# Geometric Structure and Ergodic Properties of Bony Multi-Graphs

M. Rabiee  F. H. Ghane  M. Zaj

###### Abstract

The main goal in this paper is to describe the geometric structure of
invariant graphs of a certain class of skew products. Our focus is on
attracting multi-graphs. An invariant multi-graph is an invariant compact set
which is a finite union of invariant graphs, and thus consists of a finite
number of points on each fiber. We introduce invariant bony multi-graphs and
construct an open set of skew products over an invertible base map (solenoid
map) having attracting invariant multi-graphs and bony multi-graphs which
support finitely many ergodic SRB measures. In this study some thermodynamic
properties are investigated. Finally, we extend our results to a family of
skew products over a generalized baker map.

Keywords: Skew product, Invariant graph, Bony multi-graph, Lyapunov exponent,
SRB measure, Topological pressure, Equilibrium state.

##  1 Introduction

The aim of this paper is to give a comprehensive description, from both
measure-theoretic and topological viewpoints, of the dynamics of skew product
systems that have either monotone interval maps. A skew product system is a
dynamical system $(\Theta\times\mathbb{X},F)$ of the form

$F:\Theta\times\mathbb{X}\to\Theta\times\mathbb{X},\quad(\theta,x)\mapsto(S(\theta),f_{\theta}(x)),$
(1.0.1)

that driven by a base map $S$ (which can be a solenoid map or a baker map).

In this context, describing the asymptotic behavior of the orbits and
understanding how this behaviour changes when the system is modified, are two
main targets. In the study of skew products, invariant graphs, particulary
attracting invariant graphs, play an essential role. In fact, they are the
natural substitutes of fixed points and considerably simplify the dynamics of
the forced systems. When skew product systems have uniformly contracting fiber
maps (hyperbolic setting), there exist invariant attracting sets for the
overall dynamics, which are the graph of continuous functions (see [16, 17]).
For the case that the fiber maps fail to be hyperbolic, we need to impose
specific conditions which guarantees the existence of an attracting invariant
graph that attracts orbits almost surely. This includes the skew product map
possesses a negative Lyapunov exponent in the fibre direction or it satisfies
the contraction on average condition [37, 38, 44].

Some related results on the ergodic properties and stability of attracting
graphs under deterministic perturbations have been proposed by Campbell and
Davies [8, 7]. Also, sufficient conditions for the existence and regularity of
invariant graphs has appeared in [37, 38, 40]. Stark discussed [37, 38] a
number of applications to the filtering of time series, synchronization and
quasiperiodically forced systems. In general, synchronization is the
phenomenon that different oscillations in coupled systems will converge to
oscillations that move with identical frequency. However, external forcing,
rather than coupling, can also synchronize dynamics. This phenomenon has been
widely studied in theoretical physics [31, 35] and also recently in
mathematics [18]. Another applications of invariant graphs in many branches of
nonlinear dynamics were also proposed (e.g. [5, 8, 10, 19, 30, 39] etc.).

For skew-products with monotone fiber maps there is a close relation between
(the existence of) invariant graphs, (fiber) Lyapunov exponents and ergodic
measures of $F$ (e.g. [20, 21, 44, 12] etc.). For instance, the stability of
an invariant graph is determined by its Lyapunov exponent, if it is negative
then the graph is attracting. In this literature, there are attracting
invariant graphs with more complicated dynamics. This includes bony graph
attractors which are currently object of intense study [23, 25, 44]. A bony
graph attractor is an attracting invariant graph which intersects almost every
fiber at a single point, and any other fiber at an interval which is called a
bone.

Here, a more general version of an invariant graph, the so-called invariant
multi-graph is considered. An invariant multi-graph is an invariant compact
set which is a finite union of invariant graphs, and thus consists of a finite
number of points on each fibre. In [22], Jäger and Keller provided a criteria,
in terms of Lyapunov exponents, for the existence of attracting invariant
multi-graphs. Gelfert and Oliveira [15] studied step skew-products over a
finite-state shift (base) space whose fiber maps are $C^{1}$ injective maps on
the unit interval. They provided certain invariant sets having a multi-graph
structure and can be written as graphs of one, two or more functions defined
on the base.

As an objective, we describe the geometrical structures of attracting
invariant multi-graphs for a certain class of skew product systems as defined
by (1.0.1). In particular, we construct robust invariant bony multi-graphs.

