigvee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\mathcal{A}_{\epsilon}^{(n)}]$.

Remark: Notice that $\mathcal{C}_{\epsilon}^{(n)}$ covers $\mathcal{S}_{n}$,
and that
$\\#\mathcal{C}_{\epsilon}^{(n)}\leq(\\#\mathcal{A}_{\epsilon}^{(n)})^{\frac{1}{\epsilon}}\leq
e^{\frac{1}{\epsilon}(n\epsilon h_{\mu}(f)+\epsilon^{2.5}n)}\leq
e^{nh_{\mu}(f)+\epsilon^{\frac{3}{2}}n}$.

######  Lemma 2.5.

Let $B\in\mathcal{C}_{\epsilon}^{(n)}$, then for all $n$ large enough and
$\epsilon>0$ small enough (independently of $B$),
$\frac{1}{\sqrt{\epsilon}}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(B)\leq e^{-\chi^{u}n+2\epsilon^{\frac{1}{3}}n}$, where
$e^{-\chi^{u}n}:=\prod_{\chi_{i}>0}e^{-\chi^{u}_{i}n}$.

###### Proof.

Let $y\in B$, then $B\subseteq B(y,-n(1-2\sqrt{\epsilon}),2e^{-\delta n})$.
For every $k\in[(1-\sqrt{\epsilon}n),n-1]$, $m\circ f^{-k}[B]\leq
e^{2d\sqrt{\epsilon}n\log
M_{f}}m(B(f^{-n(1-2\sqrt{\epsilon})}(y),n(1-2\sqrt{\epsilon}),2e^{-\delta
n}))$. We show that
$m(B(f^{-n(1-2\sqrt{\epsilon})}(y),n(1-2\sqrt{\epsilon}),2e^{-\delta n}))\leq
e^{-(\chi^{u}-\epsilon)n+2\epsilon^{\frac{1}{3}}n}$.

Write $x:=f^{-n(1-2\sqrt{\epsilon})}(y)$. Let $x_{i}$ s.t $f^{n\epsilon
i}(x)\in B(x_{i},n\epsilon,2e^{-\delta n})$, $x_{i}\in
f^{-n\epsilon}[E_{\epsilon}]$, $0\leq
i\leq\frac{1-2\sqrt{\epsilon}}{\epsilon}$.

Assume for contradiction that there exists a volume form
$\omega^{u}_{0}\in\wedge^{\mathrm{dim}H^{u}(\mu)}T_{y}M$ s.t
$|\omega^{u}_{0}|\geq e^{-(\chi^{u}-\epsilon)n+\epsilon^{\frac{1}{3}}n}$, and
s.t
$\sphericalangle(d_{x}\psi_{x_{0}}^{-1}\omega^{u}_{0},E^{u})\leq\epsilon^{2}$,
where $E^{u}$ is the unstable direction of $x_{0}$ in its Pesin chart
$\psi_{x_{0}}$; and finally assume that $\exp_{x}\omega^{u}_{0}\subseteq
f^{-n(1-2\sqrt{\epsilon})}[B]$ (when we think of $\omega^{u}_{0}$ as the
parallelogram it defines in $T_{x}M$). We will show a contradiction by showing
that $f^{n(1-2\sqrt{\epsilon})}[\exp_{x}\omega^{u}_{0}]$ contains a geodesic
of length greater than $2e^{-\delta n}$, which contradicts
$B(f^{n(1-2\sqrt{\epsilon})}(x),n(1-2\sqrt{\epsilon}),2e^{-\delta n})\supseteq
B\supseteq f^{n(1-2\sqrt{\epsilon})}[\exp_{x}\omega^{u}_{0}]$.

The choice of $\omega_{0}^{u}$ implies that
$|d_{x}f^{n\epsilon}\omega_{0}^{u}|\geq
e^{-(\chi^{u}-\epsilon)n(1-\epsilon)+\epsilon^{\frac{1}{3}}n}(1-\epsilon)$.
Note, since
$f^{n\epsilon}(x_{0}),f^{n\epsilon}(x_{1})\in\Lambda^{(\underline{\chi},\tau)}_{\ell}$,
we have
$f^{n\epsilon}(x_{0}),x_{1}\in\Lambda^{(\underline{\chi},\tau)}_{e^{\tau
n\epsilon}\ell}$, while also having $2e^{-\delta n}\ll e^{-\tau
n\epsilon}\frac{1}{\ell}$ for all $n$ large enough. By the Hölder continuity
of the unstable spaces of points in
$\Lambda^{(\underline{\chi},\tau)}_{e^{\tau n\epsilon}\ell}$ ([KdlLPW01,
Appendix A]), $d_{x}f^{n\epsilon}\omega^{u}_{0}$ projects to
$\omega^{u}_{1}\in\wedge^{\mathrm{dim}H^{u}(\mu)}T_{f^{n\epsilon}(x)}M$ s.t
$\sphericalangle(d_{x}\psi_{x_{1}}^{-1}\omega^{u}_{1},E^{u})\leq\epsilon^{2}$
and s.t $|\omega_{1}^{u}|\geq(1-\epsilon)\cdot
e^{-(\chi^{u}-\epsilon)n(1-\epsilon)+\epsilon^{\frac{1}{3}}n}(1-\epsilon)$.

Continuing by induction, let
$\omega^{u}_{\frac{1-2\sqrt{\epsilon}}{\epsilon}}$ be a component of
$d_{x}f^{(1-2\sqrt{\epsilon})n}\omega_{0}^{u}$ s.t
$|\omega^{u}_{\frac{1-2\sqrt{\epsilon}}{\epsilon}}|\geq
e^{\epsilon^{\frac{1}{3}}n-2\sqrt{\epsilon}(\chi^{u}-\epsilon)n}(1-\epsilon)^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}$.

Now, for all $\epsilon>0$ small enough so $2d\sqrt{\epsilon}\log
M_{f}+\epsilon>\epsilon^{\frac{1}{3}}$, we have for all $n$ large enough,
$|\omega^{u}_{\frac{1-2\sqrt{\epsilon}}{\epsilon}}|\geq 2^{d}e^{-\delta n}$,
and we are done. ∎

######  Corollary 2.6.

$h_{\mu}(f)=\chi^{u}=:\sum_{\chi_{i}(\mu)>0}\chi_{i}(\mu)$.

###### Proof.

Let $\epsilon>0$ small enough and $n$ large enough for Theorem 2.3 and Lemma
2.5. Then,

$\displaystyle e^{-\epsilon^{\frac{3}{4}}}\leq$
$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\mathcal{S}_{n})\leq\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\bigcup\mathcal{C}_{\epsilon}^{(n)})$ $\displaystyle\leq$
$\displaystyle\\#\mathcal{C}_{\epsilon}^{(n)}\cdot\max_{B\in\mathcal{C}_{\epsilon}^{(n)}}\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(B)\leq e^{nh_{\mu}(f)+\epsilon^{\frac{3}{2}}n}\cdot
e^{-\chi^{u}n+2\epsilon^{\frac{1}{3}}n}.$

Whence since this holds for arbitrarily large $n$,
$h_{\mu}(f)\geq\chi^{u}-3\epsilon^{\frac{1}{3}}$. Since $\epsilon>0$ is
arbitrarily small, and by the Ruelle inequality, $h_{\mu}(f)=\chi^{u}$. ∎

###  2.4. Physicality

Given a subsequence $(n_{k})_{k}\subseteq\mathbb{N}$, we denote by
$\overline{d}((n_{k})_{k})$ the upper-density
$\limsup_{N}\frac{1}{N}\\#\\{k:n_{k}\leq N\\}$.

######  Definition 2.7.

A Borel probability measure $\nu$ is called weakly physical with a basin

$B_{\nu}:=\left\\{y\in M:\exists n_{k}\uparrow\infty\text{ with
}\overline{d}((n_{k})_{k})=1\text{ s.t }\forall g\in
C(M),\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}g\circ f^{j}(y)\rightarrow\int
gd\nu\right\\}$

if $m(B_{\nu})>0$. If $m(B_{\nu})=1$, we say that $\nu$ is weakly physical
with a full basin.

######  Theorem 2.8 (Weak physicality).

The measure $\mu$ is weakly physical with a full basin.

