# Invariant Family of Leaf measures and The Ledrappier-Young Property for
Hyperbolic Equilibrium States

Snir Ben Ovadia

###### Abstract.

$M$ is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$,
and $f$ is a $C^{1+\beta}$ ($\beta>0$) diffeomorphism of $M$. $\varphi$ is a
Hölder continuous potential on $M$. We construct an invariant and absolutely
continuous family of measures (with transformation relations defined by
$\varphi$), which sit on local unstable leaves. We present two main
applications. First, given an ergodic homoclinic class $H_{\chi}(p)$, we prove
that $\varphi$ admits a local equilibrium state on $H_{\chi}(p)$ if and only
if $\varphi$ is “recurrent on $H_{\chi}(p)$” (a condition tested by counting
periodic points), and one of the leaf measures gives a positive measure to a
set of positively recurrent hyperbolic points; and if an equilibrium measure
exists, the said invariant and absolutely continuous family of measures
constitutes as its conditional measures. An immediate corollary is the local
product structure of hyperbolic equilibrium states. Second, we prove a
Ledrappier-Young property for hyperbolic equilibrium states- if $\varphi$
admits a conformal family of leaf measures, and a hyperbolic local equilibrium
state, then the leaf measures of the invariant family (respective to
$\varphi$) are equivalent to the conformal measures (on a full measure set).
This extends the celebrated result by Ledrappier and Young for hyperbolic SRB
measures, which states that a hyperbolic equilibrium state of the geometric
potential (with pressure 0) has conditional measures on local unstable leaves
which are absolutely continuous w.r.t the Riemannian volume of these leaves.

###### Contents

  1. 1 Introduction
  2. 2 Basic Definitions and Main Results
    1. 2.1 Basic Definitions
    2. 2.2 Main Results
  3. 3 Preliminary Constructions
    1. 3.1 Symbolic Dynamics
    2. 3.2 Maximal Dimension Unstable Leaves
    3. 3.3 Ergodic Homoclinic Classes and Maximal Irreducible Components
    4. 3.4 The Canonical Part of The Symbolic Space
  4. 4 A Markovian and Absolutely Continuous Family of Measures on the Symbolic Space
    1. 4.1 The Ruelle Operator and Sinai’s Theorem
    2. 4.2 Recurrence, Harmonic Functions, and Conformal Measures
    3. 4.3 Recurrence and Periodic Orbits
  5. 5 Absolutely Continuous Invariant Family of Leaf Measures
  6. 6 Proofs of Main Results
    1. 6.1 Local Equilibrium States and the Leaf Condition
    2. 6.2 Ledrappier-Young Property for Hyperbolic Equilibrium States

##  1\. Introduction

When studying a complex system of many states and its dynamics, one approach
is to find a measure on the space of all states of the system which is
invariant under its dynamics, to represent a distribution of possible events
when the system is in equilibrium. The collection of all possible such
invariant measures could be very big, and the field of thermodynamic formalism
offers canonical ways to single out specific invariant measures of importance.

One way to single out measures of importance, is by the notion of equilibrium
states: probability measures which maximize a dynamic quantity w.r.t some
potential (function) on the system. More explicitly, let $M$ be the space of
states of the system (a compact boundaryless Riemannian manifold), let
$f\in\mathrm{Diff}^{1+}(M)$ be the dynamics of the system, and let
$\varphi:M\rightarrow\mathbb{R}$ be a bounded measurable potential on $M$;
then an equilibrium state of $\varphi$ maximizes $\mu\mapsto
h_{\mu}(f)+\int_{M}\varphi d\mu$, where $h_{\mu}(f)$ is the metric entropy of
$\mu$ w.r.t $f$.

Two important instances of this notion are measures of maximal entropy (when
$\varphi\equiv 0$), and SRB measures (when $\varphi$ is the geometric
potential, i.e minus the logarithm of the Jacobian of $f$ when restricted to
the dynamics on unstable leaves). The fact that equilibrium states of the
geometric potential (when $\sup\\{h_{\mu}(f)+\int\varphi d\mu\\}=0$) are SRB
measures is highly non trivial, and it is due to Ledrappier and Young,
[LY85a]. It is an instance of their more general formula to compute the
entropy of general measures, via local dimension of conditional measures on
unstable leaves (see also [LY85b]). When $f$ preserves a smooth measure, the
entropy formula is due to Pesin, [Pes77].

SRB measures are measures which have conditional measures which are absolutely
continuous w.r.t the induced Riemannian volume on unstable leaves, when
disintegrated by a measurable partition subordinated to the foliation of
unstable leaves (see Rokhlin’s disintegration theorem). SRB stands for Sinai,
Ruelle, and Bowen, due to their pioneering work in the field. Hyperbolic and
ergodic SRB measures are of physical importance, as the set of points which
satisfy the forward-time Birkhoff’s ergodic theorem are of positive Riemannian
volume (and thus, by convention, are observable in an experiment or a
simulation). This property is due to Pesin’s absolute continuity theorem (see
[Pes77, KSLP86, PS89]). Tsujii showed a converse, a condition via Lyapunov
exponents on orbits such that a physical measure must be an ergodic hyperbolic
SRB, see [Tsu91].

While SRB measures are important, it is not clear in what generality systems
admit them. Sinai had shown their existence in Anosov systems, and Bowen and
Ruelle in Axiom A systems. The dual characterization through conditional leaf
measures, and as an equilibrium state, by Ledrappier and Young, offered a new
way to test if a system admits an SRB measure.

Singling out measures of importance through their conditional leaf measures,
is another important tool in thermodynamic formalism, in addition to the
notion of equilibrium states. Some important examples of this are the Margulis
construction for the measure of maximal entropy through its leaf measures
([Mar70]), and the extension of the horocyclic flow to high dimension variable
negative curvature manifolds (see [Mar75, BM77]). Unstable leaves can be
viewed as the equivalence class of an equivalence relation on orbits. The
notion of defining measures by conditional measures on unstable leaves, has
been further generalized and studied as an equivalence class of orbits on
manifolds (see [Ser80]) and symbolic spaces (see [Ser80] and [PS97]).

Given a Hölder continuous potential, the idea of an equilibrium state is well
defined, but it is not clear how to generally define the leaf measures
corresponding to a potential. The Margulis leaf measures are conformal- i.e
when pushed forward they remain invariant up to a factor of
$e^{-h_{\mathrm{top}}(f)}$, where
$h_{\mathrm{top}}(f)=\sup\\{h_{\mu}(f):\mu\text{ is an invariant probability
measure}\\}$. Similarly, the conditional leaf measures of SRB measures are the
induced Riemannian volume, which gains a factor of $\exp\Big{(}$geometric
potential$\Big{)}$. We therefore are looking for a family of conditional leaf
measures which are conformal w.r.t our potential. More explicitly, if
$\varphi$ is a bounded measurable potential, and $V^{u}$ is a local unstable
leaf, we wish to find measures which satisfy $m_{V^{u}}\circ
f^{-1}=e^{\varphi-P(\varphi)}\cdot m_{f[V^{u}]}$, where $P(\varphi)$ is the
pressure of $\varphi$ (the quantity which an equilibrium state maximizes),
which acts as a calibrating factor. In [CPZ19, CPZ], Climenhaga, Pesin, and
Zelerowicz give a construction using Carathéodory dimension. Their
construction works for a general Hölder continuous potential, in the partially
hyperbolic setup with some restrictions on the central stable foliation and on
the transitivity of the system (see the remark after Definition 2.15 for more
details).

The result of Ledrappier and Young connects the two approaches of singling out
measures of importance as equilibrium states and through conditional measures
on unstable leaves. This naturally raises the question of extending their
result for more potentials than just the geometric potential. Given a
Grassmann-Hölder continuous potential, we give a construction of an invariant
and absolutely continuous family of leaf measures which give a necessary and
sufficient condition for the existence of a hyperbolic local equilibrium state
via a leaf condition, and which act as the conditional leaf measures of the
equilibrium state when it exists. We prove a Ledrappier-Young property, such
that if the potential admits a hyperbolic local equilibrium state, and a
conformal system of measures, then the conditional measures of the equilibrium
state are equivalent to the conformal measures on a set of full measure, and
in particular the conformal measures give a positive measure to the hyperbolic
points. For the definitions of a hyperbolic local equilibrium state, the leaf
condition, and a Grassmann-Hölder continuous potential, see §2. In particular,
Grassmann-Hölder continuity applies to the family of potentials
$\\{t\cdot\varphi\\}_{t\in[0,1]}$, where $\varphi$ is the geometric potential,
and this family varies continuously between the $0$ potential and the
geometric potential.

##  2\. Basic Definitions and Main Results

Throughout this paper $M$ is a Riemannian, boundaryless, and compact manifold
of dimension $d\geq 2$, and $f\in\mathrm{Diff}^{1+\beta}(M),\beta>0$. Fix
$\chi>0$.

###  2.1. Basic Definitions

######  Definition 2.1 ($\chi$-summable points).

A point $x\in M$ is called $\chi$-summable if there is a unique splitting
$T_{x}M=H^{s}(x)\oplus H^{u}(x)$ s.t

$\displaystyle\sup_{\xi_{s}\in
H^{s}(x),|\xi_{s}|=1}\sum_{m=0}^{\infty}|d_{x}f^{m}\xi_{s}|^{2}e^{2\chi
m}<\infty,\sup_{{\xi_{u}\in
H^{u}(x),|\xi_{u}|=1}}\sum_{m=0}^{\infty}|d_{x}f^{-m}\xi_{u}|^{2}e^{2\chi
m}<\infty.$

Let $\chi$-summ define the set of $\chi$-summable points. An $f$-invariant
probability measure carried by $\chi$-$\mathrm{summ}$ is called
$\chi$-hyperbolic.

For each $x\in\chi\text{-}\mathrm{summ}$, write
$s(x):=\mathrm{dim}(H^{s}(x)),u(x):=\mathrm{dim}(H^{u}(x))$. Notice,
$\chi$-summ is $f$-invariant, and $H^{s}(\cdot),H^{u}(\cdot)$ are invariant
under $d_{\cdot}f$.

######  Theorem 2.2 (Pesin-Oseledec Reduction Theorem).

For each point $x\in\chi\text{-}\mathrm{summ}$, there exists an invertible
linear map $C_{\chi}(x):\mathbb{R}^{d}\rightarrow T_{x}M$, which depends
measurably on $x$, such that
$C_{\chi}(x)[\mathbb{R}^{s(x)}\times\\{0\\}]=H^{s}(x),C_{\chi}(x)[\\{0\\}\times\mathbb{R}^{u(x)}]=H^{u}(x)$.
$C_{\chi}(\cdot)$ are chosen measurably on $\chi\text{-}\mathrm{summ}$, and
the choice is unique up to a composition with an orthogonal mapping of the “
stable” and of the “unstable” subspaces of the tangent space. In addition,

$C_{\chi}^{-1}(f(x))\circ d_{x}f\circ C_{\chi}(x)=\begin{pmatrix}D_{s}(x)&\\\
&D_{u}(x)\end{pmatrix},$

where $D_{s}(x),D_{u}(x)$ are square matrices of dimensions $s(x),u(x)$
respectively, and $\|D_{s}(x)\|,\|D_{u}^{-1}(x)\|\leq
e^{-\chi}$,$\|D_{s}^{-1}(x)\|,\|D_{u}(x)\|\leq\kappa$ for some constant
$\kappa=\kappa(f,\chi)>1$.

The Pesin-Oseledec reduction theorem has many different versions, which are
suitable for different setups. We use the version which appears, with proof,
in [BO18, Theorem 2.4].

It is possible to show that
$\|C_{\chi}^{-1}(x)\|^{2}=\sup\limits_{\overset{|\xi_{s}+\xi_{u}|=1,}{\xi_{s}\in
H^{s}(x),\xi_{u}\in H^{u}(x)}}\left\\{2\sum\limits_{m\geq
0}|d_{x}f^{m}\xi_{s}|^{2}e^{2\chi m}+2\sum\limits_{m\geq
0}|d_{x}f^{-m}\xi_{u}|^{2}e^{2\chi m}\right\\}$. In addition,
$\|C_{\chi}(\cdot)\|\leq 1$. See [BO18, Theorem 2.4, Lemma 2.9] for details.
$\|C_{\chi}^{-1}(x)\|$ serves a measurement of the hyperbolicity of $x$\- the
greater the norm, the worse the hyperbolicity (i.e slow contraction/expansion
on stable/unstable spaces, or small angle between the stable and unstable
spaces).

######  Definition 2.3.

Let $\epsilon>0$, and let $x\in\chi\text{-}\mathrm{summ}$, then

$Q_{\epsilon}(x):=\max\left\\{Q\in\\{e^{\frac{-\ell\epsilon}{3}}\\}_{\ell\in\mathbb{N}}:Q\leq\frac{1}{3^{\frac{6}{\beta}}}\epsilon^{\frac{90}{\beta}}\|C^{-1}_{\chi}(x)\|^{\frac{-48}{\beta}}\right\\}.$

######  Definition 2.4 (Recurrent $\epsilon$-weak temperability).

Let $\epsilon>0$. A point $x\in\chi\text{-}\mathrm{summ}$ is called
$\epsilon$-weakly temperable if $\exists
q:\\{f^{n}(x)\\}_{n\in\mathbb{Z}}\rightarrow\\{e^{\frac{-\ell\epsilon}{3}}\\}_{\ell\in\mathbb{N}}$
s.t

  1. (1)

$\frac{q\circ f}{q}=e^{\pm\epsilon}$,

  2. (2)

$\forall n\in\mathbb{Z}$, $q(f^{n}(x))\leq Q_{\epsilon}(f^{n}(x))$.

If in addition to (1),(2),
$q:\\{f^{n}(x)\\}_{n\in\mathbb{Z}}\rightarrow\\{e^{-\frac{\ell\epsilon}{3}}\\}_{\ell\in\mathbb{N}}$
can be chosen to also satisfy,

  1. (3)

$\limsup\limits_{n\rightarrow\pm\infty}q(f^{n}(x))>0$,

then we say that $x$ is recurrently $\epsilon$-weakly temperable. A function
$q:\\{f^{n}(x)\\}_{n\in\mathbb{Z}}\rightarrow\\{e^{\frac{-\ell\epsilon}{3}}\\}_{\ell\in\mathbb{N}}$
which satisfies (1),(2), is called an $\epsilon$-weakly tempered kernel of
$x$. Define
$\mathrm{WT}_{\chi}^{\epsilon}:=\left\\{x\in\chi\text{-}\mathrm{summ}:x\text{
is }\epsilon\text{-weakly temperable}\right\\}$, and
$\mathrm{RWT}_{\chi}^{\epsilon}:=\left\\{x\in\chi\text{-}\mathrm{summ}:x\text{
is recurrently }\epsilon\text{-weakly temperable}\right\\}$.

Remark: Every point $x\in\mathrm{WT}_{\epsilon}^{\chi}$ (for sufficiently
small $\epsilon>0$) admits a global unstable manifold $W^{u}(x)=\\{y\in
M:d(f^{-n}(x),f^{-n}(y))\xrightarrow[n\rightarrow\infty]{}0\\}$, and a global
stable manifold $W^{s}(x)=\\{y\in
M:d(f^{n}(x),f^{n}(y))\xrightarrow[n\rightarrow\infty]{}0\\}$. See
[BS02][§5.6] for more details. In Definition 3.12 we give an alternative (yet
equivalent) notion of global stable/unstable manifolds.

######  Claim 2.5.

$\forall\epsilon>0$ and $\epsilon^{\prime}\in(0,\frac{2}{3}\epsilon]$,
$\mathrm{RWT}_{\chi}^{\epsilon^{\prime}}\subseteq\mathrm{RWT}_{\chi}^{\epsilon}$,
and
$\mathrm{WT}_{\chi}^{\epsilon^{\prime}}\subseteq\mathrm{WT}_{\chi}^{\epsilon}$.

For proof, see [BO20, Claim 2.6].

######  Lemma 2.6.

$\exists\epsilon_{\chi}>0$ which depends on $M,f,\beta$ and $\chi$ s.t
$\forall\epsilon\in(0,\epsilon_{\chi}]$ $\mathrm{RWT}_{\chi}^{\epsilon}$ has a
Markov partition and a finite-to-one coding with a Hölder continuous factor
map, where the induced coding space is locally-compact.

See [BO20] for full details.

######  Definition 2.7.

$\mathrm{RWT}_{\chi}:=\left\\{x\in\chi\text{-}\mathrm{summ}:x\text{ is
recurrently }\epsilon_{\chi}\text{-weakly temperable}\right\\}.$

Remark: $\mathrm{RWT}_{\chi}$ carries all $\chi$-hyperbolic $f$-invariant
probability measures; and its definition depends only on the quality of
hyperbolicity of the orbits of its elements. In the following parts of this
paper, when $\chi>0$ is fixed, the subscript of $\epsilon_{\chi}$ would be
omitted to ease notation.

######  Definition 2.8 (Pesin set).

$\forall N\geq 1$, the set

$\Lambda_{N}:=\\{x\in\mathrm{WT}_{\chi}^{\epsilon}:\exists\epsilon\text{-weakly
tempered kernel of }x\text{ s.t }q(x)\geq\frac{1}{N}\\},$

is called the Pesin set of level $N$ (also sometimes called level set, or
Pesin level set).

