# Equilibrium States for Center Isometries

Pablo D. Carrasco<EMAIL_ADDRESS>ICEx-UFMG, Avda. Presidente Antonio
Carlos 6627, Belo Horizonte-MG, BR31270-90  Federico Rodriguez-Hertz
<EMAIL_ADDRESS>Penn State, 227 McAllister Building, University Park, State
College, PA16802

(August 27, 2024)

###### Abstract

We develop a geometric method to establish existence of equilibrium states
associated to Hölder potentials for center isometries (as are regular elements
of Anosov actions), in particular the entropy maximizing measure and the SRB
measure. In these cases it is established the uniqueness, and it is given a
characterization of the equilibrium states in terms in terms of their
disintegrations along stable and unstable foliations. It is proven also that
the resulting system is isomorphic to a Bernoulli scheme.

###### Contents

  1. 1 Introduction and Main Results
  2. 2 Measures along foliations
    1. 2.1 Families of quasi-invariant measures
    2. 2.2 Quasi-invariant measures along leaves of foliations: the transverse measure
  3. 3 Invariant measures for center isometries
    1. 3.1 The conditional measures: potentials constant along the center and SRBs
    2. 3.2 The equilibrium state
    3. 3.3 Product structure and the Gibb’s property
    4. 3.4 Uniqueness
  4. 4 A characterization of equilibrium states
    1. 4.1 Equilibrium states have conditionals equivalent to $\mu^{u}$
    2. 4.2 Conditional measures determine equilibrium states
  5. 5 Kolmogorov and Bernoulli properties
    1. 5.1 Kolmogorov processes
    2. 5.2 Bernoulli processes
  6. 6 Applications and concluding remarks
    1. 6.1 Hyperbolic flows
    2. 6.2 Epilogue
  7. 7 Appendix: Proof of Theorem D

##  1 Introduction and Main Results

We will study some aspects of the thermodynamic formalism for maps related to
higher rank Anosov actions. Let $M$ be a compact metric space and
$f:M\rightarrow M$ an homeomorphism. For $x\in M,\epsilon>0,n\in\mathbb{N}$ we
denote by $D(x,\epsilon)$ the open disc of center $x$ and radius $\epsilon$,
and by $D(x,\epsilon,n)$ the open $(\epsilon,n)$-Bowen ball centered at $x$,

$D(x,\epsilon,n)=\\{y\in M:d(f^{j}x,f^{j}y)<\epsilon,\ j=0,\ldots,n-1\\}.$

Recall that the topological pressure of the (continuous) potential
$\varphi:M\to\mathbb{R}$ is given by

$P_{\mathit{top}}^{f}(\varphi)=P_{\scriptstyle\mathit{top}}(\varphi):=\lim_{\epsilon\mapsto
0}\limsup_{n\mapsto\infty}\frac{\log S(\epsilon,n)}{n}$

where

$S(\epsilon,n)=\inf\\{\sum_{x\in E}e^{S_{n}\varphi(x)}:M=\cup_{x\in
E}D(x,\epsilon,n)\\}\quad S_{n}\varphi=\sum_{i=0}^{n-1}\varphi\circ f^{i}.$

Denote by $\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ the set of
$f$-invariant Borel probability measures on $M$. For
$\nu\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ the pressure of $\varphi$
with respect to $\nu$ is

$P_{\nu}(\varphi):=\nu(\varphi)+h_{\nu}(f).$

[Variational Principle] It holds

$P_{\scriptstyle\mathit{top}}(\varphi)=\sup_{\mathclap{\nu\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)}}\
P_{\nu}(\varphi)$

This by now classical theorem is due to Walters [WaltersPres].

######  Definition 1.1.

A measure $\mu\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ is an
equilibrium state for the potential $\varphi$ if
$P_{\scriptstyle\mathit{top}}(\varphi)=P_{\mu}(\varphi)$.

The existence, uniqueness and general properties of equilibrium states is of
utmost relevance in ergodic theory, in particular for the differentiable one.
Indeed, one of the most important theorems in smooth ergodic theory is the
following.

Let $M$ be a closed (that is, compact and without a boundary) manifold and
consider a $\mathcal{C}^{1}$ diffeomorphism $f:M\rightarrow M$ having a
hyperbolic attractor $\Lambda$ (in particular, $f|\Lambda$ is transitive).
Then for every Hölder potential $\varphi:\Lambda\rightarrow\mathbb{R}$ there
exists a unique equilibrium state $\mathrm{m}_{\scriptstyle\varphi}$ for
$f|\Lambda$. Furthermore, is $f|\Lambda$ is topologically mixing then the
system $(f,\mathrm{m}_{\scriptstyle\varphi})$ is metrically isomorphic to a
Bernoulli shift. As stated, this theorem is a culmination of the work of
several authors. The existence and uniqueness is essentially due to Sinai,
Ruelle and Bowen [SinaiMarkov],[SRBattractor],[Bowen1974], while the Bernoulli
property is mainly due to Ornstein and Weiss [GeoBernoulli]. See also
[BowenBernoulli].

In this work we investigate the thermodynamic formalism for a generalization
of the above systems, namely when instead of a single hyperbolic map we have
an Anosov action. Let us recall that a differentiable action of a Lie Group on
a compact Riemannian manifold $\alpha:G\curvearrowright M$ is Anosov if the
$G$-orbits of $\alpha$ foliate $M$ and there exists $g\in G$ an Anosov
element, i.e. there exist an $\alpha(g)$-invariant decomposition
$TM=E^{s}\oplus E^{c}\oplus E^{u}$ and $C>0,0<\lambda<1$ such that
$E^{c}_{x}=T_{x}G\cdot x$ and with respect to the metric on $M$ it holds

$\displaystyle v\in E^{s}_{x}\Rightarrow\forall n\geq
0\quad\|D_{x}\alpha(g)^{n}(v)\|\leq C\lambda^{n}\|v\|$ $\displaystyle v\in
E^{u}_{x}\Rightarrow\forall n\leq 0\quad\|D_{x}\alpha(g)^{n}(v)\|\leq
C\lambda^{-n}\|v\|.$

In this case it is possible to modify the metric to guarantee that $\alpha(g)$
is an isometry when restricted to each $G$-orbit. Diffeomorphisms preserving a
splitting with the above properties are called _center isometries_ and will be
our main focus. See the next section for the precise definition, or the
classical reference [HPS] for further information about these systems.

Anosov flows (cf. [AnosovThesis]) constitute the most well understood example
of Anosov action, where an analogous theorem as the previous one holds
[BowenBernoulli]. The proof relies on the existence of Markov partitions for
such flows [SymbHyp]; since this technology is unavailable for Anosov actions
of rank $\geq 2$, we present a geometric construction of equilibrium states
for such maps. At this stage however we need to impose some condition on the
potential (being constant along the orbits, or being the geometrical potential
associated to a regular element). Besides the fact that these cases include
the most interesting types of potentials (corresponding to the entropy
maximizing measure and the SRB), we also believe that the techniques developed
are interesting on their own, and may be applicable to more situations. We
remark that other geometrical constructions exist in the literature
[Climenhaga2020], and the case of compact extensions of Anosov has been
treated previously [Spatzier2016]: in both cases restrictions on the potential
are imposed. The manuscript [Buzzi2019] contains some results similar to ours
for perturbations of time-one maps of Anosov flows, for the case of the
entropy maximizing measure.

For a center isometry $f$ the associated bundles $E^{s},E^{u},E^{c}$ are
integrable to continuous foliations
$\mathcal{W}^{s},\mathcal{W}^{u},\mathcal{W}^{c}$ called the stable, unstable
and center foliations. Our results require the assumption that
$\mathcal{W}^{s},\mathcal{W}^{u}$ are minimal foliations, i.e. every one of
their leaves is dense (this condition is implicit in the classical rank-one
case cited above).

Let $f:M\rightarrow M$ be a center isometry of class $\mathcal{C}^{2}$ and
$\varphi:M\rightarrow\mathbb{R}$ a Hölder potential that is either

  1. 1.

constant on center leaves, or

  2. 2.

$\varphi=-\log\det Df|E^{u}$ (SRB case).

Assume that the stable and unstable foliations of $f$ are minimal. Then there
exist
$\mathrm{m}_{\scriptstyle\varphi}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$
and families of measures $\mu^{u}=\\{\mu^{u}_{x}\\}_{x\in
M},\mu^{s}=\\{\mu^{s}_{x}\\}_{x\in M},\mu^{cu}=\\{\mu^{cu}_{x}\\}_{x\in
M},\mu^{cs}=\\{\mu^{cs}_{x}\\}_{x\in M}$ satisfying the following.

  1. 1.

The probability $\mathrm{m}_{\scriptstyle\varphi}$ is the unique equilibrium
state for the potential $\varphi$.

  2. 2.

For every $x\in M$ the measure $\mu^{\ast},\ast\in\\{u,s,cu,cs\\}$ is a Radon
measure on $W^{\ast}(x)$ which is positive on relatively open sets, and $y\in
W^{\ast}(x)$ implies $\mu^{\ast}_{x}=\mu^{\ast}_{y}$.

  3. 3.

It holds for every $x\in M$

    1. (a)

$\mu^{\sigma}_{fx}=e^{P_{\scriptstyle\mathit{top}}(\varphi)-\varphi}f_{\ast}\mu^{\sigma}_{x}\quad\sigma\in\\{u,cu\\}$.

    2. (b)

$\mu^{\sigma}_{fx}=e^{\varphi-
P_{\scriptstyle\mathit{top}}(\varphi)}f_{\ast}\mu^{\sigma}_{x}\quad\sigma\in\\{s,cs\\}$.

  4. 4.

If $\xi$ is a measurable partition that refines the partition by unstable
(stable) leaves then the conditionals
$(\mathrm{m}_{\scriptstyle\varphi})^{\xi}_{x}$ of
$\mathrm{m}_{\scriptstyle\varphi}$ are equivalent to $\mu^{u}_{x}$ (resp.
$\mu^{s}_{x})$ for $\mathrm{m}_{\scriptstyle\varphi}\operatorname{-}a.e.(x)$.

  5. 5.

For every $\epsilon>0$ sufficiently small, for every $x\in M$ the measure
$\mathrm{m}_{\scriptstyle\varphi}|D(x,\epsilon)$ has product structure with
respect to the pair $\mu^{u}_{x},\mu^{cs}_{x}$, i.e. its equivalent to
$\mu^{u}_{x}\times\mu^{cs}_{x}$.

  6. 6.

There exists $K\geq 0$ only depending on $\varphi$ such that for every
$\epsilon>0$ there exist some positive constants $a(\epsilon),b(\epsilon)$
satisfying, for every $x\in M,n\geq 0$

$a(\epsilon)e^{-Kn}\leq\frac{\mathrm{m}_{\scriptstyle\varphi}(D(x,\epsilon,n))}{e^{S_{n}\varphi(x)-nP_{\scriptstyle\mathit{top}}(\varphi)}}\leq
e^{Kn}b(\epsilon).$

If the potential is constant along centers then one can take $K=0$.

Let $\alpha:G\curvearrowright M$ be an Anosov action of class
$\mathcal{C}^{2}$. Suppose that $g\in G$ is an Anosov element such that its
associated stable and unstable foliations are minimal and denote
$f=\alpha(g)$. Let $\varphi:M\rightarrow\mathbb{R}$ be a Hölder potential that
is either constant along orbits of $\alpha$, or the geometrical potential
associated to $f$. Then there exist a unique equilibrium state
$\mathrm{m}_{\scriptstyle\varphi}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$
for $\varphi$ and families of measures $\mu^{u}=\\{\mu^{u}_{x}\\}_{x\in
M},\mu^{s}=\\{\mu^{s}_{x}\\}_{x\in M},\mu^{cu}=\\{\mu^{cu}_{x}\\}_{x\in
M},\mu^{cs}=\\{\mu^{cs}_{x}\\}_{x\in M}$ satisfying the $1-6$ of Theorem A.

In the rank-one case the previous Theorem (Corollary) was established first by
G. Margulis [TesisMarg] for the entropy maximizing measure ($\varphi\equiv
0$), and later by R. Bowen and D. Ruelle [BowenRuelle] for general potentials,
except for $2$ which appears to be new in this generality; the local product
structure is due to N. Haydn [localproductstructureHaydn] (see also
[Leplaideur2000]). The last item above is a generalization of the so called
Gibbs-property of the equilibrium states. See [EquSta] and [Haydn1992].

For the particular case of SRB measures the construction of the equilibrium
state in our context also follows from the general results in
[GibbsSinaiPesin] while the uniqueness is proven by Dolgopyat in
[lecturesugibbs], but the other parts of the theorem are new; still, we give a
different proof of existence and uniqueness for SRB measures. In this case
there exists the following characterization: an invariant measure $\mathrm{m}$
is an equilibrium state for $\varphi(x)=\log\det Df|E^{u}_{x}$ if and only if
for every measurable partition subordinated to the unstable foliation, the
conditional measures induced by $\mathrm{m}$ are absolutely continuous with
respect to Lebesgue on the corresponding leaf. Here we are able to give
similar characterization for equilibrium states, that seems to be new even in
the classical hyperbolic case (the condition of the potential being constant
on center leaves is automatic for Anosov diffeomorphisms).

In the same context of Theorem A, consider an $f$-invariant measure
$\mathrm{m}$. The following are equivalent.

  1. 1.

$\mu$ is an equilibrium state for $\varphi$.

  2. 2.

For every $\mathrm{m}$-measurable partition $\xi$ subordinated to
$\mathcal{W}^{u}$, the induced conditional measures are absolutely continuous
with respect to $\\{\mu^{u}_{x}\\}_{x}$.

  3. 3.

For every $\mathrm{m}$-measurable partition $\xi$ subordinated to
$\mathcal{W}^{s}$, the induced conditional measures are absolutely continuous
with respect to $\\{\mu^{s}_{x}\\}_{x}$.

The last part is somewhat surprising in the SRB case, as in general
$\mu^{s}_{x}$ will not be absolutely continuous with respect to Lebesgue.

We are also able to study metric properties of the equilibrium state
$\mathrm{m}_{\scriptstyle\varphi}$, namely we establish the following.

Consider the equilibrium state $\mathrm{m}_{\scriptstyle\varphi}$ given in
Theorem A. Then the system $(f,\mathrm{m}_{\scriptstyle\varphi})$ is
metrically isomorphic to a Bernoulli shift.

It is natural to inquiry about the requirements imposed to the potentials. It
turns out that in general we cannot guarantee the existence of families of
measures satisfying part $2$ of Theorem A.

There exists an analytic center isometry $f:\mathbb{T}^{3}\to\mathbb{T}^{3}$
and a $\mathcal{C}^{\infty}$ potential $\varphi:\mathbb{T}^{3}\to\mathbb{R}$
for which there is no family of measures $\\{\zeta_{x}^{u}\\}_{x\in M}$
satisfying

  * •

$f^{-1}_{\ast}\zeta_{fx}^{u}=e^{\varphi-
P_{\scriptstyle\mathit{top}}(\varphi)}\zeta_{x}^{u}$;

  * •

the measures depend continuously on the point $x$.

The methods developed in this article are of geometric flavor, and is the hope
of the authors that they can shed on light in the still under development
theory of thermodynamic formalism for partially hyperbolic systems. Evidence
of their efficacy appears in a companion article [ContributionsErgodictheory],
where we establish new results in the ergodic theory of the classical rank-one
case.

The rest of the paper is organized as follows. In section 2 we develop general
arguments to establish the existence of families of measures on foliations
with prescribed invariant properties (conformality). These results are then
used in section 3 to construct the equilibrium state for a center isometry as
in Theorem A; in this part we also discuss uniqueness. Then in section 4 we
tackle the problem of the characterization of equilibrium states in terms of
their conditional measures on stable/unstable leaves. Section 5 is devoted to
the proof the Bernoulli property of the considered system, and in the last
Section we include a discussion of the rank-one case, together with some
questions. The proof of Theorem D is given in an Appendix, since it can be
read almost independently of the rest of the work (although Theorems A and B
are used). The reader is welcomed to check directly this part after section 3.

##  2 Measures along foliations

For $r\geq 1$ we denote by $\mathscr{Diff}^{r}(M)$ the group of
$\mathcal{C}^{r}$ diffeomorphisms of $M$. Let $f\in\mathscr{Diff}^{r}(M)$ and
$\mu$ be an $f$-invariant measure. Recall that (cf. [Rokhlin]) given a
$\mu$-measurable partition $\mathtt{P}$ of $M$ we can disintegrate $\mu$ in a
family of conditional probabilities $\\{\mu_{x}^{\mathtt{P}}\\}_{x\in
M^{\prime}}$ with $\mu(M^{\prime})=1$ and such that for every
$f\in\mathcal{L}^{1}(M,\mu)$ the conditional expectation of $f$ in
$\widehat{P}$, the $\sigma$-algebra generated by $\mathtt{P}$, is given by

$\mathbb{E}_{#1}(f|\hat{\mathtt{P}})=\int fd\mu_{x}^{\mathtt{P}},\quad x\in
M^{\prime}.$

Furthermore, if $\mathtt{P}$ is countably generated (that is,
$\mathtt{P}=\bigvee_{n}\mathtt{P}_{n}$ with $\mathtt{P}_{n}$ finite partition
for every $n\in\mathbb{N}$) then for $x,y\in M^{\prime}$, if $y$ belongs to
the atom of $\mathtt{P}$ containing $x$ then $\mu^{x}=\mu^{y}$. In this case,
denote by $M_{\mathtt{P}}=M/\mathtt{P}$ the space of atoms of $\mathtt{P}$ and
consider $\pi_{\mathtt{P}}:M\rightarrow M_{\mathtt{P}}$,
$\pi_{\mathtt{P}}(x)=\mathtt{P}(x)$, the atom that contains $x$ (this map is
well defined $\mu$-a.e.). We equip $M_{\mathtt{P}}$ with the $\sigma$-algebra
induced by $\pi_{\mathtt{P}}$ and consider the _transverse measure_
$\mu_{\mathtt{P}}=(\pi_{\mathtt{P}})_{\ast}\mu$. The considerations above
imply in particular that if $\mathtt{P}$ is countably generated, then
$\displaystyle{\eta=\mathtt{P}(x)\mapsto\mu^{\eta}:=\mu_{x}^{\mathtt{P}}}$
defines a measurable function from $M_{\mathtt{P}}$ to the set of Borel
probabilities on $M$, and furthermore

$A\in\mathscr{B}_{\scriptstyle\mathrm{M}}\Rightarrow\mu(A)=\int_{M_{\mathtt{P}}}\mu^{\eta}(A)\mu_{\mathtt{P}}(d\eta).$

This shows that $\mu$ is completely determined provided that we know
$\mu_{\mathtt{P}}$ and $\\{\mu^{\eta}\\}_{\eta\in M_{\mathtt{P}}}$.

Our interest however is constructing a particular $f$-invariant measure $\mu$
(the equilibrium state); for this we will argue backwards and construct first
the transverse measure and the conditionals. Of course this presents several
technical difficulties, mainly because we do not have the partition
$\mathtt{P}$, nor there is a well defined notion of what it means to be
$\mu$-measurable.

A way to circumvent this problem is using some (continuous) invariant
structure of $f$ to define the partition. In the next part we present a
general construction that allow us to build families of measures satisfying
certain (quasi-)invariant properties.

###  2.1 Families of quasi-invariant measures

Let $M$ be a locally compact metric space and $f:M\to M$ a bi-measurable map.
From now on, by a measure on $M$ we mean a Radon measure, and denote the set
of such measures by $\mathscr{Rad}(M)$. The set of continuous functions with
compact support on $M$ is denoted by $\mathcal{C}_{c}(M)$. The set
$\mathcal{C}_{c}(M)$ is equipped with the uniform topology, while
$\mathscr{Rad}(M)$ is equipped with the vague one.

Notation: If $\phi$ is measurable we write $f\phi:=\phi\circ f$. If
$\mu\in\mathscr{Rad}(M)$ the measure $f\mu=f_{\ast}\mu$ is the unique measure
$\nu$ such that

$\forall\phi\in\mathcal{C}_{c}(M),\quad\int
f\phi\operatorname{d}\mu=\int\phi\operatorname{d}\nu$

With the above it is direct to verify that for $\mu,\nu\in\mathscr{Rad}(M)$,
$\mu=\rho\nu$ it holds $f\mu=f^{-1}\rho f\nu$. Recall that
$\mu\in\mathscr{Rad}(M)$ is quasi-invariant if $f^{-1}\mu\sim\mu$; in this
case we can write $f^{-1}\mu=\mathtt{h}_{\mu}\mu$ where
$\mathtt{h}_{\mu}=\frac{df^{-1}\mu}{d\mu}$. Similarly,
$f^{-n}\mu=\mathtt{h}_{\mu}^{(n)}\mu$.

As it can be deduced directly from the uniqueness of the Radon-Nikodym
derivative, the family $\\{\mathtt{h}_{\mu}^{(n)}\\}_{n\in\mathbb{Z}}$ is a
multiplicative cocycle over $f$ with generator $\mathtt{h}_{\mu}$.

#### Multiplicative cocycles

A family $\\{h^{(n)}:M\to\mathbb{R}_{\geq 0}\\}_{n\in\mathbb{Z}}$ is a
multiplicative cocycle over $f$ if

$\forall n,m\in\mathbb{Z},x\in M\Rightarrow h^{(n+m)}(x)=h^{(n)}(f^{m}x)\cdot
h^{(m)}(x).$

Given $h:M\to\mathbb{R}_{>0}$, it generates a multiplicative cocycle over $f$
(and in this case we call $h$ the generator of the cocycle) by defining

$h^{(n)}(x)=\begin{dcases}\prod_{k=0}^{n-1}h(f^{k}x)&k>0\\\ x&k=0\\\
\prod_{k=n}^{-1}h^{-1}(f^{k}x)&k<0.\end{dcases}$

Observe that if $\\{h^{(n)}\\}_{n\in\mathbb{Z}}$ is a positive multiplicative
cocycle over $f$ (with generator $h$) then $\\{a^{(n)}:=\log
h^{(n)}\\}_{n\in\mathbb{Z}}$ is an additive cocycle (with generator $\log h$):

$\forall n,m\in\mathbb{Z},x\in M\Rightarrow
a^{(n+m)}(x)=a^{(n)}(f^{m}x)+a^{(m)}(x).$

It will be useful to introduce the following.