###### Proof.

The idea of the proof is similar to how we prove the entropy formula for
$\mu$, where we show that the volume of points which shadow “good”
$\mu$-points is large, but we add an additional restriction to the set of
“good” $\mu$-points asking them to be $\mu$-generic; and show that
trajectories which shadow them inherit this property.

Since $C(M)$ is separable, there exists a countable set
$\mathcal{L}(M)\subseteq C(M)$ s.t $\forall h\in C(M)$, for all $\tau>0$,
there exists $g\in\mathcal{L}(M)$ s.t $\|g-h\|_{\infty}\leq\tau$.

Let $\epsilon>0$, and for $g\in\mathcal{L}(M)$ let $L(\cdot)$ be a monotone
uniform-continuity function of $g$.222That is,
$\forall\widetilde{\epsilon}>0$, $d(x,y)\leq
L(\widetilde{\epsilon})\Rightarrow|g(x)-g(y)|\leq\widetilde{\epsilon}$. Write
$\mathcal{L}(M)=\\{g_{i}\\}_{i\geq 0}$, and set
$L_{\epsilon}:=\min_{i\leq\frac{1}{\epsilon}}L_{g_{i}}$. We may assume w.l.o.g
that for all $g\in\mathcal{L}(M)$, $\|g\|_{\infty}\in(0,1]$.

Step 1: Further restricted set of “good” $\mu$-points.

Proof: Consider the set $E_{\epsilon}$ from §2.1, and assume further w.l.o.g
that for every $x\in E_{\epsilon}$, for all $n\geq n_{\epsilon}$,

$\frac{1}{n}\sum_{k=0}^{n-1}g_{i}\circ f^{k}(x)=\int
g_{i}d\mu\pm\epsilon^{2},\text{ for all }i\leq\frac{1}{\epsilon}.$ (6)

Step 2: Large volume of points shadowing “good” $\mu$-points.

Proof: Recall the definition of the set $\mathcal{S}_{n}$ from Definition 2.2,
and recall Theorem 2.3. Notice, for every
$n\geq(n_{\epsilon}\frac{1}{\epsilon^{2}}\cdot\frac{1}{1-2\sqrt{\epsilon}})^{3}$,
for every $\epsilon^{\frac{1}{3}}n\leq N\leq n(1-2\sqrt{\epsilon})$, for every
$y\in f^{-n}[\mathcal{S}_{n}]$, for all $\epsilon>0$ small and $n$ large w.r.t
$\epsilon$, for all $g\in\\{g_{i}\\}_{i\leq\frac{1}{\epsilon}}$,

$\frac{1}{N}\sum_{k=0}^{N-1}g\circ f^{k}(y)=\int
gd\mu\pm(L_{\epsilon}^{-1}(e^{-\delta
n})+3\epsilon^{\frac{1}{6}}\|g\|_{\infty})=\int
gd\mu\pm\epsilon^{\frac{1}{7}}.$ (7)

Set

$\mathcal{T}_{\epsilon}^{n}:=f^{-n}[\mathcal{S}_{n}]\text{ and
}\mathcal{T}_{\epsilon}:=\limsup_{n}\mathcal{T}_{\epsilon}^{n}.$

Notice that $m(\mathcal{T}_{\epsilon})=\lim_{n}m(\bigcup_{n^{\prime}\geq
n}\mathcal{T}_{\epsilon}^{n^{\prime}})\geq e^{-\epsilon^{\frac{3}{4}}}$, and
for every $y\in\mathcal{T}_{\epsilon}$, there exists
$(n_{k}^{\epsilon})_{k}\subseteq\mathbb{N}$ with
$\overline{d}((n_{k}^{\epsilon})_{k})\geq 1-\epsilon^{\frac{1}{3}}$ s.t for
all $k$, for all $g\in\\{g_{i}\\}_{i\leq\frac{1}{\epsilon}}$,

$\frac{1}{n_{k}^{\epsilon}}\sum_{j=0}^{n_{k}^{\epsilon}-1}g\circ f^{j}(y)=\int
gd\mu\pm\epsilon^{\frac{1}{7}}.$ (8)

Step 3: Full volume of future-generic points on an upper-density $1$
subseueqnce.

Proof: Let $\epsilon_{\ell}\downarrow 0$, and let
$\widetilde{\mathcal{T}}:=\limsup\limits_{\ell}\mathcal{T}_{\epsilon_{\ell}}$.
In particular,
$m(\mathcal{T})=\lim_{\ell_{0}}m(\bigcup\limits_{\ell\geq\ell_{0}}\mathcal{T}_{\epsilon_{\ell}})\geq\liminf\limits_{\ell_{0}}e^{-\epsilon_{\ell_{0}}^{\frac{3}{4}}}=1$.

Let $y\in\widetilde{\mathcal{T}}$ and define recursively a sequence in the
following way: Let $\epsilon_{\ell_{i}}\downarrow 0$ s.t $y\in\cap_{i\geq
0}\mathcal{T}_{\epsilon_{\ell_{i}}}$. Let
$(n_{k})_{k=0}^{N_{0}}:=(\lceil\epsilon_{\ell_{0}}^{\frac{1}{3}}n_{0}^{\epsilon_{\ell_{0}}}\rceil,\ldots,n_{0}^{\epsilon_{\ell_{0}}})$;
now assume we have $(n_{k})_{k=0}^{N_{i}}$, let $k_{\ell_{i+1}}$ be the first
$k$ s.t
$\epsilon_{\ell_{i+1}}^{\frac{1}{3}}n_{k}^{\epsilon_{\ell_{i+1}}}>n_{N_{i}}$,
and add
$(\lceil\epsilon_{\ell_{i+1}}^{\frac{1}{3}}n_{k_{\ell_{i+1}}}^{\epsilon_{\ell_{i+1}}}\rceil,\ldots,n_{k_{\ell_{i+1}}}^{\epsilon_{\ell+1}})$
to the end of $(n_{k})_{k=0}^{N_{\ell}}$. In particular,
$\overline{d}((n_{k})_{k})\geq 1-\epsilon_{\ell_{i}}^{\frac{1}{3}}$ for all
$\ell_{i}$, whence $\overline{d}((n_{k})_{k})=1$.

Now, for every $y\in\widetilde{\mathcal{T}}$, for every $i\geq 0$, for all
$k\geq 0$, for all $g\in\mathcal{L}(M)$,

$\displaystyle\int
gd\mu-\epsilon^{\frac{1}{7}}_{\ell_{i}}\leq\liminf\frac{1}{n_{k}^{\epsilon_{\ell_{i}}}}\sum_{j=0}^{n_{k}^{\epsilon_{\ell_{i}}}-1}g\circ
f^{j}(y)\leq\limsup\frac{1}{n_{k}^{\epsilon_{\ell_{i}}}}\sum_{j=0}^{n_{k}^{\epsilon_{\ell_{i}}}-1}g\circ
f^{j}(y)=\int gd\mu+\epsilon^{\frac{1}{7}}_{\ell_{i}},$
$\displaystyle\Rightarrow$
$\displaystyle\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}g\circ
f^{j}(y)\xrightarrow[k\to\infty]{}\int gd\mu.$ (9)

Step 4: $\mu$ is weakly physical with a full basin.

Proof: Let

$\mathcal{T}:=\left\\{y\in M:\exists n_{k}\uparrow\infty\text{ with
}\overline{d}((n_{k})_{k})=1\text{ s.t }\forall
g\in\mathcal{L}(M),\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}g\circ
f^{j}(y)\xrightarrow[k\to\infty]{}\int gd\mu\right\\},$

then $\widetilde{\mathcal{T}}\subseteq\mathcal{T}$ and so $m(\mathcal{T})=1$.
Given $y\in\mathcal{T}$, $\tau>0$, and $h\in C(M)$, let $g\in\mathcal{L}(M)$
s.t $\|g-h\|_{\infty}\leq\tau$. then for all $k$ large enough w.r.t $g$,

$\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}h\circ
f^{j}(y)=\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}g\circ f^{j}(y)\pm\tau=\int
gd\mu\pm 2\tau=\int hd\mu\pm 3\tau.$

Then $\frac{1}{n_{k}}\sum_{j=0}^{n_{k}-1}h\circ
f^{j}(y)\xrightarrow[k\to\infty]{}\int hd\mu$. ∎

Remark: Note that the weak physicality of $\mu$ does not depend on $\mu$ being
hyperbolic, nor on the absolute continuity of the stable foliation.