######  Definition 2.9 (Positively recurrent points).

$\forall N\geq 1$, denote by $\Lambda_{N}$ the Pesin set of level $N$. Let

$\mathrm{RWT}_{\chi}^{\mathrm{PR}}:=\left\\{x\in\mathrm{RWT}_{\chi}:\exists
N\in\mathbb{N}\text{ s.t
}\limsup_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\mathbb{1}_{\Lambda_{N}}(f^{k}(x))>0\right\\}.$

For the next definition, we assume w.l.o.g that $\exists s\in\\{1,...,d-1\\}$
s.t for all points $x\in\mathrm{RWT}_{\chi}$, $\mathrm{dim}H^{s}(x)=s$. This
is possible, since we may simply split $\mathrm{RWT}_{\chi}$ into $d-1$
disjoint sets, and all results apply to each subset.

######  Definition 2.10 (Grassmann-Hölder continuity).

Let $TM$ be the tangent bundle of $M$, and let
$x,y\in\mathrm{WT}_{\chi}^{\epsilon}$. Let $d_{s}(H^{s}(x),H^{s}(y))$ be the
Grassmannian distance (of subspaces of dimension $s$, over $TM$) between the
stable space of $x$ and stable space of $y$. Similarly, let
$d_{u}(H^{u}(x),H^{u}(y))$ be the Grassmanian (of subspaces of dimension
$d-s$, over $TM$) distance between the unstable space of $x$ and stable space
of $y$. The Lyapunov distance on $\mathrm{WT}_{\chi}^{\epsilon}$ is
$d_{GH}(x,y):=d(x,y)+d_{s}(H^{s}(x),H^{s}(y))+d_{u}(H^{u}(x),H^{u}(y))$. A
function $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ is
called Grassmann-Hölder continuous if it is Hölder continuous w.r.t
$d_{GH}(\cdot,\cdot)$.

Let $E\supseteq\mathrm{WT}_{\chi}^{\epsilon}$ be a measurable subset of
$\bigcup\\{W^{u}(x):x\in\mathrm{WT}_{\chi}^{\epsilon}\\}$. A function
$\varphi:E\rightarrow\mathbb{R}$ is called unstable-Grassmann-Hölder
continuous on $E$ if it is Hölder continuous w.r.t
$d(\cdot,\cdot)+d_{u}(\cdot,\cdot)$ on $E$, where for
$x\in\bigcup\\{W^{u}(x):x\in\mathrm{WT}_{\chi}^{\epsilon}\\}$ $H^{u}(x)$ is
defined as $T_{x}W^{u}(x)$ (which is well-defined).

Since $f\in\mathrm{Diff}^{1+\beta}(M)$, $\forall t>0$, the scaled geometric
potential $\varphi_{t}(x)=-t\cdot\log\mathrm{Jac}(d_{x}f|_{H^{u}(x)})$ is
unstable-Grassmann-Hölder continuous. Every Hölder continuous potential is
unstable-Grassmann-Hölder continuous, and every unstable-Grassmann-Hölder
continuous potential on $\mathrm{WT}_{\chi}^{\epsilon}$, is Grassmann-Hölder
continuous.

Notice also that a Grassmann-Hölder continuous potential must be bounded,
since $M$ has a finite diameter.

######  Definition 2.11 (Ergodic homoclinic class).

The ergodic homoclinic class of a hyperbolic periodic point $p$ (w.r.t $\chi$)
is

$H_{\chi}(p):=\left\\{x\in\mathrm{RWT}\chi:W^{u}(x)\pitchfork
W^{s}(o(p))\neq\varnothing,W^{s}(x)\pitchfork
W^{u}(o(p))\neq\varnothing\right\\}.$

Here $o(p)=\\{f^{k}(p)\\}$, $\pitchfork$ denotes transverse intersections of
full codimension, and $W^{s/u}(\cdot)$ are the global stable and unstable
manifolds of the point, respectively.

See Definition 3.12 for the definition of global stable and unstable leaves.
Definition 2.11 was introduced in [RHRHTU11], with a set of Lyapunov regular
points replacing $\mathrm{RWT}_{\chi}$. By a well-known argument, using the
Inclination Lemma (see [BS02, Theorem 5.7.2]), every two ergodic homoclinic
classes are either disjoint modulo all conservative measures,111Conservative
measures are measures which satisfy Poincaré’s recurrence theorem: for any set
of positive measure, almost every point in it returns to it infinitely often.
or coincide. We caution the reader that $H_{\chi}(p)$ is not closed, and that
this definition is different from the definition in [New72].

Every ergodic $\chi$-hyperbolic probability is carried by a unique ergodic
homoclinic class (see Claim 3.15).

######  Definition 2.12 (Local equilibrium state).

Let $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be an
Grassmann-Hölder continuous potential. Let $\mu$ be a $\chi$-hyperbolic
$f$-invariant ergodic probability measure. Let $H_{\chi}(p)$ be the unique
ergodic homoclinic class which carries $\mu$. $\mu$ is called a hyperbolic
local equilibrium state of $\varphi$ on $H_{\chi}(p)$, if

$h_{\mu}(f)+\int\varphi d\mu=\sup\left\\{h_{\nu}(f)+\int\varphi d\nu:\nu\text{
is an }f\text{-invariant probability measure on }H_{\chi}(p)\right\\}.$

Notice, $\int\varphi d\mu<\infty$ since $\varphi$ is bounded since it is
Grassmann-Hölder continuous.

######  Definition 2.13 (Local pressure).

Let $p$ be a periodic $\chi$-hyperbolic point, and let $H_{\chi}(p)$ be its
ergodic homoclinic class. The local pressure of $\varphi$ on $H_{\chi}(p)$ is

$P_{H_{\chi}(p)}(\varphi):=\sup\left\\{h_{\nu}(f)+\int\varphi d\nu:\nu\text{
is an }f\text{-invariant probability measures on }H_{\chi}(p)\right\\}.$

The boundedness of $\varphi$ implies $P_{H_{\chi}(p)}(\varphi)<\infty$. Let
$E$ as in Definition 2.10.

######  Definition 2.14.

Given a sequence of $C^{1}$ embedded submanifolds, $V^{u}_{n}$, we write
$V^{u}_{n}\xrightarrow[n\rightarrow\infty]{C^{1}}V^{u}$, where $V^{u}$ is an
embedded submanifold, if $\sup_{x\in V^{u}_{n}}\inf_{y\in
V^{u}}\\{d(x,y)+d_{TM}(T_{x}V^{u}_{n},T_{y}V^{u})\\}\xrightarrow[n\rightarrow\infty]{}0$.

######  Definition 2.15 ($\varphi$-conformal system of measures).

Let $\varphi:E\rightarrow\mathbb{R}$ be an unstable-Grassmann-Hölder
continuous potential. Let $H_{\chi}(p)$ be the ergodic homoclinic class of a
$\chi$-hyperbolic and periodic point $p$. A $\varphi$-conformal system of
measures on $H_{\chi}(p)$ is a collection of non-zero Radon measures on
local222“local” here means diffeomorphic to a ball, in the induced metric.
unstable leaves of points in $H_{\chi}(p)$ such that

  1. (1)

$V^{u}_{n}\xrightarrow[n\rightarrow\infty]{C^{1}}V^{u}$ implies
$m_{V^{u}_{n}}^{\varphi}(1)\xrightarrow[n\rightarrow\infty]{}m_{V^{u}}^{\varphi}(1)$,

  2. (2)

for every local unstable leaf of a point in $H_{\chi}(p)$, $V^{u}$,

$m^{\varphi}_{V^{u}}\circ f^{-1}=e^{\varphi-P_{H_{\chi}(p)}(\varphi)}\cdot
m^{\varphi}_{f[V^{u}]},$

where $m^{\varphi}_{V^{u}},m^{\varphi}_{f[V^{u}]}$ are the $\varphi$-conformal
measures on the local unstable leaves $V^{u},f[V^{u}]$, respectively.

An example of a conformal family of measures, is the Riemannian volume on
unstable leaves. It is $\varphi$-conformal with $\varphi$ being the geometric
potential $-\log\mathrm{Jac}(df|_{\text{unstable leaves}})$, when the pressure
is $0$ (e.g when an SRB exists).

It is not clear a-priori when do conformal families exist. In [CPZ],
Climenhaga, Pesin, and Zelerowicz give a construction of such families in the
partially hyperbolic setup using the Carathéodory dimension structure (which
extends their result in the uniformly hyperbolic setup [CPZ19]), under the
assumption that $\varphi$ has the Bowen property (i.e uniformly summable
variations on stable and unstable leaves, under a certain regularity
assumption).

######  Definition 2.16 ($\varphi$-invariant family of leaf measures).

Let $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a
Grassmann-Hölder continuous potential. Let $H_{\chi}(p)$ be the ergodic
homoclinic class of a periodic $\chi$-hyperbolic point $p$ s.t
$P_{H_{\chi}(p)}(\varphi)<\infty$. A $\varphi$-invariant family of leaf
measures on $H_{\chi}(p)$, is a family $\mathcal{F}_{H_{\chi}(p)}(\varphi)$ of
non-zero and finite measures s.t

  1. (1)

$\forall\mu\in\mathcal{F}_{H_{\chi}(p)}(\varphi)$, $\mu$ is carried by
$WT_{\chi}^{\epsilon}$, and its support is contained in a local unstable leaf
of a point $x\in H_{\chi}(p)$,

  2. (2)

$\exists$ a map
$A:\cup\\{\mathrm{supp}(\mu^{\prime})\\}_{\mu^{\prime}\in\mathcal{F}_{H_{\chi}(p)}(\varphi)}\rightarrow\mathbb{R}$
which is measurable when restricted to measurable sets, s.t
$\forall\mu\in\mathcal{F}_{H_{\chi}(p)}(\varphi)$,

$\mu\circ f^{-1}$
$=e^{-P_{H_{\chi}(p)}(\varphi)}\sum_{i=1}^{N_{\mu}}e^{\phi(\mu^{(i)})}\mu^{(i)}$,
where $\mu^{(i)}\in\mathcal{F}_{H_{\chi}(p)}(\varphi)$ for all $1\leq i\leq
N(\mu)<\infty$, and $\phi(\mu^{(i)})$ is a constant s.t
$\phi(\mu^{(i)})=\varphi(x)+A(x)-A(f^{-1}(x))$ for $\mu^{(i)}$-a.e $x\in M$
where $\varphi$ is defined.

  3. (3)

$\exists$ $\sigma$-compact metric space $X$ s.t
$\mathcal{F}_{H_{\chi}(p)}(\varphi)=\\{\mu_{x}\\}_{x\in X}$ and
$x_{n}\rightarrow
y\Rightarrow\mu_{x_{n}}\xrightarrow[]{\text{weak-}*}\mu_{y}$.

The measures in $\mathcal{F}_{H_{\chi}(p)}(\varphi)$ are not required to be
mutually singular.

One cannot expect uniqueness of such a family, as the Radon-Nikodym derivative
of the push-forwards can always be changed by a coboundary.

###  2.2. Main Results

######  Theorem (A).

Let $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a
Grassmann-Hölder continuous potential. Let $H_{\chi}(p)$ be the ergodic
homoclinic class of a periodic $\chi$-hyperbolic point $p$. Then $H_{\chi}(p)$
admits a $\varphi$-invariant family of leaf measures,
$\mathcal{F}_{H_{\chi}(p)}(\varphi)$.

While conformal families of measures are generally a simpler object for
calculations, their general existence is a hard problem which is only solved
in a few setups (we mention a few in §1). The main advantage of invariant
families, is that they exists with (almost) no preceding assumptions.
Furthermore, when a conformal family does exist, and so does an equilibrium
state for the respective potential, the invariant family measures must
coincide in measure-class with the conformal family. See the last result of
this section.

######  Definition 2.17 ($\varphi$-leaf condition).

Let $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a
Grassmann-Hölder continuous potential. Let $H_{\chi}(p)$ be an ergodic
homoclinic class of a $\chi$-hyperbolic periodic point $p$. We say that the
$\varphi$-leaf condition is satisfied for $H_{\chi}(p)$, if
$\exists\mu\in\mathcal{F}_{H_{\chi}(p)}(\varphi)$ s.t
$\mu(\mathrm{RWT}_{\chi}^{\mathrm{PR}})>0$.

Here $\varphi_{n}(q):=\sum_{k=0}^{n-1}\varphi(f^{-k}(q))$, and
$\mathcal{F}_{H_{\chi}(p)}(\varphi)$ is given by the theorem above. That is,
in the theorem above we construct an invariant family, and the leaf condition
applies to any invariant family as constructed by that theorem; We remind the
reader that the invariant family need not be unique, as the leaf measures can
always be changed by a coboundary factor.

######  Theorem (B).

Let $\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a
Grassmann-Hölder continuous potential. Let $H_{\chi}(p)$ be an ergodic
homoclinic class of a $\chi$-hyperbolic periodic point $p$ s.t $p$ belongs to
a Pesin level set $\Lambda_{l}$. Then $\varphi$ admits a $\chi$-hyperbolic
local equilibrium state on $H_{\chi}(p)$, if and only if $\sum_{n\geq
1}\sum_{f^{n}(q)=q,q\in H_{\chi}(p)\cap\Lambda_{l}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}=\infty$ and the $\varphi$-leaf condition is
satisfied for $H_{\chi}(p)$. If the local equilibrium state exists, then it is
unique and the $\varphi$-invariant family is its conditional measures.

Note that if the assumption above holds for one $l$, then it holds for all
$l^{\prime}\geq l$, since $\Lambda_{l^{\prime}}\supseteq\Lambda_{l}$.

######  Corollary (C).

Local equilibrium states of Grassmann-Hölder continuous potentials have local
product structure.

This follows from the absolute continuity property of the $\varphi$-invariant
family we construct. This extends previous results for the Axiom A setup (see
[Hay94, Lep00]).

######  Corollary (D).

$(M,f)$ admits a $\chi$-hyperbolic equilibrium state for $\varphi$ if and only
if there exists an ergodic homoclinic class $H_{\chi}(p)$ which satisfies the
assumptions of the previous theorem, and in addition
$P_{H_{\chi}(p)}(\varphi)=P_{\mathrm{top}}(\varphi)$
($=\sup\\{h_{\nu}(f)+\int\varphi d\nu:\nu\text{ is an }f\text{-inv. probablity
on }M\\}$).

######  Theorem (E).

Let $\varphi:WT_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a Grassmann-Hölder
continuous potential. Let $H_{\chi}(p)$ be an ergodic homoclinic class of a
$\chi$-hyperbolic periodic point $p$ which admits a (unique) $\chi$-hyperbolic
equilibrium state for $\varphi$, $\nu_{\varphi}$. Assume that $H_{\chi}(p)$
admits a $\varphi$-conformal system of measures. Let
$\mathcal{F}_{H_{\chi}(p)}(\varphi)$ be any $\varphi$-invariant family of
conditional leaf measures as in Theorem $\mathrm{(A)}$, then there is a sub-
family $\mathcal{F}^{\prime}$ which disintegrates $\nu_{\varphi}$ (as in
Theorem $\mathrm{(B)}$), and for which the following holds: For any
$\mu\in\mathcal{F}^{\prime}$, given a local unstable leaf
$V^{u}_{\mathrm{loc}}$ which carries $\mu$, and a corresponding
$\varphi$-conformal measure on $V^{u}_{\mathrm{loc}}$,
$m^{\varphi}_{V^{u}_{\mathrm{loc}}}$, there exists a measurable set of full
$\mu$-measure, $E_{\mu}$, s.t $\mu\sim
m^{\varphi}_{V^{u}_{\mathrm{loc}}}|_{E_{\mu}}$. In particular,
$m^{\varphi}_{V^{u}_{\mathrm{loc}}}(\mathrm{RWT}_{\chi}^{\mathrm{PR}})>0$.

The last property holds in particular when $\varphi$ is unstable-Grassmann-
Hölder continuous and is defined on unstable leaves, and thus we a-priori have
no reason to know that its conformal measures “see” the hyperbolic points.

If $\varphi$ is the geometric potential then the $\varphi$-conformal measures
exist and are the induced Riemannian volume of the unstable leaves, and this
theorem proves that every $\chi$-hyperbolic equilibrium state of the geometric
potential with pressure $0$, has absolutely continuous conditional measures
(and thus is an SRB measure). This is a new proof to the result by Ledrappier
and Young in [LY85a], when restricting to hyperbolic measures; and extends it
to additional potentials.

Remark: In several of the main results, we use the assumption
$\sum\limits_{n\geq 1}\sum\limits_{f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}}e^{\varphi_{n}(q)-n\cdot P_{H_{\chi}(p)}(\varphi)}$
$=\infty$ for some hyperbolic periodic point $p$ which belongs to a level set
$\Lambda_{l}$ (where
$\varphi:\mathrm{WT}_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ is a Grassmann-
Hölder continuous potential). This assumption is used in order to show that
$\varphi$ is recurrent (see §4.2) when lifted to an irreducible component
which codes $H_{\chi}(p)$ (see §3.3). This assumption is always used together
with the leaf condition (see Definition 2.17). We do not know if the first
assumption can be omitted (i.e does the leaf condition imply recurrence of the
lifted potential). We were not able to prove that using the methods of this
paper. The difference between this paper and [BO21], where the leaf condition
is sufficient, is the overlapping property of leaf measures (i.e the Lebesgue
measures of two unstable leaves which intersect, coincide on the
intersection). Here, we use a coding to construct our $\varphi$-invariant
family of leaf measures, and in order to gain a bound on the overlapping of
measures, we must use the bound on the multiplicity of the coding map
$\widehat{\pi}$; which in turn is only bounded on the recurrent part of the
symbolic space in our construction; hence the need for recurrence. There are
other constructions of codings with better multiplicity bounds for the coding
map (e.g [Buz20]), though those multiplicity improvements come on expense of
other properties which are necessary for this paper.