######  Definition 2.1.

Let $\Lambda$ be a set. We say that $M$ is a bundle over $\Lambda$ if can be
written as

$M=\bigsqcup_{i\in\Lambda}M_{i}$

where $M_{i}\subset M$ for every $i\in\Lambda$. We say that $f:M\to M$ is a
bundle isomorphism if there exists a bijection $q:\Lambda\to\Lambda$ such that
$f|:M_{i}\to M_{q(i)}$ is an homeomorphism, for every $i\in\Lambda$.

For example, if $\mathcal{F}=\\{W(x)\\}_{x\in M}$ is a foliation on $M$ then
we can write $M=\bigsqcup_{L\in\Lambda}L$ where $\Lambda=M/\mathcal{F}$, and
thus $M$ is a bundle over $\Lambda$. If furthermore $\mathcal{F}$ is
$f$-invariant, then $f$ induces a bijection $q:\Lambda\to\Lambda,q(L)=f(L)$,
and thus $f$ is a bundle isomorphism.

Consider the bundles over $\Lambda$ given as

$\displaystyle\mathrm{Con}:=\bigsqcup_{i\in\Lambda}\mathcal{C}_{c}(M_{i})$
$\displaystyle\mathrm{Rad}:=\bigsqcup_{i\in\Lambda}\mathscr{Rad}(M_{i})$

and consider also the space of sections of $\mathrm{Rad}$,

$\mathrm{Meas}:=\\{\nu:\Lambda\rightarrow\mathrm{Rad}:\nu_{i}:=\nu(i)\in\mathscr{Rad}(M_{i})\forall\
i\in\Lambda\\}.$

The natural topology of $\mathrm{Meas}$ is the weak topology determined by
$\mathrm{Con}$; it is a locally convex topology. Let
$\mathrm{Con}^{+}:=\\{\phi\in\mathrm{Con},\phi\geq 0,\phi\not\equiv 0\\}$.

######  Lemma 2.2.

Suppose that $\mathcal{A}\subset\mathrm{Meas}$ satisfies

  1. 1.

for every $\psi\in\mathrm{Con}$ there exists $c(\psi)>0$ with the property
that for every $\mu\in\mathcal{A},\mu(\psi)\leq c(\psi)$.

  2. 2.

For every $\psi\in\mathrm{Con}^{+}$ there exists $d(\psi)>0$ with the property
that for every $\mu\in\mathcal{A},\mu(\psi)\geq d(\psi)$.

Then $\mathtt{cl}(\mathcal{A})\subset\mathrm{Meas}$ is compact, and
furthermore it does not contain the zero section.

######  Proof 2.3.

This is a direct consequence of Tychonoff’s theorem.

From now we assume that $f$ is a bundle isomorphism.

######  Definition 2.4.

$\nu\in\mathrm{Meas}$ is quasi-invariant if for every $i\in\Lambda$,
$f^{-1}\nu_{qi}\sim\nu_{i}$ with positive continuous Radon-Nikodym derivative
$\rho_{i}=\frac{df^{-1}\nu_{qi}}{d\nu_{i}}$. In this case
$\rho:M\to\mathbb{R}_{>0}$ given by $\rho|M_{i}=\rho_{i}$ will be called the
Jacobian of $\nu$.

One verifies directly that if $\nu$ is quasi-invariant then $\rho$ generates a
(multiplicative) cocycle over $q$, and we write $\tilde{\rho}=\log\rho$ the
corresponding additive cocycle.

Now suppose that $\varphi:M\to\mathbb{R}$ is a continuous function and $\nu$
is quasi-invariant: we are interested in finding $\mu\in\mathrm{Meas}$ quasi-
invariant with Jacobian $e^{-\varphi}$. Consider
$\nu^{n}=e^{S_{n}\varphi}\cdot f^{-n}\nu$, i.e.

$\nu^{n}_{i}=e^{S_{n}\varphi}f^{-n}\nu_{q^{n}(i)}=e^{S_{n}\varphi+S_{n}\tilde{\rho}}\nu_{i},$
(1)

and let $\mathcal{C}^{+}$ be the positive cone generated by
$\\{\nu^{n}\\}_{n\geq 0}$,

$\mathcal{C}^{+}=\\{\sum_{i=1}^{k}a_{i}\nu^{n_{i}}:a_{i}\geq
0,k\in\mathbb{N}\\}.$ (2)

######  Theorem 2.5.

Assume that $\\{\nu^{n}\\}_{n\geq 0}$ satisfies:

  * •

if $\phi\in\mathrm{Con}^{+}$, then $\nu^{n}(\phi)>0$ for all $n$.

  * •

Given $\phi,\psi\in\mathrm{Con}$ with $\phi\not\equiv 0$ non-negative, there
exists $e(\phi,\psi)>0$ such that for every $n\geq 0$ it holds

$\frac{\nu^{n}(\psi)}{\nu^{n}(\phi)}\leq e(\phi,\psi).$

Then there exists $P\in\mathbb{R}$ and a quasi-invariant
$\mu\in\overline{\mathcal{C}^{+}}$ with Jacobian $e^{P-\varphi}$.

######  Proof 2.6.

Define $S:\mathrm{Meas}\rightarrow\mathrm{Meas}$ by

$S(\mu_{i})=e^{\varphi}\cdot f^{-1}\mu_{q(i)}.$

Then $S$ is continuous and sends $\mathcal{C}^{+}$ to itself. We compute

$S(\nu^{n}_{i})=e^{\varphi}\cdot
f^{-1}\left(e^{S_{n}\varphi}\nu_{q(i)}^{n}\right)=e^{\varphi}\cdot
f^{-1}\left(e^{S_{n}\varphi}f^{-n}\nu_{q^{n+1}(i)}\right)=e^{\varphi+S_{n}\varphi\circ
f}\cdot f^{-n+1}\nu_{q^{n+1}(i)}=\nu^{n+1}.$ (3)

Fix a non-negative, non-identically zero function $\phi_{0}\in\mathrm{Con}$
and let $\hat{\nu}^{n}:=\frac{\nu^{n}}{\nu^{n}(\phi_{0})}$; consider the set

$\displaystyle\mathcal{X}$
$\displaystyle=\mathtt{cl}(\\{\sum_{i=1}^{k}a_{i}\hat{\nu}^{n_{i}}:a_{i}\geq
0,k\in\mathbb{N},\sum_{i=1}^{k}a_{i}=1\\})$ (4)
$\displaystyle=\mathtt{cl}(\\{\nu\in\mathcal{C}^{+}:\nu(\phi_{0})=1\\}).$ (5)

By lemma 2.2 and the assumed hypotheses, $\mathcal{X}$ is a compact convex
subset of $\mathrm{Meas}$, and by eq. 3 the normalized operator
$\tilde{S}\nu:=\frac{S\nu}{S\nu(\psi_{0})}$ sends $\mathcal{X}$ to itself.
Hence, by the Schauder-Tychonoff fix point theorem, there exist
$\mu\in\mathcal{X}$ such that $\widetilde{S}(\mu)=\mu$. In other words,
$\psi\in\mathrm{Con}$ implies

$\mu_{q(i)}(\psi\circ f^{-1}e^{\varphi\circ
f^{-1}})=e^{P}\cdot\mu_{i}(\psi)\quad(\Rightarrow
f^{-1}\mu_{q(i)}=e^{P-\varphi}\cdot\mu_{i}),$ (6)

where $e^{P}=S\mu(\psi_{0})>0$.

Observe that in the theorem above, if $\psi\in\mathrm{Con}$ is non-negative,
non-identically zero then for some constant $c(\psi)>0$ it holds
$\hat{\nu}^{n}(\psi)>0$ for all $n\geq 0$, and thus the same is valid for
$\mu$. We deduce:

######  Corollary 2.7.

In the same hypotheses of the Theorem above, the section $\mu$ has full
support on each $M_{i}$.

###  2.2 Quasi-invariant measures along leaves of foliations: the transverse
measure

We will now apply the construction of the previous part to the following
setting.

  1. 1.

$M$ is a compact (closed) smooth Riemannian manifold and
$\mathcal{F}=\\{F(x)\\}_{x\in M}$ is continuous foliations with smooth leaves.
We consider the bundle structure of $M=\bigsqcup_{F\in\Lambda}F$ over
$\Lambda=M/\mathcal{F}$, and denote
$\mathrm{Con}(\mathcal{F}),\mathrm{Meas}(\mathcal{F})$ the corresponding
associated bundles.

  2. 2.

$f$ is a diffeomorphism of $M$ of differentiability class $\mathcal{C}^{2}$.

  3. 3.

$\mathcal{F}$ is $f$-invariant; the induced map by $f$ on $\Lambda$ is denoted
by $q$.

  4. 4.

There exists a complementary continuous foliation
$\mathcal{G}=\\{G(x)\\}_{x\in M}$ of $\mathcal{F}$ with smooth leaves, and
such that

    1. (a)

$\mathcal{G}$ is minimal: each leaf $G(x)$ is dense.

    2. (b)

$\mathcal{G}$ is $f$-invariant.

    3. (c)

$\mathcal{G}$ is contracting under $f$: if $d_{\mathcal{G}}$ denotes the
intrinsic distance (on the corresponding leaf of $\mathcal{G}$) we have

$\forall x,y\in M,y\in G(x)\Rightarrow d_{\mathcal{G}}(fx,fy)\leq\lambda
d_{\mathcal{G}}(x,y).$

for some $0<\lambda<1$.

The idea is that if $B$ is a foliated box corresponding to $\mathcal{F}$, we
will use the (measurable) partition $\mathcal{F}|B$ to play the role of the
quotient space where we define our transverse measure. The existence of a
complementary contracting foliation is used (crucially) to ensure the
compactness property necessary for lemma 2.2.

Notation: If $A\subset M,\epsilon>0$ let

$G(A,\epsilon):=\\{y\in M:\exists x\in A/d_{\mathcal{G}}(x,y)<\epsilon\\},$

and if $A=\\{x\\}$ we write $G(x,\epsilon)=G(\\{x\\},\epsilon)$.

######  Definition 2.8.

Let $\delta>0$.

  1. 1.

Two relatively compact sets $A_{1}\subset F_{i_{1}},A_{2}\subset F_{i_{2}}$
are $\delta$-equivalent if there exists an holonomy transport (Poincaré map)
$\mathrm{hol}^{\scriptstyle\mathcal{G}}:F_{i_{1}}\to F_{i_{2}}$ induced by
$\mathcal{G}$ satisfying:

    1. (a)

$\mathrm{hol}^{\scriptstyle\mathcal{G}}|A_{1}$ is an homeomorphism of $A_{1}$
onto $A_{2}$.

    2. (b)

For every $x\in A_{1},\mathrm{hol}^{\scriptstyle\mathcal{G}}(x)\in
G(x,\delta)$.

  2. 2.

Two functions $\psi_{1},\psi_{2}\in\mathrm{Con}(\mathcal{F})$ are
$\delta$-equivalent if $\mathtt{supp}(\psi_{1})$ is $\delta$-equivalent to
$\mathtt{supp}(\psi_{2})$ and for every $x\in\mathtt{supp}(\psi_{1})$,
$\psi_{2}(\mathrm{hol}^{\scriptstyle G}(x))=\psi_{1}(x)$. Here
$\mathrm{hol}^{\scriptstyle G}$ is the holonomy map connecting
$\mathtt{supp}(\psi_{1})$ with $\mathtt{supp}(\psi_{2})$.

Observe that given $\psi\in\mathrm{Con}(\mathcal{F}),\delta>0$ there exists
$\gamma=\gamma(\psi,\delta)>0$ such that if $x\in
G(\mathtt{supp}(\psi),\gamma)$ then

  * •

the transverse foliation defines an homeomorphism onto its image
$\mathrm{hol}^{\scriptstyle\mathcal{G}}|:\mathtt{supp}(\psi)\to F(x)$.

  * •

$\psi_{F(x)}:=\psi\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}$ is continuous,
of compact support, and $\delta$-equivalent to $\psi$.

We seek to find a quasi-invariant $\nu\in\mathrm{Meas}(\mathcal{F})$ with
regular Radon-Nikodym derivative. A natural class is obtained by fixing a
Riemannian metric on $M$ and considering
$\mathrm{Leb}\in\mathrm{Meas}(\mathcal{F})$ given by

$F\in\Lambda\Rightarrow\mathrm{Leb}_{F}=\text{ induced Lebesgue measure on the
leaf }F.$

By the basic change of variables theorem, $\mathrm{Leb}$ is quasi-invariant
its Jacobian is $\rho_{F}=\log|\det D(f|F)|$. By compactness of $M$ we have
that

$\rho\text{ is uniformly bounded from below: }\rho(x)\geq C_{\rho}>0\ \forall
x\in M,$

and since $f\in\mathscr{Diff}^{1+\alpha}(M)$ we get that $\rho$ (and
therefore, $\tilde{\rho}=\log\rho$) is $\alpha$-Hölder.

We will use the minimality of $\mathcal{G}$ to compare sets (functions)
defined on different leaves of $\mathcal{F}$. It is necessary however to
control how families of measures $\nu=\\{\nu_{F}\\}_{F\in\mathcal{F}}$ vary
with the transverse holonomy given by $\mathcal{G}$.

######  Definition 2.9.

$\nu\in\mathrm{Meas}(\mathcal{F})$ is absolutely continuous if for every
(locally defined) holonomy transport
$\mathrm{hol}^{\scriptstyle\mathcal{G}}:F_{i_{1}}\to F_{i_{2}}$,
$\mathrm{hol}^{\scriptstyle\mathcal{G}}\nu_{F_{i_{1}}}\sim\nu_{F_{i_{2}}}$. It
is strongly absolutely continuous if there exists $\delta_{0}>0$ and
$J:\\{(x,y,z)\in M\times M\times M,y\in G(x,\delta_{0}),z\in
F(y)\\}\to\mathbb{R}$ continuous such that if $A\subset F(x),B\subset F(y)$
are $\delta_{0}$-equivalent then

$\mathrm{hol}^{\scriptstyle\mathcal{G}}\nu_{F(x)}=J(x,y,\cdot)\nu_{F(y)}$

######  Proposition 2.10.

$\mathrm{Leb}$ is strongly absolutely continuous.

######  Proof 2.11.

This result is classical by now, and relies on the contracting behavior of
$\mathcal{G}$ together with the fact of $f$ being of class
$\mathcal{C}^{1+\alpha},\alpha>0$. See [ErgAnAc].

For strongly absolutely continuous sections we have the following.

######  Proposition 2.12.

Assume that $\nu$ is strongly absolutely continuous and of full support on
every $F\in\mathcal{F}$. Then given $\epsilon>0$ there exists $\delta>0$ such
that for every $\psi_{1},\psi_{2}\in\mathrm{Con}(\mathcal{F})$ that are
$\delta$-equivalent and non-identically zero, it holds

$\left|\int\psi_{1}d\nu_{F_{i_{1}}}-\int\psi_{2}d\nu_{F_{i_{2}}}\right|<\delta\int|\psi_{1}|d\nu_{F_{i_{1}}}.$

######  Proof 2.13.

We start by noting that since $\nu$ is strongly absolutely continuous, given
$\epsilon>0$ there exists $\delta$ such that if $A,B$ are $\delta$-equivalent
(with non-trivial measure) then

$\big{|}\frac{\nu(A)}{\nu(B)}-1\big{|}<\epsilon.$

The remaining part of the argument is spelled in [TesisMarg]; use the previous
remark as Lemma $2.1$ and argue as in Lemma $2.2$, Proposition $2.2$, Lemma
$2.5$ and finally Lemma $2.6$, which has the same content as our Proposition.

######  Remark 2.14.

The core of our construction in this part is due to G. Margulis (above-cited)
who considered the case where the potential $\varphi\equiv 0$ and
$\nu=\mathrm{Leb}$. Parts which follow (with minimal) mutatis-mutandis will be
directed to Margulis’ presentation.

We now complete our construction. Fix $\nu\in\mathrm{Meas}(\mathcal{F})$ that
satisfies:

  * •

for every leaf $F\in\mathcal{F}$, $\nu_{F}$ has full support,

  * •

is strongly absolutely continuous,

  * •

is quasi-invariant with Hölder Jacobian $\rho$,

and consider $\varphi:M\to\mathbb{R}$ a Hölder continuous potential. Let
$\hat{\rho}=\log\rho$ and $\hat{\varphi}=\varphi+\hat{\rho}$; this is a
$(C_{\hat{\varphi}},\theta)$-Hölder function. For $n\in\mathbb{N}$ take
$\nu^{n}=e^{S_{n}\varphi}\cdot f^{-n}\nu=e^{S_{n}\hat{\varphi}}\nu$ (cf. eq.
1).

######  Lemma 2.15.

There exists a constant $D_{1}>0$ and $\ell:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$
with the following property. For every $\delta>0$, if
$\psi_{1},\psi_{2}\in\mathrm{Con}^{+}$ are $\delta$-equivalent and $n\geq 0$,
then it holds

$\nu^{n}(\psi_{1})\leq\ell(\delta)\cdot
e^{D_{1}\delta^{\theta}}\nu^{n}(\psi_{2}).$

Furthermore, $\ell(\delta)\xrightarrow[\delta\to 0]{}1$.

######  Proof 2.16.

Define $D_{1}:=C_{\hat{\varphi}}\sum_{i=0}^{\infty}\lambda^{i\theta}$; since
$\lambda<1$, $D<+\infty$. Assume that $\psi_{1},\psi_{2}$ are
$\delta$-equivalent,
$\psi_{2}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}=\psi_{1}$. Observe that
for every $x\in\mathtt{supp}(\psi_{1})$, for every $n\geq 0$, it holds

$|S_{n}\hat{\varphi}(\mathrm{hol}^{\scriptstyle\mathcal{G}}x)-S_{n}\hat{\varphi}(x)|\leq\sum_{i=0}^{\infty}C_{\hat{\varphi}}\cdot
d_{\mathcal{G}}(f^{i}\mathrm{hol}^{\scriptstyle\mathcal{G}}x,f^{i}x)^{\theta}\leq
D_{1}\delta^{\theta}.$

It then follows that

$\displaystyle\nu^{n}(\psi_{1})$ $\displaystyle=\int
e^{S_{n}\hat{\varphi}}\psi_{2}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}d\nu=\int
e^{S_{n}\hat{\varphi}-S_{n}\hat{\varphi}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}}\psi_{2}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}e^{S_{n}\hat{\varphi}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}}d\nu$
$\displaystyle\leq
e^{D_{1}\delta^{\theta}}\int\psi_{2}e^{S_{n}\hat{\varphi}}d\mathrm{hol}^{\scriptstyle\mathcal{G}}\nu\leq\ell(\delta)e^{D_{1}\delta^{\theta}}\nu^{n}(\psi_{2})$

where $\ell$ is an upper bound of
$\frac{d\mathrm{hol}^{\scriptstyle\mathcal{G}}\nu}{d\nu}$, and therefore
converges to $1$ as $\delta\to 0$.

######  Lemma 2.17.

Let $A_{1}\subset F_{i_{1}},A_{2}\subset F_{i_{2}}$ be relatively open and
pre-compact. Then there exist $\hat{e}(A_{1},A_{2})>0$ such that for every
$n\geq 0$ it holds

$\frac{1}{\hat{e}(A_{1},A_{2})}\leq\frac{\nu^{n}(A_{1})}{\nu^{n}(A_{2})}\leq\hat{e}(A_{1},A_{2}).$

######  Proof 2.18.

Let us start noting that since $\mathcal{G}$ is minimal (on $M$ compact) we
have the following property: for any $A\subset F,F\in\mathcal{F}$ there exist
$\delta(A),r(A)>0$ such that for every $x\in M$ one can find $B_{x}\subset A$
that is $\delta(A)$-equivalent to $F(x,r(A))$.

By relative compactness of $\mathtt{cl}(A_{1})$, the closure of $A_{1}$ inside
$F_{i_{1}}$, it follows that
$\mathtt{cl}(A_{1})\subset\cup_{1}^{m}F(x_{j},r(A_{2}))$, where each plaque
$F(x_{j},r(A_{2}))$ is $\delta(A_{2})$-equivalent to some $B_{j}\subset
A_{2}$.

Fix two of these sets $E_{1}=F(x_{j},r(A_{2})),E_{2}=B_{j}$ and denote by
$\mathrm{hol}^{\scriptstyle\mathcal{G}}$ the holonomy transport such that
$\mathds{1}_{E_{1}}=\mathds{1}_{E_{2}}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}$.
By regularity of $\nu_{F_{i_{2}}}$ we can find a non-decreasing sequence
$(\phi_{k})_{k}\subset\mathcal{C}_{c}(F_{i_{2}})$ satisfying

  1. 1.

$k\in\mathbb{N}\Rightarrow 0\leq\phi_{k}\leq
1,\mathtt{supp}(\phi_{k})\subset\mathtt{cl}(A_{2})$.

  2. 2.

$\lim_{k\mapsto\infty}\phi_{k}(x)=\mathds{1}_{A_{2}}(x)$.

  3. 3.

$\nu^{n}(A)=\lim_{k}\uparrow\nu^{n}(\phi_{k})$.