Other goal is the existence of SRB measures whose supports lie on invariant
graphs. SRB stands for Sinai, Ruelle and Bowen, are the invariant measures
most compatible with volume when volume is not preserved. References [23, 42,
44] contain further results about the existence and finiteness of SRB measures
supported on attracting invariant graphs. Another main objective is to show
that the attracting multi-graphs, in our setting, carry finitely many ergodic
SRB measures.

We also investigate some thermodynamic properties for these systems.
Thermodynamic formalism, that is the formalism of equilibrium statistical
physics, was adapted to the theory of dynamical systems in the works of Sinai,
Ruelle and Bowen [4, 34, 36]. Topological pressure, topological entropy and
equilibrium states are the fundamental tools in thermodynamic formalism. The
existence and uniqueness of equilibrium states are currently object of intense
study. Here, we will provide some sufficient conditions ensuring the existence
of equilibrium states supported on invariant multi-graphs. Our approach for
existence of equilibrium states is based on the technique applied in [32]. In
this scenario, we require the graph functions. In our setting, since the
contraction on the fibers is non-uniform, graph functions are only upper
semicontinuous. For existence of equilibrium states we apply a version of
variational principle provided by Rauch [33] for discontinuous potentials.

This work is organized as follows: First, in Sect. 2, we state some
terminology concerning invariant multi-graphs and recall conditions ensuring
the existence of multi-graphs from [21, 20, 22]. In Sect. 3, we construct
(Theorem 3.0.1, Theorem 3.2.2 and Corollary 3.2.1) an open set of skew
products given by (1.0.1) over an invertible base map (solenoid map) having
attracting invariant multi-graphs or bony multi-graphs that carry finitely
many ergodic SRB measures. Then we investigate some thermodynamic properties
in Sect. 4. Sufficient conditions ensuring the existence of equilibrium states
supported on invariant multi-graphs is presented (Proposition 4.0.3). Finally,
in Sect. 5, Theorem 5.0.1, we provide a measure theoretical isomorphism
between the skew products over the solenoid map and skew products over a
generalized baker map and deduce the existence of invariant bony multi-graphs
for these systems.

##  2 Preliminaries

In this section, we introduce the concepts and the notations which are basic
for the study in the following sections.

######  Definition 2.0.1 (SRB measure).

Let $X$ be a manifold and $f:X\longrightarrow X$ a continuous map. An
$f$-invariant measure $\mu$ is called $Sinai$-$Ruelle$-$Bowen$ (SRB) measure
(or a $physical$ measure) if there exists a mesurable set $E\subset X$, of
positive Lebesgue-measure, such that for every continuous function
$\varphi:X\longrightarrow\mathbb{R}$ and every $x\in E$ we have:

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{j=0}^{n}\varphi(f^{j}(x))=\int\limits_{X}\varphi
d\mu.$ (2.0.1)

The set $E$ is called the basin of $\mu$. In other words, time averages of all
continuous functions are given by the corresponding space averages computed
with respect to $\mu$, at least for a large set of initial states $x\in X.$

###  2.1 Skew products

Consider a skew product system $(\Theta\times\mathbb{X},F)$ of the form
(1.0.1), where the dynamics on the fibre space $\mathbb{X}$ may be interpreted
as being driven by another system $(\Theta,S)$ since the transformations
$f_{\theta}:\mathbb{X}\mapsto\mathbb{X}$ depend on $\theta$. On the other
hand, the space $\mathbb{X}$ is principally considered as the fibre space over
the basis dynamics $(\Theta,S)$, i.e. the fibre map $f_{\theta}$ can be
considered as a map from $\\{\theta\\}\times\mathbb{X}$ to
$\\{S\theta\\}\times\mathbb{X}$, where $\\{\theta\\}\times\mathbb{X}$ is the
fibre space over $\theta\in\Theta$. We adopt the usual notation
$F^{n}(\theta,x)=(S^{n}\theta,f_{\theta}^{n}(x))$, for the iterates $F^{n}$ of
$F$, where $f^{n}_{\theta}=f_{S^{n-1}(\theta)}\circ\cdots\circ f_{\theta}$.
Hence, $f_{\theta}^{n+k}(x)=f_{S^{k}\theta}^{n}(f^{k}_{\theta}(x))$. For $n=1$
and $k=-1$ this includes the identity and
$f_{\theta}^{-1}(x)=(f_{S^{-1}\theta})^{-1}(x)$.