##  3\. Preliminary Constructions

###  3.1. Symbolic Dynamics

Sinai [Sin68] and Adler and Weiss [AW70] constructed Markov partitions and for
Anosov diffeomorphims and linear toral automorphisms, respectively. In [Sar13]
Sarig constructed a Markov partition for non-uniformly hyperbolic surface
diffeomorphisms. We extended his results to manifolds of any dimension greater
or equal to 2 in [BO18]. In [BO20] we modify the codings from [BO18, Sar13]
and introduce the set $\mathrm{RWT}_{\chi}$. $\mathrm{RWT}_{\chi}$ is defined
canonically, and still consists of all recurrently-codable points (see
Proposition 3.8) in the modified symbolic dynamics. In the following section
we review those results.

Since $M$ is compact, $\exists r=r(M)>0,\rho=\rho(M)>0$ s.t the exponential
map $\exp_{x}:\\{v\in T_{x}M:|v|\leq r\\}\rightarrow B_{\rho}(x)=\\{y\in
M:d(x,y)<\rho\\}$ is injective.

######  Definition 3.1 (Pesin-charts).

When $\epsilon\leq r$, the following is well defined since $C_{\chi}(\cdot)$
is a contraction (see [BO18, Lemma 2.9]):

  1. (1)

$\psi_{x}^{\eta}:=\exp_{x}\circ C_{\chi}(x):\\{v\in
T_{x}M:|v|\leq\eta\\}\rightarrow B_{\rho}(x)$, $\eta\in(0,Q_{\epsilon}(x)]$ is
called a Pesin-chart.

  2. (2)

A double Pesin-chart is an ordered couple
$\psi_{x}^{p^{s},p^{u}}:=(\psi_{x}^{p^{s}},\psi_{x}^{p^{u}})$, where
$\psi_{x}^{p^{s}}$ and $\psi_{x}^{p^{u}}$ are Pesin-charts.

######  Theorem 3.2.

$\forall\chi>0$ s.t there exists a $\chi$-hyperbolic periodic point $p$,
$\exists$ a countable and locally-finite333Finite out-going and in-going
degree at each vertex. directed graph
$\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$ which induces a topological
Markov shift
$\Sigma:=\\{\underline{u}\in\mathcal{V}^{\mathbb{Z}}:(u_{i},u_{i+1})\in\mathcal{E},\forall
i\in\mathbb{Z}\\}$ s.t $\Sigma$ admits a factor map $\pi:\Sigma\rightarrow M$
with the following properties:

  1. (1)

$\sigma:\Sigma\rightarrow\Sigma$ is defined by
$(\sigma\underline{u})_{i}:=u_{i+1}$, $i\in\mathbb{Z}$ (the left-shift); and
satifies $\pi\circ\sigma=f\circ\pi$.

  2. (2)

$\pi$ is a Hölder continuous map w.r.t the metric
$d(\underline{u},\underline{v}):=\exp\left(-\min\\{i\geq 0:u_{i}\neq
v_{i}\text{ or }u_{-i}\neq v_{-i}\\}\right)$.

  3. (3)

Let $\Sigma^{\\#}:=\left\\{\underline{u}\in\Sigma:\exists
n_{k},m_{k}\uparrow\infty\text{ s.t
}u_{n_{k}}=u_{n_{0}},u_{-m_{k}}=u_{-m_{0}},\forall k\geq 0\right\\}$, then
$\pi[\Sigma^{\\#}]$ carries all $f$-invariant, $\chi$-hyperbolic probability
measures.

This theorem is the content of [BO18, Theorem 3.13] (and similarly, the
content of [Sar13, Theorem 4.16] when $d=2$). $\mathcal{V}$ is a collection of
double Pesin-charts (see Definition 3.1), which is discrete.

######  Definition 3.3.

  1. (1)

$\forall u\in\mathcal{V}$,
$Z(u):=\pi[\\{\underline{u}\in\Sigma^{\\#}:u_{0}=u\\}]$,
$\mathcal{Z}:=\\{Z(u):u\in\mathcal{V}\\}$.

  2. (2)

$\mathcal{R}$ is defined to be a countable partition of
$\bigcup\limits_{v\in\mathcal{V}}Z(v)=\pi[\Sigma^{\\#}]$ s.t

    1. (a)

$\mathcal{R}$ is a refinement of $\mathcal{Z}$: $\forall
Z\in\mathcal{Z},R\in\mathcal{R}$, $R\cap Z\neq\varnothing\Rightarrow
R\subseteq Z$.

    2. (b)

$\forall v\in\mathcal{V}$, $\\#\\{R\in\mathcal{R}:R\subseteq Z(v)\\}<\infty$
([Sar13, § 11]).

    3. (c)

The rectangles property: $\forall R\in\mathcal{R}$,$\forall x,y\in R$
$\exists!z:=[x,y]_{R}\in R$, s.t $\forall i\geq
0,R(f^{i}(z))=R(f^{i}(y)),R(f^{-i}(z))=R(f^{-i}(x))$, where $R(t):=$the unique
partition member of $\mathcal{R}$ which contains $t$, for
$t\in\pi[\Sigma^{\\#}]$.

  3. (3)

$\forall R,S\in\mathcal{R}$, we say $R\rightarrow S$ if $R\cap
f^{-1}[S]\neq\varnothing$, and let
$\widehat{\mathcal{E}}:=\\{(R,S)\in\mathcal{R}^{2}\text{ s.t }R\to S\\}$.

  4. (4)

$\widehat{\Sigma}:=\\{\underline{R}\in\mathcal{R}^{\mathbb{Z}}:R_{i}\rightarrow
R_{i+1},\forall i\in\mathbb{Z}\\}$.

Remark: Given $\mathcal{Z}$, the definition of $\mathcal{R}$ is proper, since
such a refining partition as $\mathcal{R}$ exists by the Bowen-Sinai
refinement, see [Sar13, § 11.1]. By property (2)(b), and since $\Sigma$ is
locally-compact (see Theorem 3.2, local-finiteness of $\mathcal{G}$ implies
local-compactness of $\Sigma$), $\widehat{\Sigma}$ is also locally-compact.

######  Definition 3.4.

  1. (1)

$\widehat{\Sigma}^{\\#}:=\\{\underline{R}\in\widehat{\Sigma}:\exists
n_{k},m_{k}\uparrow\infty\text{ s.t
}R_{n_{k}}=R_{n_{0}},R_{-m_{k}}=R_{-m_{0}},\forall k\geq 0\\}$.

  2. (2)

Two partition members $R,S\in\mathcal{R}$ are said to be affiliated if
$\exists u,v\in\mathcal{V}$ s.t $R\subseteq Z(u),S\subseteq Z(v)$ and
$Z(u)\cap Z(v)\neq\varnothing$ (this terminology is due to O. Sarig, [Sar13, §
12.3]).

######  Claim 3.5 (Local finiteness of the cover $\mathcal{Z}$).

$\forall Z\in\mathcal{Z}$,$\\#\\{Z^{\prime}\in\mathcal{Z}:Z^{\prime}\cap
Z\neq\varnothing\\}<\infty$.

This claim is the content of [BO18, Theorem 5.2] (and similarly [Sar13,
Theorem 10.2] when $d=2$). Remark: By Claim 3.5 and Definition 3.3(2)(b), it
follows that every partition member of $\mathcal{R}$ has only a finite number
of partition members affiliated to it.

######  Definition 3.6.

Let $R\in\mathcal{R}$, $N(R):=\\#\\{S\in\mathcal{R}:S\text{ is affiliated to
}R\\}$.

######  Theorem 3.7.

Given $\widehat{\Sigma}$ from Definition 3.3, there exists a factor map
$\widehat{\pi}:\widehat{\Sigma}\rightarrow M$ s.t

  1. (1)

$\widehat{\pi}$ is Hölder continuous w.r.t the metric
$d(\underline{R},\underline{S})=\exp\left(-\min\\{i\geq 0:R_{i}\neq
S_{i}\text{ or }R_{-i}\neq S_{-i}\\}\right)$.

  2. (2)

$f\circ\widehat{\pi}=\widehat{\pi}\circ\sigma$, where $\sigma$ denotes the
left-shift on $\widehat{\Sigma}$.

  3. (3)

$\widehat{\pi}|_{\widehat{\Sigma}^{\\#}}$ is finite-to-one.

  4. (4)

$\forall\underline{R}\in\widehat{\Sigma}$,
$\widehat{\pi}(\underline{R})\in\overline{R_{0}}$.

  5. (5)

$\widehat{\pi}[\widehat{\Sigma}^{\\#}]$ carries all $\chi$-hyperbolic
invariant probability measures.

This theorem is the content of the main theorem of [BO18], Theorem 1.1 (and
similarly the content of [Sar13, Theorem 1.3] when $d=2$). Notice, the fibers
of $\widehat{\pi}$ being finite in $\widehat{\Sigma}^{\\#}$ does not imply
being uniformly bounded.

######  Proposition 3.8.

$\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\pi[\Sigma^{\\#}]=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}=\mathrm{RWT}_{\chi}$.

This is the content of [BO20, Proposition 4.11, Corollary 4.12].

###  3.2. Maximal Dimension Unstable Leaves

######  Definition 3.9.

An unstable leaf (of $f$) in $M$, $V^{u}$, is a $C^{1+\frac{\beta}{3}}$
embedded, open, Riemannian submanifold of $M$, such that $\forall x,y\in
V^{u}$, $\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log
d(f^{-n}(x),f^{-n}(y))<0$. Similarly, a stable leaf is an unstable leaf of
$f^{-1}$.

######  Definition 3.10.

An unstable leaf is called an unstable leaf of maximal dimension, if it is not
contained in any unstable leaf of a greater dimension.

Notice that if $x\in\mathrm{RWT}_{\chi}$ belongs to an unstable leaf of
maximal dimension $V^{u}$, then $\mathrm{dim}H^{u}(x)=\mathrm{dim}V^{u}$. This
can be seen using the following claim.

######  Claim 3.11.

$\forall\underline{u}\in\Sigma$, there exists a maximal dimension unstable
leaf $V^{u}(\underline{u})$, which depends only on $(u_{i})_{i\leq 0}$, and a
maximal dimension stable leaf $V^{s}(\underline{u})$, which depends only on
$(u_{i})_{i\geq 0}$, s.t $\\{\pi(\underline{u})\\}=V^{u}(\underline{u})\cap
V^{s}(\underline{u})$, and $f[V^{s}(\underline{u})]\subset
V^{s}(\sigma\underline{u})$ and $f^{-1}[V^{u}(\underline{u})]\subset
V^{u}(\sigma^{-1}\underline{u})$.

This is the content of [BO18, Proposition 3.12,Theorem 3.13, Proposition 4.4]
(and similarly [Sar13, Proposition 4.15,Theorem 4.16,Proposition 6.3] when
$d=2$). By construction, $V^{s}(\underline{u}),V^{u}(\underline{u})$ are
local, in the sense that they have finite (intrinsic) diameter.

###  3.3. Ergodic Homoclinic Classes and Maximal Irreducible Components

######  Definition 3.12.

Let $x\in\mathrm{RWT}_{\chi}$, and let $\underline{u}\in\Sigma^{\\#}$ s.t
$\pi(\underline{u})=x$. The global stable (resp. unstable) manifold of $x$ is
$W^{s}(x):=\bigcup_{n\geq 0}f^{-n}[V^{s}(\sigma^{n}\underline{u})]$ (resp.
$W^{u}(x):=\bigcup_{n\geq 0}f^{n}[V^{u}(\sigma^{-n}\underline{u})]$ ).

This definition is proper and is independent of the choice of $\underline{u}$,
by the construction of $V^{u}(\underline{u})$. For more details see [BO18,
Definition 2.23, Definition 3.2]. Recall the remark after Definition 2.4.

Let $p$ be a periodic point in $\chi\text{-}\mathrm{summ}$, i.e hyperbolic
periodic point. Since $p$ is periodic, $\|C^{-1}_{\chi}(\cdot)\|$ is bounded
along the orbit of $p$, and therefore $p\in\mathrm{RWT}_{\chi}$.

######  Definition 3.13.

Let $\Sigma^{\prime}$ be a topological Markov shift. A cylinder
$[R_{0},...,R_{n}]$, $n\geq 0$, is the set
$\\{\underline{R}^{\prime}\in\Sigma^{\prime}:R_{i}^{\prime}=R_{i}\forall
i=0,...,n\\}$. An element of a topological Markov shift is called a chain.

######  Definition 3.14.

Consider the Markov partition $\mathcal{R}$ from Definition 3.3.

  1. (1)

For any $a,b\in\mathcal{R}$, $n\in\mathbb{N}$, we write $a\xrightarrow[]{n}b$
if there exists a non-empty cylinder $[W_{1},...,W_{n}]$ s.t $W_{1}=a$ and
$W_{n}=b$.

  2. (2)

Define $\sim\subseteq\mathcal{R}\times\mathcal{R}$ by $R\sim S\iff\exists
n_{RS},n_{SR}\in\mathbb{N}\text{ s.t
}R\xrightarrow[]{n_{RS}}S,S\xrightarrow[]{n_{SR}}R$. The relation $\sim$ is
transitive and symmetric. When restricted to $\\{R\in\mathcal{R}:R\sim R\\}$,
it is also reflexive, and thus an equivalence relation. Denote the
corresponding equivalence class of some representative $R\in\mathcal{R}$,
$R\sim R$, by $\langle R\rangle$.

  3. (3)

A maximal irreducible component in $\widehat{\Sigma}$, corresponding to
$R\in\mathcal{R}$ s.t $R\sim R$, is
$\\{\underline{R}\in\widehat{\Sigma}:\underline{R}\in\langle
R\rangle^{\mathbb{Z}}\\}$.

  4. (4)

If a topological Markov shift is a maximal irreducible component of itself, we
call it irredcible.

If $\widetilde{\Sigma}$ is a maximal irreducible component of
$\widehat{\Sigma}$, then $\widehat{\pi}[\widetilde{\Sigma}^{\\#}]$ is a subset
of an ergodic homoclinic class, where
$\widetilde{\Sigma}^{\\#}:=\widehat{\Sigma}^{\\#}\cap\widetilde{\Sigma}$.

Recall Definition 2.11.

######  Claim 3.15.

Let $\mu$ be an $f$-invariant ergodic probability measure which is carried by
$\mathrm{RWT}_{\chi}$. Then there exists a unique ergodic homoclinic class
$H_{\chi}(p)$, where $p$ is a $\chi$-hyperbolic periodic point, which carries
$\mu$.

For proof, see [RHRHTU11].

######  Proposition 3.16.

Let $p$ be a periodic $\chi$-hyperbolic point. Then, there exists a maximal
irreducible component, $\widetilde{\Sigma}\subseteq\widehat{\Sigma}$, s.t
$\widehat{\pi}[\widetilde{\Sigma}^{\\#}]\supseteq
H_{\chi}(p)\cap\widehat{\pi}[\widehat{\Sigma}^{\\#\\#}]$, where

$\widehat{\Sigma}^{\\#\\#}=\\{\underline{R}\in\widehat{\Sigma}:\exists R\text{
s.t }\\#\\{i\geq 0:R_{i}=R\\}=\\#\\{i\leq 0:R_{i}=R\\}=\infty\\}.$

In particular, $\widehat{\pi}[\widetilde{\Sigma}^{\\#}]=H_{\chi}(p)$ modulo
all conservative measures.

This is the content of [BO20, Theorem 5.9], and is based on the result of
Buzzi, Crovisier and Sarig in [BCS] for homoclinic classes of the type of
Newhouse [New72], and the $f$-invariant, $\chi$-hyperbolic, probability
measures which they carry.

###  3.4. The Canonical Part of The Symbolic Space

######  Definition 3.17.

$\displaystyle\widehat{\Sigma}_{L}:=$ $\displaystyle\\{(R_{i})_{i\leq
0}\in\mathcal{R}^{-\mathbb{N}}:\forall i\leq 0,R_{i-1}\rightarrow
R_{i}\\},\text{
}\sigma_{R}:\widehat{\Sigma}_{L}\rightarrow\widehat{\Sigma}_{L},\sigma_{R}((R_{i})_{i\leq
0})=(R_{i-1})_{i\leq 0}.$

Notice, $\sigma_{R}$ is the right-shift, not the left-shift. In order to
prevent any confusion, we will always notate $\sigma_{R}$ with a subscript $R$
(for “right”), when considering the right-shift.

######  Definition 3.18 (The canonical coding $\underline{R}(\cdot)$).

Recall that $R(x)$ is the unique $\mathcal{R}$-element which contains $x$.
Given

$x\in\pi[\Sigma^{\\#}]=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}\text{,
let }(\underline{R}(x))_{i}:=R(f^{i}(x)),i\in\mathbb{Z}.$

One should notice that
$\widehat{\pi}(\underline{R}(x))=x,\underline{R}(x)\in\widehat{\Sigma}^{\circ}$
(see Definition 3.21 for $\widehat{\Sigma}^{\circ}$).

######  Definition 3.19.

$\forall\underline{R}\in\widehat{\Sigma}_{L},\text{
}W^{u}(\underline{R}):=\bigcap_{j=0}^{\infty}f^{j}[R_{-j}].$

$W^{u}(\underline{R})$ might be empty, and is not expected to be an immersed
submanifold. It is a subset of $\mathrm{RWT}_{\chi}$. We have the following
important property.

######  Corollary 3.20.

$\forall\underline{R}\in\widehat{\Sigma}_{L}$,

$f[W^{u}(\underline{R})]=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{\sigma_{R}\underline{S}=\underline{R}}W^{u}(\underline{S}).$

See [BO21, Corollary 3.19] for proof.