######  Corollary 2.9.

There exists at most one physical measure for $(M,f)$, and if it exists it
must be $\mu$.

##  3\. The non-ergodic case

###  3.1. Entropy formula via entropy shadowing

Up until now we treated the case where $\mu$ is ergodic. This simplification
serves as a didactic tool to make the proof more intuitive and easy to follow;
The proof in the non-ergodic case is more complicated, since we wish to
eventually prove that $\int h_{\mu_{x}}(f)d\mu(x)=\int\chi^{u}(x)d\mu(x)$
(where $\mu=\int\mu_{x}d\mu(x)$ is the ergodic decomposition of $\mu$ and
$\chi^{u}(x):=\sum\chi^{+}(x)$), however it may be that neither of these
functions are constant $\mu$-a.e; And so in particular we can neither control
$\\#\vee_{i=0}^{\frac{1}{\epsilon}}f^{in\epsilon}[\mathcal{A}_{\epsilon}^{(n)}]$
by $e^{nh_{\mu}(f)}=e^{n\int h_{\mu_{x}}(f)d\mu(x)}$, nor is the volume of
each element in
$\vee_{i=0}^{\frac{1}{\epsilon}}f^{in\epsilon}[\mathcal{A}_{\epsilon}^{(n)}]$
controlled by $e^{-n\int\chi^{u}d\mu}$. While $\chi^{u}(\cdot)$ is continuous
on Pesin blocks, $h_{\mu_{x}}$ is merely measurable.

We treat this added difficulty by restricting to Lusin sets, and use a sort of
“entropy shadowing” property, where if a trajectory remains close to different
points with good local entropy estimates, then all said shadowed points must
have similar local entropy.

######  Theorem 3.1.

$h_{\mu}(f)=\int\chi^{u}(x)d\mu(x)$.

###### Proof.

Let $\mu=\int\mu_{x}d\mu(x)$ be the ergodic decomposition of $\mu$. Let
$\epsilon>0$, and let
$E_{j_{h},j_{\chi_{1}},\ldots,j_{\chi_{d}}}:=\\{x:h_{\mu_{x}}(f)\in[j_{h}\cdot\epsilon^{5}-\frac{\epsilon^{5}}{2},j_{h}\cdot\epsilon^{5}+\frac{\epsilon^{5}}{2}),\chi_{i}(x)\in(j_{\chi_{i}}\cdot\epsilon^{5}-\frac{\epsilon^{5}}{2},j_{\chi_{i}}\cdot\epsilon^{5}+\frac{\epsilon^{5}}{2}],i\leq
d\\}$, $\underline{j}\in\\{0,\ldots,\frac{2d\log
M_{f}}{\epsilon^{5}}\\}^{d+1}$. Assume w.l.o.g that $\mu(E_{\underline{j}})>0$
for all $\underline{j}$, and let
$\rho_{\epsilon}:=\epsilon\cdot(\min_{\underline{j}}\\{\mu(E_{\underline{j}})\\})^{4}>0$.

For each $\underline{j}$, we define the set
$E_{\rho_{\epsilon}}^{\underline{j}}$ as in §2.1, for the measure
$\mu_{\underline{j}}:=\frac{1}{\mu(E_{\underline{j}})}\int_{E_{\underline{j}}}\mu_{x}d\mu(x)$.
Then it follows that
$\mu(\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{\underline{j}}E^{\underline{j}}_{\epsilon})\geq
e^{-\rho_{\epsilon}^{3}}$.

Let $L_{\epsilon}$ be a Lusin set for the function $x\mapsto h_{\mu_{x}}(f)$
s.t $\forall\underline{j}$,
$\mu_{\underline{j}}(L^{\underline{j}}_{\epsilon})\geq
e^{-2\rho_{\epsilon}^{3}}$, where
$L^{\underline{j}}_{\epsilon}:=L_{\epsilon}\cap E_{\underline{j}}$. Since the
Lusin theorem tells us that $L_{\epsilon}$ can be chosen to be closed, there
exists $0<r_{\epsilon}:=\frac{1}{2}\sup\\{r>0:\forall x,y\in
L_{\epsilon},d(x,y)\leq
r\Rightarrow|h_{\mu_{x}}(f)-h_{\mu_{y}}(f)|\leq\epsilon^{5}\\}$. Similarly,
assume that $L_{\epsilon}$ is a Lusin set for the function
$x\mapsto\underline{\chi}(x)$, with the same estimates w.r.t the
$|\cdot|_{\infty}$-norm.

Finally, given $n\in\mathbb{N}$, set
$G_{\epsilon}^{\underline{j},n}:=L^{\underline{j}}_{\epsilon}\cap
f^{n\epsilon}[L^{\underline{j}}_{\epsilon}]$, and so
$\mu(\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{\underline{j}}G_{\epsilon}^{\underline{j},n})\geq
e^{-\rho_{\epsilon}^{2}}$.

Cover each $G_{\epsilon}^{\underline{j},n}$ with
$\widetilde{\mathcal{A}}_{\epsilon}^{\underline{j},n}$\- a cover by
exponential Bowen balls $B(\cdot,-n\epsilon,e^{-2\delta n})$, as in §2.1.
Hence
$\\#\mathcal{A}_{\epsilon}^{\underline{j},n}=e^{n\epsilon(h_{\mu_{j}}(f)\pm
2\epsilon^{2})}$, where
$\mathcal{A}_{\epsilon}^{\underline{j},n}:=\\{B(x,n\epsilon,e^{-\delta
n}):B(x,n\epsilon,e^{-2\delta
n})\in\widetilde{\mathcal{A}}_{\epsilon}^{\underline{j},n}\\}$, similarly to
§2.1.

Set
$\mathcal{S}_{n}:=\bigcap_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\bigcup_{\underline{j}}\bigcup\mathcal{A}_{\epsilon}^{\underline{j},n}]$.
As in Theorem 2.3,
$\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\mathcal{S}_{n})\geq e^{-\rho_{\epsilon}^{\frac{3}{4}}}$ (for all
$\epsilon>0$ sufficiently small).

Then, for any $\underline{j}$, as in Lemma 2.1, for all $n$ large enough (s.t
$\epsilon=\frac{1}{N^{6}}$ and $N^{6}|n$),
$\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\bigcup\mathcal{A}_{\epsilon}^{\underline{j},n})\geq\frac{1}{2}\mu(G_{\epsilon}^{\underline{j},n})\geq\frac{1}{2}\mu(E_{\underline{j}})e^{-\rho_{\epsilon}^{2}}\gg
2\rho_{\epsilon}^{\frac{3}{4}}$, whence

$\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\bigcup\mathcal{A}_{\epsilon}^{\underline{j},n}\cap\mathcal{S}_{n})\geq\frac{1}{5}\mu(E_{\underline{j}})\geq\rho_{\epsilon}.$
(10)

Notice, given
$B\in\vee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\bigcup_{\underline{j}}\mathcal{A}_{\epsilon}^{\underline{j},n}]$
s.t
$B=\bigcap_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[B_{i}]$
with $B_{i}\in\mathcal{A}_{\epsilon}^{\underline{j}^{i},n}$, and given
$D\in\mathcal{A}_{\epsilon}^{\underline{j},n}$ s.t $D\cap B\neq\varnothing$,
we have

$|h_{\mu_{\underline{j}}}(f)-h_{\mu_{\underline{j}^{i}}}(f)|\leq\epsilon^{3}\text{
and
}|\int\chi^{u}d\mu_{\underline{j}}-\int\chi^{u}d\mu_{\underline{j}^{i}}|\leq\epsilon^{3}\text{
for all }i\leq\frac{1-2\sqrt{\epsilon}}{\epsilon}$ (11)

(as long as $n$ is large enough so $2e^{-\delta n}\leq r_{\epsilon}$, since
$h_{\mu_{\cdot}}(f)=h_{\mu_{f^{n\epsilon}(\cdot)}}(f)$ and
$\underline{\chi}(\cdot)=\underline{\chi}(f^{n\epsilon}(\cdot))$).