In this article, we are interested in skew product dynamical systems $F$,
defined as below:

######  Definition 2.1.1.

$\mathcal{F}$ denotes the family of all skew product transformations $F$ given
by (1.0.1) on $\Theta\times\mathbb{I}$ having the following properties:

  1. (1)

The base $\Theta$ is equipped with a $\sigma$-algebra $\mathcal{B}$ and a
probability measure $m$, such that $(\Theta,\mathcal{B},m)$ becomes a
probability measurable space, and the base transformation $S:\Theta\to\Theta$
is a bi-measurable and ergodic measure-preserving bijection. In most
situations we will assume that $\Theta$ is a compact metric space, in which
case $\mathcal{B}$ is always the (completed) Borel $\sigma$-algebra and $S$ is
a homeomorphism.

  2. (2)

Let $\mathbb{I}=[0,1]$ and $\text{int}(\mathbb{I})=(0,1)$. The fibre maps
$f_{\theta}:\mathbb{I}\to\text{int}(\mathbb{I})$ are given by
$f_{\theta}(x)=\pi_{2}\circ F(\theta,x)$ with $\pi_{2}$ the natural projection
from $\Theta\times\mathbb{I}$ to $\mathbb{I}$. We will assume that the fiber
maps $f_{\theta}$ are $C^{2}$ increasing interval maps.

  3. (3)

For each $x\in\mathbb{I}$ the map $\theta\mapsto F(\theta,x)$ is measurable.
If $\Theta$ is a compact metric space and $S$ is a homeomorphism then the map
$\theta\mapsto F(\theta,x)$ is continuous.

######  Remark 2.1.1.

By definition, the dynamic contracts the whole space, i.e.
$F(\Theta\times\mathbb{I})\subset\Theta\times\text{int}(\mathbb{I})$.

We equip $\mathcal{F}$ with the following metric:

$\displaystyle\text{dist}_{\mathcal{F}}(F,G):=\sup\limits_{\theta\in\Theta}\text{dist}_{C^{2}}(f^{\pm
1}_{\theta},g^{\pm 1}_{\theta}),\ \ \text{for each}\ \ F,G\in\mathcal{F},$
(2.1.1)

where $f_{\theta}$ and $g_{\theta}$ are the fiber maps of $F$ and $G$
respectively.

###  2.2 Invariant graphs and Lyapunov exponents

Invariant graphs are fundamental objects in the study of skew product systems,
and they are of major interest.

######  Definition 2.2.1 (Invariant graph).

Let $F\in\mathcal{F}$ be a skew product as Definition 2.1.1. A measurable
function $\gamma:\Theta\to\mathbb{I}$ is called an invariant graph (with
respect to $F$) if for all $\theta\in\Theta$:

$F(\theta,\gamma(\theta))=(S\theta,\gamma(S\theta)),\text{ equivalently}\quad
f_{\theta}(\gamma(\theta))=\gamma(S\theta).$ (2.2.1)

The point set $\Gamma:=\\{(\theta,\gamma(\theta)):\theta\in\Theta\\}$ will be
called invariant graph as well, but it is labeled with the corresponding
capital letter. Denote by $\text{Cl}(\Gamma)$ the closure of $\Gamma$ in
$\Theta\times\mathbb{I}$.

Let $m$ be an invariant measure for the base map $S$. We say that
$\gamma:\Theta\to\mathbb{I}$ is an $(F,m)$ invariant graph if

$f_{\theta}(\gamma(\theta))=\gamma(S(\theta)),\ \text{for}\ m-a.e.\
\theta\in\Theta.$ (2.2.2)

In the sequel, we will denote by $\pi_{1}:\Theta\times\mathbb{I}\to\Theta$ and
$\pi_{2}:\Theta\times\mathbb{I}\to\mathbb{I}$ the canonical projections onto
the first and second coordinates, respectively. To any $(F,m)$-invariant graph
$\gamma$, an $F$-invariant measure $m_{\gamma}$ can be assigned by defining

$m_{\gamma}(A)=m(\pi_{1}(A\cap\Gamma))$ (2.2.3)

for any measurable set $A\subseteq\Theta\times\mathbb{I}$. Note that the
measure $m_{\gamma}$ is ergodic if and only if $m$ is ergodic.