######  Definition 3.21.

$\widehat{\Sigma}^{\circ}:=\\{\underline{R}\in\widehat{\Sigma}:\forall
n\in\mathbb{Z},f^{n}(\widehat{\pi}(\underline{R}))\in R_{n}\\},$
$\widehat{\Sigma}^{\circ}_{L}:=\\{\underline{R}\in\widehat{\Sigma}^{\\#}_{L}:W^{u}(\underline{R})\neq\varnothing\\},$

where $\widehat{\Sigma}^{\\#}_{L}:=\\{(R_{i})_{i\leq
0}:(R_{i})_{i\in\mathbb{Z}}\in\widehat{\Sigma}^{\\#}\\}$. We call
$\widehat{\Sigma}^{\circ},\widehat{\Sigma}^{\circ}_{L}$ the canonical parts of
the respective symbolic spaces.

Notice that $\underline{R}(\cdot)$ is the inverse of
$\widehat{\pi}|_{\widehat{\Sigma}^{\circ}}$; and that
$\widehat{\Sigma}^{\circ}_{L}=(\widehat{\Sigma}^{\circ})_{L}:=\\{(R_{i})_{i\leq
0}:(R_{i})_{\in\mathbb{Z}}\in\widehat{\Sigma}^{\circ}\\}$.

######  Claim 3.22.

$\widehat{\pi}[\widehat{\Sigma}^{\circ}]=\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\pi[\Sigma^{\\#}]=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m<EMAIL_ADDRESS>
See [BO21, Corollary 3.21] for proof.

######  Lemma 3.23.

If $R_{0}\rightarrow R_{1}$, where $R_{0},R_{1}\in\mathcal{R}$, and
$R_{0}\subseteq Z(u_{0})$, $u_{0}\in\mathcal{V}$ then $\exists
u_{1}\in\mathcal{V}$ s.t $u_{0}\rightarrow u_{1}$ and $R_{1}\subseteq
Z(u_{1})$. Analogously, if $R_{0}\rightarrow R_{1}$ and $R_{1}\subseteq
Z(v_{1})$, then $\exists v_{0}$ s.t $v_{0}\rightarrow v_{1}$ and
$R_{0}\subseteq Z(v_{0})$.

See [BO21, Lemma 4.1] for proof.

######  Definition 3.24.

Given a chain $\underline{R}\in\widehat{\Sigma}_{L}$ we say that a chain
$\underline{u}\in\mathcal{V}^{-\mathbb{N}}$ covers the chain $\underline{R}$
if $\underline{u}$ is admissible and $R_{i}\subseteq Z(u_{i})$ for all $i\leq
0$. We write $\underline{u}\curvearrowright\underline{R}$.

By Lemma 3.23, for every chain $\underline{R}\in\widehat{\Sigma}_{L}$, the
collection of chains which cover $\underline{R}$ is not empty.

######  Definition 3.25.

$\Sigma_{L}:=\\{(u_{i})_{i\leq 0}:\underline{u}\in\Sigma\\}$.

######  Proposition 3.26.

$\forall\underline{R}\in\widehat{\Sigma}_{L}$, $\exists$ a local unstable leaf
of maximal dimension $V^{u}(\underline{R})$ s.t
$f^{-1}[V^{u}(\underline{R})]\subseteq V^{u}(\sigma_{R}\underline{R})$, and
$V^{u}(\underline{R})$ is an open submanifold of $M$, with finite and positive
induced Riemannian volume. In addition,
$V^{u}(\underline{R})\subseteq\psi_{x}[\\{v:|v|_{\infty}\leq p^{u}\\}]$ for
all $u=\psi_{x}^{p^{s},p^{u}}\in\mathcal{V}$ s.t $Z(u)\supseteq R_{0}$, and
$\underline{R}\mapsto V^{u}(\underline{R})$ is continuous in $C^{1}$-norm in
any local choice of coordinates. Moreover, $V^{u}(\underline{R})\subseteq
V^{u}(\underline{u})$ $\forall\underline{u}\in\Sigma_{L}$ s.t
$\underline{u}\curvearrowright\underline{R}$.

See [BO21, § 4.1] for proof.

##  4\. A Markovian and Absolutely Continuous Family of Measures on the
Symbolic Space

###  4.1. The Ruelle Operator and Sinai’s Theorem

######  Definition 4.1 (Ruelle operator).

Let $\widetilde{\Sigma}_{L}$ be a one-sided irreducible locally compact
topological Markov shift (of negative chains), and let
$\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a Hölder continuous
potential. The associated Ruelle operator
$L_{\phi}:C(\widetilde{\Sigma}_{L})\rightarrow C(\widetilde{\Sigma}_{L})$, is
defined by

$(L_{\phi}h)(\underline{R}):=\sum_{\widetilde{\sigma}_{R}\underline{S}=\underline{R}}e^{\phi(\underline{S})}h(\underline{S}),$

where $h\in C(\widetilde{\Sigma}_{L})$, and
$\widetilde{\sigma}_{R}:\widetilde{\Sigma}_{L}\rightarrow\widetilde{\Sigma}_{L}$
is the right-shift.

######  Definition 4.2.

Let $X$ be a topological Markov shift (either one-sided or two-sided). A
function $\widehat{\varphi}:X\rightarrow\mathbb{R}$ is called weakly Hölder
continuous, if $\exists C>0,\alpha>0$, s.t
$\forall\underline{x},\underline{y}\in X$ s.t
$d(\underline{x},\underline{y})\leq e^{-1}$, we have
$|\widehat{\varphi}(\underline{x})-\widehat{\varphi}(\underline{y})|\leq
C\cdot d(\underline{x},\underline{y})^{\alpha}$.

This notion is introduced in [Sar09], with the weaker assumption of
$d(\underline{x},\underline{y})\leq e^{-2}$.

Recall Definition 2.10.

######  Claim 4.3.

Let $\varphi:WT_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a potential which
is Grassmann-Hölder continuous. Let
$\widehat{\varphi}:\widehat{\Sigma}\rightarrow\mathbb{R}$,
$\widehat{\varphi}:=\varphi\circ\widehat{\pi}$, be the lift of $\varphi$. Then
$\widehat{\varphi}$ is weakly Hölder continuous on $\widehat{\Sigma}$.

This follows from [BO18, Proposition 6.1].

######  Theorem 4.4 (Sinai).

Let $\varphi:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ be a Grassmann-Hölder
continuous potential. Then there exist two functions
$\varphi^{*},A:\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}\rightarrow\mathbb{R}$
such that $\varphi^{*}(x)=\varphi^{*}((R(f^{-i}(x)))_{i\geq 0})$ depends only
on the past of the itinerary of $x$, $\varphi^{*}=\varphi+A-A\circ f^{-1}$,
$A$ is bounded, and
$\widehat{A}:=A\circ\widehat{\pi},\phi:=\varphi^{*}\circ\widehat{\pi}$ are
weakly Hölder continuous.

This theorem appears in Sinai’s fundamental paper [Sin72]. He proves it in the
context of a topological Markov shift with a finite alphabet (for the
countable case see [Dao13]). We give the proof in the case when a countable
Markov partition exists.

###### Proof.

First, $\forall a\in\mathcal{R}$, fix a point $y_{a}\in a$. $\forall
x\in\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}$,
define $x^{*}:=[x,y_{R(x)}]_{R(x)}\in R(x)$, where $R(x)$ is unique partition
element of $\mathcal{R}$ which contains $x$, and $[\cdot,\cdot]_{R(x)}$
denotes the Smale bracket of two points in $R(x)$ (the right-hand term yields
its future, and the left-hand terms yields its past). Define
$A:\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}\rightarrow\mathbb{R}$,

$A(x):=\sum_{n\geq 0}\varphi(f^{-n}(x^{*}))-\varphi(f^{-n}(x)).$

By Claim 4.3, $\widehat{\varphi}:=\varphi\circ\widehat{\pi}$ is weakly Hölder
continuous. In addition,
$d(\underline{R}(f^{-n}(x)),\underline{R}(f^{-n}(x^{*})))\leq e^{-n}$. Let
$C>0,\theta\in(0,1)$ be the weak Hölder constants of $\widehat{\varphi}$ s.t
$d(\underline{R}^{(1)},\underline{R}^{(2)})\leq e^{-n}\Rightarrow$
$|\widehat{\varphi}(\underline{R}^{(1)})-\widehat{\varphi}(\underline{R}^{(2)})|\leq
C\cdot\theta^{n}$. Then $\widehat{A}$ is a uniformly convergent series with
uniformly bounded and equicontinuous summands, and so $\widehat{A}$ is bounded
and continuous. In particular, $A$ is also bounded.

Let $\varphi^{*}:=\varphi+A-A\circ f^{-1}$ and
$\phi:=\varphi^{*}\circ\widehat{\pi}$. The weak Hölder continuity of
$\widehat{A}=A\circ\widehat{\pi}$ and of $\phi$ follows from the weak Hölder
continuity of $\widehat{\varphi}$ and the proof of the finite alphabet case in
[Sin72].

It remains to show that $\varphi(x)=\varphi((R(f^{-i}(x)))_{i\geq 0})$.

$\displaystyle\varphi^{*}(x)$ $\displaystyle=\varphi(x)+A(x)-A(f^{-1}(x))=$
$\displaystyle=\varphi(x)+\sum_{n\geq
0}\varphi(f^{-n}(x^{*}))-\varphi(f^{-n}(x))-\sum_{n\geq
0}\varphi(f^{-n}((f^{-1}(x))^{*}))-\varphi(f^{-n-1}(x))$
$\displaystyle=\varphi(x^{*})+\sum_{n\geq
0}\varphi\left(f^{-n}\left(f^{-1}(x^{*})\right)\right)-\varphi\left(f^{-n}\left(\left(f^{-1}(x)\right)^{*}\right)\right).$

The last expression depends only on $(R(f^{-i}(x)))_{i\geq 0}$.

∎

Remark: Although $\widehat{\varphi},\widehat{A},\phi$ are defined on
$\widehat{\Sigma}^{\circ}$, they extend continuously to $\widehat{\Sigma}$
because of their uniform moduli of continuity and the fact that
$\widehat{\Sigma}^{\circ}$ is dense in $\widehat{\Sigma}$. Thus
$\phi:\widehat{\Sigma}_{L}\rightarrow\mathbb{R}$ is well defined and weakly
Hölder continuous.

###  4.2. Recurrence, Harmonic Functions, and Conformal Measures

As before, $\widetilde{\Sigma}_{L}$ is a maximal irreducible component of
$\widehat{\Sigma}_{L}$, and $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$
is a weakly Hölder continuous potential.

In this section we use the Gurevich pressure, and we begin by recalling its
definition. For a weakly Hölder potential
$\zeta:\widehat{\Sigma}_{L}\rightarrow\mathbb{R}$ , the local partition
functions are for $n\geq 1$,

$Z_{n}(\zeta,R):=\sum\limits_{\underline{S}\in\widehat{\Sigma}_{L}\cap[R],\sigma_{R}^{n}\underline{S}=\underline{S}}e^{\sum_{k=0}^{n-1}\zeta(\sigma_{R}^{k}\underline{S})},$

where the sum has finitely many terms since $\widehat{\Sigma}_{L}$ is locally
compact. The Gurevich pressure of the potential $\zeta$ is

$P_{G}(\zeta):=\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log
Z_{n}(\zeta,R)\in(-\infty,\infty].$

When $\widehat{\Sigma}_{L}$ is irreducible, the limit is independent of the
choice of the symbol $R$ (see [Sar09, Proposition 3.2]). We say that the
potential $\zeta$ is recurrent if for some symbol $R$,

$\sum_{n\geq 1}e^{-nP_{G}(\zeta)}Z_{n}(\zeta,R)=\infty.$

In this case the sum diverges for all $R$, see [Sar09, Corollary 3.1]. We say
that the potential $\zeta$ is positive recurrent if it is recurrent, and for
some symbol $R$,

$\sum_{n\geq 1}n\cdot
e^{-nP_{G}(\zeta)}\sum\limits_{\begin{subarray}{c}\underline{S}\in\widehat{\Sigma}_{L}\cap[R]\text{,
s.t }\sigma_{R}^{n}\underline{S}=\underline{S},\\\ \text{ and }\forall
0<l<n,S_{-l}\neq
R\end{subarray}}e^{\sum_{k=0}^{n-1}\zeta(\sigma_{R}^{k}\underline{S})}<\infty.$

Again, this property turns out to be independent of $R$. For more details, see
[Sar09, § 3.1.3]. For a detailed review of the properties of recurrent
potentials, see [Sar09].

######  Definition 4.5.

Let $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a weakly Hölder
continuous potential s.t $P_{G}(\phi)<\infty$, where $\widetilde{\Sigma}_{L}$
is a maximal irreducible component of $\widehat{\Sigma}_{L}$. A positive and
continuous function $\psi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}^{+}$ is
called $\phi$-harmonic if $L_{\phi}\psi=e^{P_{G}(\phi)}\psi$. A non-zero Radon
measure $p$ on $\widetilde{\Sigma}_{L}$ is called $\phi$-conformal if
$L_{\phi}^{*}p=e^{P_{G}(\phi)}p$, where $L_{\phi}^{*}$ is the dual operator of
$L_{\phi}$.

Notice, $\widetilde{\Sigma}_{L}$ may be non-compact, and thus $p$ may be
infinite.

######  Theorem 4.6.

Let $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a weakly Hölder
continuous potential, where $\widetilde{\Sigma}_{L}$ is an irreducible locally
compact one-sided topological Markov shift. If $P_{G}(\phi)<\infty$ then there
exist a $\phi$-harmonic function $\psi$, and a $\phi$-conformal measure $p$.

This theorem is due to work of Sarig [Sar99, Sar01], Cyr [Cyr10], and Shwartz
[Shw19]. Sarig had proven that when $\phi$ is recurrent, $\psi$ and $p$ exist
and are unique up to scaling. In the transient case, Cyr showed the existence
of a $\phi$-conformal measure, and Shwartz proved the existence of a
$\phi$-harmonic function with a weakly Hölder logarithm. In the transient
case, the $\phi$-harmonic functions and the $\phi$-conformal measures are not
always unique up to scaling. For the structure of the set of these objects,
see [Shw19].

######  Lemma 4.7.

Let $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a weakly Hölder
continuous potential, where $\widetilde{\Sigma}_{L}$ is an irreducible locally
compact one-sided topological Markov shift. Let
$\psi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}^{+}$ be a $\phi$-harmonic
function. Then $\log\psi$ is weakly Hölder continuous.

In the recurrent case, Sarig proves it in [Sar09, Proposition 3.4]. In the
transient case, it follows from the proof of Theorem 5.7 in [Shw19]. Shwartz
shows in the proof that $\log\psi$ is uniformly continuous. A close inspection
of his argument tells that in fact, if $\phi$ is weakly Hölder continuous,
then so is $\log\psi$.

###  4.3. Recurrence and Periodic Orbits

######  Lemma 4.8.

Let $\varphi:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ be a Grassmann-Hölder
continuous potential, and let $\phi:\widehat{\Sigma}_{L}\rightarrow\mathbb{R}$
be a corresponding weakly Hölder continuous one-sided potential given by
Theorem 4.4 and the remark following it. Let $H_{\chi}(p)$ be an ergodic
homoclinic class of a periodic and $\chi$-hyperbolic point $p$, and let
$\widetilde{\Sigma}$ be a maximal irreducible component of $\widehat{\Sigma}$
s.t $\widehat{\pi}[\widetilde{\Sigma}^{\\#}]=H_{\chi}(p)$ modulo conservative
measures (see Proposition 3.16). Then
$\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ is recurrent if and only if

$\sum_{n\geq 1}\sum_{\begin{subarray}{c}f^{n}(q)=q,\\\ q\in
H_{\chi}(p)\cap\Lambda_{l}\end{subarray}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}=\infty,$

where $\Lambda_{l}$ is a level set s.t $p\in\Lambda_{l\cdot
e^{-2\sqrt[3]{\epsilon}}}$ and
$\varphi_{n}(q)=\sum_{k=0}^{n-1}\varphi(f^{-k}(q))$.

###### Proof.

We start with the “if” part. Let $q\in H_{\chi}(p)\cap\Lambda_{l}$ be a
periodic $\chi$-hyperbolic point of period $n$. By Proposition 3.16, $q$ has a
(periodic444Let
$\underline{R}\in\widehat{\Sigma}^{\\#}\cap\widehat{\pi}^{-1}[\\{q\\}]$. Then
for any $m\geq 0$, $\widehat{\pi}(\sigma^{m\cdot n}\underline{R})=q$, whence
by the finiteness of $\widehat{\Sigma}^{\\#}\cap\widehat{\pi}^{-1}[\\{q\\}]$,
$\exists m_{1},m_{2}\in\mathbb{N}$ s.t $\sigma^{m_{1}\cdot
n}\underline{R}=\sigma^{m_{2}\cdot n}\underline{R}$, and so $\sigma^{m\cdot
n}\underline{R}=\underline{R}$ with
$m=\max\\{m_{1},m_{2}\\}-\min\\{m_{1},m_{2}\\}$.) coding in
$\widetilde{\Sigma}$. Let $\underline{S}^{q}$ be such a periodic coding.

By [BO21, Claim 7.6], there is a finite collection of symbols
$\mathcal{R}_{l}\subseteq\mathcal{R}$, which depends only on $l$, and which
contains $S_{0}^{q}$.