Define $\psi_{k}=\phi_{k}\circ\mathrm{hol}^{\scriptstyle\mathcal{G}}$ and
apply the previous lemma to deduce that for every $n$,

$\displaystyle\frac{\nu^{n}(E_{1})}{\nu^{n}(E_{2})}=\frac{\lim_{k}\nu^{n}(\psi_{k})}{\sup_{k}\nu^{n}(\phi_{k})}\leq\sup_{k}\frac{\psi_{k}}{\phi_{k}}\leq\ell(\delta(A_{2}))e^{D_{1}\delta(A_{2})^{\theta}}:=e_{0}.$

Therefore,

$\displaystyle\frac{\nu^{n}(A_{1})}{\nu^{n}(A_{2})}\leq\frac{\nu^{n}(F(x_{j},r(A_{2}))}{\nu^{n}(A_{2})}\leq
m\cdot\max_{j}\frac{\nu^{n}(F(x_{j},r(A_{2})))}{\nu^{n}(B_{j})}\leq m\cdot
e_{0}.$

From here the conclusion follows.

######  Corollary 2.19.

Fix $\psi\in\mathrm{Con}^{+}(\mathcal{F})$. Then for every
$\phi\in\mathrm{Con}(\mathcal{F})$ there exists $\hat{e}(\phi,\psi)>0$ such
that for every $n\geq 0$ it holds

$\frac{\nu^{n}(\phi)}{\nu^{n}(\psi)}\leq\hat{e}(\phi,\varphi).$

######  Proof 2.20.

This is essentially Lemma $2.4$ of [TesisMarg]. For $r>0$ let
$A_{r}:=\psi^{-1}(r,\infty)$ and note that since $\psi\in\mathrm{Con}$ is non-
negative and non-identically zero, there exists $r>0$ such that $A_{r}$ is
relatively open and pre-compact inside the leaf of $\mathcal{F}$ that contains
$\mathtt{supp}(\psi)$. Take $A$ open, relatively compact containing
$\mathtt{supp}(\phi)$ and use the previous Lemma to deduce

$\frac{\nu^{n}(\phi)}{\nu^{n}(\psi)}<\frac{\|\phi\|_{\infty}\cdot\nu^{n}(A)}{r\cdot\nu^{n}(A_{r})}<\frac{\|\phi\|_{\infty}}{r}\
\hat{e}(A,A_{r}).$

We have thus shown that the in the assumed conditions, the hypotheses of
theorem 2.5 are satisfied, and thus we conclude the existence of
$P\in\mathbb{R}$ and $\mu\in\mathrm{Meas}(\mathcal{F})$ of full support on
each leaf of $\mathcal{F}$, that is quasi-invariant with Jacobian
$e^{P-\varphi}$.

#### Strong absolute continuity of the quasi-invariant section

It turns out that the quasi-invariance of $\mu$ implies that is strongly
absolutely continuous. We keep the hypotheses of the previous part and denote
for $x\in M$, $\mu_{x}=\mu_{F(x)}$. By quasi-invariance we get

$\forall n\in\mathbb{N},\quad f^{-n}\mu_{f^{n}x}=e^{nP-S_{n}\varphi}\mu_{x}.$
(7)

######  Lemma 2.21.

Given $\epsilon>0$ there exists $\delta>0$ such that for every
$\psi_{1},\psi_{2}\in\mathrm{Con}^{+}$ that are $\delta$-equivalent, then

$\left|\frac{\mu(\psi_{1})}{\mu(\psi_{2})}-1\right|<\epsilon.$

######  Proof 2.22.

By lemma 2.15, for every $n\in\mathbb{N}$ we have
$\hat{\nu}^{n}(\psi_{1})\leq\ell(\delta)\cdot
e^{D_{1}\delta^{\theta}}\hat{\nu}^{n}(\psi_{2})$; the inequality extends to
convex combinations of $\\{\hat{\nu}^{n}\\}_{n}$ and therefore, to elements of
$\mathcal{X}$ (cf. eq. 4). In particular $\mu$ satisfies the claim.

We now fix $x_{0},y_{0}$ in the same leaf of $\mathcal{G}$ and consider
$\mathrm{hol}^{\scriptstyle\mathcal{G}}:A(x_{0})\subset F(x_{0})\to
B(y_{0})\subset F(y_{0})$ the corresponding Poincaré map. For $x\in A(x_{0})$
define

$\mathrm{Jac}^{\mathcal{G}}_{x_{0},y_{0}}(x)=\prod_{j=0}^{\infty}\frac{e^{\varphi\circ
f^{j}(\mathrm{hol}^{\scriptstyle\mathcal{G}}x)}}{e^{\varphi\circ
f^{j}(x)}}=\lim_{n}e^{S_{n}\varphi(\mathrm{hol}^{\scriptstyle\mathcal{G}}x)-S_{n}\varphi(x)}.$
(8)

Since $\varphi$ is Hölder and $\mathcal{G}$ is contracting, the previous
formula defines a continuous function
$\mathrm{Jac}^{\mathcal{G}}_{x_{0},y_{0}}:A(x_{0})\rightarrow\mathbb{R}$.

######  Proposition 2.23.

It holds
$(\mathrm{hol}^{\scriptstyle\mathcal{G}})^{-1}\mu_{y_{0}}=\mathrm{Jac}^{\mathcal{G}}_{x_{0},y_{0}}\cdot\mu_{x_{0}}$.

######  Proof 2.24.

Denote $h=\mathrm{hol}^{\scriptstyle\mathcal{G}}$, and for $n\geq 0$ let
$h_{n}$ be the Poincaré map defined on a neighborhood of $f^{n}x$ inside
$f^{n}(A(x_{0}))$. As $f$ preserves $\mathcal{F},\mathcal{G}$, $h=f^{-n}\circ
h_{n}\circ f^{n}$ and therefore

$\displaystyle h^{-1}\mu_{y_{0}}=(f^{-n}\circ h_{n}^{-1})e^{S_{n}\varphi\circ
f^{-n}-nP}\mu_{f^{n}y_{0}}=f^{-n}\left(e^{S_{n}\varphi\circ h\circ
f^{-n}-nP}h_{n}^{-1}\mu_{f^{n}y_{0}}\right)$
$\displaystyle=f^{-n}\left(e^{S_{n}\varphi\circ h\circ
f^{-n}-nP}\left(\mu_{f^{n}x_{0}}+\left(h_{n}^{-1}\mu_{f^{n}y_{0}}-\mu_{f^{n}x_{0}}\right)\right)\right)$
$\displaystyle=e^{S_{n}\varphi\circ
h-S_{n}\varphi}\mu_{x_{0}}+f^{-n}e^{S_{n}\varphi\circ h\circ
f^{-n}-nP}\left(h_{n}^{-1}\mu_{f^{n}y_{0}}-\mu_{f^{n}x_{0}}\right).$

By the lemma above, given $\epsilon>0$ there exists $n_{0}$ such that for
$n\geq n_{0}$ it holds: for any $\psi\in\mathrm{Con}^{+}$ supported in
$f^{n}(A(x_{0}))$,

$|h_{n}^{-1}\mu_{f^{n}y_{0}}(\psi)-\mu_{f^{n}x_{0}}(\psi)|<\epsilon
h_{n}^{-1}\mu_{f^{n}y_{0}}(\psi)$

which in turn implies

$\displaystyle|f^{-n}e^{S_{n}\varphi\circ h\circ
f^{-n}-nP}\left(h_{n}^{-1}\mu_{f^{n}y_{0}}-\mu_{f^{n}x_{0}}\right)(\psi\circ
f^{n})|$ $\displaystyle<\epsilon\mu_{f^{n}y_{0}}(\psi\circ
h_{n}^{-1}e^{S_{n}\varphi\circ f^{-n}-nP})$ $\displaystyle=\epsilon
h^{-1}\mu_{y_{0}}(\psi\circ f^{n}).$

Thus we can write $h^{-1}\mu_{y_{0}}=J_{n}\cdot\mu_{x_{0}}+\upsilon_{n}$,
where $\upsilon_{n}$ converges to zero in the vague topology, and
$\\{J_{n}\\}_{n}$ is a sequence of continuous functions that converges
uniformly to $\mathrm{Jac}^{\mathcal{G}}_{x_{0},y_{0}}$. This concludes the
proof.

##  3 Invariant measures for center isometries

We will use the previous construction in the case when $f$ is a center
isometry. For convenience of the reader we recall the definition and some
basic facts below.

######  Definition 3.1.

Let $M$ be a closed manifold. A diffeomorphism
$f\in\mathscr{Diff}^{M}(r),r\geq 1$ is a center isometry if there exist a
continuous splitting of the tangent bundle of the form

$TM=E^{u}\oplus E^{c}\oplus E^{s}$

where both bundles $E^{s},E^{u}$ are non-trivial, a (continuous) Riemannian
metric $\|\cdot\|$ on $M$ and a constant $0<\lambda<1$ such that

  1. 1.

All bundles $E^{u},E^{s},E^{c}$ are $Df$-invariant.

  2. 2.

For every $x\in M$, for every unit vector $v^{\ast}\in
E^{\ast}_{x},\ast=s,c,u$,

$\displaystyle\|D_{x}f(v^{c})\|=1$
$\displaystyle\|D_{x}f^{n}(v^{s})\|,\|D_{x}f^{-1}(v^{u})\|<\lambda.$

Center isometries are a subclass of the so called _Partially Hyperbolic
Systems_. From their theory we need the following.

######  Theorem 3.2.

If $f\in\mathscr{Diff}^{r}(M)$ is a center isometry then the bundles
$E^{s},E^{u},E^{c}$, $E^{cs}=E^{c}\oplus E^{s},E^{cu}=E^{c}\oplus E^{u}$ are
(uniquely) integrable to continuous foliations
$\mathcal{W}^{s}=\\{W^{s}(m)\\}_{x\in M}$,
$\mathcal{W}^{u}=\\{W^{u}(m)\\}_{x\in M},\mathcal{W}^{c}=\\{W^{c}(m)\\}_{x\in
M},\mathcal{W}^{cs}=\\{W^{cs}(m)\\}_{x\in
M},\mathcal{W}^{cu}=\\{W^{cu}(m)\\}_{x\in M}$ respectively called the _stable_
,_unstable_ ,_center_ , _center stable_ and _center unstable_ foliations,
whose leaves are $\mathcal{C}^{r}$ immersed submanifolds. Moreover, leaves of
$\mathcal{W}^{cs}$ are saturated by leaves of
$\mathcal{W}^{s},\mathcal{W}^{c}$, and leaves of $\mathcal{W}^{cu}$ are
saturated by leaves of $\mathcal{W}^{u},\mathcal{W}^{c}$.

######  Proof 3.3.

Integrability of the bundles $E^{s},E^{u}$ is consequence of the classical
Stable Manifold theorem (see for example Theorem 4.1 in [PesinLect]), while
the rest of the assertions can be deduced from theorem 7.5 in [PartSurv].

Standing hypotheses for the rest of the section: $f:M\rightarrow M$ is center
isometry of class $\mathcal{C}^{2}$ and $\varphi:M\to\mathbb{R}$ is a Hölder
potential. The metric in $M$ is assumed to make $E^{s},E^{c},E^{u}$ mutually
perpendicular.111This can always be achieved by passing to an equivalent
metric. Both foliations $\mathcal{W}^{s},\mathcal{W}^{u}$ are minimal.

Applying the results of section 2 with
$\mathcal{F}=\mathcal{W}^{cu},\mathcal{G}=\mathcal{W}^{s}$ we get the
following.

######  Proposition 3.4.

There exists a section
$\mu^{cu}=\mu^{cu}_{\varphi}\in\mathrm{Meas}(\mathcal{W}^{cu})$ and
$P\in\mathbb{R}$ such that:

  1. 1.

For every $\psi\in\mathrm{Con}^{+}(\mathcal{W}^{cu},)$ it holds
$\mu^{cu}(\psi)>0$.

  2. 2.

For every $x\in M$, $f^{-1}\mu^{cu}_{fx}=e^{P-\varphi}\mu^{cu}_{x}$.

  3. 3.

If $\mathrm{hol}^{s}=\mathrm{hol}^{s}_{x_{0},y_{0}}:A(x_{0})\subset
W^{cu}(x_{0})\rightarrow B(y_{0})\subset W^{cu}(y_{0})$ is the Poincaré map
determined by the stable holonomy that sends $x_{0}$ to $y_{0}$, then

$(\mathrm{hol}^{s})^{-1}\mu^{cu}_{y_{0}}=\mathrm{Jac}^{s}_{x_{0},y_{0}}\cdot\mu^{cu}_{x_{0}}$

where

$\mathrm{Jac}^{s}_{x_{0},y_{0}}(x)=\prod_{j=0}^{\infty}\frac{e^{\varphi\circ
f^{j}(\mathrm{hol}^{s}x)}}{e^{\varphi\circ f^{j}(x)}}.$ (9)

Note that $f^{-1}$ is also a center isometry whose stable foliation coincides
with the unstable of $f$, and likewise its unstable foliation coincides with
the stable of $f$. By applying the previous arguments to $f^{-1}$ and the
potential $\varphi\circ f^{-1}$ we conclude the following.

######  Proposition 3.5.

There exist $\mu^{cs}=\mu^{cs}_{\varphi}\in\mathrm{Meas}(\mathcal{W}^{cu})$
and $P^{\prime}\in\mathbb{R}$ such that:

  1. 1.

For every $\psi\in\mathrm{Con}^{+}(\mathcal{W}^{cs})$ it holds
$\mu^{cs}(\psi)>0$.

  2. 2.

For every $x\in M$, $f^{-1}\mu^{cs}_{fx}=e^{\varphi-P^{\prime}}\mu^{cs}_{x}$.

  3. 3.

If $\mathrm{hol}^{u}=\mathrm{hol}^{u}_{x_{0},y_{0}}:A(x_{0})\subset
W^{cs}(x_{0})\rightarrow B(y_{0})\subset W^{cs}(y_{0})$ is the Poincaré map
determined by the stable holonomy that sends $x_{0}$ to $y_{0}$, then

$(\mathrm{hol}^{u})^{-1}\mu^{cs}_{y_{0}}=\mathrm{Jac}^{u}_{x_{0},y_{0}}\cdot\mu^{cs}_{x_{0}}$

where

$\mathrm{Jac}^{u}_{x_{0},y_{0}}(x)=\prod_{j=1}^{\infty}\frac{e^{\varphi\circ
f^{-j}(\mathrm{hol}^{u}x)}}{e^{\varphi\circ f^{-j}(x)}}.$ (10)

###  3.1 The conditional measures: potentials constant along the center and
SRBs

We are interested in constructing $\mu^{u}\in\mathrm{Meas}(\mathcal{W}^{u})$
satisfying analogous properties as $\mu^{cu}$. Our previous construction
however does not apply in this case since the transverse foliation to
$\mathcal{W}^{u}$ is not contracting. To bypass this problem we assume some
condition on $\varphi$, namely that it is either

  * •

constant on center leaves ($c$-constant case), or

  * •

$\varphi=-\log\det Df|E^{u}$ (SRB case).

In the SRB case we define

$\mu^{u}_{x}=\text{Lebesgue measure on }W^{u}(x);$

by the change of variables theorem $f^{-1}\mu^{u}_{fx}=\det
Df|E^{u}\cdot\mu^{u}_{x}=e^{-\varphi}\mu^{u}_{x}$. For the $c$-constant case,
note that by compactness of $M$ there exists some $\epsilon_{0}>0$ such that
for every $x\in M$ there exists a well defined projection
$\pi_{x}^{c}:W^{c}(W^{u}(x),\epsilon_{0})$ by sliding along local center
plaques. Given $x$ we define a measure on $W^{u}(x)$ by setting

$\mu^{u}_{x}=\pi^{c}_{x}\mu^{cu}_{x}$

Note that $\mu\in\mathrm{Meas}(\mathcal{W}^{u})$ is quasi-invariant with
Jacobian $e^{P-\varphi}$; indeed by using that $\varphi$ is constant on center
leaves and that $f$ is an isometry on these, we deduce

$\displaystyle
f^{-1}\mu^{u}_{fx}=f^{-1}\pi^{c}_{fx}\mu^{cu}_{fx}=\pi^{c}_{x}f^{-1}\mu^{cu}_{fx}=\pi^{c}_{x}e^{P-\varphi}\mu^{cu}_{x}=e^{P-\varphi}\pi^{c}_{x}\mu^{cu}_{x}=e^{P-\varphi}\mu^{u}_{x}.$
(11)

An important case of $c$-constant potential is when $\varphi\equiv 0$; in this
case we denote $m^{\ast}\in\mathrm{Meas}(\mathcal{F}^{\ast}),\
\ast\in\\{s,u,cs,cu\\}$ the corresponding sections.

######  Definition 3.6.

$m^{\ast}$ are the Margulis measures on $\mathcal{F}^{\ast}$.

It follows that

$\displaystyle f^{-1}m^{cu}_{fx}=e^{h}m^{cu}_{x}$ (12) $\displaystyle
f^{-1}m^{u}_{fx}=e^{h}m^{u}_{x}$ (13)

for some $h\in\mathbb{R}$.

######  Proposition 3.7.

  1. 1.

$m^{cu}$ ($m^{cs}$) is invariant under $\mathrm{hol}^{s}$ holonomy
(respectively, $\mathrm{hol}^{u}$ holonomy).

  2. 2.

The constant $h$ is positive.

######  Proof 3.8.

By part 3 of proposition 3.4 it holds $\mathrm{Jac}^{s}_{x_{0},y_{0}}\equiv
1$, thus implying the first part.

Recall the construction of $m^{cu}$ on page 2.4, with $\nu=\mathrm{Leb}$ and
$\nu^{n}=f^{-n}\nu$ and a fixed normalized
$\psi_{0}\in\mathrm{Con}^{+}(\mathcal{W}^{cu})$. Thus we have for every $n\geq
0$,

$\displaystyle f^{-1}\nu^{n}(\psi_{0})\geq\min\\{\det
Df|E^{cu}\\}\nu^{n}(\psi_{0})\Rightarrow
f^{-1}\hat{\nu}^{n}(\psi_{0})\geq\min\\{\det Df|E^{cu}\\}.$

This implies the same inequality for every member of $\mathcal{X}$, and in
particular

$e^{h}=m^{cu}(\psi_{0}\circ f^{-1})\geq\min\\{\det Df|E^{cu}\\}>1.$

######  Remark 3.9.

The reader can perceive that in principle $h$ (and $P,P^{\prime}$) depend on
$\psi_{0}$. We will show in the next section that $h=h_{\mathrm{top}}(f)$ and
$P=P^{\prime}=P_{\mathrm{top}}(\varphi)$.

We return to the general case; we will use the family $\\{\mu^{u}_{x}\\}_{x}$
to define some new measures $\\{\nu^{u}_{x}\\}_{x}$ that will be serve as
conditional measures on local unstable manifolds for the equilibrium state.

For each $x\in M,y\in W^{u}(x)$ let

$\Delta_{x}^{u}(y):=\prod_{k=1}^{\infty}\frac{e^{\varphi\circ
f^{-k}(y)}}{e^{\varphi\circ f^{-k}(x)}},$ (14)

and note that $\Delta_{x}^{u}:W^{u}(x)\to\mathbb{R}$ is continuous. Define

$\nu^{u}_{x}=\Delta_{x}^{u}\ \mu^{u}_{x}.$ (15)

One verifies easily that $\nu^{u}_{x}$ is a Radon measure on $W^{u}(x)$ which
is positive on open sets, and furthermore

$\displaystyle f^{-1}\nu^{u}_{fx}=\Delta_{fx}^{u}\circ f\cdot
f^{-1}\mu^{u}_{fx}=\Delta_{fx}^{u}\circ f\cdot
e^{P-\varphi}\mu^{u}_{x}=e^{P-\varphi(x)}\nu^{u}_{x}.$ (16)

For $y\in W^{u}(x)$ we have $\nu^{u}_{y}=c(y,x)\cdot\nu^{u}_{x}$ with
$c(y,x)>0$, hence $\\{\nu^{u}_{y}\\}_{y\in W^{u}(x)}$ defines a projective
class of measures $[\nu^{u}_{x}]$. In particular if $\xi$ is a
$\mu^{cu}_{x}$-measurable partition subordinated to $\mathcal{W}^{u}$ we have
that the probability measures

$\nu^{\xi}_{x}:=\nu^{u}_{x}(\ \cdot\ |\xi(x))$ (17)

satisfy $\nu^{\xi}_{x}=\nu^{\xi}_{y}$ for $y\in\xi(x)$.

Before moving on we establish the following lemma that will allow us to
compare different unstable measures. Consider the compact space

$N=\\{(x_{0},y_{0},z)\in
M^{3}:y_{0}\in\mathtt{cl}(W^{u}(x_{0},\epsilon/2)),z\in\mathtt{cl}(W^{c}(x_{0},\epsilon/2))\\}$

and for $(x_{0},y_{0},z)\in N$ define

$T_{x_{0},y_{0}}(z)=\prod_{k=1}^{+\infty}\frac{e^{\varphi\circ
f^{-k}(\mathrm{hol}^{u}_{x_{0},y_{0}})}}{e^{\varphi\circ f^{-k}(z)}}.$ (18)

######  Lemma 3.10.

It holds that:

  1. 1.

the constant $c(y,x)$ converges uniformly to $1$ as $y\to x$, and

  2. 2.

$T:N\to\mathbb{R}$ is (uniformly) continuous.

######  Proof 3.11.

The first follows directly since $c(y,x)=\Delta_{y}(x)$. For the second, for
ever $k\geq 0$ consider the continuous function $T_{k}:N\to\mathbb{R}$ defined
as

$T_{k}(x_{0},y_{0},z)=\sum_{i=1}^{k}\varphi\circ
f^{-i}(\mathrm{hol}^{u}_{x_{0},y_{0}}(z))-\varphi\circ f^{-i}(z)$

and note that by the Weirstrass’ test, $T_{k}$ converges uniformly to

$\sum_{k=1}^{+\infty}\varphi\circ
f^{-k}(\mathrm{hol}^{u}_{x_{0},y_{0}}z)-\varphi\circ f^{-k}(z)$

which implies that $\log T$ is continuous, and thus $T$ is continuous.