Throughout the paper, the set of all $F$-invariant probability measures on
$\Theta\times\mathbb{I}$ is denoted by $\mathcal{M}(F)$, and the set of
$\mu\in\mathcal{M}(F)$ which project to $m$ by $\mathcal{M}_{m}(F)$.

The next theorem is a counterpart of Theorem 1.8.4 in [2] to our setting, see
also [20, Theorem 1.1] and Furstenberg Theorem [14]. It provides that there is
a one-to-one correspondence between invariant graphs and invariant ergodic
measures of skew products forced by monotone interval maps.

######  Theorem 2.2.1.

Suppose $F\in\mathcal{F}$ is a skew product as Definition 2.1.1,
$m\in\mathcal{M}(S)$ and $\mu\in\mathcal{M}_{m}(F)$ is ergodic. Then there
exists an $(F,m)$-invariant graph $\gamma$ such that $\mu=m_{\gamma}$.

In the investigation of skew product systems, attracting invariant graphs are
often useful characteristics, which we simply call attractors. Formally, we
introduce the following definition. Note that the effect of the attraction is
observed only in the fibre space $\mathbb{I}$.

######  Definition 2.2.2 (Attracting invariant graphs).

A point $(\theta,x)\in\Theta\times\mathbb{I}$ is attracting, if there is a
constant $\delta>0$ such that

$\lim_{n\to\infty}|{f}_{\theta}^{n}(x)-{f}_{\theta}^{n}(z)|=0,$

for all $z\in(x-\delta,x+\delta)$. Furthermore, an invariant graph $\gamma$ is
called an attracting invariant graph or attractor with respect to the
invariant probability $\mu$ if $\mu$-almost every point is attracting.

In a weaker form, we say that $A$ is an _attractor_ for $F$ (in the sense of
Milnor [28]) if there is a set of points in the phase space with positive
probability whose future orbits tend to $A$, as the number of iterates tends
to infinite. The set of orbits attracted to $A$ in the future is called its
_basin_ and denoted by $B(A)$.

######  Definition 2.2.3 (Maximal attractor).

Given a continuous skew product $F$, a compact set
$D\subset\Theta\times\mathbb{I}$ is trapping if $F(D)\subset int{D}$. Then the
closed $F$-invariant set

$A_{max}:=\bigcap\limits_{n\geq 0}F^{n}(D)$

is said to be a maximal attractor for $F$.

######  Definition 2.2.4 (Maximal Lyapunov exponent).

The maximal Lyapunov exponent of an $F$-invariant probability measure $\mu$ is
defined by

$\lambda(\mu,F)=\lim_{n\to\infty}\frac{1}{n}\int_{\Theta\times\mathbb{I}}\log|Df_{\theta}^{n}(x)|d\mu(\theta,x).$
(2.2.4)

The maximal Lyapunov exponent of an invariant graph $\gamma$ with respect to a
$S$-invariant probablity measure $m$ is defined as

$\lambda(m,\gamma)=\lim_{n\to\infty}\frac{1}{n}\int_{\Theta}\log|Df_{\theta}^{n}(x)|dm(\theta).$
(2.2.5)

Note that (2.2.5) is a special case of (2.2.4), with $\mu$ given by
$\mu(A)=m(\\{\theta\in\Theta:(\theta,\gamma(\theta))\in A\\}$.

######  Definition 2.2.5 (Upper Lyapunov exponent).

Following [12, 20], for each skew product $F$ given by Definition 2.1.1, the
_upper Lyapunov exponent of_ $(\theta,x)\in\Theta\times\mathbb{I}$ is

$\lambda_{max}(\theta,x)=\lim_{n\to\infty}\frac{1}{n}\log|Df_{\theta}^{n}(x)|,$
(2.2.6)

when the limit exists, where $Df_{\theta}(x)$ is the derivative of
$f_{\theta}$ in $x$.

Given any $F$-invariant probability measure $\mu$, we define the _upper
Lyapunov exponent of_ $\mu$ by

$\lambda_{max}(\mu)=\int\lambda_{max}(\theta,x)d\mu(\theta,x).$ (2.2.7)

Note that since $f_{\theta}^{n}$ is increasing interval maps,
$|Df_{\theta}^{n}(x)|=Df_{\theta}^{n}(x)$.