By [BO18, Theorem 1.3] there exists
$C_{l}=\max\\{N(R)^{2}:R\in\mathcal{R}_{l}\\}\in\mathbb{N}$ s.t
$|\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#}|\leq C_{l}$ (see
Definition 3.6 for the definition of $N(R)$). In addition, by Theorem 4.4,
$\exists C:=2\|A\|_{\infty}<\infty$ s.t

$\phi_{n}(\underline{S}^{q})=\pm C+\varphi_{n}(q),\text{ }\forall
n\in\mathbb{N}.$ (1)

Since $\varphi$ is bounded (recall remark after Definition 2.10), so is
$\phi$. By the spectral decomposition (see [Sar15, Theorem 2.5]), we may
assume w.l.o.g that $(\widetilde{\Sigma},\sigma)$ is topologically mixing
(otherwise apply the proof to each mixing component of a suitable period of
$\sigma_{R}$). By the variational principle (see [Sar09, Theorem 4.4], and
[IJT15] when the potential is not bounded), the Gurevich pressure of $\phi$ on
$\widetilde{\Sigma}_{L}$, $P_{G}(\phi)$, satisfies

$P_{G}(\phi)=\sup\\{h_{\nu}(\widetilde{\sigma}_{R})+\int_{\widetilde{\Sigma}_{L}}\phi
d\nu:\nu\text{ is an invariant probability measure on
}\widetilde{\Sigma}_{L}\\}.$

Every invariant probability measure on $\widetilde{\Sigma}_{L}$, $\nu$,
extends uniquely to an invariant probability measure on $\widetilde{\Sigma}$,
$\nu^{\prime}$, s.t $\nu^{\prime}\circ\tau^{-1}=\nu$ and
$h_{\nu}(\widetilde{\sigma}_{R})=h_{\nu^{\prime}}(\sigma)$, where
$\tau:\widetilde{\Sigma}\rightarrow\widetilde{\Sigma}_{L}$ is the projection
onto the non-positive coordinates. Thus,

$P_{G}(\phi)=\sup\\{h_{\nu^{\prime}}(\sigma)+\int_{\widetilde{\Sigma}}\phi
d\nu^{\prime}:\nu^{\prime}\text{ is an invariant probability measure on
}\widetilde{\Sigma}\\}.$

In addition, every invariant probability measure on $H_{\chi}(p)$, $\mu$,
lifts to a an invariant probability measure on $\widetilde{\Sigma}$,
$\mu^{\prime}$, s.t $\mu^{\prime}\circ\widehat{\pi}^{-1}=\mu$ and
$h_{\mu}(f)=h_{\mu^{\prime}}(\sigma)$. Then, we get

$P_{G}(\phi)=P_{H_{\chi}(p)}(\varphi).$ (2)

By the Ruelle inequality the topological entropy of $f$ is finite, and since
$\varphi$ is bounded, $P_{H_{\chi}(p)}(\varphi)<\infty$.

For every $n\in\mathbb{N}$ $\exists a_{n},b_{n}\in\mathcal{R}_{l}$ s.t

$\displaystyle\sum_{\begin{subarray}{c}f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}\\\
\underline{S}^{q}\in\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#},S^{q}_{0}=a_{n},S^{q}_{n}={b_{n}}\end{subarray}}e^{\phi_{n}(\underline{S}^{q})}\geq\frac{1}{|\mathcal{R}_{l}|^{2}}\cdot\sum_{\begin{subarray}{c}f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}\\\
\underline{S}^{q}\in\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#}\end{subarray}}e^{\phi_{n}(\underline{S}^{q})}.$

For every $n\geq 1$, $\forall q\in H_{\chi}(p)\cap\Lambda_{l}$ s.t
$f^{n}(q)=q$, $|\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#}|\leq
C_{l}$. By assumption, $\infty=\sum\limits_{n\geq
1}\sum\limits_{f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}$. Recall (1), then by the pigeonhole principle
$\exists a,b\in\mathcal{R}_{l}$ and $n_{k}\uparrow\infty$ s.t $a_{n_{k}}=a$
and $b_{n_{k}}=b$, and so that the following sum is infinite:

$\displaystyle\infty=\sum_{k\geq
0}\sum_{\begin{subarray}{c}f^{n_{k}}(q)=q,q\in H_{\chi}(p)\cap\Lambda_{l}\\\
\underline{S}^{q}\in\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#},S^{q}_{0}=a,S^{q}_{n_{k}}=b\end{subarray}}e^{\phi_{n_{k}}(S^{q})-n_{k}\cdot
P_{G}(\phi)}.$

Let $C_{H}^{\prime}>0,\gamma\in(0,1)$ be the Hölder constant and Hölder
exponent of $\phi$, respectively. Let $C_{H}:=C_{H}^{\prime}\cdot\sum_{n\geq
0}\gamma^{n}<\infty$.

Let $\underline{W}$ be an admissible word s.t $W_{0}=b$ and $W_{m-1}=a$, where
$m:=|\underline{W}|$. For every finite word $\underline{W}^{\prime}$ s.t
$W^{\prime}_{0}=W^{\prime}_{|\underline{W}^{\prime}|-1}$, let
$\underline{S}^{\underline{W}^{\prime}}$ denote the periodic extension of
$\underline{W}^{\prime}$ to a bi-infinite chain (in
$\widetilde{\Sigma}^{\\#}$).

For every $n\geq 0$, for every finite word $\underline{W}^{\prime}$ s.t
$|\underline{W}^{\prime}|=n$, every extension of $\underline{W}^{\prime}$ to
$\widetilde{\Sigma}^{\\#}$ which codes a periodic point in
$H_{\chi}(p)\cap\Lambda_{l}$ of period $n$, must code the same point, by the
shadowing lemma (see [BO18, Proposition 3.12(4)]). Therefore

$\displaystyle\infty=$ $\displaystyle\sum_{k\geq
0}\sum_{\begin{subarray}{c}f^{n_{k}}(q)=q,q\in H_{\chi}(p)\cap\Lambda_{l}\\\
\underline{S}^{q}\in\widehat{\pi}^{-1}[\\{q\\}]\cap\widetilde{\Sigma}^{\\#},S^{q}_{0}=a,S^{q}_{n_{k}}=b\end{subarray}}e^{\phi_{n_{k}}(\underline{S}^{q})-n_{k}\cdot
P_{G}(\phi)}$ $\displaystyle\leq$ $\displaystyle
C_{l}e^{C_{H}^{\prime}\sum_{j=1}^{n_{k}}\gamma^{j}}\sum_{k\geq
0}\sum_{\overset{}{\widetilde{\underline{W}}:|\widetilde{\underline{W}}|=n_{k},\widetilde{W}_{0}=a,\widetilde{W}_{n_{k}-1}=b}}e^{\phi_{n_{k}}(\underline{S}^{(\widetilde{\underline{W}}\cdot\underline{W})})-n_{k}\cdot
P_{G}(\phi)}$ $\displaystyle\leq$ $\displaystyle
C_{l}e^{m\cdot(\|\varphi\|_{\infty}+P_{G}(\phi))+C_{H}}\cdot\sum_{k\geq
0}\sum_{\underline{S}\in\widetilde{\Sigma}:\sigma^{n_{k}+m}\underline{S}=\underline{S},S_{0}=a}e^{\phi_{n_{k}+m}(\underline{S})-(n_{k}+m)\cdot
P_{G}(\phi)}$ $\displaystyle\leq$ $\displaystyle
C_{l}e^{m\cdot(\|\varphi\|_{\infty}+P_{G}(\phi))+C_{H}}\cdot\sum_{n\geq
1}\sum_{\underline{S}\in\widetilde{\Sigma}:\sigma^{n}\underline{S}=\underline{S},S_{0}=a}e^{\phi_{n}(\underline{S})-n\cdot
P_{G}(\phi)}.$

Therefore $\phi$ is recurrent on $\widetilde{\Sigma}$.

It remains to show the “only if” part. Assume that $\phi$ is recurrent on
$\widetilde{\Sigma}$, then $\forall a\in\mathcal{R}_{l}$,

$\sum\limits_{n\geq
1}\sum\limits_{\overset{\underline{S}\in\widetilde{\Sigma}:}{\sigma^{n}\underline{S}=\underline{S},S_{0}=a}}e^{\phi_{n}(\underline{S})-n\cdot
P_{G}(\phi)}=\infty.$

Choose $a\in\mathcal{R}_{l}$ s.t
$[a]\cap\widehat{\pi}^{-1}[\\{p\\}]\cap\widetilde{\Sigma}^{\\#}\neq\varnothing$.
$\forall n\geq 1$, every periodic chain
$\underline{S}\in\widetilde{\Sigma}\cap[a]$ of period $n$, codes a periodic
point of period $n$ in $H_{\chi}(p)\cap\Lambda_{l}$ (see [BO18, Theorem
4.13]). There can be no more than $C_{l}$ chains which code the same point,
thus

$\infty=\sum_{n\geq
1}\sum_{\underline{S}\in\widetilde{\Sigma}:\sigma^{n}\underline{S}=\underline{S},S_{0}=a}e^{\phi_{n}(\underline{S})-n\cdot
P_{G}(\phi)}\leq C_{l}e^{C}e^{C_{H}}\sum_{n\geq 1}\sum_{f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}.$

∎

##  5\. Absolutely Continuous Invariant Family of Leaf Measures

The following theorem is a version of the Ledrappier theorem ([Led74],[LLS16,
Theorem 3.3]). We give a new proof here which is good for our needs, while the
previous proofs provide a formula for the measures of cylinders.

######  Theorem 5.1.

Let $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a weakly Hölder
continuous potential s.t $P_{G}(\phi)<\infty$, where $\widetilde{\Sigma}_{L}$
is an irreducible locally compact one-sided topological Markov shift. Let
$\psi$ be a $\phi$-harmonic function on $\widetilde{\Sigma}_{L}$. Then,
$\exists$ a family of probability measures on $\widetilde{\Sigma}$,
$\\{\widehat{p}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$,
s.t $\forall\underline{R}\in\widetilde{\Sigma}_{L}$,
$\widehat{p}_{\underline{R}}$ is carried by
$\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall i\leq 0\\}$ and
$\widehat{p}_{\underline{R}}\circ\sigma^{-1}=\sum_{\widetilde{\sigma}_{R}\underline{S}=\underline{R}}e^{\phi(\underline{S})+\log\psi(\underline{S})-\log\psi\circ\widetilde{\sigma}_{R}(\underline{S})-P_{G}(\phi)}\widehat{p}_{\underline{S}}$,
where $\widetilde{\sigma}_{R}:=\sigma_{R}|_{\widetilde{\Sigma}_{L}}$.

###### Proof.

Fix $[R]\subseteq\widetilde{\Sigma}_{L}$, and let
$\underline{R}\in\widetilde{\Sigma}_{L}\cap[R]$. Let
$\tau:\widehat{\Sigma}\rightarrow\widehat{\Sigma}_{L}$ be the projection to
the non-positive coordinates. Write
$\widetilde{\phi}:=\phi+\log\psi-\log\psi\circ\widetilde{\sigma}_{R}-P_{G}(\phi)$,
and
$\widetilde{\phi}_{n}:=\sum_{k=0}^{n-1}\widetilde{\phi}\circ\widetilde{\sigma}_{R}^{k}$.
It follows that $L_{\widetilde{\phi}}1=1$. Define,

$\widehat{p}_{\underline{R}}^{(n)}:=\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{n}(\underline{S})}\delta_{\sigma^{-n}\underline{S}^{\pm}},$
(3)

where $\delta_{\underline{S}^{\pm}}$ is a Dirac measure with mass at the chain
$\underline{S}^{\pm}$, and $\underline{S}^{\pm}$ is some (any) choice of a
chain in $\tau^{-1}[\\{\underline{S}\\}]\cap\widetilde{\Sigma}$. We will show
that the limit
$\widehat{p}_{\underline{R}}^{(n)}\xrightarrow[n\rightarrow\infty]{\text{weak-}*}\widehat{p}_{\underline{R}}$
exists, and is independent of the choice of the representatives
$\underline{S}^{\pm}\in\tau^{-1}[\\{\underline{S}\\}]\cap\widetilde{\Sigma}$.

Step 1: $\forall n\geq 0$,
$\widehat{p}_{\underline{R}}^{(n)}(1)=(L_{\widetilde{\phi}}^{n}1)(\underline{R})=1$.
$\\{\widetilde{\underline{R}}\in\widetilde{\Sigma}\cap[R]:\widetilde{R}_{i}=R_{i}\forall
i\leq 0\\}$ is a compact set which carries
$\widehat{p}_{\underline{R}}^{(n)}$, $\forall n\geq 1$. Thus, $\exists
n_{k}\uparrow\infty$ and a probability $\widehat{p}_{\underline{R}}$ on
$\\{\widetilde{\underline{R}}\in\widetilde{\Sigma}\cap[R]:\widetilde{R}_{i}=R_{i},\forall
i\leq 0\\}$ s.t
$\widehat{p}_{\underline{R}}^{(n_{k})}\xrightarrow[k\rightarrow\infty]{}\widehat{p}_{\underline{R}}$.

Step 2: Let
$[\underline{w}]_{a}:=\sigma^{a}[w_{a},w_{a+1},...,w_{b-1},w_{b}]$, where
$a,b\in\mathbb{Z}$ and $a\leq b$. Let $n,m\geq 0$. Observe:

  1. (1)

If $b\leq 0$,
$\widehat{p}_{\underline{R}}^{(n)}(\mathbb{1}_{[\underline{w}]_{a}})=1$ if
$\underline{R}\in[\underline{w}]_{a}$, and
$\widehat{p}_{\underline{R}}^{(n)}(\mathbb{1}_{[\underline{w}]_{a}})=0$
otherwise.

  2. (2)

If $a\leq 0\leq b$,
$\widehat{p}_{\underline{R}}^{(n)}(\mathbb{1}_{[\underline{w}]_{a}})=\mathbb{1}_{\sigma^{a}[w_{a},\ldots,w_{0}]}(\underline{R})\cdot\widehat{p}_{\underline{R}}^{(n)}(\mathbb{1}_{[w_{0},...,w_{b}]})$
(and in particular, this equals $0$ if $w_{0}\neq R$).

  3. (3)

Assume $\underline{w}=R,w_{1},...,w_{b}$, then
$[\underline{w}]_{a}=[\underline{w}]$. If $n>b$,

$\displaystyle\widehat{p}_{\underline{R}}^{(n+m)}(\mathbb{1}_{[\underline{w}]})=$
$\displaystyle\sum_{\widetilde{\sigma}_{R}^{m}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{m}(\underline{S})}\left(\sum_{\widetilde{\sigma}_{R}^{n}\underline{Q}=\underline{S}}e^{\widetilde{\phi}_{n}(\underline{Q})}\delta_{\sigma^{-n-m}\underline{Q}^{\pm}}(\mathbb{1}_{[\underline{w}]})\right)$
$\displaystyle=$
$\displaystyle\sum_{\widetilde{\sigma}_{R}^{m}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{m}(\underline{S})}\left(\sum_{\widetilde{\sigma}_{R}^{n}\underline{Q}=\underline{S}}e^{\widetilde{\phi}_{n}(\underline{Q})}\delta_{\sigma^{-n}\underline{Q}^{\pm}}(\mathbb{1}_{[\underline{w}]}\circ\sigma^{-m})\right)$
$\displaystyle=$
$\displaystyle\sum_{\widetilde{\sigma}_{R}^{m}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{m}(\underline{S})}\widehat{p}_{\underline{S}}^{(n)}(\mathbb{1}_{[\underline{w}]}\circ\sigma^{-m})=\sum_{\widetilde{\sigma}_{R}^{m}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{m}(\underline{S})}\left(\widehat{p}_{\underline{S}}^{(n)}\circ\sigma^{m}\right)(\mathbb{1}_{[\underline{w}]}).$
(4)

Notice that (3) holds regardless of the choice of
$\underline{S}^{\pm},\underline{Q}^{\pm}$.

Step 3: Let $\underline{w}=(R,w_{1},...,w_{b-1},w_{b})$, $b\geq 1$, $l\geq 1$.
For all $n=b+l>b$:

$\widehat{p}^{(n)}_{\underline{R}}(\mathbb{1}_{[\underline{w}]})=\sum_{\widetilde{\sigma}_{R}^{b}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{b}(\underline{S})}\left(\widehat{p}_{\underline{S}}^{(l)}\circ\sigma^{b}\right)(\mathbb{1}_{[\underline{w}]})=\sum_{\widetilde{\sigma}_{R}^{b}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{b}(\underline{S})}\mathbb{1}_{[\underline{w}]}(\underline{S}).$
(5)

This is a constant sequence for all $n>b$.555Having a consistent explicit
formula for the limit measure of shifted cylinders allows us to use it as a
definition and apply the Carathéodory extension theorem; however we choose to
present the argument which uses weak-$*$ limits for future computations.
Therefore, since the cylinders (and their shfits) generate the Borel sigma
algebra, and
$\widehat{p}_{\underline{R}}^{(n_{k})}\xrightarrow[k\rightarrow\infty]{\text{weak-}*}\widehat{p}_{\underline{R}}$,
we get that
$\widehat{p}_{\underline{R}}^{(n)}\xrightarrow[n\rightarrow\infty]{\text{weak-}*}\widehat{p}_{\underline{R}}$;
and the limit is independent of the choice of
$\underline{S}^{\pm}\in\tau^{-1}[\\{\underline{S}\\}]\cap\widetilde{\Sigma}$
for a chain $\underline{S}\in\widetilde{\Sigma}_{L}$ in (3). In addition it
follows that, $\forall n\geq 0$,

$\widehat{p}_{\underline{R}}\circ\sigma^{-n}=\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}=\underline{R}}e^{\widetilde{\phi}_{n}(\underline{S})}\widehat{p}_{\underline{S}}.$
(6)

∎

This construction could a-priori depend on the choice of $\psi$; and in the
case that $\phi$ is transient, $\psi$ may not be unique. We will later show
applications in setups which imply the recurrence of $\phi$, and thus remove
the need to choose $\psi$.