###  3.2 The equilibrium state

It is seldom the case that for a given $\mu$ invariant measure for $f$, the
partition by unstable manifolds is $\mu$\- measurable, therefore we will have
to work in sets $B$ where $\mathcal{W}^{u}|B$ defines a sufficiently nice
partition. Since we are trying to use the family $\\{\mu^{cs}_{x}\\}_{x\in M}$
as a substitute for the transverse measure of the equilibrium state, we will
have to deal with the technical difficulty of having to show coherence between
different choices $B,B^{\prime}$; this will be bypassed using our control on
the behavior of $\\{\mu^{cs}_{x}\\}_{x\in M}$ under unstable (local)
holonomies. We will now make some geometrical preparations to deal with this
problem.

#### Dynamical boxes

To take advantage of the local product structure between $\mathcal{W}^{cs}$
and $\mathcal{W}^{u}$, we introduce some new notation. Due to the fact that
the invariant bundles are mutually perpendicular there exists $\epsilon_{0}>0$
such that for any $0<\epsilon\leq\epsilon_{0}$ it holds for every $x,y\in
M,d(x,y)<\epsilon$ implies

$\displaystyle\\#W^{s}(x,2\epsilon)\cap W^{cu}(y,2\epsilon)=1$ (19)
$\displaystyle\\#W^{u}(x,2\epsilon)\cap W^{cs}(y,2\epsilon)=1$ (20)

######  Definition 3.12.

For $0<\epsilon\leq\epsilon_{0}$ and $x\in M$ we consider the sets

$\displaystyle P^{cs}(x,\epsilon)$
$\displaystyle:=W^{c}(W^{s}(x,\epsilon),\epsilon)$ (21) $\displaystyle
P^{cu}(x,\epsilon)$ $\displaystyle:=W^{c}(W^{u}(x,\epsilon),\epsilon)$ (22)
$\displaystyle B(x,\epsilon)$ $\displaystyle:=\bigcup_{\mathclap{y\in
W^{u}(x,\epsilon)}}\ \mathrm{hol}^{u}_{x,y}(P^{cs}(x,\epsilon))$ (23)

where $\mathrm{hol}^{u}_{x,y}:W^{cs}(x)\rightarrow W^{cs}(y)$ is the locally
defined Poincaré map sending $x$ to $y$.

Sets $B(x,\epsilon)$ constructed in this fashion will be referred as
_dynamical boxes_ , while $\mathrm{hol}^{u}_{x,y}(P^{cs}(x,\epsilon))$ will be
referred as the $cs$-plaques of $B(x,\epsilon)$.

By shrinking $\epsilon_{0}$ if necessary can guarantee also that for
$0<\epsilon\leq\epsilon_{0}$, for every $x,x^{\prime}\in M$,

  * •

$P^{cu}(x,\epsilon)\cap P^{cu}(x^{\prime},\epsilon)\neq\emptyset\Rightarrow
P^{cu}(x,\epsilon)\cap P^{cu}(x^{\prime},\epsilon)\subset
P^{cu}(x^{\prime\prime},3\epsilon)$ for some $x^{\prime\prime}\in M$.

  * •

$B(x,\epsilon)\cap B(x^{\prime},\epsilon)\neq\emptyset\Rightarrow
B(x,\epsilon)\cap B(x^{\prime},\epsilon)\subset B(x^{\prime\prime},5\epsilon)$
for some $x^{\prime\prime}\in M$.

To understand better the structure of dynamical boxes we note the following.

######  Lemma 3.13.

Fix $x\in M$ and for $y\in W^{s}(y)$ consider the locally defined holonomy
$\mathrm{hol}^{s}_{x,y}:W^{cu}(x)\to W^{cu}(y)$. Then $\mathrm{hol}^{s}$ sends
center leaves into center leaves and $\mathrm{hol}^{s}|W^{c}(x)$ is an
isometry. Similarly for unstable holonomies.

######  Proof 3.14.

Let $\delta>0$ sufficiently small so that
$\mathrm{hol}^{s}_{x,y}:A=W^{cu}(x,\delta)\to B\subset W^{cu}(y)$ is a well
defined homeomorphism. Since $f$ preserves
$\mathcal{W}^{cu},\mathcal{W}^{c},\mathcal{W}^{s}$ it follows

$\mathrm{hol}^{s}_{x,y}=f^{-n}\circ\mathrm{hol}^{s}_{f^{n}x,f^{n}y}\circ
f^{n}\quad\forall n\geq 0$

where $\mathrm{hol}^{s}_{f^{n}x,f^{n}y}:f^{n}(A)\to f^{n}(B)\subset
W^{c}(f^{n}y)$. Using that $f|\mathcal{W}^{c}$ is an isometry and
$f|\mathcal{W}^{u}$ expands distances, it follows that $\mathrm{hol}^{s}$
sends $W^{c}(x,\delta)$ to $W^{c}(y)$ and that the differentiable map
$\mathrm{hol}^{s}_{f^{n}x,f^{n}y}|W^{c}(f^{n}x)$ approaches uniformly the
identity for large $n$, hence the claim.

We remark that in general $\mathrm{hol}^{s}$ does not send $\mathcal{W}^{u}$
to itself. See figure 1.

Figure 1: Comparison between $P^{cu}(x,\epsilon),P^{cu}(y,\epsilon)$ for $y\in
W^{s}(x)$. The stable holonomy sends each center plaque
$W^{c}(x^{\prime},\epsilon)\in P^{cu}(x,\epsilon)$ to a center plaque
$W^{c}(y^{\prime},\epsilon)\in P^{cu}(y,\epsilon)$, but the image of unstable
plaques in $P^{cu}(x,\epsilon)$ are only Hölder submanifolds in
$P^{cu}(y,\epsilon)$. Nevertheless,
$\mathrm{hol}^{s}_{x,y}(W^{u}(x,\epsilon))\to W^{u}(x,\epsilon)$ in the
$\mathcal{C}^{0}$ topology as $y\to x$

######  Corollary 3.15.

For every $0<\epsilon<\epsilon_{0}$ and $x\in M$ it holds

$B(x,\epsilon)=\bigcup_{y\in
W^{s}(x,\epsilon)}\mathrm{hol}^{s}_{x,y}(P^{cu}(x,\epsilon))$

where $\mathrm{hol}^{s}_{x,y}:W^{cu}(x)\rightarrow W^{cu}(y)$ is the locally
defined Poincaré map sending $x$ to $y$.

######  Proof 3.16.

This is direct consequence of Lemma 3.13 above plus the fact that
$\mathcal{W}^{c}$ sub-foliates $\mathcal{W}^{cu},\mathcal{W}^{cs}$.

In view of this Corollary, sets $\mathrm{hol}^{s}_{x,y}(P^{cu}(x,\epsilon))$
will be referred as the $cu$-plaques of $B(x,\epsilon)$.

#### The equilibrium state.

We will now construct the equilibrium state. Fix $B=B(x,\epsilon)$ a dynamical
box with $0<\epsilon<\epsilon_{0}/2$. By the choice of $\epsilon$ the set
$W^{u}(x,B)=W^{u}(x,2\epsilon)\cap B$ consists of a unique unstable plaque.
Similar considerations hold for the other foliations.

If $W$ is a $cs$-plaque of $B$ and $x\in B$ denote $x_{w}=W^{u}(x,B)\cap W$.
For $U\subset B$ open sub-box, define the function
$\alpha_{W,B}^{U}:W\rightarrow\mathbb{R}$ by

$\alpha_{W,B}^{U}(w)=\nu^{u}_{w}(U\cap W^{u}(w,2\epsilon)).$ (24)

######  Lemma 3.17.

$\alpha_{W,B}^{U}$ is upper semi-continuous, hence measurable.

######  Proof 3.18.

The SRB case is direct, so we will consider only the case when $\varphi$ is
constant along center leaves.

We use the following notation: for $\gamma>0$ and $C$ contained in a leaf of
$\mathcal{W}^{cs}|B$ we write

$\mathrm{Col}_{\gamma}(C):=\\{y:\exists y^{\prime}\in C\text{ s.t.
}d_{\mathcal{W}^{cu}}(y,y^{\prime})<\gamma\\}.$

Consider $y\in B$ and a sequence $(y_{m})_{m}\subset B$,
$y_{m}\xrightarrow[m\mapsto\infty]{}y$. Let $A=W^{u}(y,B)\cap
U,A_{m}=W^{u}(y_{m},B)\cap U$ and fix $\gamma>0$. By Lemma 3.10 the function
$y\mapsto\Delta_{y_{w}}(y)$ is positive and uniformly continuous on $B$, hence
given $\tau>0$ there exists $0<\rho<\tau$ such that $y,z\in B$,
$d(y,z)<\rho\Rightarrow\left|\frac{\Delta_{y_{w}}(y)}{\Delta_{z_{w}}(z)}-1\right|<\tau$.
Moreover, by Proposition 3.4 we can assume that if
$\phi_{1},\phi_{2}\in\mathrm{Con}^{+}(\mathcal{W}^{cu})$ are $\rho$-equivalent
then $|\frac{\mu^{cu}(\phi_{1})}{\mu^{cu}(\phi_{2})}-1|<\tau$; this is turn
implies that if $Q_{1},Q_{2}\in\Sigma(\mathcal{W}^{cu})$ are $\rho$-equivalent
and $\phi_{1},\phi_{2}$ as before then

$\left|\frac{\mu^{cu}(\phi_{1}\mathds{1}_{Q_{1}})}{\mu^{cu}(\phi_{2}\mathds{1}_{Q_{2}})}-1\right|<\tau.$

By continuity of the unstable and center unstable foliations there exists
$m_{0}$ such that for every $m\geq m_{0}$ the set $W^{c}(A_{m},\epsilon)$ is
$\rho$-equivalent to a subset $E_{m}$ of
$\mathrm{Col}_{\gamma}(W^{c}(A,\epsilon))$ and furthermore the Hausdorff
distance between $A,A_{m}$ is less than $\rho$. Let
$g_{m}:=\mathrm{hol}^{u}|:E_{m}\rightarrow W^{c}(A_{m},\epsilon)$ the
corresponding holonomy and observe that for such $m$, $|\pi^{c}_{y_{m}}\circ
g_{m}-\pi^{c}_{y}|<\rho$. Hence,

$\displaystyle\nu^{u}_{y_{m}}(A_{m})=\int_{W^{c}(A_{m},\epsilon)}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}d\mu^{cu}_{y_{m}}$
$\displaystyle=\int_{E_{m}}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}\circ
g_{m}d\mu^{cu}_{y}+\left(\int_{W^{c}(A_{m},\epsilon)}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}d\mu^{cu}_{y_{m}}-\int_{E_{m}}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}\circ
g_{m}d\mu^{cu}_{y}\right)$
$\displaystyle<(1+\tau)\int_{E_{m}}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}\circ
g_{m}d\mu^{cu}_{y}$
$\displaystyle=(1+\tau)\int_{E_{m}}J^{u}_{y}\circ\pi^{c}_{y}d\mu^{cu}_{y}+(1+\tau)\left(\int_{E_{m}}J^{u}_{y_{m}}\circ\pi^{c}_{y_{m}}\circ
g_{m}-J^{u}_{y}\circ\pi^{c}_{y}d\mu^{cu}_{y}\right)$
$\displaystyle<(1+\tau)(1+\tau)\int_{E_{m}}J^{u}_{y}\circ\pi^{c}_{y}d\mu^{cu}_{y}\leq(1+\tau)^{2}\int_{Col_{\gamma}(W^{c}(A,\epsilon))}J^{u}_{y}\circ\pi^{c}_{y}d\mu^{cu}_{y}.$

This implies that
$\limsup_{m}\alpha_{W,B}^{U}(y_{m})=\limsup_{m}\nu^{u}_{y_{m}}(A_{m})\leq\int_{Col_{\gamma}(W^{c}(A,\epsilon))}J^{u}_{y}\circ\pi_{y}^{c}d\mu^{cu}_{y}.$
Taking $\gamma\mapsto 0$ and using regularity of the measure $\mu^{cu}_{y}$ we
conclude $\limsup_{m}\alpha_{W,B}^{U}(y_{m})\leq\alpha_{W,B}^{U}(y)$, as we
wanted to show.

Standard arguments of measure theory together with the previous Lemma permit
us to define a Borel measure $\mathrm{m}^{W,B}$ on $B$ by the condition

$U\subset B\text{ open sub-box
}\Rightarrow\mathrm{m}^{W,B}(U):=\int\alpha_{W,B}^{U}(w)\mu^{cs}_{w_{0}}(dw)$

where $w_{0}\in W$.

Let $W^{\prime}$ be another $cs$-plaque of $\mathcal{W}^{cs}|B$ and consider
$\mathrm{hol}^{u}_{w_{0},w_{0}^{\prime}}:W\rightarrow W^{\prime}$ the
corresponding Poincaré map. Fix $U\subset B$ open, and to simplify the
notation write $U_{w}^{u}:=U\cap W^{u}(w,B)$. We compute

$\displaystyle\mathrm{m}^{W^{\prime},B}(U)$
$\displaystyle=\int_{W^{\prime}}\nu^{u}_{w^{\prime}}(U^{u}_{w^{\prime}})\mu^{cs}_{w_{0}^{\prime}}(dw^{\prime})=\int_{W}\nu^{u}_{\mathrm{hol}^{u}_{w_{0},w_{0}^{\prime}}(w)}(U^{u}_{\mathrm{hol}^{u}_{w_{0},w_{0}^{\prime}}(w)})\mathrm{Jac}^{u}_{w_{0},w_{0}^{\prime}}(w)\mu^{cs}_{w_{0}}(dw)$
$\displaystyle=\int_{W}\left(\nu^{u}_{\mathrm{hol}^{u}_{w_{0},w_{0}^{\prime}}(w)}(U^{u}_{w})\prod_{k=1}^{+\infty}\frac{e^{\varphi\circ
f^{-k}\mathrm{hol}^{u}_{w_{0},w_{0}^{\prime}}(w)}}{e^{\varphi\circ
f^{-k}(w)}}\right)\mu^{cs}_{w_{0}}(dw)$
$\displaystyle=\int_{W}\nu^{u}_{w}(U_{w})\mu^{cs}_{w_{0}}(dw)=\mathrm{m}^{W,B}(U),$

hence $\mathrm{m}^{W^{\prime},B}=\mathrm{m}^{W,B}$: we write
$\mathrm{m}^{B}=\mathrm{m}^{W,B}$ for any plaque $W\in\mathcal{W}^{cs}|B$. The
following is now clear.

######  Lemma 3.19.

Consider $0<\epsilon<\epsilon^{\prime}<\epsilon_{0}$ and $x,x^{\prime}\in M$
such that $B=B(x,\epsilon)\subset B(x^{\prime},\epsilon^{\prime})$. Then
$\mathrm{m}^{B}=\mathrm{m}^{B^{\prime}}|B$.

Take a finite covering
$\mathscr{B}=\\{B_{1}=B(x_{1},\epsilon_{1}),\ldots,B_{R}=B(x_{R},\epsilon_{1})\\}$
of $M$ with $0<\epsilon_{1}<\frac{\epsilon_{0}}{3}$ and define the measure
$\mathrm{m}$ by the condition: $A\subset B_{i}$ is Borel $\Rightarrow$
$\mathrm{m}(A):=\mathrm{m}^{B_{i}}(A)$. If $A\subset B_{i}\cap B_{j}$ then by
our choice of $\epsilon_{1}$ there exists $x\in M$ with $A\subset
B(x,\epsilon_{0})$, and thus by the previous lemma
$\mathrm{m}^{B_{i}}(A)=\mathrm{m}^{B_{j}}(A)$, hence $\mathrm{m}$ is a well
defined Borel measure on $M$. With no loss of generality we assume further
that for every $i$ there exists $x_{i}^{\prime}$ such that $f(B_{i})\subset
B(x_{i}^{\prime},\epsilon_{0})$.

######  Proposition 3.20.

The measure $\mathrm{m}$ is $f$-invariant.

######  Proof 3.21.

Fix $\mathrm{m}^{B_{i}}=\mathrm{m}^{W_{i},B_{i}}$ and $U\subset f(B_{i})$
open. Then

$\displaystyle f_{\ast}\mathrm{m}^{W_{i},B_{i}}(U)$
$\displaystyle=\int_{W}\nu^{u}_{w}(f^{-1}U_{fw}^{u})\mu^{cs}_{w_{0}}(dw)=\int_{W}e^{\varphi(w)-P}\nu^{u}_{fw}(U_{fw}^{u})\mu^{cs}_{w_{0}}(dw)\quad$
$\displaystyle\text{ by }\eqref{eq:invariancianu}$
$\displaystyle=e^{P^{\prime}-P}\int\nu^{u}_{z}(U_{z}^{u})\mu^{cs}_{fw_{0}}(dz)\quad$
by Prop. 3.5 $\displaystyle=e^{P^{\prime}-P}\mathrm{m}^{fW_{i},fB_{i}}(U).$

Hence
$f_{\ast}\mathrm{m}^{W_{i},B_{i}}=e^{P^{\prime}-P}\mathrm{m}^{fW_{i},fB_{i}}$,
and this implies $f_{\ast}\mathrm{m}=e^{P^{\prime}-P}\mathrm{m}$. The measure
$\mathrm{m}$ is finite on compact sets and $fM=M$, thus $P=P^{\prime}$ and
$\mathrm{m}$ is $f$-invariant.

######  Corollary 3.22.

It holds $P=P^{\prime}$.

######  Definition 3.23.

We denote $\mathrm{m}_{\scriptstyle\varphi}$ the probability measure
$\frac{\mathrm{m}}{\mathrm{m}(M)}$.

Interchanging $\mu^{u}$ by $\mu^{s}$ and $\mu^{cs}$ by $\mu^{cu}$ we construct
another invariant measure $\tilde{\mathrm{m}}_{\scriptstyle\varphi}$ which is
given locally as

$\tilde{\mathrm{m}}_{\scriptstyle\varphi}(U)=\int_{V}\nu_{v}^{s}(U_{v}^{s})\mu^{cs}_{V}(dv).$

Convention: The family $\mathscr{B}=\\{B_{1},\ldots B_{R}\\}$ together with
the corresponding center stable plaques $W_{1},\ldots W_{R}$ is considered to
be fixed.

###  3.3 Product structure and the Gibb’s property

We start establishing the following.

######  Proposition 3.24 (Product structure of
$\mathrm{m}_{\scriptstyle\varphi}$).

There exists $0<\epsilon_{2}\leq\epsilon_{1}$ such that for every
$0<\epsilon\leq\epsilon_{2}$ there exists $c(\epsilon)>0$ satisfying for every
$x\in M$,

$\frac{1}{c}\leq\frac{\mathrm{m}_{\scriptstyle\varphi}(B(x,\epsilon))}{\mu^{u}_{x}(W^{u}(x,B(x,\epsilon)))\cdot\mu^{cs}_{x}{(P^{cs}(x,\epsilon)})}\leq
c.$

The proof follows directly by the definition of
$\mathrm{m}_{\scriptstyle\varphi}$ with the following lemma.

######  Lemma 3.25.

There exists $0<\epsilon_{2}\leq\epsilon_{1}$ such that for
$0<\epsilon\leq\epsilon_{2}$ we can find $c(\epsilon)>0$ such that for all
$x,x^{\prime}\in M$ the following holds.

$\displaystyle(\ast)\quad
c(\epsilon)^{-1}\leq\frac{\mu^{cu}_{x}(W^{cu}(x,\epsilon))}{\mu^{cu}_{x^{\prime}}(W^{cu}(x^{\prime},\epsilon))},\frac{\mu^{cs}_{x}(W^{cs}(x,\epsilon))}{\mu^{cs}_{x^{\prime}}(W^{cs}(x^{\prime},\epsilon))}\leq
c(\epsilon)$ $\displaystyle(\ast\ast)\quad
c(\epsilon)^{-1}\leq\frac{\mu^{u}_{x}(W^{u}(x,\epsilon))}{\mu^{u}_{x^{\prime}}(W^{u}(x^{\prime},\epsilon))}\leq
c(\epsilon)$ $\displaystyle(\ast\ast\ast)\quad
c(\epsilon)^{-1}\leq\frac{\nu^{u}_{x}(W^{u}(x,\epsilon))}{\nu^{u}_{x^{\prime}}(W^{u}(x^{\prime},\epsilon))}\leq
c(\epsilon)$

######  Proof 3.26.

Let $\epsilon_{2}$ be the Lebesgue number associated to the covering
$\mathscr{B}$. We start with $(\ast)$; it suffices to consider the first set
of inequalities. Since $\mu^{cu}_{x}$ is a Radon measure of full support, we
have the following: for $\epsilon>0$ fixed, there exists $\gamma>0$ and
$c_{0}(x,\epsilon)>0$ such that

$x^{\prime}\in W^{cu}(x,\rho)\Rightarrow
c_{0}(\epsilon,x)^{-1}\leq\frac{\mu^{cu}_{x}(W^{cu}(x,\epsilon))}{\mu^{cu}_{x^{\prime}}(W^{cu}(x^{\prime},\epsilon))}\leq
c_{0}(\epsilon,x).$

By lemma 3.10 it is no loss of generality to assume that the same holds for
$x^{\prime}\in D(x,\gamma)$, and hence due to the compactness of $M$ we get a
uniform $c(\epsilon)$ independent of $x$.