Write
$\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}:=\\{D\in\mathcal{A}_{\epsilon}^{\underline{j},n}:\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(D)\leq
e^{\epsilon^{\frac{3}{2}}n}\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(D\cap\mathcal{S}_{n})\\}$ and
$\check{\mathcal{A}}_{\epsilon}^{\underline{j},n}:=\mathcal{A}_{\epsilon}^{\underline{j},n}\setminus\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}$.
Then $\\#\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}\geq
e^{n\epsilon(h_{\mu_{j}}(f)-2\epsilon^{2})}e^{-\epsilon^{\frac{3}{2}}n}$;
otherwise

$\displaystyle 0<\rho_{\epsilon}\leq$
$\displaystyle\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(\bigcup\mathcal{A}_{\epsilon}^{\underline{j},n}\cap\mathcal{S}_{n})$
$\displaystyle\leq$
$\displaystyle\\#\check{\mathcal{A}}_{\epsilon}^{\underline{j},n}\cdot
e^{-n\epsilon(h_{\mu_{\underline{j}}}(f)-\epsilon^{2})}e^{-n\epsilon^{\frac{3}{2}}}+e^{n\epsilon(h_{\mu_{\underline{j}}}(f)-\epsilon^{2})-n\epsilon^{\frac{1}{3}}}\cdot
e^{-n\epsilon^{\frac{1}{3}}(h_{\mu_{\underline{j}}}(f)-\epsilon^{2})}\leq
2e^{-\frac{1}{2}n\epsilon^{\frac{3}{2}}}\xrightarrow[n\to\infty]{}0,$

a contradiction!

Now, recall that $\mathcal{A}_{\epsilon}^{\underline{j},n}$ is a cover of
multiplicity bounded by $e^{2\epsilon^{2}n}$, and hence so is
$\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}$. As in [BORH, Lemma 2.2],
there exists a pair-wise disjoint sub-cover
$\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}\subseteq\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}$
s.t
$\\#\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}\geq\\#\hat{\mathcal{A}}_{\epsilon}^{\underline{j},n}e^{-2\epsilon^{2}n}$.
Finally, notice that
$\vee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\vee_{\underline{j}^{\prime}}\mathcal{A}_{\epsilon}^{\underline{j}^{\prime},n}]$
refines $\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}$. Therefore it
follows that for any $\underline{j}$, there exists
$D_{\underline{j}}\in\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}$ s.t

$\displaystyle\\#\\{B\in\vee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\vee_{\underline{j}^{\prime}}\mathcal{A}_{\epsilon}^{\underline{j}^{\prime},n}]:B\cap
D_{\underline{j}}\neq\varnothing\\}\leq$
$\displaystyle\frac{\\#\\{B\in\vee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\vee_{\underline{j}^{\prime}}\mathcal{A}_{\epsilon}^{\underline{j}^{\prime},n}]:\exists
D^{\prime}\in\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}\text{ s.t
}B\cap
D^{\prime}\neq\varnothing\\}}{\\#\overline{\mathcal{A}}_{\epsilon}^{\underline{j},n}}$
$\displaystyle\leq$ $\displaystyle\frac{C_{\epsilon}\cdot
e^{n(1-2\sqrt{\epsilon})(h_{\mu_{\underline{j}}(f)}+\epsilon^{3})}}{e^{n\epsilon(h_{\mu_{\underline{j}}(f)}-2\epsilon^{2})-n\epsilon^{\frac{3}{2}}-n\epsilon^{2}}},$

where $C_{\epsilon}:=(\frac{2d\log
M_{f}}{\epsilon^{5}})^{\frac{d+1}{\epsilon}}$. For all $n$ large enough s.t
$e^{-n\epsilon^{3}}\leq\frac{1}{C_{\epsilon}}$, we have

$\\#\\{B\in\vee_{i=0}^{\frac{1-2\sqrt{\epsilon}}{\epsilon}}f^{in\epsilon}[\vee_{\underline{j}^{\prime}}\mathcal{A}_{\epsilon}^{\underline{j}^{\prime},n}]:B\cap
D_{\underline{j}}\neq\varnothing\\}\leq
e^{nh_{\mu_{\underline{j}}}(f)+2\epsilon^{\frac{3}{2}}n}.$ (12)

Therefore, in total,

$\displaystyle
e^{-n\epsilon(h_{\mu_{\underline{j}}}(f)-\epsilon^{2})}\leq\mu(D_{\underline{j}})\leq$
$\displaystyle
2\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(D_{\underline{j}})\leq
2e^{n\epsilon^{\frac{3}{2}}}\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(D_{\underline{j}}\cap\mathcal{S}_{n})$ $\displaystyle\leq$
$\displaystyle 2e^{n\epsilon^{\frac{3}{2}}}\cdot
e^{nh_{\mu_{\underline{j}}}(f)+2\epsilon^{\frac{3}{2}}n}\cdot\max_{B\cap
D_{\underline{j}}\neq\varnothing}\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-k}(B)$ $\displaystyle\leq$ $\displaystyle
2e^{n\epsilon^{\frac{3}{2}}}\cdot
e^{nh_{\mu_{\underline{j}}}(f)+2\epsilon^{\frac{3}{2}}n}\cdot\max_{B\cap
D_{\underline{j}}\neq\varnothing}\frac{1}{\sqrt{\epsilon}n}\sum_{k=(1-\sqrt{\epsilon})n}^{n-1}m\circ
f^{-(k-(1-2\sqrt{\epsilon})n)}(f^{-(1-2\sqrt{\epsilon})n}[B])$
$\displaystyle\leq$ $\displaystyle 2e^{n\epsilon^{\frac{3}{2}}}\cdot
e^{nh_{\mu_{\underline{j}}}(f)+2\epsilon^{\frac{3}{2}}n}\cdot\max_{B\cap
D_{\underline{j}}\neq\varnothing}e^{2\sqrt{\epsilon}d\log
M_{f}n}m(f^{-(1-2\sqrt{\epsilon})n}[B])$ $\displaystyle\leq$ $\displaystyle
2e^{n\epsilon^{\frac{3}{2}}}\cdot
e^{nh_{\mu_{\underline{j}}}(f)+2\epsilon^{\frac{3}{2}}n}\cdot
e^{2\sqrt{\epsilon}d\log
M_{f}n}e^{-n(1-2\sqrt{\epsilon})(\chi^{u}(\mu_{\underline{j}})-\epsilon^{2})}e^{2\epsilon^{\frac{1}{3}}n}.$

Where the last inequality is by (11) similarly to Lemma 2.5. Then,

$\displaystyle e^{-nh_{\mu_{\underline{j}}}(f)}\leq
e^{-n\chi^{u}(\mu_{\underline{j}})}e^{10d\log M_{f}\epsilon^{\frac{1}{3}}n},$

and since $\epsilon>0$ is allowed to be arbitrarily small, for $\mu$-a.e $x$
$h_{\mu_{x}}(f)\geq\chi^{u}(x)$, and we are done. ∎

##  4\. Volume is almost exponentially mixing

###  4.1. Exponential decay of correlations implies exponential convergence

While checking the condition at (5) may seem difficult since one has to
a-priori know the measure $\mu$, here we present a condition which implies (5)
without comparing to an explicit $f$-invariant measure.

######  Proposition 4.1.

Assume that there exist $C,\gamma,\alpha>0$ s.t for all
$g,h\in\mathrm{H\ddot{o}l}_{\alpha}(M)$ s.t $\int hdm=0$,

$\left|\int g\circ f^{n}\cdot hdm\right|\leq
C\|g\|_{\alpha}\|h\|_{\alpha}e^{-\gamma n}.$

Then there exists an $f$-invariant Borel probability $\mu$ s.t $m\circ
f^{-n}\xrightarrow[n\to\infty]{}\mu$ exponentially fast, and moreover (3)
holds.

###### Proof.

We start by proving that the exponential convergence property applies to the
Jacobian of $f$ (recall that $d_{\cdot}f$, $d_{\cdot}f^{-1}$ are
$\beta$-Hölder continuous):

Step 1: $\exists C^{\prime\prime}\geq C$, $\gamma^{\prime}\in(0,\gamma)$ s.t
$\forall g,h\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with $\int hdm=0$, $\left|\int
g\circ f^{n}\cdot hdm\right|\leq
C^{\prime\prime}\|g\|_{\beta}\|h\|_{\beta}e^{-\gamma^{\prime}n}$.