######  Corollary 5.2 (Absolute Continuity).

Let $\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be a weakly Hölder
continuous potential s.t $P_{G}(\phi)<\infty$, where $\widetilde{\Sigma}_{L}$
is an irreducible locally compact one-sided topological Markov shift. Let
$\psi$ be a $\phi$-harmonic function on $\widetilde{\Sigma}_{L}$. Let
$\\{\widehat{p}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
be the measures constructed in Theorem 5.1. Let
$[R]\subseteq\widetilde{\Sigma}_{L}$, let
$\underline{R},\widetilde{\underline{R}}\in\widetilde{\Sigma}_{L}\cap[R]$, and
let

$\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}:\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq
0\\}\rightarrow\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=\widetilde{R}_{i},\forall
i\leq 0\\}$

be the holonomy map $\underline{v}\mapsto(\widetilde{R}_{i})_{i\leq
0}\cdot(v_{i})_{i\geq 0}$, where $\cdot$ denotes an admissible concatenation.
Then $\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}$ is a
continuous and invertible map s.t
$\widehat{p}_{\underline{R}}\circ\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}^{-1}\sim\widehat{p}_{\widetilde{\underline{R}}}$.

###### Proof.

Let $\psi$ be the $\phi$-harmonic function which is used in Theorem 5.1, and
write
$\widetilde{\phi}:=\phi+\log\psi-\log\psi\circ\widetilde{\sigma}_{R}-P_{G}(\phi)$
and
$\widetilde{\phi}_{n}:=\sum_{k=0}^{n-1}\widetilde{\phi}\circ\widetilde{\sigma}_{R}^{k}$.
By Lemma 4.7 by the local compactness of $\widetilde{\Sigma}_{L}$, the weak
Hölder continuity of $\phi$, implies that $\widetilde{\phi}$ is weakly Hölder
continuous.

Let $[\underline{w}]_{a}:=\sigma^{a}[w_{a},w_{a+1},...,w_{b-1},w_{b}]$, where
$a,b\in\mathbb{Z}$ and $a\leq b$, and write
$g:=\mathbb{1}_{[\underline{w}]_{a}}$. Let $j\geq 0$, and fix an extension
$\underline{S}^{\pm,(j)}\in\tau^{-1}[\\{\underline{S}\\}]$ for each
$\underline{S}\in\widetilde{\sigma}_{R}^{-j}[\\{\widetilde{\underline{R}}\\}]$,
where $\tau:\widetilde{\Sigma}\rightarrow\widetilde{\Sigma}_{L}$ is the
projection to the non-positive coordinates. Let
$\\{\widehat{p}_{\underline{R}}\\}_{\widetilde{\Sigma}_{L}}$ be the measures
constructed in Theorem 5.1. By steps 2 and 3 in Theorem 5.1,

$\widehat{p}_{\widetilde{\underline{R}}}(g)=\lim_{j\rightarrow\infty}\sum_{\widetilde{\sigma}_{R}^{j}\underline{S}=\widetilde{\underline{R}}}e^{\widetilde{\phi}_{j}(\underline{S})}\delta_{\sigma^{-j}\underline{S}^{\pm,(j)}}(g).$
(7)

Moreover, by step 3 in Theorem 5.1, this limit is independent of the choice of
$\\{\underline{S}^{\pm,(j)}\\}_{j\geq 0}$. Thus, similarly,

$\widehat{p}_{\underline{R}}(g\circ\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}})=\lim_{j\rightarrow\infty}\sum_{\widetilde{\sigma}_{R}^{j}\widetilde{\underline{S}}=\underline{R}}e^{\widetilde{\phi}_{j}(\widetilde{\underline{S}})}\delta_{\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}\left(\sigma^{-j}\underline{S}^{\pm,(j)}\right)}(g),$
(8)

where $\forall j\geq 0$,
$\forall\widetilde{\underline{S}}\in\widetilde{\sigma}_{R}^{-j}[\\{\underline{R}\\}]$,
$\tau\left(\sigma^{j}\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}\left(\sigma^{-j}\underline{S}^{\pm,(j)}\right)\right)=\widetilde{\underline{S}}$.

Let $C>0,\gamma\in(0,1)$ be the Hölder constant and Hölder exponent of
$\widetilde{\phi}$, respectively. $\forall j\geq 0$,
$\forall\underline{S}\in\widetilde{\sigma}_{R}^{-j}[\\{\widetilde{\underline{R}}\\}]$,
write
$\widetilde{\underline{S}}:=\tau\left(\sigma^{j}\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}\left(\sigma^{-j}\underline{S}^{\pm,(j)}\right)\right)\in\widetilde{\sigma}_{R}^{-j}[\\{\underline{R}\\}]$.
Write $d(\underline{R},\widetilde{\underline{R}})=e^{-n}$, $n\geq 1$. Then,

$e^{\widetilde{\phi}_{j}(\underline{S})}=e^{\pm C\sum_{k\geq
0}\gamma^{k+n}}\cdot e^{\widetilde{\phi}_{j}(\widetilde{\underline{S}})}.$

Write $\widetilde{C}_{\phi}:=\exp(C\cdot\sum_{k\geq
0}\gamma^{k})\in(0,\infty)$. Then,

$\forall h\in
C(\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=\widetilde{R}_{i},\forall i\leq
0\\}),\text{
}\widehat{p}_{\widetilde{\underline{R}}}(h)=\left(\widetilde{C}_{\phi}\right)^{\pm\gamma^{n}}\cdot\left(\widehat{p}_{\underline{R}}\circ\widehat{\Gamma}_{\underline{R}\,\widetilde{\underline{R}}}^{-1}\right)(h).$
(9)

∎

Remark: The proof of Corollary 5.1 shows something a little stronger than
absolute continuity, it shows that
$\\{\widehat{p}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
is a continuous family with a uniform modulus of continuity for the
holonomies.

######  Definition 5.3.

Let $\widetilde{\Sigma}$ be a maximal irreducible component of
$\widehat{\Sigma}$, and let
$\\{\widehat{p}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
be the family of measures given by Theorem 5.1, when
$P_{G}(\phi|_{\widetilde{\Sigma}_{L}})<\infty$. Define
$\forall\underline{R}\in\widetilde{\Sigma}_{L}$,
$\widehat{\mu}_{\underline{R}}:=\psi(\underline{R})\cdot\widehat{p}_{\underline{R}}$,
where $\psi$ is the $\phi$-harmonic function on $\widetilde{\Sigma}_{L}$ which
is fixed in Theorem 5.1.

######  Corollary 5.4.

The family of measures
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
from Definition 5.3 satisfies:

$\widehat{\mu}_{\underline{R}}\circ\sigma^{-1}=e^{-P_{G}(\phi)}\sum\limits_{\widetilde{\sigma}_{R}\underline{S}=\underline{R}}e^{\phi(\underline{S})}\widehat{\mu}_{\underline{S}}$.
In addition,
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
is an absolutely continuous family of measures.

The invariance follows from Definition 5.3 and from (6). The absolute
continuity follows from Corollary 5.2.

######  Corollary 5.5.

When $\phi$ is recurrent, the family
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
is the unique (up to scaling) continuous family of measures which satisfies
$\widehat{\mu}_{\underline{R}}\circ\sigma^{-1}=e^{-P_{G}(\phi)}\sum_{\widetilde{\sigma}_{R}\underline{S}=\underline{R}}e^{\phi(\underline{S})}\widehat{\mu}_{\underline{S}}$.

###### Proof.

Let $p$ and $\psi$ be version of the (unique up to scaling) $\phi$-conformal
measure and $\phi$-harmonic function on $\widetilde{\Sigma}_{L}$,
respectively. Then,
$\widehat{\mu}=\int\widehat{\mu}_{\underline{R}}dp(\underline{R})$ is a
$\sigma$-invariant measure on $\widetilde{\Sigma}$. By [Sar01, Theorem 1], the
recurrence of $\phi$ implies that $p$ is conservative and finite on cylinders.
Thus, $\widehat{\mu}$ must be conservative as well. Fix
$[R]\subseteq\widetilde{\Sigma}_{L}$, and let $[R,R_{-n+2},...,R_{-1},R]$ be
an admissible cylinder. Then,

$\displaystyle\widehat{\mu}([R,R_{-n+2},...,R_{-1},R])=\int\widehat{\mu}_{\underline{R}}([R,R_{-n+2},...,R_{-1},R])dp\propto(\psi\cdot
p)([R,R_{-n+2},...,R_{-1},R]).$ (10)

This determines $\widehat{\mu}_{\underline{R}}$ for $p$-a.e $\underline{R}$
(up to a scaling constant), and since $p$ gives a positive volume to every
cylinder ([Sar99, Sar01]), the continuity of the family
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
determines the family uniquely. ∎

######  Definition 5.6 (Leaf Measures).

Let $\varphi$ be a Grassmann-Hölder continuous potential, and let
$\widetilde{\Sigma}_{L}$ be a maximal irreducible component of
$\widehat{\Sigma}_{L}$, and let $\underline{R}\in\widetilde{\Sigma}_{L}$. The
corresponding leaf measure is
$\mu_{\underline{R}}:=\widehat{\mu}_{\underline{R}}\circ\widehat{\pi}^{-1}$,
where $\widehat{\mu}_{\underline{R}}$ is as in Definition 5.3.

######  Theorem 5.7.

Let $\varphi:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ be a Grassmann-Hölder
continuous potential. Let $H_{\chi}(p)$ be an ergodic homoclinic class of a
periodic and $\chi$-hyperbolic point $p$. Then there exists a corresponding
family of leaf measures which is $\varphi$-invariant, and is absolutely
continuous, that is,

  1. (1)

Leaf measures: $\forall\underline{R}\in\widetilde{\Sigma}_{L}$,
$\mu_{\underline{R}}$ is carried by
$\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall i\leq
0\\}]\subseteq V^{u}(\underline{R})$.

  2. (2)

$\varphi$-invariance up to a bounded coboundary:
$\forall\underline{R}\in\widetilde{\Sigma}_{L}$,

$\mu_{\underline{R}}\circ
f^{-1}=e^{-P_{H_{\chi}(p)}(\varphi)}\cdot\sum_{\widetilde{\sigma}_{R}\underline{S}=\underline{R}}e^{A-A\circ
f^{-1}}\cdot e^{\varphi}\cdot\mu_{\underline{S}},$

where $A:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ is bounded and given by
Theorem 4.4.

  3. (3)

Absolute continuity: for any cylinder $[R]\subseteq\widetilde{\Sigma}_{L}$,
and for any two chains
$\underline{R},\underline{S}\in\widetilde{\Sigma}_{L}\cap[R]$, let

$\Gamma_{\underline{R}\,\underline{S}}:\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq
0\\}]\rightarrow\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=S_{i},\forall
i\leq 0\\}]$

be the holonomy map along the stable leaves of
$\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall i\leq
0\\}]$ (it is well-defined on the rectangle $\widehat{\pi}[[R]]$). Then
$\left(\mu_{\underline{R}}\circ\Gamma_{\underline{R}\,\underline{S}}^{-1}\right)(g)=C_{\phi}^{\pm
1}\cdot\mu_{\underline{S}}(g)$ for every $g\in C(V^{u}(\underline{S}))$, where
$C_{\phi}>0$ is a global constant depending only on $\phi$ (and a fixed
corresponding harmonic function).

$\mathcal{F}_{H_{\chi}(p)}(\varphi):=\\{\mu_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
is a $\varphi$-invariant family of leaf measures (see Definition 2.16).

###### Proof.

Let $A,\phi:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ be given by Theorem 4.4
s.t $\varphi=\varphi^{*}+A-A\circ f^{-1}$, where $A$ is bounded and
$\phi=\varphi^{*}\circ\widehat{\pi}:\widehat{\Sigma}\rightarrow\mathbb{R}$ is
well defined and weakly Hölder continuous by the remark after Theorem 4.4; and
$\phi(\underline{R})$ depends only on the negative coordinates of
$\underline{R}$.

Let $\widetilde{\Sigma}$ be a maximal irreducible component of
$\widehat{\Sigma}$ s.t $\widehat{\pi}[\widetilde{\Sigma}^{\\#}]=H_{\chi}(p)$
modulo all conservative measures (see Proposition 3.16). Let
$[R]\subseteq\widetilde{\Sigma}_{L}$, and let
$\underline{R},\underline{S}\in[R]\cap\widetilde{\Sigma}_{L}$.

Since $\varphi$ is bounded, so is $\phi$. W.l.o.g
$P_{G}(\phi|_{\widetilde{\Sigma}_{L}})=0$, otherwise replace $\phi$ by $\phi-
P_{G}(\phi|_{\widetilde{\Sigma}_{L}})$. By the spectral decomposition, we may
assume w.l.o.g that $(\widetilde{\Sigma},\sigma)$ is topologically mixing (see
[Sar15, Theorem 2.5]). Thus, by the variational principle see [Sar09, Theorem
4.4] (and the fact that $\widehat{\pi}|_{\widetilde{\Sigma}^{\\#}}$ is finite-
to-one which allows to lift probabilities while preserving the entropy666Every
invariant probability measure can be decomposed into its ergodic components,
and each component lifts to an invariant probability measure. Then this lift
is $N$-to-one for some $N\in\mathbb{N}$ since the cardinality of fibers is an
invariant function. Finite extensions preserve entropy, and entropy is
affine.), $P_{G}(\phi|_{\widetilde{\Sigma}})=P_{H_{\chi}(p)}(\varphi)<\infty$.

  1. (1)

$\displaystyle\mu_{\underline{R}}(\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq 0\\}])=$
$\displaystyle\widehat{\mu}_{\underline{R}}\circ\widehat{\pi}^{-1}(\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq 0\\}])$ $\displaystyle\geq$
$\displaystyle\widehat{\mu}_{\underline{R}}(\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq 0\\})=\widehat{\mu}_{\underline{R}}(1)=\mu_{\underline{R}}(1).$

So $\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq 0\\}]\subseteq V^{u}(\underline{R})$ has full measure.

  2. (2)

By the relation $\widehat{\pi}\circ\sigma=f\circ\widehat{\pi}$ and Corollary
5.4,

$\displaystyle\mu_{\underline{R}}\circ
f^{-1}=\widehat{\mu}_{\underline{R}}\circ\widehat{\pi}^{-1}\circ
f^{-1}=\left(\widehat{\mu}_{\underline{R}}\circ\sigma^{-1}\right)\circ\widehat{\pi}^{-1}=\left(\sum_{\widetilde{\sigma}_{R}\underline{Q}=\underline{R}}e^{\phi(\underline{Q})}\widehat{\mu}_{\underline{Q}}\right)\circ\widehat{\pi}^{-1}=\sum_{\widetilde{\sigma}_{R}\underline{Q}=\underline{R}}e^{\phi(\underline{Q})}\mu_{\underline{Q}}.$

For a continuous test function $g\in C(V^{u}(\underline{Q}))$,

$\displaystyle
e^{\phi}\cdot\mu_{\underline{Q}}(g)=\int\limits_{\begin{subarray}{c}\underline{Q}^{\prime}\in\widetilde{\Sigma}:\\\
Q_{i}=Q_{i}^{\prime},\forall i\leq
0\end{subarray}}e^{\varphi^{*}\circ\widehat{\pi}}g\circ\widehat{\pi}d\widehat{\mu}_{\underline{Q}}=\int\limits_{\begin{subarray}{c}\underline{Q}^{\prime}\in\widetilde{\Sigma}:\\\
Q_{i}=Q_{i}^{\prime},\forall i\leq
0\end{subarray}}\left(e^{\varphi^{*}}g\right)\circ\widehat{\pi}d\widehat{\mu}_{\underline{Q}}=\left(e^{\varphi+A-A\circ
f^{-1}}\cdot\mu_{\underline{Q}}\right)(g).$

This completes the $\varphi$-invariance up to a bounded coboundary.

  3. (3)

Let
$\Gamma_{\underline{R}\,\underline{S}}:\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq
0\\}]\rightarrow\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=S_{i},\forall
i\leq 0\\}]$ be the holonomy map along the stable leaves of points in
$\widehat{\pi}[\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall i\leq
0\\}]$. Let
$\widehat{\Gamma}_{\underline{R}\,\underline{S}}:\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=R_{i},\forall
i\leq 0\\}\rightarrow\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=S_{i},\forall
i\leq 0\\}$ be a continuous and invertible map s.t
$\underline{v}\mapsto(S_{i})_{i\leq 0}\cdot(v_{i})_{i\geq 0}$, where $\cdot$
denotes an admissible concatenation. Let $g\in C(V^{u}(\underline{S}))$, and
write $\widehat{g}:=g\circ\widehat{\pi}\in
C(\\{\underline{v}\in\widetilde{\Sigma}:v_{i}=S_{i},\forall i\leq 0\\})$. It
follows that
$\Gamma_{\underline{R}\,\underline{S}}\circ\widehat{\pi}=\widehat{\pi}\circ\widehat{\Gamma}_{\underline{R}\,\underline{S}}$,
and so
$\left(\mu_{\underline{R}}\circ\Gamma_{\underline{R}\,\underline{S}}^{-1}\right)(g)=\left(\widehat{\mu}_{\underline{R}}\circ\widehat{\Gamma}_{\underline{R}\,\underline{S}}^{-1}\right)(\widehat{g})=\psi(\underline{R})\cdot\left(\widehat{p}_{\underline{R}}\circ\widehat{\Gamma}_{\underline{R}\,\underline{S}}^{-1}\right)(\widehat{g})$
and
$\mu_{\underline{S}}(g)=\psi(\underline{S})\cdot\widehat{p}_{\underline{S}}(\widehat{g})$.
By Lemma 4.7, $\mathrm{Var}_{1}(\log\psi)<\infty$. Thus, by (9), we are done.