For $(\ast\ast)$, the claim is immediate for the SRB case, so we just need to
consider when $\varphi$ is constant along centrals. But this follows from as
in the previous part since
$\mu^{u}_{x}(A)=\mu^{cu}_{x}(W^{c}(A,\epsilon_{0}))$.

Finally, $(\ast\ast\ast)$ is consequence of $(\ast\ast)$ together with lemma
3.10.

######  Remark 3.27.

Arguing similarly and re-defining $c,\epsilon_{2}$ if necessary we can
guarantee that for every $x\in M$,

$\frac{1}{c(\epsilon)}\leq\frac{\tilde{\mathrm{m}}_{\scriptstyle\varphi}(B(x,\epsilon))}{\mu^{s}_{x}(W^{s}(x,B(x,\epsilon)))\cdot\mu^{cu}_{x}{(P^{cu}(x,\epsilon)})}\leq
c(\epsilon).$

Next we establish a Gibbs’s type property for
$\mathrm{m}_{\scriptstyle\varphi}$ (resp.
$\tilde{\mathrm{m}}_{\scriptstyle\varphi}$) that will allow us to show that it
is an equilibrium state for the potential $\varphi$. By reducing
$\epsilon_{2}$ if necessary, we can assume that for every
$0<\epsilon<\epsilon_{2}$, for every $\epsilon$-ball $D\subset M$ the distance
between $x,y\in D$ is given by

$\displaystyle d(x,y)=\inf\\{$
$\displaystyle\mathrm{length}(\alpha):\alpha:[0,1]\rightarrow D\text{
piecewise }\mathcal{C}^{1},\alpha(0)=x,\alpha(1)=y,\alpha^{\prime}\in
E^{\ast}\ \ast=s,c,u\\}.$

With this metric and using that $f$ is a center isometry we have

$\displaystyle
D^{u}(x,\epsilon,n)=D(x,\epsilon,n)|W^{u}(x)=f^{-n}W^{u}(f^{n}x,\epsilon)$
(25) $\displaystyle D(x,\epsilon,n)=\bigcup_{y\in
W^{cs}(x,\epsilon)}D^{u}(y,\epsilon,n)$

We note the following classical remark.

######  Lemma 3.28.

Given $\epsilon>0$ there exists $K(\epsilon)>0$ with the property that for
every $x,y\in M,n\geq 0$ it holds

  1. 1.

$y\in D^{u}(x,\epsilon,n)\Rightarrow|S_{n}\varphi(x)-S_{n}\varphi(y)|<K$.

  2. 2.

$y\in D(x,\epsilon,n)\Rightarrow|S_{n}\varphi(x)-S_{n}\varphi(y)|<nK$. If
$\varphi$ does not depend on centrals then $y\in
D(x,\epsilon,n)\Rightarrow|S_{n}\varphi(x)-S_{n}\varphi(y)|<K$.

Moreover, $K(\epsilon)\mapsto 0$ as $\epsilon\mapsto 0$.

######  Proof 3.29.

We will prove only the second part, as the arguments for the first are
included in the proof of this case. Let $y_{s}:=W^{s}(y,\epsilon)\cap
f^{-n}W^{cu}(f^{n}x,\epsilon)$. Points in the same stable leaf are
exponentially contracted and $\varphi$ is Hölder, thus

$|S_{n}\varphi(x)-S_{n}\varphi(y)|\leq|S_{n}\varphi(x)-S_{n}\varphi(y_{s})|+C_{1}(\epsilon)$

for some constant $C_{1}(\epsilon)$ that converges to zero as $\epsilon$ tends
to zero. Since $f^{n}y_{s}\in W^{cu}(f^{n}x,\epsilon)$, the point
$z_{u}:=W^{u}(f^{n}y,\epsilon)\cap W^{c}(f^{n}x,\epsilon)$ is well defined.
Observe then that the distance $d(f^{-j}z_{u},f^{n-j}y_{s})$ is exponentially
contracted under $f^{-1}$, and thus for some other similar constant
$C_{2}(\epsilon)$ we have

$|S_{n}\varphi(x)-S_{n}\varphi(y)\leq|S_{n}\varphi(x)-S_{n}\varphi(f^{-n}z_{u})|+C_{2}(\epsilon).$

Since $f$ is a center isometry and $\varphi$ is Hölder it follows that

$|S_{n}\varphi(x)-S_{n}\varphi(f^{-n}z_{u})|\begin{dcases}=0&c\text{-constant
case}\\\ \leq C_{\varphi}n\epsilon^{\theta}:=C_{3}(\epsilon)&\text{ in
general}.\end{dcases}$

Defining $K(\epsilon):=C_{1}(\epsilon)+C_{2}(\epsilon)+C_{3}(\epsilon)$ we
finish the proof.

Now we can deduce:

######  Corollary 3.30 (Gibbs-like property of
$\mathrm{m}_{\scriptstyle\varphi}$).

Given $\epsilon>0$ there exists $d(\epsilon)>0$ such that for every
$n\in\mathbb{N},x\in M$ it holds

$\displaystyle d^{-1}(\epsilon)\leq$
$\displaystyle\frac{\nu^{u}_{x}(D^{u}(x,\epsilon,n))}{e^{S_{n}\varphi(x)-nP}}\leq
d(\epsilon)$ $\displaystyle d^{-1}(\epsilon)e^{-K(\epsilon)n}\leq$
$\displaystyle\frac{\mathrm{m}_{\scriptstyle\varphi}(D(x,\epsilon,n))}{e^{S_{n}\varphi(x)-nP}}\leq
d(\epsilon)e^{K(\epsilon)n}$

where $K(\epsilon)$ is the constant given in lemma 3.28. If $\varphi$ does not
depend on centrals then in the second line of inequalities we can substitute
$nK(\epsilon)$ by $K(\epsilon)$.

######  Proof 3.31.

To simplify the notation we will write $a\simeq b$ if the quotient
$\frac{a}{b}$ is bounded above and below with constants only depending on
$\epsilon$. The first part is direct consequence of lemma 3.25 and (16). For
the second part, we have
$\mathrm{m}^{W_{i},B_{i}}(D(x,\epsilon,n))=\int_{W^{cs}(x,\epsilon)}\nu^{u}_{y_{w}}(D^{u}(y,\epsilon,n))\mu^{cs}_{x}(dy)$,
therefore

$\displaystyle\mathrm{m}^{W_{i},B_{i}}(D(x,\epsilon,n))$
$\displaystyle\simeq\int_{W^{cs}(x,\epsilon)}e^{S_{n}\varphi\circ
f^{-n}(y)-nP}\mu^{cs}_{x}(dy)\quad$ (by the first part) $\displaystyle\simeq
e^{S_{n}\varphi(x)-nP\pm nK}\mu^{cs}_{x}(W^{cs}(x,\epsilon))\quad$ (by lemma
3.28) $\displaystyle\simeq e^{S_{n}\varphi(x)-nP\pm nK}\quad$
$\displaystyle\text{(by
\lx@cref{creftype~refnum}{lem:comparacionentretamanhos})}.$

The last claim follows from the corresponding one in lemma 3.28.

We are ready to show that $\mathrm{m}_{\scriptstyle\varphi}$ is an equilibrium
state.

######  Proposition 3.32.

It holds $P=P_{\scriptstyle\mathit{top}}(\varphi)=P_{\mu_{\varphi}}(\varphi).$

######  Proof 3.33.

Fix $0<\epsilon<\epsilon_{2}$ small and consider a set $E$ such that
$\\{B(x,\epsilon)\\}_{x\in E}$ is pairwise disjoint, maximal with this
property. Observe that $M=\cup_{x\in E}B(x,2\epsilon)$. Take a partition by
Borel sets $A=\\{A_{x}:x\in E\\}$ such that for every $x\in E$,
$D(x,\epsilon)\subset A_{x}\subset D(x,2\epsilon)$. Denote
$A^{n}=\bigvee_{i=0}^{n-1}f^{-i}A$ and choose a set of representatives $E_{n}$
such that $A^{n}=\\{A^{n}_{x}:x\in E_{n}\\}$. It follows that for every $x\in
E_{n}$, $D(x,\epsilon,n)\subset A_{x}^{n}\subset D(x,4\epsilon,n)$. Then by
lemma 3.28 and the previous corollary,

$\displaystyle-\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})\log\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})+\int_{A_{x}^{n}}S_{n}\varphi
d\mathrm{m}_{\scriptstyle\varphi}$
$\displaystyle\geq-\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})(\log\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})-S_{n}\varphi(x)+nK(4\epsilon))$
$\displaystyle\geq-\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})(-nP+2nK(4\epsilon)+\log
d(4\epsilon))$ thus,
$\displaystyle\frac{H_{\mathrm{m}_{\scriptstyle\varphi}}(\bigvee_{i=0}^{n-1}f^{-i}A)}{n}+\int\varphi
d\mu_{\varphi}$ $\displaystyle=\frac{1}{n}\sum_{x\in
E_{n}}\big{(}-\mathrm{m}_{\scriptstyle\varphi}(A^{n}_{x})\log\mathrm{m}_{\scriptstyle\varphi}(A^{n}_{x})+\int_{A_{x}^{n}}S_{n}d\mathrm{m}_{\scriptstyle\varphi}\big{)}$
$\displaystyle\geq P-2K(4\epsilon)+\frac{\log d(4\epsilon)}{n}.$

Taking limit first as $n\to\infty$ and then as $\epsilon\to 0$ we obtain

$\displaystyle P_{\mathrm{m}_{\scriptstyle\varphi}}(\varphi)\geq
h_{\mathrm{m}_{\scriptstyle\varphi}}(f;A)+\int\varphi
d\mathrm{m}_{\scriptstyle\varphi}\geq P.$

On the other hand, by the same construction

$\displaystyle S(4\epsilon,n)\leq\sum_{x\in
E_{n}}e^{S_{n}\varphi(x)}\leq\sum_{x\in
E_{n}}\mathrm{m}_{\scriptstyle\varphi}(D(x,\epsilon,n))e^{nP\pm
nK(\epsilon)}\leq e^{nP\pm nK(\epsilon)}\sum_{x\in
E_{n}}\mathrm{m}_{\scriptstyle\varphi}(A_{x}^{n})=e^{nP\pm nK(\epsilon)}$

and thus $P_{\scriptstyle\mathit{top}}(\varphi)\leq P$. The variational
principle then implies that
$P=P_{\scriptstyle\mathit{top}}(\varphi)=P_{\mathrm{m}_{\scriptstyle\varphi}}(f)$
and we are done.

Since $P=0$ by construction in the SRB case, we get.

######  Corollary 3.34.

If $\mathrm{m}_{\scriptscriptstyle SRB}$ is the measure associated to
$\varphi=-\log\det Df|E^{u}$, then $P_{\mathrm{m}_{\scriptscriptstyle
SRB}}(f)=0$.

###  3.4 Uniqueness

As was indicated in the introduction, the uniqueness for the SRB case is
proven in [lecturesugibbs]. Here we present a different argument that
establishes uniqueness in both the $c$-constant and the SRB case; it is based
in arguments of Bowen [Bowen1974] for expansive systems with the specification
property. We will use that $\mathrm{m}_{\scriptstyle\varphi}$ is ergodic; this
will be established in section 5.

The proof is by contradiction: we suppose that there exists
$\mathrm{m}\neq\mathrm{m}_{\scriptstyle\varphi}$ another equilibrium state for
$\varphi$, that without loss of generality we assume to be ergodic. Since
$\mathrm{m}\perp\mathrm{m}_{\scriptstyle\varphi}$, we can find an invariant
Borel set $X\subset M$ such that
$\nu(X)=1,\mathrm{m}_{\scriptstyle\varphi}(X)=0$. Fix $\epsilon>0$ small
($0<\epsilon<\epsilon_{2}$) and construct, as in the proof of proposition
3.32, a finite partition by Borel sets $\mathscr{A}^{n}=\\{A_{x}^{n}:x\in
E^{n}\\}$, with $D(x,\epsilon,n)\subset A_{x}^{n}\subset D(x,4\epsilon,n)$ for
every $x\in E$. Recall that by (25), $D(x,\epsilon,n)=\bigcup_{y\in
W^{cs}(x,\epsilon)}D^{u}(y,\epsilon,n)$ therefore each $A_{x}^{n}$ contains a
center stable plaque $W^{cs}(x,\epsilon)\subset P_{x}^{n}\subset
W^{cs}(x,4\epsilon)$. We will now subdivide further each $A^{n}_{x}$, and for
this we will subdivide the corresponding $W^{cs}(x,4\epsilon)$ in the form

$W^{cs}(x,4\epsilon)=\bigcup_{y\in I_{x}}W^{s}(J_{y}^{x},4\epsilon)$

where $I_{x}\subset W^{c}(x,4\epsilon)$ is finite, and $J_{y}^{x}\subset
W^{c}(y,(4\epsilon)^{n})$. This way we can induce a partition on $A^{n}_{x}$,
and get a finite partition $\mathscr{C}^{n}=\\{C^{n}_{z}:z\in F^{n}\\}$ such
that

  * •

for each $z\in F$ there exists $x\in E$ such that $D(x,\epsilon,n)\subset
C_{z}^{n}\subset D(x,4\epsilon,n)$.

  * •

$\sup_{z}\operatorname{diam}(f^{[n/2]}C^{n}_{z})\xrightarrow[n\to\infty]{}0$.

######  Definition 3.35.

A set of the form $S(x,\epsilon,n)=W^{s}(W^{c}(x,\epsilon^{n}),\epsilon)$ will
be referred as a $(\epsilon,n)$-slice.

We need the following refinement of lemma 3.25.

######  Lemma 3.36.

For $\epsilon>0$ sufficiently small there exists $e(\epsilon)>0$ such that for
every $x,x^{\prime}\in M$, it holds

$e(\epsilon)^{-1}\leq\frac{\mu^{cs}_{x}(S(x,\epsilon,n))}{\mu^{cs}_{x^{\prime}}(S(x^{\prime},\epsilon,n))}\leq
e(\epsilon).$

######  Proof 3.37.

Fix $0<\epsilon<\epsilon_{2}$ and
$\eta=\mathrm{Leb}\in\mathrm{Meas}(\mathcal{W}^{cs})$. Consider in
$Q(x,\epsilon)=\bigcup_{y\in
W^{s}(x,\epsilon)}\mathrm{hol}^{s}_{x,y}(W^{c}(x,\epsilon))$ and observe that
by lemma 3.13 the set $\mathbb{Q}(x,\epsilon)$ is saturated by local center
leaves $W^{c}(y,\epsilon)$. Define the transverse measure on
$W^{c}(x,\epsilon)$:

$\eta^{c}_{x,T}(A)=\frac{\eta(A)}{\eta(Q(x,\epsilon))}$

The foliation $\mathcal{W}^{s}|\mathcal{W}^{cs}$ is $\mathcal{C}^{1}$
[HolFol], therefore $\eta^{c}_{x}$ is the Lebesgue measure. The same holds for
$\eta^{n}$ (1) due to Fubini’s theorem and the fact that
$f^{n}(Q(x,\epsilon))$ is a set saturated by center plaques
$W^{c}(y,\epsilon)$ (see the proof of Theorem $7.8$ in [PesinLect]), and $\det
Df|E^{c}=1$. By passing to the limit we deduce that $\mu^{c}_{x,T}$ is the
Lebesgue measure (the one induced by the metric on $E^{c}$ that is invariant
by $f$).

In particular we there exists $e_{0}(\epsilon)>0$ such that for every
$x,x^{\prime}\in M,n\in\mathbb{N}$,

$e_{0}(\epsilon)^{-1}\leq\frac{\mu^{c}_{x,T}(W^{c}(x,\epsilon^{n}))}{\mu^{c}_{x^{\prime},T}(W^{c}(x^{\prime},\epsilon^{n}))}\leq
e_{0}(\epsilon).$

On the other hand by the same token, the measure
$\mu^{cs}_{x}{(S(x,\epsilon,n))}$ is comparable, modulo constants that only
depend on $\epsilon$, to
$\frac{\mu^{cs}_{x}{(Q(x,\epsilon)})}{\mu^{c}_{x,T}(W^{c}(x,\epsilon^{n}))}$
and
$\mu^{cs}_{x}{(Q(x,\epsilon))},\mu^{cs}_{x^{\prime}}{(Q(x^{\prime},\epsilon))}$
are uniformly comparable (lemma 3.25) the claim follows.

######  Corollary 3.38.

Given $\epsilon>0$ there exists $l(\epsilon)>0$ such that for every
$C^{n}_{z}$,

$e^{S_{n}\varphi(z)-nP_{\scriptstyle\mathit{top}}(\varphi)}\leq
l(\epsilon)\cdot\mathrm{m}_{\scriptstyle\varphi}(C^{n}_{z}).$

######  Proof 3.39.

Arguing exactly as in the proof of corollary 3.30, using that the size of
$C^{n}_{z}$ decays exponentially fast with $n$, we deduce using the previous
lemma and the product structure of $\mathrm{m}_{\scriptstyle\varphi}$,
proposition 3.24,

$\mathrm{m}_{\scriptstyle\varphi}(C^{n}_{z})\sim\nu^{u}_{z}(D(x,\epsilon,n))\cdot\mu^{cs}_{x}(S(z,\epsilon,n))\sim
d(4\epsilon)e(\epsilon)\cdot
e^{S_{n}\varphi(z)-nP_{\scriptstyle\mathit{top}}(\varphi)}.$

We continue with the proof of uniqueness. Observe that the partition
$f^{[n/2]}\mathscr{C}^{n}$ generates $\mathscr{B}_{\scriptstyle\mathrm{M}}$,
and therefore (see for example lemma $2$ in [BowenMaxHyp]) there exists a
Borel set $X_{n}$ that is union of atoms of $\mathscr{C}^{n}$ satisfying

$\displaystyle\mathrm{m}_{\scriptstyle\varphi}(f^{[n/2]}X_{n}\triangle
X)\xrightarrow[n\to\infty]{}0\quad\nu(f^{[n/2]}X_{n}\triangle
X)\xrightarrow[n\to\infty]{}0$ which, by invariance of $X$ implies
$\displaystyle\mathrm{m}_{\scriptstyle\varphi}(X_{n}\triangle
X)\xrightarrow[n\to\infty]{}0\quad\nu(X_{n}\triangle
X)\xrightarrow[n\to\infty]{}0$ therefore,
$\displaystyle\mathrm{m}_{\scriptstyle\varphi}(X_{n})\xrightarrow[n\to\infty]{}0,\quad\nu(X_{n})\xrightarrow[n\to\infty]{}1.$

Note that for any $x$ it holds
$D(x,\epsilon,\infty):=\cap_{n=0}^{\infty}D(x,\epsilon,n)=W^{c}(x,\epsilon)$,
and since $f$ is a center isometry
$h_{\scriptstyle\mathit{top}}(f,D(x,\epsilon,\infty))=0$. It follows that $f$
is what is called an $h$-expansive map; from their theory we will only use
lemma $1$ of [BowenMaxHyp] which tells us that for $\epsilon$ sufficiently
small,

$h_{\nu}(f)\leq\frac{1}{n}H_{\nu}(\mathscr{C}^{n}),$

hence

$\displaystyle P_{\scriptstyle\mathit{top}}(\varphi)=h_{\nu}(f)+\int\varphi
d\nu\leq\frac{1}{n}(-\sum_{z\in F^{n}}\nu(C_{z}^{n})\log\nu(C_{z}^{n})+\int
S_{n}\varphi d\nu)$ $\displaystyle\Rightarrow
nP_{\scriptstyle\mathit{top}}(\varphi)\leq\sum_{z\in
F^{n}}(-\nu(C_{z}^{n})\log\nu(C_{z}^{n})+\int
S_{n}\varphi\cdot\mathds{1}_{C_{z}^{n}}d\nu)\leq\sum_{z\in
F^{n}}\log\nu(C_{z}^{n})+(K+S_{n}\varphi(z))\nu(C_{z}^{n})$

where $C_{z}^{n}\subset A^{n}_{x}\subset D(x,4\epsilon,n)$, and
$K=K(4\epsilon)$ is given in lemma 3.28. It follows that

$nP_{\scriptstyle\mathit{top}}(\varphi)\leq K+\sum_{C_{z}^{n}\subset
X_{n}}\nu(C_{z}^{n})(S_{n}\varphi(z)-\log\nu(C_{z}^{n}))+\sum_{C_{z}^{n}\subset
X_{n}^{c}}\nu(C_{z}^{n})(S_{n}\varphi(z)-\log\nu(C_{z}^{n})).$ (26)

Next we use the following Calculus fact (cf. Lemma $7$ [Bowen1974]).