Proof: Let $\\{\psi_{i}\\}_{i=1}^{N}$ be a partition of unity of $M$, s.t for
all $i\leq N$, $0\leq\psi_{i}\leq 1$ is a $C^{\infty}$ function which is
supported on a ball $U_{i}$ with a $C^{\infty}$ diffeomorphism
$\Theta_{i}:B_{\mathrm{R}^{d}}(0,1)\to U_{i}$. Let $g$ be a $\beta$-Hölder
function (w.l.o.g $\|g\|_{\infty}\leq 1$), and assume that $g$ is supported on
some set $U_{i}$.

Let $\eta>0$ to be determined later, and let
$K_{n}:B_{\mathrm{R}^{d}}(0,e^{-\eta n})\to\mathbb{R}$ be the cone function
kernel: $K_{n}(t)=(e^{-\eta n}-|t|)\cdot\frac{H_{n}}{e^{-\eta n}}$ where
$H_{n}=\frac{C_{d}}{e^{-\eta nd}}$ satisfies $\int K_{n}(t)dt=1$. Extend
$K_{n}$ naturally to $\mathbb{R}^{d}$ by taking the value $0$ outside
$B_{\mathrm{R}^{d}}(0,e^{-\eta n})$. Set
$g_{n}:=(g\circ\Theta_{i}*K_{n})\circ\Theta_{i}^{-1}$. Then one can check that
the following properties hold:

  1. (1)

$\|K_{n}\|_{\alpha}\leq\|K_{n}\|_{\mathrm{Lip}}=\frac{H_{n}}{e^{-\eta
n}}=C_{d}\cdot e^{\eta n(d+1)}$ where $C_{d}$ is a constant depending only on
$d$,

  2. (2)

$\|g_{n}\|_{\alpha}\leq\mathrm{Vol}(B_{\mathrm{R}^{d}}(0,1))\cdot\mathrm{Lip}(\Theta_{i}^{-1})\cdot\|K_{n}\|_{\alpha}$,

  3. (3)

$|g_{n}(x)-g(x)|\leq\mathrm{Lip}(\Theta_{i})\cdot\|g\|_{\beta}\cdot e^{-\eta
n\beta}$.

Notice, given $g,h\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with $\int hdm=0$
(w.l.o.g $\|g\|_{\infty},\|h\|_{\infty}\leq 1$), we can write
$g=\sum_{i=1}^{N}\psi_{i}\cdot g$, $h=\sum_{i=1}^{N}\psi_{i}\cdot h$, and
$\sum_{i=1}^{N}\int h\cdot psi_{i}dm=0$, where $\psi_{i}g$ and $\psi_{i}h$ are
$\beta$-Hölder for all $i\leq N$. Therefore,

$\int g\circ fhdm=\sum_{i=1}^{N}\sum_{j=1}^{N}\int(\psi_{i}g)\circ
f^{n}(\psi_{j}h-\int\psi_{j}hdm)dm.$

Set $g^{i}:=\psi_{i}\cdot g$ and $h^{j}:=\psi_{j}\cdot h-\int\psi_{j}\cdot
hdm$, then

$\displaystyle\int g^{i}\circ f^{n}h^{j}dm=$
$\displaystyle\int(g^{i}-g^{i}_{n})\circ f^{n}h^{j}dm+\int g^{i}_{n}\circ
f^{n}(h^{j}-h^{j}_{n})dm+\int g^{i}_{n}\circ f^{n}\cdot(\int h^{j}_{n}dm)dm$
$\displaystyle+$ $\displaystyle\int g^{i}_{n}\circ f^{n}\cdot(h^{j}_{n}-\int
h^{j}_{n}dm)dm.$

On the r.h.s, all three first summands are exponentially small by item (3).
The last summand on the r.h.s is bounded by $Ce^{-\gamma
n}\|g^{i}_{n}\|_{\alpha}\|h^{j}_{n}\|_{\alpha}\leq Ce^{-\gamma n}\cdot
C_{d}^{2}\mathrm{Vol}(B_{\mathrm{R}^{d}}(0,1))^{2}\mathrm{Lip}(\Theta_{i}^{-1})\mathrm{Lip}(\Theta_{j}^{-1})e^{2\eta
n(d+1)}\leq C^{\prime}_{d}C_{0}e^{-\frac{\gamma}{2}n}$ whenever
$\eta:=\frac{\gamma}{4(d+1)}$ and $C_{0}:=\max_{i\leq
N}\\{\mathrm{Lip}(\Theta_{i}),\mathrm{Lip}(\Theta_{i}^{-1})\\}$. Therefore for
$\gamma^{\prime}:=\eta\beta\in(0,\frac{\gamma}{2})$,

$\displaystyle\left|\int g\circ f^{n}hdm\right|\leq$ $\displaystyle
N^{2}\cdot\left(C_{0}\|g\|_{\beta}e^{-\eta\beta
n}+(1+C_{0}\|g\|_{\beta}e^{-\eta n\beta})C_{0}\|h\|_{\beta}e^{-\eta
n\beta}+C_{0}\|h\|_{\beta}e^{-\eta
n\beta}+C^{\prime}_{d}C_{0}e^{-\frac{\gamma}{2}n}\right)$ $\displaystyle\leq$
$\displaystyle
C^{\prime\prime}\|g\|_{\beta}\|h\|_{\beta}e^{-\gamma^{\prime}n}.$

Step 2: There exist a constant $C_{f}^{\prime\prime}>0$ and an $f$-invariant
Borel probability $\mu$ s.t $\forall g,h\in\mathrm{H\ddot{o}l}_{\beta}(M)$,

$\left|\int g\circ f^{n}hdm-\int gd\mu\int hdm\right|\leq
C_{f}^{\prime\prime}e^{-\gamma^{\prime}n}\|g\|_{\beta}\|h\|_{\beta}.$

Proof: Fix $h\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with $h\geq 0$ and
$h\not\equiv 0$, and let $g\in\mathrm{H\ddot{o}l}_{\beta}(M)$. Define the
sequence $a_{n}^{h}(g):=\int g\circ f^{n}hdm$, and notice

$\left|a_{n+1}^{h}(g)-a_{n}^{h}(g)\right|=\left|\int g\circ
f^{n}\mathrm{Jac}(f^{-1})h\circ f^{-1}dm-\int g\circ
f^{n}hdm\right|=\left|\int g\circ f^{n}Hdm\right|,$

where $H:=\mathrm{Jac}(f^{-1})h\circ
f^{-1}-h\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with $\int Hdm=0$. Therefore,
$|a_{n+1}^{h}(g)-a_{n}^{h}(g)|\leq
C_{f}\|h\|_{\beta}\|g\|_{\beta}e^{-\gamma^{\prime}n}$, for a constant $C_{f}$
depending on $f$. Thus the sequence $\\{a_{n}^{h}(g)\\}_{n\geq 0}$ is a Cauchy
sequence and has a limit defined $\int hdm\cdot\mu_{h}(g)$. Notice:

  1. (1)

$\mu_{h}(a_{1}g_{1}+a_{2}g_{2})=a_{1}\mu_{h}(g_{1})+a_{2}\mu_{h}(g_{2})$,

  2. (2)

$\mu_{h}(1)=1$,

  3. (3)

$g\geq 0\Rightarrow\mu_{h}(g)\geq 0$,

  4. (4)

$\mu_{h}(g)\leq\|g\|_{\infty}$.

Therefore by the Riesz representation theorem, $\mu_{h}(\cdot)$ defined a
Borel probability measure on $M$. Moreover, it is easy to check that
$\mu_{h}(g)=\mu_{h}(g\circ f)$ from definition, therefore $\mu_{h}$ is
$f$-invariant.