∎

Remark: Absolute continuity implies a local product structure. Other related
important properties of hyperbolic equilibrium states of Hölder continuous
potentials are true, such as being Bernoulli up to a period. This has been
proven when $\dim M=2$ in [Sar11] and for the setup of countable Markov shifts
in [Dao13]. Sarig’s proof extends, as it only uses the fact that the
equilibrium state can be coded as an equilibrium state on a countable Markov
shift, of a weakly Hölder continuous potential. In [Dao13] Daon relaxes the
assumption on the regularity of the potential to the Walters property.

##  6\. Proofs of Main Results

###  6.1. Local Equilibrium States and the Leaf Condition

######  Theorem 6.1.

Let $M$ be a compact Riemannian manifold without boundary and of dimension
$d\geq 2$. Fix $\chi>0$ and $\epsilon=\epsilon_{\chi}$ as in Lemma 2.6. Let
$f\in\mathrm{Diff}^{1+\beta}(M)$, $\beta>0$, let $\chi>0$, and let
$\varphi:M\rightarrow\mathbb{R}$ be a Grassmann-Hölder continuous potential.
Let $p$ be a $\chi$-hyperbolic periodic point. Then, there exists a
$\chi$-hyperbolic local equilibrium state of $\varphi$ on $H_{\chi}(p)$ if and
only if the following two conditions hold:

  1. (1)

$\sum_{n\geq 1}\sum_{f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{l}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}=\infty,$

where $\Lambda_{l}$ is a level set s.t $p\in\Lambda_{l\cdot
e^{-2\sqrt[3]{\epsilon}}}$; and

  2. (2)

there exists a leaf measure $\mu\in\mathcal{F}_{H_{\chi}(p)}(\varphi)$ s.t

$\mu(\mathrm{RWT}_{\chi}^{\mathrm{PR}})>0,$

where $\mathcal{F}_{H_{\chi}(p)}(\varphi)$ is any $\varphi$-invariant family
of measures given by Theorem 5.7.

If a $\chi$-hyperbolic local equilibrium state of $\varphi$ exists on
$H_{\chi}(p)$, then its conditional measures on unstable leaves are
proportional to the members of the $\varphi$-invariant family of measures; and
it is unique.

###### Proof.

The “only if” direction is straightforward: Let $\mu$ be a $\chi$-hyperbolic
local equilibrium state of $\varphi$ on $H_{\chi}(p)$. Let
$\widetilde{\Sigma}$ be a maximal irreducible component which lifts all
conservative measures on $H_{\chi}(p)$ (see Proposition 3.16). Lift $\mu$ to
an ergodic, $\sigma$-invariant, probability measure on $\widetilde{\Sigma}$,
$\widehat{\mu}$, s.t $\widehat{\mu}\circ\widehat{\pi}^{-1}=\mu$, and
$h_{\widehat{\mu}}(\sigma)+\int\varphi\circ\widehat{\pi}d\widehat{\mu}=h_{\mu}(f)+\int\varphi
d\mu$. This is possible by choosing a suitable ergodic component of
$\int\frac{1}{|N_{x}|}\sum_{\underline{v}\in
N_{x}}\delta_{\underline{v}}d\mu(x)$, where
$N_{x}:=\widehat{\pi}^{-1}[\\{x\\}]\cap\widehat{\Sigma}^{\\#}$. Let
$\phi:\widetilde{\Sigma}_{L}\rightarrow\mathbb{R}$ be given by Theorem 4.4 and
the remark which follows it, and let
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$
be given by Definition 5.3.

By the spectral decomposition (see [Sar15, Theorem 2.5]), we may assume
w.l.o.g that $(\widetilde{\Sigma}_{L},\widetilde{\sigma}_{R})$ is
topologically mixing. It follows that

$\displaystyle h_{\widehat{\mu}}(\sigma)+\int\phi d\widehat{\mu}=$
$\displaystyle\sup\\{h_{\widehat{\nu}}(\sigma)+\int\phi
d\widehat{\nu}:\widehat{\nu}\text{ is an invariant probability measure on
}\widetilde{\Sigma}\\}$ $\displaystyle=$
$\displaystyle\sup\\{h_{\widehat{\nu}\circ\tau^{-1}}(\widetilde{\sigma}_{R})+\int\phi
d\widehat{\nu}\circ\tau^{-1}:\widehat{\nu}\text{ is an invariant probability
measure on }\widetilde{\Sigma}\\},$

where $\tau:\widetilde{\Sigma}\rightarrow\widetilde{\Sigma}_{L}$ is the
projection onto the non-positive coordinates. Every invariant probability
$\nu$ on $\widetilde{\Sigma}_{L}$ can be lifted to an invariant probability
measure on $\widetilde{\Sigma}$, $\widehat{\nu}$, s.t
$\widehat{\nu}\circ\tau^{-1}=\nu$ and
$h_{\nu}(\widetilde{\sigma}_{R})=h_{\widehat{\nu}}(\sigma)$. So,

$h_{\widehat{\mu}\circ\tau^{-1}}(\widetilde{\sigma}_{R})+\int\phi
d\widehat{\mu}\circ\tau^{-1}=\sup\\{h_{\nu}(\sigma)+\int\phi d\nu:\nu\text{ is
an invariant probability measure on }\widetilde{\Sigma}_{L}\\}.$

Therefore, by [BS03], $\phi$ is recurrent on $\widetilde{\Sigma}_{L}$. Thus,
by Lemma 4.8, $\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}f^{n}(q)=q,\\\ q\in
H_{\chi}(p)\cap\Lambda_{l}\end{subarray}}e^{\varphi_{n}(q)-n\cdot
P_{H_{\chi}(p)}(\varphi)}=\infty$.

By [Sar09, Theorem 4.5], $\widehat{\mu}\circ\tau^{-1}=\psi\cdot p$, where
$\psi$ is the $\phi$-harmonic function and $p$ is the $\phi$-conformal
measure, scaled s.t $\psi\cdot p(1)=1$, and $\psi,p$ are unique up to a
scaling. Then, by (10),
$\widehat{\mu}=\int_{\widetilde{\Sigma}_{L}}\widehat{\mu}_{\underline{R}}dp(\underline{R})$.
Thus, $\mu=\int\mu_{\underline{R}}dp$, where
$\\{\mu_{\underline{S}}\\}_{\underline{S}\in\widetilde{\Sigma}_{L}}$ is the
$\varphi$-invariant family of leaf measures as in Theorem 5.7. In addition,
$\mu$ is carried by $\mathrm{RWT}_{\chi}^{\mathrm{PR}}$ since it is an
invariant probability measure. This concludes the “only if” direction.

The “if” direction: $\phi$ is recurrent on $\widetilde{\Sigma}_{L}$ by Lemma
4.8. Let $p$ be a $\phi$-conformal measure carried by
$\widetilde{\Sigma}_{L}^{\\#}$, and let
$\\{\mu_{\underline{S}}\\}_{\underline{S}\in\widetilde{\Sigma}_{L}}$ be the
$\phi$-invariant family of leaf measures given by Theorem 5.7. By [Sar01,
Theorem 1], $p$ is ergodic and conservative. Let

$\mu:=\int_{\widetilde{\Sigma}_{L}}\mu_{\underline{R}}dp(\underline{R}).$

It follows that $\mu$ is an ergodic, $f$-invariant, and conservative measure
on $H_{\chi}(p)$, and that $\mu$ is finite on Pesin level sets. This measure
is finite if and only if $\phi$ is positive recurrent. For full details of the
proof of these properties, see [BO21, Theorem 6.2]. (In that proof $\phi$
denotes the geometric potential, but the proof is identical for general weakly
Hölder continuous functions).

Notice that $\mu(1)=p(\psi)$, where $\psi$ is the unique (up to scaling)
$\phi$-harmonic function on $\widetilde{\Sigma}_{L}$. By [Sar99, Theorem 8] if
$h_{\mathrm{top}}(\sigma),\infty$ then $\mu(1)<\infty$ if and only if
$\psi\cdot p$ is an equilibrium state of $\phi$ on $\widetilde{\Sigma}_{L}$
(and thus $\mu$ is a local $\chi$-hyperbolic equilibrium state of $\varphi$ on
$H_{\chi}(p)$).

By assumption $\exists\underline{R}\in\widetilde{\Sigma}_{L}$ s.t
$\mu_{\underline{R}}(\mathrm{RWT}_{\chi}^{\mathrm{PR}})>0$. By [BO21, Theorem
6.8], $\mu$ is finite. This concludes the “if” direction.

The proof shows that the local equilibrium state can always be disintegrated
into conditional measures on unstable leaves, where the conditional measures
are $\\{\mu_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$.

We proceed to show the uniqueness of the $\chi$-hyperbolic local equilibrium
state, when it exists. We saw that $\mu=\widehat{\mu}\circ\widehat{\pi}^{-1}$
where $\widehat{\mu}$ is an equilibrium state for $\phi$ on
$\widetilde{\Sigma}$. By [BS03], $\widehat{\mu}$ is unique and satisfies a
variational principle on $\widetilde{\Sigma}$. It follows that a
$\chi$-hyperbolic local equilibrium states is unique on $H_{\chi}(p)$. ∎

Remark: If $\exists p\in\Lambda_{\ell}$ s.t $\sum_{n\geq
1}\sum_{f^{n}(q)=q,q\in
H_{\chi}(p)\cap\Lambda_{\ell}}e^{\varphi_{n}(q)-nP_{H_{\chi}(p)}(\varphi)}=\infty$,
then $\sum_{n\geq 1}\sum_{f^{n}(q)=q,q\in H_{\chi}(p)\cap\Lambda_{\ell
e^{2\sqrt[3]{\epsilon}}}}e^{\varphi_{n}(q)-nP_{H_{\chi}(p)}(\varphi)}=\infty$.
Similarly, if $p\in\Lambda_{\ell e^{-2\sqrt[3]{\epsilon}}}$, then
$p\in\Lambda_{\ell}$. Then the condition in Theorem 6.1(2) can be relaxed so
$p\in\Lambda_{l}$.

###  6.2. Ledrappier-Young Property for Hyperbolic Equilibrium States

In the celebrated results of [LY85a, LY85b], Ledrappier and Young prove a
general formula for the entropy of invariant probability measures of
diffeomorphisms in terms of the local dimensions of their conditional
measures. One very important application of their theory, is the
characterization of SRB measures as those which satisfy Pesin’s entropy
formula, extending the previous same result of Ledrappier for hyperbolic SRB
measures [Led84]. That is, an ergodic and hyperbolic invariant probability
measure, $\mu$, which satisfies the following two properties must have
conditional measures on unstable local manifolds which are absolutely
continuous w.r.t the induced Riemannian volume measure: Let
$\varphi(x):=-\log\mathrm{Jac}(d_{x}f|_{H^{u}(x)}):\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$
(the geometric potential), then

  1. (1)

$h_{\mu}(f)+\int\varphi d\mu=\sup\Big{\\{}h_{\nu}(f)+\int\varphi
d\nu:\nu\text{ is an }\begin{array}[]{ll}\text{erg.}\\\ \text{hyp.}\\\
f\text{-inv.}\end{array}\text{prob.}\Big{\\}},$

where the geometric potential in the integrand is well defined almost
everywhere for every hyperbolic invariant probability measure,

  2. (2)

$\sup\left\\{h_{\nu}(f)+\int\varphi d\nu:\nu\text{ is an erg. hyp.
}f\text{-inv. prob.}\right\\}=0$,

Note that $P_{G}(\phi|_{\widetilde{\Sigma}_{L}})\leq 0$ is always true,
because of the Ruelle-Margulis inequality [Rue78]. The fact that SRB measures
satisfy (1),(2) is due to Ledrappier and Strelcyn [LS82]. A different way to
write this characterization, is the following: Let $\mu$ be an ergodic
hyperbolic invariant probability measure. Then $\mu$ is an SRB measure if and
only if,

  1. (1)

$\phi$ is positive recurrent when lifted to an irreducible component
$\widetilde{\Sigma}_{L}\subseteq\widehat{\Sigma}_{L}$ (see §4.2), where $\phi$
is given by Theorem 4.4 applied to $\varphi(x)$,

  2. (2)

$P_{G}(\phi|_{\widetilde{\Sigma}_{L}})=0$.

It follows that conditions (1) and (2) above are satisfied $\Leftrightarrow$
$\mu_{V^{u}}\ll m_{V^{u}}$, where $\mu_{V^{u}}$ is a conditional measure of
$\mu$ w.r.t a measurable partition of local unstable leaves $\\{V^{u}\\}$, and
$m_{V^{u}}$ is the induced Riemannian volume of $V^{u}$ (the conformal measure
of the geometric potential $\varphi$ on $V^{u}$).

In this section, we extend this result by replacing the geometric potential
$\varphi$ by a general potential which is Grassmann-Hölder continuous, and by
replacing the induced Riemannian volume $m_{V^{u}}$ by any $\varphi$-conformal
family. We obtain a new proof different to that of Ledrappier in the
hyperbolic case [Led84], and to that of Ledrappier and Young in the general
case [LY85a].

######  Theorem 6.2.

Let $M$ be a compact Riemannian manifold without boundary, and of dimension
$d\geq 2$. Let $f\in\mathrm{Diff}^{1+\beta}(M)$, $\beta>0$, let $\chi>0$ and
$\epsilon=\epsilon_{\chi}>0$ as in Lemma 2.6. Let
$\varphi:WT_{\chi}^{\epsilon}\rightarrow\mathbb{R}$ be a Grassmann-Hölder
continuous potential (recall Definition 2.10). Let $q$ be a $\chi$-hyperbolic
periodic point. Assume that $H_{\chi}(q)$ admits a (unique) $\chi$-hyperbolic
equilibrium state of $\varphi$, $\nu$. For any family of conditional measures
as in Theorem 6.1, $\mathcal{F}_{H_{\chi}(q)}(\varphi)$, there is a sub-family
$\mathcal{F}^{\prime}$ which disintegrates $\mu$ as in Theorem 6.1, and for
which the following holds: Fix $\mu\in\mathcal{F}^{\prime}$ which is carried
by a local unstable leaf $V^{u}$. Assume that $H_{\chi}(q)$ admits a
$\varphi$-conformal system of measures
$\mathcal{C}:=\\{m^{\varphi}_{W^{u}}:W^{u}\text{ is a local unstable leaf of
}H_{\chi}(q)\\}$. Then, $\mu\ll m^{\varphi}_{V^{u}}$. In particular,
$m_{V^{u}}^{\varphi}$ gives a positive measure to
$\mathrm{RWT}_{\chi}^{\mathrm{PR}}$.

###### Proof.

Let $\phi:\mathrm{RWT}_{\chi}\rightarrow\mathbb{R}$ be given by Theorem 4.4.
As in the proof of Theorem 5.7, let $\widetilde{\Sigma}$ be a maximal
irreducible component of $\widehat{\Sigma}$ s.t
$\widehat{\pi}[\widetilde{\Sigma}^{\\#}]=H_{\chi}(q)$ modulo conservative
measures; and write
$\mathcal{F}_{H_{\chi}(q)}(\varphi)=\\{\mu_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}}$.

Denote the lift of $\nu$ to $\widetilde{\Sigma}$ by $\widehat{\nu}$. As in the
proof of Theorem 6.1, $\phi$ is positive recurrent on $\widetilde{\Sigma}$ and
$\widehat{\nu}=\int_{\widetilde{\Sigma}_{L}}\widehat{\mu}_{\underline{R}}dp(\underline{R})$,
where $p$ is the unique (up to scaling) $\phi$-conformal measure on
$\widetilde{\Sigma}_{L}$. In [Sar99], Sarig showed that when $\phi$ is
positive recurrent, $p$ is conservative, and is carried by
$\widetilde{\Sigma}_{L}^{\\#}$. Set
$\mathcal{F}^{\prime}:=\\{\mu_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}_{L}^{\\#}}$.
Write $\mu=\mu_{\underline{R}}$, where
$\underline{R}\in\widetilde{\Sigma}_{L}^{\\#}$. Let $S\in\mathcal{R}$ be a
symbol which repeats infinitely often in $\underline{R}$.

Let $K\in\mathbb{N}$ be $N(R_{0})\cdot N(S)$ (see Definition 3.6). This is a
bound on the number of sequences in $\widehat{\Sigma}^{\\#}$ which code the
same point and s.t this point can be coded by a sequence which has $R_{0}$
repeat infinitely often in the future and $S$ repeat infinitely often in the
past (see [BO18, Theorem 1.3]). Choose $n\geq 1$ and
$\underline{S},\underline{Q}\in\widetilde{\Sigma}_{L}$ s.t
$\sigma_{R}^{n}\underline{S}=\sigma_{R}^{n}\underline{Q}=\underline{R}$ and
$S_{0}=Q_{0}=R_{0}$. Let
$\underline{S}^{\pm},\underline{Q}^{\pm}\in\widetilde{\Sigma}$ which return to
$[R_{0}]$ infinitely often in the future and s.t $S^{\pm}_{i}=S_{i},\forall
i\leq 0$ and $Q^{\pm}_{i}=Q_{i},\forall i\leq 0$. Define
$x:=\widehat{\pi}(\underline{S}^{\pm}),y:=\widehat{\pi}(\underline{Q}^{\pm}),z:=\widehat{\pi}([\underline{S}^{\pm},\underline{Q}^{\pm}]_{R_{0}})$,
where $[\underline{S}^{\pm},\underline{Q}^{\pm}]_{R}$ is the Smale bracket of
$\underline{S}^{\pm}$ and $\underline{Q}^{\pm}$ in $[R_{0}]$. If
$V^{u}(\underline{S})\cap V^{u}(\underline{Q})\neq\varnothing$, then by [BO18,
Proposition 3.12] $x=z=y$.