######  Claim 1.

if $a_{1},\cdots,a_{k}\in[0,1]$ are such that $S=\sum_{i=1}^{k}a_{i}\leq 1$
then for any list of real numbers $b_{1},\cdots,b_{k}$ it holds

$\sum_{i=1}^{k}a_{i}(b_{i}-\log a_{i})\leq S(\log\sum_{i=1}^{k}e^{b_{i}}-\log
S).$

We get, using the above claim in (26) with
$a_{i}=\nu(C_{z}^{n}),b_{i}=S_{n}\varphi(z)$,

$\displaystyle
nP_{\scriptstyle\mathit{top}}(\varphi)-K\leq\nu(X_{n})\log\sum_{C_{z}^{n}\subset
X_{n}}e^{S_{n}\varphi(z)}+\nu(X_{n}^{c})\log\sum_{C_{z}^{n}\subset
X_{n}^{c}}e^{S_{n}\varphi(z)}+K^{\prime}$

where $K^{\prime}=\frac{1}{2}\max\\{-t\log t:t\in[0,1]\\}$. Write
$-nP_{\scriptstyle\mathit{top}}(\varphi)=\nu(X_{n})\log
e^{-nP_{\scriptstyle\mathit{top}}(\varphi)}+\nu(X_{n}^{c})\log
e^{-nP_{\scriptstyle\mathit{top}}(\varphi)}$ and appply corollary 3.30 to get

$\displaystyle-(K+K^{\prime})$
$\displaystyle\leq\nu(X_{n})\log\sum_{C_{z}^{n}\subset
X_{n}}e^{S_{n}\varphi(z)-nP_{\scriptstyle\mathit{top}}(\varphi)}+\nu(X_{n}^{c})\log\sum_{C_{z}^{n}\subset
X_{n}^{c}}e^{S_{n}\varphi(z)-nP_{\scriptstyle\mathit{top}}(\varphi)}$
$\displaystyle\leq\nu(X_{n})\log\sum_{C_{z}^{n}\subset
X_{n}}l(\epsilon,n)\cdot\mathrm{m}_{\scriptstyle\varphi}(C_{z}^{n})+\nu(X_{n}^{c})\log\sum_{C_{z}^{n}\subset
X_{n}^{c}}l(\epsilon,n)\cdot\mathrm{m}_{\scriptstyle\varphi}(C_{z}^{n})$
$\displaystyle\leq 2\log
l(\epsilon)+\nu(X_{n})\log\mathrm{m}_{\scriptstyle\varphi}(X_{n})+\nu(X_{n}^{c})\log\mathrm{m}_{\scriptstyle\varphi}(X_{n}^{c})\xrightarrow[n\to\infty]{}-\infty$

which is a contradiction.

##  4 A characterization of equilibrium states

The aim of this section is to characterize equilibrium states in terms of
their conditional measures along strong unstable leaves, very much as the case
of SRB measures, where they are characterized by the property of their
conditionals being absolutely continuous with respect to Lebesgue measures
along unstables. We will rely on the seminal work of Ledrappier-Young
[LedYoungI].

Hypotheses of this section: $f:M\rightarrow M$ is a $\mathcal{C}^{2}$ center
isometry preserving a Borel probability measure $\mathrm{m}$. We assume the
existence of a family of measures $\mu^{\ast},\ast\in\\{s,u,cs,cu\\}$
satisfying $2,3$ of Theorem A.

Of course the existence of such families is guaranteed only when $\varphi$ is
either constant along centrals or $\varphi=-\log\det Df|E^{u}$, but since the
results only depend on the existence of these families of measures, there is
value in writing this part abstractly.

Let $\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ and
$\nu^{\sigma}\in\mathrm{Meas}(\mathcal{F}^{\ast}),\ast=s,u$. We say that
$\mathrm{m}$ has conditionals equivalent to $\nu^{\ast}$ along
$\mathcal{F}^{\sigma}$ if for every $\mathrm{m}$-measurable partition $\xi$
subordinated to the (possibly non-measurable partition) $\mathcal{F}^{\ast}$
it holds $\mathrm{m}^{\xi}_{x}\sim\nu_{x}^{\ast}$ for
$\mu\operatorname{-}a.e.(x)$. See [Rokhlin] for a discussion of measurable
partitions and conditional measures.

Define,

$\displaystyle\mathscr{Pr}(M,\mu^{u})$
$\displaystyle=\\{\mathrm{m}\in\mathscr{Pr}(M):\mathrm{m}\text{ has
conditionals equivalent to }\mu^{u}\\}$ (27)
$\displaystyle\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M,\mu^{u})$
$\displaystyle=\mathscr{Pr}(M,\mu^{u})\cap\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M).$
(28)

We will prove the following.

######  Theorem 4.1.

The set $\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M\,\mu^{u})$ coincides with
the set of equilibrium states for $\varphi$.

#### Increasing partitions

A $\mathrm{m}$-measurable partition $\xi$ is called increasing if it refines
its image ($f\xi<\xi$). If $\xi$ is a partition we denote by
$\mathcal{B}_{\xi}(M)$ the sub-$\sigma$ algebra of
$\mathscr{B}_{\scriptstyle\mathrm{M}}$ that consists of $\xi$-saturated sets.

######  Definition 4.2.

We call a measurable partition $\xi$ a SLY partition if it is increasing,
subordinated to $\mathcal{W}^{u}$ and furthermore for
$\mathrm{m}\operatorname{-}a.e.(x)\in M$ the atom $\xi(x)$ contains a
neighborhood of $x$ inside $W^{u}(x)$.

In our restricted setting, a SLY partition automatically satisfies the
following additional properties:

  1. 1.

$\bigvee_{n\geq 0}f^{-n}\xi=\mathscr{B}_{\scriptstyle\mathrm{M}}$.

  2. 2.

$\bigwedge_{n\geq 0}f^{n}\xi=\mathcal{B}^{u}$

where $\mathcal{B}^{u}$ is the $\sigma$-algebra of $\mathcal{W}^{u}$ saturated
sets. A key result is the following.

######  Theorem 4.3.

There exist SLY partitions of arbitrarily small mesh. If $\xi$ is a SLY
partition then

$h_{\mathrm{m}}(f)=h_{\mathrm{m}}(f,\xi)=H_{\mathrm{m}}(f^{-1}\xi|\xi).$

The existence part is essentially due to Sinai [SinaiMarkov] (compare Section
3 in [LedStrelcyn] and Lemma 2.4.2 in [LedYoungI], where the construction is
done in more generality). The second part is a deep result of Ledrappier and
Young - Corollary 5.3 in [LedYoungI]. We remark that although in this last
article the result is presented for ergodic measures, it extends easily to non
- ergodic ones by using the ergodic decomposition. We have tacitly used that
the strong unstable manifolds coincide with the Pesin’s unstable manifolds, as
consequence of the fact that $f$ does not have any positive Lyapunov exponents
along center directions, independently of the measure.

###  4.1 Equilibrium states have conditionals equivalent to $\mu^{u}$

Now we fix $\xi$ a SLY partition for the measure $\mathrm{m}$, and denote by
$\mathrm{m}_{x}=\mathrm{m}^{\xi}_{x}$ the conditional measure of $\mathrm{m}$
on $\xi(x)$. The main result of [LedYoungI] admits the following
generalization in our setting.

######  Theorem 4.4.

If $\mathrm{m}$ is any equilibrium state corresponding to $\varphi$ then

$\mathrm{m}_{x}\sim\mu^{u}_{x}\text{ on }\xi(x)\quad\text{ for
}\mathrm{m}\operatorname{-}a.e.(x).$

To give a self contained presentation we spell the complete proof instead of
just citing the relevant parts of Ledrappier-Young work. Assume for the moment
that the conclusion of the theorem was true. Then for
$\mathrm{m}\operatorname{-}a.e.(x)$ we have

$d\mathrm{m}_{x}=\rho_{x}d\mu^{u}_{x},$

where $\rho_{x}$ is a non-negative function such that
$\int_{\xi(x)}\rho_{x}d\mu^{u}_{x}=1$. Using that the partition $\xi$ is
measurable we can assemble all $\\{\rho_{x}\\}$ into one measurable function
$\rho:M\rightarrow\mathbb{R}$. Since
$\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ we have

$f\mathrm{m}_{x}^{f^{-1}\xi}=\mathrm{m}_{fx}(=\mathrm{m}_{fx}^{\xi})\quad\mathrm{m}\operatorname{-}a.e.(x).$

On the other hand $f^{-1}\xi>\xi$, thus for any
$A\in\mathscr{B}_{\scriptstyle\mathrm{M}}$ we can write

$\displaystyle\frac{\int_{A\cap f^{-1}\xi(x)}\rho
d\mu^{u}_{x}}{\mathrm{m}_{x}(f^{-1}\xi(x))}$
$\displaystyle=\frac{\mathrm{m}_{x}(A\cap
f^{-1}\xi(x))}{\mathrm{m}_{x}(f^{-1}\xi(x))}=\mathrm{m}_{x}^{f^{-1}\xi}(A)=\mathrm{m}_{f_{x}}(fA\cap\xi(fx))$
$\displaystyle=\int_{f(A\cap f^{-1}\xi(x))}\rho d\mu^{u}_{fx}=\int_{A\cap
f^{-1}\xi(x)}\rho\circ fe^{P-\varphi}d\mu^{u}_{x},$

hence by uniqueness of the conditional measures
$\mathrm{m}\operatorname{-}a.e.(x)$ it holds

$\frac{\rho(y)}{\mathrm{m}_{x}(f^{-1}\xi(x))}=\rho(fy)\cdot
e^{P-\varphi(y)}\quad\text{ for }\mu_{x}\operatorname{-}a.e.(y)\in
f^{-1}\xi(x).$

As the denominator appearing in the previous equation is $f^{-1}\xi$\-
measurable, we finally conclude that the function

$x\mapsto\frac{\rho(x)}{\rho(f^{-1}x)}e^{P-\varphi(f^{-1}x)}$

is $\xi$ \- measurable (i.e. constant on the atoms of $\xi$). By a simple
inductive argument one is led to conclude that $\rho$ satisfies

$y\in\xi(x)\Rightarrow\frac{\rho(y)}{\rho(x)}=\prod_{k=1}^{\infty}\frac{e^{\varphi\circ
f^{-k}(y)}}{e^{\varphi\circ f^{-k}(x)}}=\Delta_{x}^{u}(y).$ (29)

######  Lemma 4.5.

For every $x\in M$ the function $W^{u}(x)\ni y\mapsto\log\Delta_{x}^{u}(y)$ is
locally Lipschitz and hence uniformly bounded away from $0$ and $\infty$ on
the atom $\xi(x)$.

The proof is straightforward using that for every $x\in M$ the set $W^{u}(x)$
is an immersed submanifold and the fact that $\varphi$ is globally Hölder.

From (29) we deduce that $\rho$ should have the form

$\rho(y)=\frac{\Delta_{x}(y)}{L(x)},\quad y\in\xi(x)$ (30)

with $L(x)=\int_{\xi(x)}\Delta_{x}(y)d\mu^{u}_{x}(y)$.

We now define a measure $\tilde{\mathrm{m}}$ by requiring
$\tilde{\mathrm{m}}=\mathrm{m}$ on $\mathcal{B}_{\xi}$ and such its
conditionals on $\xi$ are given by
$d\tilde{\mathrm{m}}_{x}=\frac{\Delta_{x}}{L(x)}d\mu_{x}=\rho d\mu_{x}$ (cf.
(17)). The proof of theorem 4.4 will be achieved by establishing that
$\mathrm{m}=\tilde{\mathrm{m}}$, and for this it suffices to show that
$\mathrm{m}=\tilde{\mathrm{m}}$ on every $\mathcal{B}_{f^{-n}\xi},n\geq 0$,
since the previous $\sigma$ \- algebras generate
$\mathscr{B}_{\scriptstyle\mathrm{M}}$.

We start by noticing the following: for
$q(x)=\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))$ we have

$\displaystyle q(x)$
$\displaystyle=\frac{1}{L(x)}\int_{f^{-1}(\xi(fx))}\Delta_{x}(f^{-1}fy)\frac{e^{P-\varphi(y)}}{e^{P-\varphi(f^{-1}fy)}}d\mu^{u}_{x}$
$\displaystyle=\frac{1}{L(x)}\int_{\xi(fx)}J_{x}^{u}(f^{-1}z)e^{\varphi(f^{-1}z)-P}d\mu^{u}_{fx}(z)=\frac{L(fx)}{L(x)}e^{\varphi(x)-P}\leq
1.$

Thus $\frac{L(fx)}{L(x)}\leq e^{\varphi(x)-P}$ is in
$\mathcal{L}^{1}(\mathrm{m})$. A classic lemma of measure theory (cf.
proposition $2.2$ in [LedStrelcyn]) then implies

$\int\log\frac{L\circ f}{L}d\mathrm{m}=0\Rightarrow\int-\log
q(x)d\mathrm{m}(x)=P-\int\varphi d\mathrm{m}.$

We have shown the following.

######  Lemma 4.6.

It holds,

$P-\int\varphi
d\mathrm{m}=\int-\log\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))d\mathrm{m}(x)=\int-\log\left(\int_{f^{-1}\xi(x)}\rho(y)d\mu^{u}_{x}(y)\right)d\mathrm{m}(x).$

By definition, the measures $\mathrm{m}$ and $\tilde{\mathrm{m}}$ coincide on
$\mathcal{B}_{\eta}$; to prove that these two measures coincide it suffices to
establish that $\mathrm{m}$ and $\nu$ coincide on $\mathcal{B}_{f^{-1}\xi}$
provided that $\mathrm{m}$ is an equilibrium state, and argue by induction.

Start by noting that since the partition $f^{-1}\xi$ restricted to each atom
$\xi(x)$ is countable, we have for $\mathrm{m}\operatorname{-}a.e.(x)$

$\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)=\frac{\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))}{m_{x}(f^{-1}\xi(x))}.$

To prove that $\mathrm{m}=\tilde{\mathrm{m}}$ on $\mathcal{B}_{f^{-1}\xi}$ we
will show that $\frac{d\nu}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)\equiv 1$ for
$\mathrm{m}\operatorname{-}a.e.(x)$. Observe that for
$\mathrm{m}\operatorname{-}a.e.(x)$ one has $\mathrm{m}_{x}(f^{-1}\xi(x))>0$,
hence the function $\frac{d\nu}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)$ is well
defined for $\mathrm{m}\operatorname{-}a.e.(x)$, but not necessarily for
$\tilde{\mathrm{m}}\operatorname{-}a.e.(x)$. This point is not emphasized in
[LedYoungI], and some minor extra considerations are required to complete the
argument222This was remarked some time ago by the second author in some
private notes.. For the sake of exactness we will be precise in what follows.

For an atom $\xi(x)$ we write
$f^{-1}\xi|\xi(x)=\\{A^{x}_{j}\\}_{j=1}^{\infty}$. We compute

$\displaystyle\int_{\xi(x)}\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(y)d\mathrm{m}_{x}(y)=\int_{\xi(x)}\frac{\tilde{\mathrm{m}}_{y}(f^{-1}\xi(y))}{\mathrm{m}_{y}(f^{-1}\xi(y))}d\mathrm{m}_{x}(y)=\int_{\xi(x)}\frac{\tilde{\mathrm{m}}_{x}(f^{-1}\xi(y))}{\mathrm{m}_{x}(f^{-1}\xi(y))}d\mathrm{m}_{x}(y)$
$\displaystyle=\sum_{\begin{subarray}{c}j=1\\\
\mathrm{m}_{x}(A^{x}_{j})>0\end{subarray}}^{\infty}\int_{A^{x}_{j}}\frac{\tilde{\mathrm{m}}_{x}(A^{x}_{j})}{\mathrm{m}_{x}(A^{x}_{j})}d\mathrm{m}_{x}(y)=\sum_{\begin{subarray}{c}j=1\\\
\mathrm{m}_{x}(A^{x}_{j})>0\end{subarray}}^{\infty}\nu_{x}(A^{x}_{j})\leq\sum_{j=1}^{\infty}\nu_{x}(A^{x}_{j})=\nu_{x}(\xi(x))=1$
(31)
$\displaystyle\Rightarrow\int\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)d\mathrm{m}(x)\leq
1$ (32)

Now we use our hypothesis that $\mathrm{m}$ is an equilibrium state: by lemma
4.6,

$\int-\log\mathrm{m}_{x}(f^{-1}\xi(x))dm(x)=H_{\mathrm{m}}(f^{-1}\xi|\xi)=P-\int\varphi
d\mathrm{m}=\int-\log\tilde{\mathrm{m}}_{x}(f^{-1}\xi(x))d\mathrm{m}(x)$

and thus
$\int\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)d\mathrm{m}(x)=0$.
Using Jensen’s inequality and (32) we conclude

$\displaystyle
0=\int\log\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}d\mathrm{m}\leq\int\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\log\bigg{|}_{f^{-1}\xi}(x)d\mathrm{m}(x)\leq\log
1=0.$

Strict convexity of the logarithm function implies then

$\frac{d\tilde{\mathrm{m}}}{d\mathrm{m}}\bigg{|}_{f^{-1}\xi}(x)=1\quad\mathrm{m}\operatorname{-}a.e.$

We conclude that the chain of inequalities appearing are equalities, in
particular for equation (31). As a result we have
$\mu_{x}(A_{j}^{x})>0\Rightarrow\mathrm{m}_{x}(A^{x}_{j})>0$, and this in turn
imply that $\mathrm{m}\operatorname{-}a.e.(x)$ it holds
$\mathrm{m}_{x}=\tilde{\mathrm{m}}_{x}$ on $f^{-1}\xi|\xi(x)$, since these
measures coincide on all the atoms and not only on the ones of
$\mathrm{m}_{x}$ positive measure. Consequently,
$\mathrm{m}=\tilde{\mathrm{m}}$ on $\mathcal{B}_{f^{-1}\xi}$, and this
finishes the general inductive step necessary to prove theorem 4.4.

######  Corollary 4.7.

Suppose that $\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ is an
equilibrium state for the potential $\varphi$ and $\xi$ is a SLY partition as
above. Then for $\mathrm{m}\operatorname{-}a.e.(x)$ the conditional measures
of $\mathrm{m}$ with respect to $\xi$ are given by

$\mathrm{m}^{\xi}_{x}=\tilde{\mathrm{m}}^{\xi}_{x}=\frac{\Delta_{x}}{L(x)}d\mu^{u}_{x}|\xi(x)\quad
L(x)=\int_{\xi(x)}\Delta_{x}d\mu^{u}_{x}.$

In particular the disintegration can be extended to every leaf of
$\mathcal{W}^{u}$ and
$\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M,\mu^{u})$.

We remark that above we did not assume that
$\mathrm{m}=\mathrm{m}_{\scriptstyle\varphi}$, and only used the existence of
an equilibrium state, which in our context follows from general arguments
(since $f$ is h-expansive the function
$\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)\to
h_{\mathrm{m}}(f)$ is upper semi-continuous, therefore has a maximum). Note
however that these abstract method do not shield information about the
equilibrium state besides its existence.

###  4.2 Conditional measures determine equilibrium states

We are interested now in the following converse of theorem 4.4.

######  Theorem 4.8.

Let $\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ and assume that
with respect to some SLY partition $\xi$ we have that
$\mathrm{m}_{x}<<\mu_{x}^{u}$ for $\mathrm{m}\operatorname{-}a.e.(x)$. Then
$\mathrm{m}$ is an equilibrium state.

######  Proof 4.9.

The proof uses similar arguments to the ones above, and is based in
[LedStrelcyn]. Define the measure $\tilde{\mathrm{m}}$ on $M$ by requiring
that for $A\subset M$ Borel,

$\tilde{\mathrm{m}}(A)=\int\mu^{u}_{x}(A)dm(x).$ (33)

Observe that $\tilde{\mathrm{m}}$ is not necessarily finite, but only
$\sigma$-finite. By hypothesis $d\mathrm{m}_{x}=\rho d\mu_{x}$, thus
$\mathrm{m}<<\tilde{\mathrm{m}}$ and moreover
$\mathrm{m}\operatorname{-}a.e.(x)$ it holds

$\frac{d\mathrm{m}}{d\tilde{\mathrm{m}}}=\rho\quad\tilde{\mathrm{m}}\operatorname{-}a.e.(x)$
(34)

(see proposition $4.1$ in [LedStrelcyn]). Note that
$\mathrm{m}^{f^{-1}\xi}_{x}=\frac{\mathrm{m}_{x}(\cdot|f^{-1}\xi)}{\mathrm{m}_{x}(f^{-1}\xi(x)}$;
by an analogous computation as the one carried in the proof of lemma 4.6 we
get

$\mathrm{m}_{x}(f^{-1}\xi(x))=\frac{\rho\circ f(x)}{\rho(x)}e^{\varphi(x)-P},$

Finally, we compute

$\displaystyle 0\leq h_{\mathrm{m}}(f)$
$\displaystyle=H(f^{-1}\xi|\xi)=\int-\log\mathrm{m}_{x}(f^{-1}\xi(x))d\mathrm{m}(x)$
$\displaystyle=P-\int\varphi d\mathrm{m}+\int-\log\frac{\rho\circ
f(x)}{\rho(x)}d\mathrm{m}(x)$

and since $\varphi$ is integrable (bounded), the function
$-\log^{-}\frac{\rho\circ f}{\rho}$ is integrable as well. Using again
Proposition $2.2$ of [LedStrelcyn] we conclude that $\log\frac{\rho\circ
f}{\rho}$ has zero integral, hence

$h_{\mathrm{m}}(f)=P-\int\varphi d\mathrm{m}$

and $\mathrm{m}$ is an equilibrium state. The proof is complete.

Joining theorems 4.4 and 4.8 we get theorem 4.1. Below we list some of its
consequences.

######  Corollary 4.10.

If $\mathrm{m}$ is an equilibrium state then $\mathrm{m}$ has conditionals
equivalent to $\mu^{s},\mu^{u}$.

######  Proof 4.11.

The set of equilibrium states for $f$ coincides with the one for $f^{-1}$.

Since the measures $\mu^{u}_{x}$ are positive on open sets, proposition $4.1$
in [Katok1996] implies:

######  Corollary 4.12.

For every equilibrium state $\mathrm{m}$ for $\varphi$ it holds
$h_{\mathrm{m}}(f)>0$

Finally, we note that the previous theorem gives another proof of the fact
that $\mathrm{m}_{\scriptstyle\varphi}$ is an equilibrium state (and $P$ is
the topological pressure), as the conditionals of
$\mathrm{m}_{\scriptstyle\varphi}$ are absolutely continuous with respect to
$\mu^{u}$.

##  5 Kolmogorov and Bernoulli properties

The aim of this part is to establish the Bernoulli property for equilibrium
states. Our arguments are written in some more generality than the context
that we have been working, with the intention that they may be applicable in
other situations.