Given $h_{1},h_{2}\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with $h_{1},h_{2}\geq 0$
and $h_{1},h_{2}\not\equiv 0$, then

$\left|\frac{1}{\int h_{1}dm}a_{n}^{h_{1}}(g)-\frac{1}{\int
h_{2}dm}a_{n}^{h_{2}}(g)\right|=\left|\int g\circ
f^{n}\cdot\left(\frac{h_{1}}{\int h_{1}dm}-\frac{h_{2}}{\int
h_{2}dm}\right)\right|\xrightarrow[n\to\infty]{}0.$

Therefore $\mu_{h}(g)$ is independent of the choice of $h$, and can be denoted
by $\mu(g)$. Given $h\in\mathrm{H\ddot{o}l}_{\beta}(M)$ with
$\|h\|_{\infty}\leq 1$ and $h\not\equiv 0$, we can write $h=h^{+}-h^{-}$,
where $0\leq h^{+},h^{-}\leq 1$ and
$\|h^{+}\|_{\beta},\|h^{-}\|_{\beta}\leq\|h\|_{\beta}$. Assume w.l.o.g
$h^{+},h^{-}\not\equiv 0$. Then for $\sigma\in\\{-,+\\}$,

$\displaystyle\left|a_{n}^{h^{\sigma}}(g)-\int
h^{\sigma}dm\cdot\mu(g)\right|\leq
C_{f}^{\prime}\|h\|_{\beta}\|g\|_{\beta}e^{-\gamma^{\prime}n}\Rightarrow\left|a_{n}^{h}(g)-\int
hdm\cdot\mu(g)\right|\leq
2C_{f}^{\prime}\|h\|_{\beta}\|g\|_{\beta}e^{-\gamma^{\prime}n}.$

In particular, for $h\equiv 1$, we get $m\circ f^{-n}\xrightarrow[\exp]{}\mu$.
∎

###  4.2. Positive entropy, ergodicity, and uniqueness

In this section we assume the following strong notion of exponential
convergence:

$\exists C>0,\alpha\in(0,1],\gamma>0:\forall
g,h\in\mathrm{H\ddot{o}l}_{\alpha}(M),\left|\int g\circ f^{n}\cdot
hdm-\mu(g)\cdot m(h)\right|\leq C\cdot\|g\|_{\alpha}\cdot\|h\|_{\alpha}\cdot
e^{-\gamma n}.$ (13)

Recall Proposition 4.1, where we show that (13) holds whenever the volume is
almost exponentially mixing (recall (4)). The condition of almost exponential
mixing of the volume is inspired by the setup studied by Dolgopyat, Kangowski,
and Rodriguez-Hertz in [DKRH] (however notice that the volume need not be
invariant in our setup).

We continue to show that under the assumption of (13), unless $\mu$ is a Dirac
delta measure at a fixed point (a necessary condition, see the remark
following Theorem 4.2), indeed $\mu$ must be an ergodic SRB measure with at
least one positive Lyapunov exponent almost everywhere, and it is the unique
SRB measure of $(M,f)$. A nice corollary of our proof is that every
$f$-invariant Borel probability measure on $M$ has a uniform bound form below
on its maximal Lyapunov exponent in terms of the rate of mixing, aside at most
for $\mu$ in the case where $\mu$ is a Dirac delta measure.

######  Theorem 4.2 (Positive exponents).

The following dichotomy holds:

  1. (1)

for every ergodic $f$-invariant Borel probability $\nu$,
$\max_{i}\chi^{+}_{i}(x)>\frac{\gamma}{2d}$ $\nu$-a.e,

  2. (2)

$\mu$ is a Dirac delta measure at a fixed point with $\chi^{u}(\mu)=0$, and
every other ergodic $f$-invariant Borel probability $\nu$ has
$\max_{i}\chi^{+}_{i}(x)>\frac{\gamma}{2d}$ $\nu$-a.e.

In particular, if $\mu$ is not a Dirac delta measure at a fixed point, then
$\max_{i}\chi^{+}_{i}(x)\geq\frac{\gamma}{2d}$ $\mu$-a.e.

###### Proof.

Let $\nu$ be an ergodic $f$-invariant Borel probability s.t
$\max_{i}\chi_{i}^{+}(x)\leq\frac{\gamma}{2d}-8(d+1)\epsilon$ $\nu$-a.e, where
w.l.o.g $0<\epsilon\ll\frac{\gamma}{2d}$. Let $x$ be a $\nu$-typical point.
Let $n$ large s.t $f^{i}[B(x,n,e^{-\epsilon n})]$ is contained in the Pesin
chart of $f^{i}(x)$ for all $0\leq i\leq n$. Let $g_{x}$ be a Lipschitz
function s.t $g_{x}|_{B(x,n,e^{-2\epsilon n})}=1$, $g_{x}|_{B(x,n,e^{-\epsilon
n})^{c}}=0$, and $\mathrm{Lip}(g_{x})\leq
e^{(\frac{\gamma}{2d}-3(d+1)\epsilon)n}$.

Let $p$ and $q$ s.t $\mu(B(p,e^{-\epsilon n})),\mu(B(q,e^{-\epsilon n}))\geq
e^{-n(d+1)\epsilon}$ for all $n$ large enough, and let
$g_{t}|_{B(t,e^{-2\epsilon n})}=1$, $g_{t}|_{B(t,e^{-\epsilon n})^{c}}=0$, and
$\mathrm{Lip}(g_{t})\leq e^{2\epsilon n}$, $t\in\\{p,q\\}$. Then by (13), for
$t\in\\{p,q\\}$ and all $n$ large enough,

$\displaystyle\int g_{t}\circ f^{n}g_{x}dm=$
$\displaystyle\mu(g_{t})m(g_{x})\pm 4Ce^{-\gamma n}e^{2\epsilon
n}e^{(\frac{\gamma}{2d}-3(d+1)\epsilon)n}$ $\displaystyle=$ $\displaystyle
e^{\pm\epsilon}\mu(g_{t})m(g_{x})>0\text{ }(\because\mu(g_{t})m(g_{x})\geq
e^{-n(d+1)2\epsilon-(\frac{\gamma}{2d}+\epsilon)dn}).$

Thus in particular $B(p,e^{-\epsilon n})\cap B(f^{n}(x),e^{-\epsilon
n})\neq\varnothing$ and $B(q,e^{-\epsilon n})\cap B(f^{n}(x),e^{-\epsilon
n})\neq\varnothing$, and so $d(p,q)\leq 4e^{-\epsilon
n}\xrightarrow[n\to\infty]{}0$. Therefore $\mu=\delta_{p}=\delta_{q}$.

Moreover, if we assume further that $x$ is $\nu$-generic, for any
$h\in\mathrm{Lip}_{+}(M)$,

$\displaystyle m(g_{x})(h\circ f^{n}(x)\pm\mathrm{Lip}(h)e^{-\epsilon n})=$
$\displaystyle\int h\circ f^{n}g_{x}dm=\mu(h)m(g_{x})\pm 4Ce^{-\gamma
n}e^{\epsilon n}e^{(\frac{\gamma}{2d}-3(d+1)\epsilon)n}$ $\displaystyle=$
$\displaystyle m(g_{x})(\mu(h)\pm 4Ce^{-\frac{\gamma}{2}n}e^{\epsilon
n}e^{(\frac{\gamma}{2d}-3(d+1)\epsilon)n}e^{\epsilon dn}).$

Averaging over $n=N,\ldots,2N$, for $N\in\mathbb{N}$ large,

$\displaystyle\nu(h)\xleftarrow[\infty\leftarrow
N]{}\frac{1}{N}\sum_{n=N}^{2N-1}h\circ f^{n}(x)\pm\mathrm{Lip}(h)e^{-\epsilon
N}=\mu(h)\pm 4Ce^{-\frac{\gamma}{2}N}e^{\epsilon
N}e^{(\frac{\gamma}{2d}-3(d+1)\epsilon)N}e^{\epsilon
dN}\xrightarrow[N\to\infty]{}\mu(h).$

Therefore $\mu(h)=\nu(h)$, and so by the Riesz representation theorem,
$\nu=\mu$ (since $\overline{\mathrm{Lip}_{+}(M)}^{C(M)}=C_{+}(M)$). ∎

Remark: The assumption that $\mu$ is not a Dirac delta measure is necessary,
as can be seen by the north-pole south-pole example: Let $\mathbb{S}^{1}$ be
the unit circle, let $f\in\mathrm{Diff}^{1+\beta}(\mathbb{S}^{1})$, and let
two fixed points, $N\in\mathbb{S}^{1}$ with a derivative bigger than $1$, and
$S\in\mathbb{S}^{1}$ with a derivative smaller than $1$. One can check that in
this case $|\int_{\mathbb{S}^{1}}g\circ f^{n}hdm-g(S)m(h)|$,
$g,h\in\mathrm{Lip}(\mathbb{S}^{1})$, is exponentially small as in (13), by
using a partition of unity which separates $N$ and $S$. This example extends
to a closed surface using a D-A system with two repellers, and a fixed
attracting point.