Then for each
$\underline{S}\in\widetilde{\sigma}_{R}^{-n}[\\{\underline{R}\\}]\cap[R_{0}]$,
the number of
$\underline{Q}\in\widetilde{\sigma}_{R}^{-n}[\\{\underline{R}\\}]\cap[R_{0}]$
s.t $V^{u}(\underline{S})\cap V^{u}(\underline{Q})\neq\varnothing$ is bounded
by $K$. That is,

$\\#\\{\underline{S}^{\prime}\in\widetilde{\Sigma}_{L}:S^{\prime}_{0}=R_{0},\sigma_{R}^{n}\underline{S}^{\prime}=\underline{R},V^{u}(\underline{S}^{\prime})\cap
V^{u}(\underline{S})\neq\varnothing\\}\leq K.$

Define on $V^{u}(\underline{R})$ the density function
$\rho_{n}:=\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}^{\prime}=\underline{R},S^{\prime}_{0}=R}\mathbb{1}_{V^{u}(\underline{S}^{\prime})}\circ
f^{n}$, then for any $n\geq 1$,

$0\leq\rho_{n}\leq K.$ (11)

Let
$c_{R_{0}}:=\min\\{\sup\\{m_{V^{u}(\underline{R}^{\prime})}^{\varphi}(1):\underline{R}^{\prime}\in[R_{0}]\\}^{-1},\inf\\{m_{V^{u}(\underline{R}^{\prime})}^{\varphi}(1):\underline{R}^{\prime}\in[R_{0}]\\}\\}$.
This is positive by the continuity of $\mathcal{C}$, since $[R_{0}]$ is
compact and $\underline{R}^{\prime}\mapsto V^{u}(\underline{R}^{\prime})$ is
continuous in $C^{1}$-norm (see Definition 2.15).

Let $g\in C(V^{u}(\underline{R}))$ s.t $g$ is $L$-Lipschitz, and
$\|g\|_{\infty}\leq 1$. Write $\widehat{g}:=g\circ\widehat{\pi}$, and so
$\mu_{\underline{R}}(g)=\widehat{\mu}_{\underline{R}}(\widehat{g})$. Let

$\widehat{\omega}_{n}:=\frac{1}{n}\sum_{k=0}^{n-1}\mathbb{1}_{[R_{0}]}\circ\sigma^{k}.$
(12)

Notice, $|\widehat{\omega}_{n}|\leq 1$. By Birkhoff’s ergodic theorem, and the
absolute continuity of
$\\{\widehat{\mu}_{\underline{R}}\\}_{\underline{R}\in\widetilde{\Sigma}}$
w.r.t holonomies (see Corollary 5.2), for $\widehat{\mu}_{\underline{R}}$-a.e
$\underline{R}^{\pm}$,
$\widehat{\omega}_{n}(\underline{R}^{\pm})\xrightarrow[n\rightarrow\infty]{}\widehat{\nu}([R_{0}])$,
and so
$\widehat{g}(\underline{R}^{\pm})\cdot\widehat{\omega}_{n}(\underline{R}^{\pm})\xrightarrow[n\rightarrow\infty]{}\widehat{\nu}([R_{0}])\cdot\widehat{g}(\underline{R}^{\pm})$.
Then, by Lebesgue’s dominated convergence theorem,

$\widehat{\mu}_{\underline{R}}(\widehat{g}\cdot\widehat{\omega}_{n})\xrightarrow[n\rightarrow\infty]{}\widehat{\nu}([R_{0}])\cdot\widehat{\mu}_{\underline{R}}(\widehat{g})=\widehat{\nu}([R_{0}])\cdot\mu_{\underline{R}}(g).$
(13)

Since $\widehat{\pi}$ is Hölder continuous, and $g$ is $L$-Lipschitz,
$\widehat{g}$ is Hölder continuous as well. Let $H_{L}>0,\theta\in(0,1)$ s.t
$d(\underline{R}^{\pm},\underline{S}^{\pm})\leq
e^{-n}\Rightarrow|\widehat{g}(\underline{R}^{\pm})-\widehat{g}(\underline{S}^{\pm})|\leq
H_{L}\cdot\theta^{n}$.

For every $n\geq 1$ and $\underline{S}\in\widetilde{\Sigma}_{L}$ s.t
$\sigma_{R}^{n}\underline{S}=\underline{R}$, fix
$\underline{S}^{\pm}\in\widetilde{\Sigma}$ s.t $S^{\pm}_{i}=S_{i}$ $\forall
i\leq 0$. By [BO18, Proposition 4.4],

$\mathrm{diam}_{V^{u}(\underline{R})}(f^{-n}[V^{u}(\underline{S})])\leq
4e^{-\frac{\chi\cdot n}{2}},$ (14)

where $\mathrm{diam}_{V^{u}(\underline{R})}$ denotes the diameter w.r.t the
induced Riemannian metric on $V^{u}(\underline{R})$.

Let $\underline{W}=(R_{0},W_{1},\ldots,W_{n-2},R_{0})$ be an admissible word
of length $n\geq 1$. We estimate
$\widehat{\mu}_{\underline{R}}([\underline{W}])$: First we write
$[\underline{W}]=\sigma^{-n}\sigma^{n}[\underline{W}]$. Next, by Corollary
5.5,

$\displaystyle\widehat{\mu}_{\underline{R}}([\underline{W}])=$
$\displaystyle\widehat{\mu}_{\underline{R}}(\sigma^{-n}\sigma^{n}[\underline{W}])=e^{-nP_{G}(\phi)}\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}=\underline{R}}e^{\phi_{n}(\underline{S})}\widehat{\mu}_{\underline{S}}(\sigma^{n}[\underline{W}])=e^{-nP_{G}(\phi)+\phi_{n}(\underline{R}\cdot\underline{W})}\widehat{\mu}_{\underline{R}\cdot\underline{W}}(1),$

where $\cdot$ denotes an admissible concatenation. Recall that
$\widehat{\mu}_{\underline{R}^{\prime}}(1)=\psi(\underline{R}^{\prime})$,
where $\psi$ is the unique (up to scaling) $\phi$-harmonic function on
$\widetilde{\Sigma}_{L}$. Then for
$C_{R_{0}}:=\max_{[R_{0}]}\\{\psi,\psi^{-1}\\}$,

$\widehat{\mu}_{\underline{R}}([\underline{W}])=C_{R_{0}}^{\pm 1}\cdot
e^{-nP_{G}(\phi)+\phi_{n}(\underline{R}\cdot\underline{W})}.$ (15)

Since $\mathcal{C}$ is a $\varphi$-conformal family, we get that for all
$n\geq 1$ and for every
$\underline{S}\in\widetilde{\sigma}_{R}^{-n}[\\{\underline{R}\\}]$,

$m^{\varphi}_{V^{u}(\underline{R})}(f^{-n}[V^{u}(\underline{S})])=\int
e^{\varphi_{n}(x)-nP_{H_{\chi}(q)}(\varphi)}dm^{\varphi}_{V^{u}(\underline{S})}.$

We wish to estimate $\varphi_{n}(\cdot)$ on $V^{u}(\underline{S})$. For that
we need the non-trivial fact that $d_{\mathrm{TM}}(f^{-i}(x),f^{-i}(y))$
decreases exponentially fast in $i\geq 0$. This is true since
$V^{u}(\underline{S})$ is contained in the graph of a function with a Hölder
continuous derivative (and $d(f^{-i}(x),f^{-i}(y))\leq
4e^{-\frac{\chi}{2}i}$), and the Hölder constant and exponent do not depend on
the choice of $\underline{S}\in[R_{0}]$. For more details see [BO18,
Definition 3.1]. Thus, since $\varphi$ is Grassmann-Hölder continuous, we get
that $\exists C_{\varphi}>0$ s.t $\forall x,y\in V^{u}(\underline{S})$,
$\sup_{n\geq 0}|\varphi_{n}(x)-\varphi_{n}(y)|\leq C_{\varphi}$. Using the
fact that $V^{u}(\underline{S})$ contains a point with a coding in
$\widetilde{\Sigma}^{\\#}$, together with Theorem 4.4, where
$\varphi=\phi+A-A\circ f^{-1}$ with $\|A\|_{\infty}<\infty$, we get for all
$n\geq 1$,

$m^{\varphi}_{V^{u}(\underline{R})}(f^{-n}[V^{u}(\underline{S})])=e^{\pm(C_{\varphi}+2\|A\|)}e^{\phi_{n}(\underline{S})-nP_{H_{\chi}(q)}(\varphi)}\cdot
m_{V^{u}(\underline{S})}(1)=(c_{R_{0}}^{-1}e^{C_{\varphi}+2\|A\|})^{\pm
1}e^{\phi_{n}(\underline{S})-nP_{G}(\phi)}.$ (16)

In the last equality we used the fact that
$P_{G}(\phi)=P_{H_{\chi}(q)}(\varphi)$ (recall (2)).

The last identity we need before the main computation of the proof is the
following: for any $n\geq 1$,

$(\mathbb{1}_{[R_{0}]}\circ\sigma^{n})\cdot\widehat{\mu}_{\underline{R}}=\sum_{\begin{subarray}{c}|\underline{W}|=n\\\
W_{0}=W_{n-1}=R_{0}\end{subarray}}\widehat{\mu}_{\underline{R}}|_{[\underline{W}]}.$
(17)

On the left-hand-side, $\mathbb{1}_{[R_{0}]}\circ\sigma^{n}$ acts as a density
for $\widehat{\mu}_{\underline{R}}$, and on the right-hand-side,
$\widehat{\mu}_{\underline{R}}|_{[\underline{W}]}$ is the restriction of the
measure to the respective cylinder.

We are now able to use the estimates and identities from (11), (14),(15),
(16), and (17) to get $\forall n\geq 1$,

$\displaystyle K\cdot m_{V^{u}(\underline{R})}^{\varphi}(g)\geq$
$\displaystyle(\rho_{n}\cdot m_{V^{u}(\underline{R})}^{\varphi})(g)\text{
}(\because\eqref{forVolLemma})$ $\displaystyle(\because g\text{ is
Lip},\eqref{weakercontraction})\text{ }=$ $\displaystyle\pm 4LK\cdot
e^{-\frac{\chi\cdot
n}{2}}m_{V^{u}(\underline{R})}^{\varphi}(1)+\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}=\underline{R},S_{0}=R_{0}}m_{V^{u}(\underline{R})}^{\varphi}(f^{-n}[V^{u}(\underline{S})])\cdot
g(f^{-n}\circ\widehat{\pi}(\underline{S}^{\pm}))$
$\displaystyle(\because\eqref{eqNoel3})\text{ }=$ $\displaystyle\pm 4LK\cdot
e^{-\frac{\chi\cdot
n}{2}}c_{R_{0}}^{-1}+(c_{R_{0}}^{-1}e^{C\varphi+2\|A\|})^{\pm
1}\sum_{\widetilde{\sigma}_{R}^{n}\underline{S}=\underline{R},S_{0}=R_{0}}e^{\phi_{n}(\underline{S})-n\cdot
P_{G}(\phi)}\cdot\widehat{g}(\sigma^{-n}(\underline{S}^{\pm}))$
$\displaystyle(\because\eqref{eqNoel})\text{ }=$ $\displaystyle\pm 4KL\cdot
c_{R_{0}}^{-1}e^{-\frac{\chi\cdot
n}{2}}+(c_{R_{0}}^{-1}e^{C\varphi+2\|A\|}C_{R_{0}})^{\pm
1}\sum_{|\underline{W}|=n,W_{n-1}=R_{0}}\widehat{\mu}_{\underline{R}}([\underline{W}])\cdot\widehat{g}(\sigma^{-n}(\underline{S}^{\pm}))$
$\displaystyle(\because\widehat{g}\text{ is Lip},\eqref{eqNoel2})\text{ }=$
$\displaystyle\pm 4LKc_{R_{0}}^{-1}\cdot e^{-\frac{\chi\cdot n}{2}}\pm
e^{C_{\varphi}+2\|A\|}c_{R_{0}}^{-1}C_{R_{0}}\cdot H_{L}\
\theta^{n}+(c_{R_{0}}^{-1}e^{C\varphi+2\|A\|}C_{R_{0}})^{\pm
1}\cdot\widehat{\mu}_{\underline{R}}(\mathbb{1}_{[R_{0}]}\circ\sigma^{n}\cdot\widehat{g}).$

Write $\theta_{1}:=\max\\{\theta,e^{-\frac{\chi}{2}}\\}\in(0,1)$,
$\widetilde{C}_{R_{0}}:=4e^{C_{\varphi}+2\|A\|}c_{R_{0}}^{-1}C_{R_{0}}\cdot\sum_{k\geq
0}\theta_{1}^{k}<\infty$. Thus, $\forall n\geq 1$,

$K\cdot m_{V^{u}(\underline{R})}^{\varphi}(g)\geq 2LK\cdot
H_{L}\cdot\widetilde{C}_{R_{0}}+\widetilde{C}_{R_{0}}^{\pm
1}\cdot\widehat{\mu}_{\underline{R}}(\mathbb{1}_{[R_{0}]}\circ\sigma^{n}\cdot\widehat{g}).$
(18)

By summing and averaging (18) $N$ times, and by (13) (recall the definition of
$\widehat{\omega}_{N}$ in (12)),

$\displaystyle K\cdot
m_{V^{u}(\underline{R})}^{\varphi}(g)\geq\frac{1}{N}\sum_{n=1}^{N}(\rho_{n}\cdot
m_{V^{u}(\underline{R})}^{\varphi})(g)=\pm\frac{1}{N}2LK\cdot
H_{L}\cdot\widetilde{C}_{R_{0}}+\widetilde{C}_{R_{0}}^{\pm
1}\cdot\widehat{\mu}_{\underline{R}}(\widehat{\omega}_{N}\widehat{g})\xrightarrow[N\rightarrow\infty]{}\widehat{\nu}([R_{0}])\widetilde{C}_{R_{0}}^{\pm
1}\mu_{\underline{R}}(g).$

Since $\mathrm{Lip}(V^{u}(\underline{R}))$ is dense in
$\|\cdot\|_{\infty}$-norm in $C(V^{u}(\underline{R}))$, by the Riesz-Kakutani-
Markov representation theorem,
$\mu_{\underline{R}}\leq\widehat{\nu}([R_{0}])^{-1}\cdot
K\cdot\widetilde{C}_{R_{0}}\cdot m_{V^{u}(\underline{R})}^{\varphi}$. ∎

Remarks:

  1. (1)

In fact, the proof shows that not only $\mu_{\underline{R}}\ll
m_{V^{u}(\underline{R})}^{\varphi}$, but that
$\mu_{\underline{R}}=(\widehat{\nu}([R_{0}])^{-1}\cdot
K\cdot\widetilde{C}_{R_{0}})^{\pm 1}\cdot
m_{V^{u}(\underline{R})}^{\varphi}|_{E_{\underline{R}}^{R_{0}}}$, where
$E_{\underline{R}}^{R_{0}}:=\widehat{\pi}[\\{\underline{R}^{\pm}\in\widetilde{\Sigma}:\forall
i\leq 0,R^{\pm}_{i}=R_{i}\text{ and }\\#\\{j\geq
0:R^{\pm}_{j}=R_{0}\\}=\infty\\}]$ carries $\mu_{\underline{R}}$.

  2. (2)

In addition, the proof shows that the $\varphi$-conformal family of measures
is unique up to equivalence of the leaf measures, when restricted to sets like
$E_{\underline{R}}^{R_{0}}$; and it works for a $\varphi$-conformal family
where the transformation law is not exactly $e^{\varphi_{n}-n\cdot
P_{H_{\chi}(p)}(\varphi)}$, but merely up to a multiplicative constant uniform
in $n$.

## Acknowledgements

I would like to thank the Eberly College of Science in the Pennsylvania State
University for excellent working conditions. I am grateful to the referee who
read this manuscript with care and who provided many helpful suggestions which
improved this manuscript and the presentation of the results in it. I would
also like to thank Prof. Omri Sarig for reading this manuscript and for his
wise and useful input.

## Special Notation

$\mathrm{WT}$ | weakly temperable points (Definition 2.4)
---|---
$\mathrm{RWT}$ | recurrently weakly temperable points (Definition 2.4, Definition 2.7)
$\Sigma$ | infinite-to-one Markov extension (Theorem 3.2)
$\widehat{\Sigma}$ | finite-to-one Markov extension (Definition 3.3)
$\mathcal{R}$ | Markov partiton (Definition 3.3)
$\widetilde{\Sigma}$ | maximal irreducible component of $\widehat{\Sigma}$ (Definition 3.14)
$\widehat{\Sigma}^{\circ}$ | itineraries of orbits in the Markov partition (Definition 3.18)
$X_{L}$, $X\in\\{\Sigma,\widehat{\Sigma},\widetilde{\Sigma},\Sigma^{\circ}\\}$ | $\\{(x_{i})_{i\leq 0}:(x_{i})_{i\in\mathbb{Z}}\in X\\}$
$X^{\\#}$, $X\in\\{\Sigma,\widehat{\Sigma},\widetilde{\Sigma}\\}$ | $\\{x\in X:\exists a,b:x_{i}=a,x_{-j}=b\text{ for infinitely many positive }i\text{ and positive }j\\}$
$X^{\\#}_{L}$, $X\in\\{\Sigma,\widehat{\Sigma},\widetilde{\Sigma}\\}$ | $\\{x\in X_{L}:\exists a:x_{-i}=a\text{ for infinitely many positive }i\\}$
$\sigma_{R}$ | the right-shift (Definition 3.17)
$\widetilde{\sigma}_{R}$ | the restriction of the right-shift to $\widetilde{\Sigma}_{L}$ (Definition 4.1)

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