###  5.1 Kolmogorov processes

We will establish first that $(f,\mathrm{m}_{\scriptstyle\varphi})$ has the
Kolmogorov property, and in particular is mixing of all orders. The proof is
based on the methods originally developed by Grayson, Pugh and Shub for
establishing the stable ergodicity of the time-one map of the geodesic flow on
manifolds of constant sectional curvature [StErgGeo]. We will rely heavily on
the technology developed in [StableJulienne] and its subsequent refinements.
As a guide, we use the general lines of Section $4$ in [AccStaErg].

Fix $\mathrm{m}\in\mathscr{Pr}_{\cramped[\scriptstyle]{f}}(M)$ and denote
$\mathrm{Pin}(f,\mathrm{m})$ the Pinsker $\sigma$-algebra of the system, i.e.
the $\sigma$-algebra generated by

$\\{\xi\text{ countable measurable partition
}:H_{\mathrm{m}}(\xi)<\infty,h_{\mathrm{m}}(f;\xi)=0\\}.$

We use the following characterization of Kolmogorov-systems, although the
reader may want to take it as the definition of the Kolmogorov property.

######  Theorem 5.1 (Rohklin-Sinai).

The system $(f,\mathrm{m})$ has the Kolmogorov property if and only if
$\mathrm{Pin}(f,\mathrm{m})=\\{\emptyset,M\\}\operatorname{mod}\,0$.

We say that a subset $X\subset M$ is $s$-saturated ($u$-saturated) if it
consists of whole stable (respectively, unstable) leaves. If it is both $s$\-
and $u$-saturated we say that it is bi-saturated. Given a point $x\in M$ its
accessibility class $\mathrm{ACC}(x)$ is the largest bi-saturated set
containing $x$. Analogously, we say that $X\subset M$ is
$\mathrm{m}$-essentially $s$-saturated ($u$-saturated) if there exists a
$s$-saturated ($u$-saturated) Borel set $X_{0}$ so that $\mathrm{m}(X\Delta
X_{0})=0$. By [LedYoungI] we have:

######  Theorem 5.2 (Ledrappier-Young).

If $X\in\mathrm{Pin}(f,\mathrm{m})$ then $X$ is both $\mathrm{m}$-essentially
$s$-saturated and $\mathrm{m}$-essentially $u$-saturated.

What we will show is that given $X\in\mathscr{B}_{\scriptstyle\mathrm{M}}$ is
$\mathrm{m}$-essentially $s$-saturated, then it coincides almost everywhere
with a $s$-saturated set $D(X)$ which is relevant from the point of view of
the measure (it consists of some sort of density points, see below), and then
use the minimality to conclude that this set is either full or null for
$\mathrm{m}$. As a remark for the reader versed in Pugh-Shub’s program, we
point out we are not assuming $\mathrm{m}$-essentially accessibility333The map
$f$ is $\mathrm{m}$-essentially accessible if any bi-saturated set has measure
zero or one., and we will not establish that if $X$ is both
$\mathrm{m}$-essentially $s$ and $u$ saturated then $D(X)$ is bi-saturated.
Instead, we will use the minimality of the strong foliations to spread $D(X)$
to the whole manifold.

From now on we will assume that $\mathrm{m}$ is positive on open sets.

######  Definition 5.3.

We say that the family
$\mathscr{E}=\\{\\{E_{n}(x)\\}_{n\in\mathbb{N}}\\}_{x\in M}$ is a regular
Vitali444More general definitions of Vitali bases (not regular) are available
in the literature (cf. [StableJulienne]). The definition presented here is
enough for our purposes. basis if for every $x\in M$ the family
$\\{E_{n}(x)\\}_{n\in\mathbb{N}}$ is a local basis of neighborhoods at $x$
such that

  1. 1.

each $E_{n}(x)$ is a pre-compact open set.

  2. 2.

$\operatorname{diam}(E_{n}(x))\xrightarrow[n\mapsto\infty]{}0$.

######  Definition 5.4.

Let $\mathscr{E}$ be a regular Vitali basis and
$X\in\mathscr{B}_{\scriptstyle\mathrm{M}}$. We say that $x\in M$ is a
$\mathscr{E}$-density point of $X$ if

$\displaystyle\lim_{n\rightarrow\infty}\frac{m(X\cap
E_{n}(x))}{m(E_{n}(x))}=1.$

We denote by $\mathscr{D}(X;\mathscr{E})$ the set of $\mathscr{E}$-density
points of $X$.

In particular, density points for the basis $\\{D(x;\frac{1}{2^{n}})\\}_{n}$
will be called $B\operatorname{-}L$ density points555$B-L$ stands for
Besitcovich-Lebesgue. It is standard that for any Borel set $X$,
$\mathrm{m}$-almost every point of $X$ is a $B\operatorname{-}L$ density point
of $X$. To prove the Kolmogorov-property in the essentially accessible
setting, it suffices to establish that if $X\subset M$ is both
$\mathrm{m}$-essentially $s$-saturated and $m$-essentially $u$-saturated then
the set of its $B-L$ density points is bi-saturated. This is the approach
followed in [AccStaErg], where the Lebesgue measure is considered.

In our case however, we do not have as much control on the properties of the
$\varphi$ equilibrium state as one has with the Lebesgue measure, and in
particular we do not know of to control $B\operatorname{-}L$ density points.
Instead, we will work with density points corresponding to some dynamical
defined Vitali bases called _juliennes_ , and show that under suitable
hypotheses

  1. 1.

any $X\in\mathscr{B}_{\scriptstyle\mathrm{M}}$ coincides $\mathrm{m}$-almost
everywhere with the set of its julienne density points.

  2. 2.

If $X$ is $\mathrm{m}$-essentially $s$-saturated then the set of its julienne
density points is $s$-saturated.

Convention: From now on we specialize in the case
$\mathrm{m}=\tilde{\mathrm{m}}_{\scriptstyle\varphi}$.

For some simplifications in the notation we choose to work with
$\tilde{\mathrm{m}}_{\scriptstyle\varphi}$, but as previously noted the
properties of $\tilde{\mathrm{m}}_{\scriptstyle\varphi}$ are completely
analogous to the ones of $\mathrm{m}_{\scriptstyle\varphi}$, and we will refer
directly to them without further clarification.

Some finer control in the local holonomies are now required, specifically that
they are Hölder continuous. See Theorem A of [HolFol].

######  Theorem 5.5 (Pugh-Shub-Wilkinson).

There exists $C_{H}>0,\vartheta\in(0,1)$ such that for every $x,y,z\in M$ with
$y\in W^{s}(x;\epsilon),z\in W^{u}(x,\epsilon)$ the locally defined holonomy
maps $g^{s}:W^{cu}(x)\rightarrow W^{cu}(y),g^{u}:W^{cs}(x)\rightarrow
W^{cs}(y)$ are $(C_{H},\vartheta)$-Hölder.

Choose numbers $0<\varepsilon<\epsilon,0<\sigma<1$ such that
$d(x,y)<\varepsilon\Rightarrow B(y,\varepsilon)\subset B(x,\epsilon)$,
$\lambda^{\vartheta}<\sigma$ and define the $u,s,c,cu,cs,ucs,scu$-juliennes as

$\displaystyle J^{u}_{n}(x)$
$\displaystyle:=f^{-n}(W^{u}(f^{n}x;\varepsilon))$ $\displaystyle
J^{s}_{n}(x)$ $\displaystyle:=f^{n}(W^{s}(f^{-n}x;\varepsilon))$
$\displaystyle B^{c}_{n}(x)$ $\displaystyle:=W^{c}(x;\sigma^{n})$
$\displaystyle J^{cu}_{n}(x)$ $\displaystyle:=\bigcup_{\mathclap{y\in
B^{c}_{n}(x)}}\ J^{u}_{n}(y)$ $\displaystyle J^{cs}_{n}(x)$
$\displaystyle:=\bigcup_{\mathclap{y\in B^{c}_{n}(x)}}\ J^{s}_{n}(y)$
$\displaystyle J^{ucs}_{n}(x)$ $\displaystyle:=\bigcup_{\mathclap{y\in
J_{n}^{cs}(x)}}\ J^{u}_{n}(y)$ $\displaystyle J^{scu}_{n}(x)$
$\displaystyle:=\bigcup_{\mathclap{y\in J_{n}^{cu}(x)}}\ J^{s}_{n}(y).$

######  Remark 5.6.

The definition of juliennes for general partially hyperbolic maps is more
involved, since no a priori knowledge of the action of $f$ on the center
direction is assumed. In our case, the fact that $f$ is a center isometry
simplifies all the definitions.

By corollary 3.30 we obtain.

######  Lemma 5.7.

There exists constants $c_{1}(\epsilon),c_{2}(\epsilon)>0$ such that for every
$x\in M$ it holds

$c_{1}\leq\frac{\nu^{s}_{x}(J_{n}^{s}(x))}{e^{S_{n}\varphi\circ
f^{-n}(x)-nP}}\leq c_{2}.$

From now on we omit the reference of the constants respect to $\epsilon$,
which is considered to be fixed and small with respect to $\epsilon_{0}$. We
then have the following.

######  Corollary 5.8.

There exist $L_{1}(\epsilon),L_{2}(\epsilon)>0$ such that for every $x\in M$,
for every $n\geq 0$ it holds that $y,z\in J^{cu}_{n}(x)$ implies

$L_{1}\leq\frac{\nu^{s}_{x}(J_{n}^{s}(y))}{\nu^{s}_{x}(J_{n}^{s}(z))}\leq
L_{2}.$

######  Proof 5.9.

Arguing as in the proof of lemma 3.28 we deduce the existence of some
$K^{\prime}>0$ such that for every $x\in M$ and $n\geq 0$,

$y,z\in J^{cu}_{n}(x)\Rightarrow|S_{n}\varphi\circ f^{-n}y-S_{n}\varphi\circ
f^{-n}z|<K^{\prime};$

this relies on the fact the central size of $J^{cu}$ is exponentially small.
The claimed property follows now from the previous lemma.

Using the above and the product structure of $\mathrm{m}$ (cf. proposition
3.24) we deduce the following.

######  Corollary 5.10.

There exist $L_{3}(\epsilon),L_{4}(\epsilon)>0$ such that for every $x\in M$,
for every $n\geq 0$ it holds

$L_{3}\leq\frac{m(J^{scu}_{n}(x))}{\mu^{s}_{x}(J^{s}_{n}(x))\cdot\mu^{cu}_{x}(J^{cu}_{n}(x))}\leq
L_{4}.$

The key property of juliennes is that they behave in a controlled matter under
holonomies.

######  Proposition 5.11.

There exists $k\geq 1$ such that for every $x,x^{\prime}\in M,x^{\prime}\in
W^{s}(x,\epsilon)$ the corresponding stable holonomy
$\mathrm{hol}^{s}:W^{cu}(x,\epsilon_{0})\rightarrow
W^{cu}(x^{\prime},\epsilon_{0})$ satisfies

$J_{n+k}^{cu}(x^{\prime})\subset\mathrm{hol}^{s}(J_{n}^{cu}(x))\subset
J_{n-k}^{cu}(x)$

This is completely analogous to proposition $B.8$ of [AccStaErg]. As our
juliennes differ from the ones used in that work, we present the proof.

######  Proof 5.12.

Take $z\in B^{c}_{n}(x),y\in J^{u}_{n}(x)$ and consider
$z^{\prime}=\mathrm{hol}^{s}(z),y^{\prime}=\mathrm{hol}^{s}(y)$. Then
$d(f^{n}y,f^{n}z)<\varepsilon$, and thus

$\displaystyle d(f^{n}y^{\prime},f^{n}z^{\prime})\leq
d(f^{n}y^{\prime},f^{n}y)+d(f^{n}y,f^{n}z)+d(f^{n}z,f^{n}z^{\prime})\leq\lambda^{n}\left(d(y,y^{\prime})+d(z,z^{\prime})\right)+d(f^{n}y,f^{n}z)\leq
2\epsilon$

if $n$ is sufficiently large. Define
$w^{\prime}=W^{u}(y^{\prime},\epsilon_{0})\cap
W^{c}(x^{\prime},\epsilon_{0})$. Since the bundles mutually orthogonal, and
since $\epsilon_{0}$ is small, there exists a constant $C_{a}>0$ such that

$d(f^{n}y^{\prime},f^{n}w^{\prime}),d(f^{n}w^{\prime},f^{n}z^{\prime})\leq
C_{a}\epsilon.$

This implies at once that there exists $k\geq 1$ independent of $x$ such that
if $n\geq k$, $y^{\prime}\in J^{u}_{n-k}(w^{\prime})$. On the other hand,
$d(y,z)\leq\lambda^{n}\varepsilon$ hence by theorem 5.5

$d(z^{\prime},w^{\prime})\leq C_{a}\varepsilon^{\theta}\lambda^{n\vartheta}.$

Using lemma 3.13 we obtain that for sufficiently large $k$

$d(x^{\prime},w^{\prime})\leq
d(x,z)+d(z^{\prime},w^{\prime})<\sigma^{n}+C_{a}\varepsilon^{\vartheta}\lambda^{n\vartheta}<\sigma^{n-k}.$

This shows that $y^{\prime}=\mathrm{hol}^{s}(y)\in J^{cu}_{n-k}(x^{\prime})$
and finishes the proof of the second inclusion. The first one is obtained
similarly by using $(\mathrm{hol}^{s})^{-1}$.

Now we can establish a central proposition.

######  Proposition 5.13.

If $X$ is $\mathrm{m}$-essentially $s$-saturated , then the set of its
$J^{scu}$-density points is $s$-saturated.

For the proof, we will employ the following Lemma.

######  Lemma 5.14.

Let $X_{s}$ be an s-saturated set and $x\in M$. Then

$\lim_{n\mapsto\infty}\frac{\mathrm{m}(X_{s}\cap
J_{n}^{scu}(x))}{\mathrm{m}(J_{n}^{scu}(x))}=1\Leftrightarrow\lim_{n\mapsto\infty}\frac{\mu^{cu}_{x}(X_{s}\cap
J_{n}^{cu}(x))}{\mu^{cu}_{x}(J_{n}^{cu}(x))}=1.$

The proof is not too hard, and is direct consequence of corollary 5.10. We
omit the proof, and refer the reader to proposition $2.7$ of [ErgPH], where a
completely analogous statement is proven.

######  Proof 5.15 (Proof of proposition 5.13).

Consider $X_{s}$ an $s$-saturated set such that $\mathrm{m}(X\Delta X_{s})=0$.
Let $x$ be a $J^{scu}$-density point of $X$ and let $\mathrm{hol}^{s}$ be a
(local) stable holonomy sending $x$ to $y$. Using the previous proposition we
deduce

$\mu^{cu}_{y}(\mathrm{hol}^{s}(J_{n-k}^{scu}(x)\cap
X_{s}))\leq\mu^{cu}_{y}(J_{n}^{scu}(y)\cap
X_{s})\leq\mu^{cu}_{y}(\mathrm{hol}^{s}(J_{n+k}^{scu}(x)\cap X_{s})),$

and since at this scale $(\mathrm{hol}^{s})^{-1}\mu^{cu}_{y}$ is uniformly
comparable to $\mu^{cu}_{x}$, we have

$\displaystyle 1$
$\displaystyle=\lim_{n\mapsto\infty}\frac{\mathrm{m}(X_{s}\cap
J_{n}^{scu}(x))}{\mathrm{m}(J_{n}^{scu}(x))}=\lim_{n\mapsto\infty}\frac{\mu^{cu}_{x}(X_{s}\cap
J_{n}^{scu}(x))}{\mu^{cu}_{x}(J_{n}^{scu}(x))}=\lim_{n\mapsto\infty}\frac{\mu^{cu}_{y}(\mathrm{hol}^{s}(X_{s}\cap
J_{n}^{scu}(x))}{\mu^{cu}_{y}(\mathrm{hol}^{s}(J_{n}^{scu}(x))}$
$\displaystyle=\lim_{n\mapsto\infty}\frac{\mu^{cu}_{y}(X_{s}\cap
J_{n}^{scu}(y))}{\mu^{cu}_{y}(J_{n}^{scu}(y))}=\lim_{n\mapsto\infty}\frac{\mathrm{m}(X_{s}\cap
J_{n}^{scu}(y))}{\mathrm{m}(J_{n}^{scu}(y))},$

i.e. $y$ is also a point of $J^{scu}$-density of $X_{s}$.

We will show now that any measurable $X$ coincides $\mathrm{m}$-almost
everywhere with $D(X;J^{scu})$.

######  Definition 5.16.

Let $\mathrm{m}\in\mathscr{Pr}(M)$ be positive on open sets. A regular Vitali
basis $\mathscr{E}$ is $\mathrm{m}$-engulfing if there exists $L>0$ with the
following property: for every $x\in M,n\in\mathbb{N}$ there exists a closed
set $\widehat{E}_{n}(x)$ such that

  1. 1.

$E_{n}(x)\subset\widehat{E}_{n}(x)$ and $\mathrm{m}(\widehat{E}_{n}(x))\leq
L\cdot\mathrm{m}(E_{n}(x))$.

  2. 2.

If $n^{\prime}\geq n,y\in M$ and $E_{n^{\prime}}(y)\cap E_{n}(x)\neq\emptyset$
then $E_{n^{\prime}}(y)\subset\widehat{E}_{n}(x)$.

  3. 3.

Given $N_{0}$ there exists $\delta>0$ such that
$\mathrm{m}(\mathtt{cl}(E_{n}(x)))<\delta$ implies $n\geq N_{0}$.

######  Theorem 5.17.

If $\mathscr{E}$ is a $\mathrm{m}$-engulfing basis then for every
$X\in\mathscr{B}_{\scriptstyle\mathrm{M}}$ the set of its
$\mathscr{E}$-density points coincide almost everywhere with $X$.

See theorem $3.1$ in [StableJulienne].

######  Proposition 5.18.

$J^{scu}$ is $\mathrm{m}$-engulfing.

######  Proof 5.19.

By uniformity of the contraction-expansion rates and the definition of
$ucs$-juliennes, one sees that there exists $l>0$ such that for $n\geq l$ it
holds $y\in J^{scu}_{n}(x)$ then $\mathtt{cl}(J^{scu}_{n+l}(y))\subset
J^{scu}_{n}(x)$. Arguing as in corollary 5.8 we deduce the existence of
$D=D(l)$ such that for every $n\geq l$,

$\frac{\mu^{cu}_{x}((J^{cu}_{n}(x))}{\mu^{cu}_{x}((J^{cu}_{n+l}(x))}\leq
D(l).$

Define then $\widehat{J}^{ucs}_{n}(x):=\mathtt{cl}(J^{ucs}_{n-l}(x))$ and
observe that due to corollary 5.10 and the inequality above one can deduce the
existence of $\hat{D}>0$ such that

$\mathrm{m}(\widehat{J}^{ucs}_{n}(x))\leq D\mathrm{m}(J^{ucs}_{n}(x)).$

This implies the $\mathrm{m}$-engulfing property.

The following is now clear.

######  Corollary 5.20.

Every measurable set $X$ coincides $m$-almost everywhere with $D(X;J^{scu})$.

We have thus established that if $X$ is $\mathrm{m}$-essentially $s$-saturated
then it coincides almost everywhere with the $s$-saturated set $D(X;J^{scu})$;
This allow us to conclude the $K$-property

######  Theorem 5.21.

The system $(f,\mathrm{m}_{\scriptstyle\varphi})$ is Kolmogorov.

######  Proof 5.22.

Take $X\in\mathrm{Pin}(f,\mathrm{m})$ of positive measure; then $D(X;J^{scu})$
has positive measure and is $s$-saturated. Assuming that $X^{c}$ also has
positive measure, we deduce that $D(X^{c};J^{scu})$ also has positive measure.
But since $\mathcal{W}^{s}$ is minimal arbitrarily close to points in
$D(X^{c};J^{scu})$ one can find points in $D(X;J^{scu})$, which contradicts
the definition of density point. Therefore $\mathrm{m}(X)=1$ and
$\mathrm{Pin}(f,\mathrm{m})=\\{\emptyset,M\\}\operatorname{mod}\,0$, and the
system is Kolmogorov, by theorem 5.1.

###  5.2 Bernoulli processes

Using the previous theorem, we will prove now that
$(f,\mathrm{m}_{\scriptstyle\varphi})$ is isomorphic to a Bernoulli shift. We
rely heavily on the machinery developed by D. Ornstein and B. Weiss to
establish the similar statement for the case where $f$ is the time-one map of
an Anosov flow and $\mathrm{m}_{\scriptstyle\varphi}$ is the Lebesgue measure
[GeoBernoulli] (see also [RatGibbs]). For the sake of completeness we repeat
here the main concepts and arguments from that work that we will use.
Nonetheless, some familiarity with the Ornstein and Weiss article is desirable
to follow this section.

Throughout this part all probability spaces are standard, and all partitions
of them are considered to be finite and ordered. If $\mathtt{P}$ is a
partition its $i$-th atom will be denoted $P_{i}$. For partitions
$\mathtt{P},\mathtt{P^{\prime}}$ of the probability space
$(X,\mathcal{B}_{X},\mu)$ having the same number $k$ of atoms, its partition
distance is defined as

$|\mathtt{P}-\mathtt{P^{\prime}}|:=\sum_{i=1}^{k}\mu(P_{i}\vartriangle
P_{i}^{\prime}).$

######  Definition 5.23.

Let $\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n}$ be partitions
of the probability spaces $(X,\mathcal{B}_{X},\mu_{X})$,
$(Y,\mathcal{B}_{Y},\mu_{Y})$ having the same number of atoms. Then its
$\overline{d}$-distance is

$\overline{d}(\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n}):=\inf_{g_{1},g_{2}}\frac{1}{n}\sum_{1}^{n}|g_{1}\mathtt{P_{j}}-g_{2}\mathtt{Q_{j}}|$

where $g_{1},g_{2}$ are isomorphisms between $X,Y$ and some fixed Lebesgue
space $(Z,\mathcal{B}_{Z},\mu_{Z})$ (say, the interval).