By Theorem 3.1 and Theorem 4.2, it follows from [LY85] that $\mu$ has
absolutely continuous conditionals on unstable leaves a.e. The following
theorem is a corollary of this fact together with (13). The proof uses
absolutely continuous fake $cs$-foliations in exponentially small charts,
constructed in [DKRH]. These foliations are used to treat the trajectory of an
exponentially small ball as the trajectory of single unstable leaf, where the
conditional measure of $\mu$ is equivalent to the induced Riemannian volume,
which lets us compare the two measures.

######  Proposition 4.3.

Assume that there exists an ergodic SRB measure $\nu$ with $h_{\nu}(f)>0$. For
every $\epsilon\in(0,\frac{\gamma}{4\log M_{f}})$ there is a set
$G_{\epsilon}$ with $\nu(G_{\epsilon})\geq e^{-\epsilon}$ s.t $\forall x\in
G_{\epsilon}$, $\forall\delta\in(0,\epsilon)$, $\forall n\geq
n_{\epsilon,\delta}$, $\forall g\in\mathrm{H\ddot{o}l}_{\alpha}^{+}(M)$,

$\mu(g)\geq\frac{e^{-7\delta^{2}d}}{\nu_{\xi^{u}(x)}(B^{u}(x,e^{-\delta
n})\cap
K_{\epsilon})\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int\limits_{B^{u}(x,e^{-\delta
n})\cap K_{\epsilon}}(g\circ
f^{n}-\|g\|_{\alpha\mathrm{-H\ddot{o}l}}e^{-\frac{\delta}{2}n\alpha})d\nu_{\xi^{u}(x)}-C\|g\|_{\alpha\mathrm{-H\ddot{o}l}}e^{-(\gamma-2\delta)n}$
(14)

where $\xi^{u}$ is a measurable partition subordinated to the unstable
foliation of $\nu$, $\nu_{\xi^{u}(\cdot)}$ are the respective conditional
measures, and $K_{\epsilon}$ is a Pesin block with $\nu(K_{\epsilon})\geq
e^{-\epsilon^{2}}$.

For the definition of a measurable partition subordinated to the unstable
foliation of $\nu$, see [LY85], and the respective conditional measures exists
$\nu$-a.e by the Rokhlin disintegration theorem.

###### Proof.

Step 1: $\nu_{\xi^{u}(x)}=C^{\pm 1}\frac{1}{m_{\xi^{u}(x)}(1)}m_{\xi^{u}(x)}$
on a large set, where $m_{\xi^{u}(x)}$ is the induced Riemannian volume on
$\xi^{u}(x)$ and $C>0$ is a constant close to $1$.

Proof: Let $\xi^{u}$ be a partition measurable partition subordinate to the
unstable foliation of $\nu$ s.t $\xi^{u}(x)\supseteq B^{u}(x,r_{x})$ for
$\nu$-a.e $x$ (see [LY85]). Let $\nu=\int\nu_{\xi^{u}(x)}d\nu(x)$ be the
corresponding disintegration of $\nu$ given by the Rokhlin disintegration
theorem. By Theorem 4.2, and [LY85], for $\nu$-a.e $x$, $\nu_{\xi^{u}(x)}\sim
m_{\xi^{u}(x)}$.

Denote by $\rho_{x}$ the Radon-Nikodym derivative
$\frac{d\nu_{\xi^{u}(x)}}{dm_{\xi^{u}(x)}}$. By the construction of $\xi^{u}$,
$\rho_{x}(x)$ is uniformly bounded a.e, since the elements of $\xi^{u}$ are
contained in local unstable leaves, and moreover $\log\rho_{x}$ is
$\frac{\beta}{3}$-Hölder continuous with a uniform Hölder constant (see [LY85]
for more details of this classical result).

Therefore, given $\epsilon>0$ and a small $\delta\in(0,\epsilon)$, there
exists $\ell_{\epsilon}=\ell_{\epsilon}(\delta)$ s.t
$\nu\left(\Lambda_{\ell_{\epsilon}}^{(\underline{\chi}(\nu),\delta^{3}\tau_{\underline{\chi}(\nu)})}\right)\geq
e^{-\epsilon^{2}}$. Let $0<\chi\leq\min\\{\chi_{i}(\nu):\chi_{i}(\nu)\neq
0,i\leq d\\}$ (w.l.o.g $\delta\leq\frac{\chi}{2}$), and set
$K_{\epsilon}:=\Lambda_{\ell_{\epsilon}}^{(\underline{\chi}(\nu),\delta^{3}\tau_{\underline{\chi}(\nu)})}$.
For all $x\in K_{\epsilon}$, the local unstable leaf of $x$ contains a
relative open ball of radius at least $\frac{1}{2\ell_{\epsilon}}$.

Step 2: Absolutely continuous fake $cs$-foliation in $B(x,n\epsilon,e^{-\delta
n})$, for $x\in K_{\epsilon}$, by [DKRH, § 5,6].

Proof: Given $x\in K_{\epsilon}$, and $n$ large enough so
$e^{-\frac{\delta}{3}n}\ll\frac{1}{\ell_{\epsilon}}$, let
$V^{\mathrm{cs}}_{n}(x)$ be a “fake central-stable leaf” at $x$, constructed
in [DKRH, Lemma 2.6]. That is, for every $x\in K_{\epsilon}$ there exists a
local submanifold of $x$ transversal to $\xi^{u}(x)$,
$V^{\mathrm{cs}}_{n}(x)$, s.t $\forall 0\leq i\leq n$,

  1. (1)

$f^{i}[B_{V^{\mathrm{cs}}_{n}(x)}(x,e^{-\delta n})]\subseteq
B(f^{i}(x),e^{-\frac{\delta}{2}n})$,

  2. (2)

$f^{i}[V^{\mathrm{cs}}_{n}(x)]$ is a graph of a function with a Lipschitz
constant smaller or equal to $\frac{2\tau}{\chi}\leq\delta^{2}$ over
$\psi_{f^{i}(x)}[\mathbb{R}^{\mathrm{cs}}\cap B(0,e^{-\frac{\delta}{2}n})]$
(where $\psi_{y}$ is the Pesin chart of $y$),

  3. (3)

$\\{V^{\mathrm{cs}}(y):y\in B(x,e^{-\delta n})\cap K_{\epsilon}\\}$ is a
foliation of $B(x,e^{-\delta n})\cap K_{\epsilon}$ ([DKRH, § 5]).

Moreover, by [DKRH, Proposition 6.4],

  1. (4)

For all $n$ large enough (when $\delta>0$ is small enough), the holonomy map
$\pi$ along $\\{V^{\mathrm{cs}}(x^{\prime})\\}_{x^{\prime}\in
K_{\epsilon}\cap\xi^{u}(x)}$ from $\xi^{u}(x)\cap K_{\epsilon}$ to
$\xi^{u}(y)$, $y\in K_{\epsilon}\cap B(x,e^{-\delta n})$, has a Jacobian
$\mathrm{Jac}(\pi)=e^{\pm\delta^{2}}$.

In fact it follows that $\mathrm{Jac}(\pi|_{B^{u}(x,e^{-\delta
n})})=e^{o(1)}$.

Step 3: For every $x\in K_{\epsilon}$ and $n$ large enough, and for every
$g\in\mathrm{H\ddot{o}l}_{\alpha}^{+}(M)$,
$\frac{1}{\nu_{\xi^{u}(x)}(\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}g\circ
f^{n}d\nu_{\xi^{u}(x)}=(\mu(g)\pm
e^{-\frac{\gamma}{2}n}\|g\|_{\alpha-\mathrm{H\ddot{o}l}})e^{\pm\delta}$, where
$\mathcal{W}^{\mathrm{cs}}_{n}(x)$ is a foliation box in the chart of $x$.