To clarify the the meaning of this definition we observe the following. Given
a partition $\mathtt{P}$ of $X$ and $x\in X$ the $\mathtt{P}$-name of $x$ is
simply the atom where is contained. Similarly, for a sequence of partitions
$\\{\mathtt{P_{j}}\\}_{1}^{n}$ having the same number of elements the
$\\{\mathtt{P_{j}}\\}_{1}^{n}$-name of $x\in X$ is the $n$-tuple
$(P_{j_{1}(x)},\ldots,P_{j_{n}(x)})$ such that
$x\in\cap_{k=1}^{n}P_{j_{k}(x)}$. It is straightforward to verify that for
partitions $\mathtt{P},\mathtt{P^{\prime}}$ of $X$ the partition distance
$|\mathtt{P}-\mathtt{P^{\prime}}|$ is simply (twice) the expected value of the
function $g_{\mathtt{P},\mathtt{P^{\prime}}}:X\rightarrow\\{0,1\\}$ such that

$g_{\mathtt{P},\mathtt{P^{\prime}}}(x)=1\Leftrightarrow\text{the
}\mathtt{P},\mathtt{P^{\prime}}\text{ names of }x\text{ are different}.$

Consequently, we have that
$\overline{d}(\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n})\leq
r$ if and only if there is a way to re-code simultaneously
$\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n}$ such that the
average of the expected value of points having different $n$-name with respect
to the encoding is less than equal to $r$. The following definition and lemma
give an important tool to estimate the size of the $\overline{d}$ distance
between two sequence of partitions.

######  Definition 5.24.

A measurable map $g:(X,\mu_{X})\rightarrow(Y,\mu_{Y})$ is said to be
$\epsilon$-measure preserving if there exists $\mathscr{Ex}\subset X$ such
that $\mu_{X}(\mathscr{Ex})<\epsilon$ and with the property that $B\subset
Y,g^{-1}B\cap\mathscr{Ex}=\emptyset$ implies

$\left|\frac{\mu_{Y}(B)}{\mu_{X}(g^{-1}B)}-1\right|<\epsilon.$

In this case $\mathscr{Ex}$ is the exceptional set of $g$.

######  Lemma 5.25.

Let $\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n}$ be partitions
of the probability spaces $(X,\mathcal{B}_{X},\mu_{X})$,
$(Y,\mathcal{B}_{Y},\mu_{Y})$ having the same number of atoms, and assume that
there exists $g:(X,\mu_{X})\rightarrow(Y,\mu_{Y})$ $\epsilon$-measure
preserving with exceptional set $\mathscr{Ex}$ such that

$x\not\in\mathscr{Ex}\Rightarrow\frac{1}{n}\\#\\{1\leq i\leq
n:\mathtt{P_{j}}-\text{name of }x\neq\mathtt{Q_{j}}-\text{name of
}gx\\}\leq\epsilon.$

Then
$\overline{d}(\\{\mathtt{P_{j}}\\}_{1}^{n},\\{\mathtt{Q_{j}}\\}_{1}^{n})\leq
16\epsilon$.

See lemma $1.3$ in [GeoBernoulli] for the proof. We will usually apply the
previous lemma with $Y\subset X$ a subspace, where $\mu_{Y}:=\mu_{X}(\cdot|Y)$
and $\mathtt{Q_{j}}=\mathtt{P_{j}}|Y$.

The following definition and theorem are central for what follows.

######  Definition 5.26.

Let $f:(X,\mathcal{B}_{X},\mu_{X})\rightarrow(X,\mathcal{B}_{X},\mu_{X})$ be
an automorphism and $\mathtt{P}$ a finite partition of $X$. We say that
$\mathtt{P}$ is very weak Bernoullian (VWB)666In [GeoBernoulli] the VWB
condition is defined with the roles of past and future interchanged. This
however is a minor difference, since if a system is Bernoulli then its inverse
is Bernoulli as well. if given $\epsilon>0$ there exists $N_{0}$ such that for
all $N^{\prime}\geq N\geq N_{0}$ there exist a family of atoms
$\mathscr{G}\subset\bigvee_{N^{\prime}}^{N}f^{-i}\mathtt{P}$ satisfying

  * •

$\mu_{X}(\mathscr{G})>1-\epsilon$;

  * •

for $n\geq 0$ it holds
$A\in\mathscr{G}\Rightarrow\overline{d}(\\{f^{j}\mathtt{P}\\}_{1}^{n},\\{f^{j}\mathtt{P}|A\\}_{1}^{n})\leq\epsilon$.

######  Theorem 5.27 (Ornstein).

Let $f:(X,\mathcal{B}_{X},\mu_{X})\rightarrow(X,\mathcal{B}_{X},\mu_{X})$ be
an automorphism.

  1. 1.

If $\mathtt{P}$ is VWB then
$(X,\bigvee_{-\infty}^{\infty}f^{n}\mathtt{P},\mu_{X})$ is isomorphic to a
Bernoulli shift.

  2. 2.

Suppose that there exists an increasing family of $f$-invariant
$\sigma$-algebras $\\{\mathcal{A}_{n}\\}_{n=1}^{\infty}$ satisfying:

    1. (a)

$\mathcal{A}_{n}\nearrow\mathcal{B}_{X}$.

    2. (b)

For every $n$, $(X,\mathcal{A}_{n},\mu_{X})$ is isomorphic to a Bernoulli
shift.

Then $(X,\mathcal{B}_{X},\mu_{X})$ is isomorphic to a Bernoulli shift.

From now on we will specialize in our case of interest
$(X,\mathcal{B}_{X},\mu_{X})=(M,\mathcal{B}_{M},m=\tilde{\mathrm{m}}_{\scriptstyle\varphi})$.
The idea is to show that there exist VWB partitions of arbitrarily small
diameter and use the previous theorem to conclude that the process
$(f,\tilde{\mathrm{m}}_{\scriptstyle\varphi})$ is Bernoulli. We fix a family
of $cu$-boxes $\mathscr{V}=\\{V_{i}\\}_{i=1}^{R}$ together with their
corresponding $cu$-transversals $\\{W_{i}\\}_{i=1}^{R}$, and define
$\tilde{\mathrm{m}}_{\scriptstyle\varphi}$ using these families.

Consider a partition $\mathtt{P}$ such that each atom $P\in\mathtt{P}$ is a
proper set with diameter smaller than the Lebesgue number of
$\\{V_{i}\\}_{i=1}^{R}$, and furthermore its boundary $\partial P_{i}$ is
piecewise differentiable. Clearly there exists such partitions of arbitrarily
small diameter and thus it is enough to prove that $\mathtt{P}$ as above is
VWB.

######  Definition 5.28.

Let $A,B\subset M$ with $B$ a $cu$-box. We say that $x\in A\cap B$ is a point
of ($s-$)tubular intersection if the stable plaque of $B$ containing $x$ is
completely contained in $A\cap B$. If every point of $A$ is a point of tubular
intersection with $B$ we say that $A$ is a $s$-band.

######  Lemma 5.29.

Let $C$ be a $cu$-box and $\delta>0$. Then there exists $N_{1}$ such that for
all $N^{\prime}>N\geq N_{1}$ one can find
$\mathscr{H}\subset\bigvee_{N}^{N^{\prime}}f^{n}\mathcal{P}$ with
$\mathrm{m}(\bigcup_{H\in\mathscr{H}}H)\geq 1-\delta$ satisfying that if $A\in
H$ there exists $E\subset A$ with the properties

  1. 1.

$\mathrm{m}(E|A)\geq 1-\delta$.

  2. 2.

$E\cap C$ is a $s$-band.

Compare lemma $2.1$ in [GeoBernoulli]. Note that in general the disintegration
of $\mathrm{m}$ into local stable manifolds is much worse behaved that the
case considered by Ornstein and Weiss (where $\mathrm{m}$ is Lebesgue).

######  Proof 5.30.

Define the set

$\mathcal{H}:=\\{x:\lim_{n\mapsto\infty}\frac{S_{n}\varphi(x)}{n}=\int\varphi
dm\\}.$

The system $(f,\mathrm{m})$ is ergodic and $\varphi$ is continuous, thus
$\mathcal{H}$ is a full measure $s$-saturated set in $M$. By Egoroff’s theorem
we can find $\mathcal{H}_{0}\subset\mathcal{H}$ with
$\mathrm{m}(\mathcal{H}_{0})\geq 1-\frac{\delta^{2}}{4}$ where the above
convergence is uniform, hence there exists $N_{0}$ such that for all $N\geq
N_{0}$, for all $x\in\mathcal{H}_{0}$ it holds

$S_{n}\varphi(x)-nP=S_{n}\varphi(x)-n(\int\varphi
d\mathrm{m}+h_{\mathrm{m}}(f))<-\frac{nh_{\mathrm{m}}(f)}{2}$

(recall that $h_{\mathrm{m}}(f)>0$ by corollary 4.12). Let
$\zeta:=\max\\{\operatorname{diam}W^{s}(x,C):x\in C\\}$, and for $n\geq N_{1}$
define

$\displaystyle B_{n}$ $\displaystyle=\cup_{i}\\{x\in C\cap f^{-n}P_{i}:x\text{
is not of tubular intersection}\\}$ $\displaystyle=\cup_{i}\\{x\in f^{n}C\cap
P_{i}:x\text{ is not of tubular intersection}\\}.$

As $f^{n}C$ is a $cu$-box, $B_{n}\subset\bigcup_{i}\bigcup_{x\in\partial
P_{i}}f^{n}W^{s}(f^{-n}x,\zeta)$. Now by formula (16), lemma 3.25 and the fact
that $\mathcal{H}_{0}$ is $s$-saturated we deduce the existence of a constant
$d=d(\zeta)>0$ such that

$\forall x\in\partial
P_{i}\cap\mathcal{H}_{0},\quad\nu^{s}_{x_{w}}(f^{n}W^{s}(f^{-n}x,\zeta))\leq
e^{S_{n}\varphi(x_{w})-nP}\leq
de^{-n\frac{h_{\mathrm{m}_{\scriptstyle\varphi}}(f)}{2}}$

where $P_{i}\subset V=V(P_{i})$. On the other hand, each $\partial P_{i}$
consists of finitely many differentiable pieces, and each piece has a well
defined angle with the $cu$-plaques of $V$. By theorem 4.4, each piece
intersects almost every $cu$-plaque in a set of zero measure, and since there
are finitely many atoms we conclude that for some constant $d^{\prime}$ it
holds

$\mathrm{m}(B_{n}\cap\mathcal{H}_{0})\leq d^{\prime}e^{-n\frac{h_{m}(f)}{2}}.$

Define $\mathcal{B}:=\cup_{N_{1}}^{\infty}\mathcal{B}_{n}$ where $N_{1}>N_{0}$
is such that
$\mathrm{m}(\mathcal{B}\cap\mathcal{H}_{0})<\frac{\delta^{2}}{4}$.

Claim 1: For all $N^{\prime}>N\geq N_{1}$ there exists
$\mathscr{H}_{1}\subset\bigvee_{N}^{N^{\prime}}f^{-n}\mathcal{P}$ such that
$\mathrm{m}(\cup_{A\not\in\mathscr{H}_{1}}A)\leq\frac{\delta}{2}$ and
$A\in\mathscr{H}_{1}\Rightarrow\mathrm{m}(\mathcal{H}_{0}^{c}|A)<\frac{\delta}{2}$.

Otherwise $\mathrm{m}(\mathcal{H}_{0}^{c})\geq\frac{\delta^{2}}{4}$,
contradicting the choice of $\mathcal{H}_{0}$.

Claim 2: For all $N^{\prime}>N\geq N_{1}$ there exists
$\mathscr{H}_{2}\subset\bigvee_{N}^{N^{\prime}}f^{-n}\mathcal{P}$ such that
$\mathrm{m}(\cup_{A\not\in\mathscr{H}_{2}}A)\leq\frac{\delta}{2}$ and
$A\in\mathscr{H}_{2}\Rightarrow\mathrm{m}(\mathcal{B}\cap\mathcal{H}_{0}|A)<\frac{\delta}{2}$.

Otherwise
$\mathrm{m}(\mathcal{B}\cap\mathcal{H}_{0})\geq\frac{\delta^{2}}{4}$,
contradicting the definition of $\mathcal{B}$.

Let
$\mathscr{H}=\bigvee_{N}^{N^{\prime}}f^{n}\mathcal{P}\cap\mathscr{H}_{1}^{c}\cap\mathscr{H}_{2}$.
Then $\mathrm{m}(\mathscr{H})\geq 1-\delta$ and for $A\in\mathscr{H}$ the
existence of the $cu$-band $E\subset A$ satisfying the conclusion of the lemma
is easily deduced from the above two claims.

Since $(f,\mathrm{m})$ has the $K$-property we have the following.

######  Lemma 5.31.

Let $C$ be a $cu$-box and $\delta>0$. Then there exists $N_{2}\geq N_{1}$ such
that for any $N^{\prime}\geq N\geq N_{2}$ there exists
$\mathscr{G}\subset\bigvee_{N}^{N^{\prime}}f^{n}\mathtt{P}$ satisfying

  * •

$\mathrm{m}(\cup_{A\in\mathscr{G}})\geq 1-\delta$;

  * •

$A\in\mathscr{G}\Rightarrow|\mathrm{m}(C|A)-\mathrm{m}(C)|<\delta$.

We are ready to prove the analogue of the main lemma of [GeoBernoulli].

######  Proposition 5.32.

Given $\delta>0$ there exists $\rho>0$ such that if $C$ is a $cu$-box with
diameter less than $\rho$ and $E\subset C$ is a $s$-band then there exists a
one-to-one mapping $\theta:E\rightarrow C$ such that

  1. 1.

$\theta$ is measure preserving.

  2. 2.

For all $n\geq 0,x\in E$ it holds $d(f^{-n}x,f^{-n}\theta(x))\leq\delta$.

######  Proof 5.33.

Consider $\rho>0$ smaller than the Lebesgue number of the covering
$\\{V_{i}\\}_{i=1}^{R}$. If $C$ is a $cu$-box with $diamC\leq\rho$ then it is
contained in some $V=V_{i}$, and hence using lemma 3.19 it is no loss of
generality to assume $C=V$.

Now observe that if $g^{s}=\mathrm{hol}^{s}_{x,y}:W_{x}\rightarrow W_{y}$
denotes the Poincaré transport along stables for $x,y\in V,y\in W^{s}(x,B)$ we
can multiply $g^{s}$ by a non-negative continuous function to obtain a map
$\widehat{g}^{s}:W_{x}\rightarrow W_{y}$ which sends $\mu^{cu}_{x}|W_{x}$ to
$\mu^{cu}_{y}|W_{y}$ (cf. proposition 2.23). Given $E\subset B$ an $s$-band
define $\theta|E\cap W_{x}\rightarrow W_{x}$ by any measure preserving map,
and $\theta|E\cap W_{y}:=\widehat{g}^{s}_{x,y}\circ\theta|E\cap
W_{x}\circ\widehat{g}^{s}_{y,x}$. The map $\theta:E\rightarrow B$ is a well
defined one to one map that sends $\mathrm{m}|E$ to $\mathrm{m}|B$. Moreover
it preserves the $cu$-plaques of $B$, thus it suffices to take $C$ such that
their $cu$-plaques have diameter smaller than $\delta$.

Now we can apply lemma $2.3$ of [GeoBernoulli] to obtain the following.

######  Lemma 5.34.

Let $\varepsilon,\delta>0$ be given. Then there exists $N$ with the property
that for all $N^{\prime}\geq N$ one can find
$\mathscr{G}\subset\bigvee_{N}^{N^{\prime}}f^{-n}\mathtt{P}$ satisfying:

  1. 1.

$\mathrm{m}(\mathscr{G})\geq 1-\varepsilon$;

  2. 2.

for $A\in\mathscr{G}$ there exists $E\subset A$ and a one-to-one mapping
$\theta:A\rightarrow M$ such that the following is true

    1. (a)

$\mathrm{m}(E|A)\geq 1-\varepsilon$;

    2. (b)

$\theta$ is $\varepsilon$-measure preserving;

    3. (c)

for all $n\geq 0,x\in E$ it holds $d(f^{-n}x,f^{-n}\theta(x))\leq\delta$.

For convenience of the reader we spell the proof.

######  Proof 5.35.

Consider $0<\rho$ corresponding to $\delta$ in the previous proposition. Using
Betsicovich’s covering lemma we construct a partition
$\\{U_{1},\ldots,U_{l}\\}$ such that

  1. 1.

each $U_{j}$ is a cu-box, $1\leq j\leq u$ of diameter less than $\rho$.

  2. 2.

$m(U_{0})\leq\frac{\varepsilon}{10}$.

Then using lemmas 5.29 and 5.31 we deduce that there exists $N$ such that for
all $N^{\prime}\geq N$ we can find
$\mathscr{G}\subset\bigvee_{N}^{N^{\prime}}f^{-n}\mathtt{P}$ with
$\mathrm{m}(\cup_{A\in\mathscr{G}}A)\geq 1-\varepsilon$ and such that for
$A\in\mathscr{G},1\leq j\leq l$ there exists an $s$-band $E\subset A$ of
relative measure bigger than $1-\varepsilon$ with the property that if $1\leq
j\leq u$ it holds

  1. 1.

$\sum_{1}^{l}|\mathrm{m}(B_{j}|E)-\mathrm{m}(B_{j})|<\varepsilon$.

  2. 2.

$E\cap U_{j}$ s-band, $E\cap U_{0}=\emptyset$.

Finally we can use the previous proposition in each $E\cap U_{j}$ to define
$\theta:E\rightarrow M$ satisfying the conclusion of the lemma.

We are ready to finish the proof.

######  Theorem 5.36.

The partition $\mathcal{P}$ is VWB.

######  Proof 5.37.

Let $\epsilon>0$ be given. Since for every $s$-saturated set $E$ the
$f^{j}\mathtt{P}$-name and the $f^{j}\mathtt{P}|E$-name of a point $x$ are
determined by the $\mathtt{P}$-name of $f^{-j}x$, and since all atoms
$A\in\mathtt{P}$ have (piecewise) smooth boundaries, we can find $\delta>0$
such that the conclusion of the previous lemma implies the hypotheses of lemma
5.25. The argument is similar to the one employed in the proof of lemma 5.29,
as points $x,y$ such that $d(x,y)<\delta$ and $x,y$ are in different atoms are
in the $\delta$-neighborhood of the union of the boundary of the atoms.

Choose $N$ sufficiently large such that the previous lemma holds for
$\varepsilon$ replaced with $\frac{\epsilon}{16}$. If $N^{\prime}\geq N$ and
$G\subset\bigvee_{N}^{N^{\prime}}f^{n}\mathtt{P}$ is the set of atoms given
above, for $A\in\mathscr{G}$ we can define $\theta:E\subset A\rightarrow E$
$\frac{\epsilon}{16}$-measure preserving, so extending $\theta$ arbitrarily to
$A\setminus E$ and applying lemma 5.25 we conclude that for all $n\geq 0$ it
holds

$\overline{d}(\\{f^{j}\mathtt{P}\\}_{1}^{n},\\{f^{j}\mathtt{P}|A\\}_{1}^{n})\leq\epsilon.$

Hence $\mathtt{P}$ is VWB.

Theorems 5.36 and 5.27 imply Theorem C.

##  6 Applications and concluding remarks

In this last section we apply our results, particularly of section 4, to the
rank-one case. We finish with some questions.

###  6.1 Hyperbolic flows

Let $\Phi=(f^{t})_{t\in\mathbb{R}}$ be an Anosov flow with invariant splitting
$TM=E^{s}\oplus E^{c}\oplus E^{u}$, where $E^{c}$ is generated by tangents to
flow direction. If this flow is transitive there is a dichotomy: either

  1. 1.

$\Phi$ is a suspension of an Anosov diffeomorphism, and therefore $M$ is a
fiber bundle over $S^{1}$, with fibers transverse to the flow line, or

  2. 2.

both $\mathcal{W}^{s},\mathcal{W}^{u}$ are minimal foliations.

See [AnosovFlow]. In what follows we will focus in this last case.

For Anosov flows the powerful of symbolic representations is available
[SymbHyp, RatnerMarkov]. With this one can establish the following.

######  Proposition 6.1.

Given a Hölder potential $\varphi:M\to\mathbb{R}$ there exists
$\mu^{u}\in\mathrm{Meas}(\mathcal{W}^{u})$ satisfying:

  1. a)

$\mu^{u}_{x}$ has full support inside $W^{u}(x)$, and is non-atomic.

  2. b)

$\forall t\in\mathbb{R},x\in M$,
$f^{-t}\mu^{u}_{f^{t}(x)}=e^{P_{\scriptstyle\mathit{top}}(\varphi)t-\int_{0}^{t}\varphi(f^{s}(x))ds}\mu^{u}_{x}$.

  3. c)

The map $\mu^{u}:x\to\mu^{u}_{x}$ is weakly continuous in the following sense:
given $y\in W^{u}(x,\epsilon_{0})$ and $\mathrm{hol}^{s}_{y,x}:W^{u}(y)\to
W^{u}(x)$ the locally defined Poincaré map, it follows that for any $A\subset
W^{u}(x)$ relatively open and pre-compact it holds

$\mu_{y}^{u}(\mathrm{hol}^{s}_{y,x}(A))\xrightarrow[y\to x]{}\mu^{u}_{x}(A).$

Moreover the convergence is uniform in $x$.

######  Proof 6.2.

The construction is due to N. Haydn [localproductstructureHaydn], based in the
original construction of Gibbs measures from Sinai [Sinai_1972]. Details are
given in the Appendix of the paper of M. Pollicot and C. Waldken
[Pollicott2001]. Part $c)$ is not explicitly stated by Haydn who only states
(a stronger version of) this fact for holonomies corresponding to the strong