Proof: We start by defining $\mathcal{W}^{\mathrm{cs}}_{n}(x)$ for $x\in
K_{\epsilon}$. Let $R(x,e^{-\delta n}e^{2\delta^{2}}):=\psi_{x}(R(0,e^{-\delta
n}e^{2\delta^{2}}))$, where $\psi_{x}$ is the Pesin chart of $x$, and
$R(\cdot,r)$ is a ball of radius $r$ w.r.t to the metric
$|\cdot|^{\prime}:=\max\\{|\cdot_{\mathrm{u}}|_{2},|\cdot_{\mathrm{cs}}|_{2}\\}$,
where $\mathrm{u},\mathrm{cs}$ are the corresponding components in the chart
of $x$. In particular, given $y\in K_{\epsilon}\cap B^{u}(x,e^{-\delta n})$,
$B_{V^{\mathrm{cs}}_{n}(y)}(y,e^{-\delta n})\subseteq R(x,e^{-\delta
n}e^{2\delta^{2}})$, since $V^{\mathrm{cs}}_{n}(y)$ is the graph of a
$\delta^{2}$-Lipschitz function in the chart of $x$. We define
$\mathcal{W}^{\mathrm{cs}}_{n}(x):=\bigcup_{y\in K_{\epsilon}\cap
B^{u}(x,e^{-\delta n})}B_{V^{\mathrm{cs}}_{n}(y)}(y,e^{-\delta n})$.

Let $x$ be a $\nu_{\xi^{u}(x)}$-density point of $K_{\epsilon}$ s.t
$\frac{\nu_{\xi^{u}(x)}(K_{\epsilon}\cap
B^{u}(x,r))}{\nu_{\xi^{u}(x)}(B^{u}(x,r))}\geq e^{-\delta^{2}}$, $\forall
r\in(0,2e^{-\delta n})$. By the Hölder continuity of $\log\rho_{x}$,
$m_{\xi^{u}(x)}=(\rho_{x}(x))^{-1}e^{\pm 2e^{-\frac{\beta}{3}\delta
n}}\mu_{\xi^{u}(x)}$ on $B^{u}(x,2e^{-\delta n})$. Therefore,
$\frac{m_{\xi^{u}(x)}(K_{\epsilon}\cap B(x,e^{-\delta
n}))}{m_{\xi^{u}(x)}(B(x,e^{-\delta n}))}\geq e^{-2\delta^{2}}$ for all $n$
large enough. Finally, let $h$ be a Lipschitz function s.t $h|_{R(x,e^{-\delta
n}e^{\delta^{2}})}=1$, $h|_{R(x,e^{-\delta n}e^{2\delta^{2}})^{c}}=0$,
$\mathrm{Lip(h)}\leq 2e^{2\delta n}$. In particular,
$m(\mathcal{W}^{\mathrm{cs}}(x))\geq e^{-2\delta^{2}d}m(h)$.

In addition, by the absolute continuity of the foliation
$\mathcal{W}^{\mathrm{cs}}(x)$ and since the induced leaf volume of each
laminate in $\mathcal{W}^{\mathrm{cs}}(x)$ is comparable up to a $e^{\pm
2\delta^{2}d}$ factor,
$\frac{1}{m(\mathcal{W}^{\mathrm{cs}}_{n}(x))}m|_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}=e^{\pm
3\delta^{2}d}\frac{1}{m_{\xi^{u}(x)}(\mathcal{W}^{\mathrm{cs}}_{n}(x))}m_{\xi^{u}(x)}|_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}$
for sets saturated by $W^{\mathrm{cs}}(x)$ for all $n$ large enough. Thus by
Step 2,

$\displaystyle\frac{1}{m(h)}\int h\cdot g\circ f^{n}dm=$
$\displaystyle\frac{m(\mathcal{W}^{\mathrm{cs}}_{n}(x))}{m(h)}\cdot\frac{1}{m(\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int
h\cdot g\circ f^{n}dm$ $\displaystyle\geq$
$\displaystyle\frac{m(\mathcal{W}^{\mathrm{cs}}_{n}(x))}{m(B(x,e^{-\delta
n}))}\cdot\frac{m(B(x,e^{-\delta
n}))}{m(h)}\cdot\frac{1}{m(\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int\mathbb{1}_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}\cdot
g\circ f^{n}dm$ $\displaystyle\geq$ $\displaystyle e^{-2\delta^{2}d}\cdot
e^{-2\delta^{2}d}\cdot
e^{-3\delta^{2}d}\frac{1}{m_{\xi^{u}(x)}(\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int\limits_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}(g\circ
f^{n}-\|g\|_{\alpha\mathrm{-H\ddot{o}l}}e^{-\frac{\delta}{2}n\alpha})dm_{\xi^{u}(x)}$
$\displaystyle\geq$ $\displaystyle
e^{-7\delta^{2}d}\frac{1}{\nu_{\xi^{u}(x)}(\mathcal{W}^{\mathrm{cs}}_{n}(x))}\int\limits_{\mathcal{W}^{\mathrm{cs}}_{n}(x)}(g\circ
f^{n}-\|g\|_{\alpha\mathrm{-H\ddot{o}l}}e^{-\frac{\delta}{2}n\alpha})d\nu_{\xi^{u}(x)}.$
(15)

By (13), the l.h.s equals to $\mu(g)\pm
C\cdot\|g\|_{\alpha\mathrm{-H\ddot{o}l}}e^{-(\gamma-2\delta)n}$, and for all
$\delta>0$ small enough, we are done by choosing $G_{\epsilon}$ to be a
density set of $K_{\epsilon}$ with uniform estimates as in step 3. ∎

Remark: An upper bound for (14) can be achieved similarly through (4.2),
although the error term will not be exponentially small in $n$; however the
error term is to the already averaged quantity.

######  Corollary 4.4 (Uniqueness).

The system $(M,f)$ admits exactly one ergodic SRB measure, and it is the
measure $\mu$. In particular, $\mu$ is ergodic.

###### Proof.

Let $\nu$ be an ergodic SRB measure (in the sense of the entropy formula), we
wish to show that $\nu=\mu$. Therefore assume that $\nu\neq\mu$, and by
Theorem 4.2, $h_{\nu}(f)=\sum\chi^{+}(\nu)>0$. Let $g\in\mathrm{Lip}(M)$ s.t
$0\leq g\leq 1$. Let $\epsilon>0$, and $x\in G_{\epsilon}$ which is
$\nu$-generic for $g$. Assume further that $x$ is a $\nu_{\xi^{u}(x)}$-density
point of $E_{n^{\prime}}:=\\{x^{\prime}\in K_{\epsilon}:\forall n\geq
n^{\prime},\frac{1}{n}\sum_{k=n}^{2n-1}g\circ f^{k}=e^{\pm\delta}\nu(g)\\}$
s.t $\frac{\nu_{\xi^{u}(x)}(B^{u}(x,e^{-\delta n})\cap
E_{n^{\prime}})}{\nu_{\xi^{u}(x)}(B^{u}(x,e^{-\delta n}))}\geq e^{-\delta}$,
for some large $n^{\prime}$. Then, for all $n$ large enough, $\mu(g)\geq
e^{-\delta}(e^{-7d\delta^{2}}\nu(g)-\|g\|_{\mathrm{Lip}}e^{-\frac{n\alpha\delta}{3}})$.
If $\nu(g)=0$, then $\mu(g)=0$, otherwise for all $n$ large enough w.r.t $g$,
$\mu(g)\geq e^{-8d\delta^{2}}\nu(g)$, hence $\nu=\mu$. In particular, $\mu\geq
e^{-8d\delta^{2}}\nu$ (by the Riesz representation theorem, and since
$\overline{\mathrm{Lip}^{+}(M)}^{C(M)}=C(M)$). Since $\delta>0$ is arbitrary,
$\mu\geq\nu$ for every ergodic SRB measure $\nu$ s.t $h_{\nu}(f)>0$.

Assume that $\mu$ can be written as
$\mu=a\mu_{1}+(1-a)\frac{\mu-\mu_{1}}{1-a}$ with $a\in(0,1)$ and
$\mu_{1}\perp(\mu-\mu_{1})$. If $\mu$ admitted an ergodic component with no
positive Lyapunov exponents, then by Theorem 4.2 $\mu$ is a Dirac mass at a
fixed point, which contradicts the fact that $\mu\geq\nu$ with $h_{\nu}(f)>0$.
Therefore $\mu_{1}$ admits a positive Lyapunov exponent a.e. Therefore
$\mu\geq\mu_{1}$.

Let $G$ be a set s.t $\mu_{1}(G)=1$ and $(\mu-\mu_{1})(G)=0$. Then
$1>a=a\mu_{1}(G)=\mu(G)\geq\mu_{1}(G)=1$, a contradiction! Thus $\mu$ is
ergodic and has positive entropy. ∎

Remark: The weak physicality with full Basin of $\mu$ (Theorem 2.8) also
implies that $(M,f)$ may admit no physical measures aside for at most $\mu$.

## References

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