######  Definition 2.2.6 (Fiber Lyapunov exponent).

The (fiber) Lyapunov exponent of an $(F,m)$-invariant graph $\gamma$ is given
by

$\lambda_{m}(\gamma)=\int_{\Theta}\log Df_{\theta}(\gamma(\theta))dm(\theta).$
(2.2.8)

######  Remark 2.2.1.

Let $\gamma$ be an $(F,m)$-invariant graph with $\lambda_{m}(\gamma)<0$. Then,
by [20, Lemma 1.10], $\gamma$ is an attracting graph with respect to the
invariant probability $m_{\gamma}$.

Note that by the Birkhoff ergodic theorem

$\displaystyle\lambda_{max}(\theta,\gamma(\theta))$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\log
Df_{\theta}^{n}(\gamma(\theta))=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\log
Df_{S^{k}\theta}(f_{\theta}^{k}(\gamma(\theta)))$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\log
Df_{S^{k}\theta}(\gamma(S^{k}\theta))=\int_{\Theta}\log
Df_{\theta}(\gamma(\theta))dm(\theta)$ $\displaystyle=\lambda_{m}(\gamma)$

for $m$-a.e. $\theta\in\Theta$. So the average Lyapunov exponent of an
invariant graph equals its point-wise Lyapunov exponent for $m$-a.e.
$\theta\in\Theta$.

Here, we focus on skew products having the non-uniformly contraction rates
along the fiber. We address the situation where we only have information about
average rates of contraction. This is a weaker form of contraction which is a
necessary condition for synchronization, see [38]. When we talk about average
rates of contraction, we need an invariant measure. In our setting, take the
$S$-invariant measure $m$, as in Definition 2.1.1.

######  Definition 2.2.7 (Non-uniform contraction condition).

Suppose that the limit

$\lambda(\theta)=\lim_{n\to\infty}\frac{1}{n}\sup_{x,x^{\prime}\in
I}\log\frac{d(f_{\theta}^{n}(x),f_{\theta}^{n}(x^{\prime}))}{d(x,x^{\prime})}$
(2.2.9)

exists for almost every $\theta$ and is a measurable function of $\theta$
(this is the maximal Lyapunov exponent in the fibre direction of the skew
product $F$). We say that the skew-product $F$ contracts non-uniformly if
there exists $\lambda<0$ such that

$\lambda(\theta)\leq\lambda<0$ (2.2.10)

for a.e. $\theta\in\Theta$.

######  Remark 2.2.2.

In a special case, when $\mu$ is given by
$\mu(A)=m(\\{\theta\in\Theta:(\theta,\gamma(\theta))\in A\\}$, if the fiber
Lyapunov exponent of $m$ has negative upper bound then the skew product $F$
satisfies (2.2.10) and hence it contracts non-uniformly.

As a consequence of non-uniform contraction condition we have the following
result of Stark [38], which applies in a more general setting.

######  Theorem 2.2.2.

([38, Theorem 1.4]) Suppose $\Theta$ is a compact metric space,
$S:\Theta\to\Theta$ a homeomorphism, $m$ an invariant measure, $X$ a complete
metric space and the fiber map $f:\Theta\times X\to X$ is a continuous map
satisfying (2.2.10). Then there exists an $S$-invariant set
$\Lambda\subset\Theta$ such that $m(\Lambda)=1$ and a function
$\gamma:\Lambda\to X$ such that the graph of $\gamma$ is invariant and
attracting under $F$.

###  2.3 Multi graphs

We concentrate on the case that a compact invariant set is just a finite union
of invariant graphs, and thus consists of a finite number of points on each
fibre.

Let $F\in\mathcal{F}$ be a skew product over a base map $S$ as Definition
2.1.1.

######  Definition 2.3.1 (Multi-graph).

Following [15], given $F\in\mathcal{F}$, a multi-function
$\psi:D\subset\Theta\to\mathbb{I}$ is a relation that associates to every
point $\theta\in D$ a nonempty subset $\psi(\theta)\subset\mathbb{I}$. A
multi-function $\psi:D\subset\Theta\to\mathbb{I}$ is uniformly finite if there
exists $k\geq 1$ such that $\\#\psi(\theta)\leq k$ for all $\theta\in D$.
Given a uniformly finite multi-function $\psi$, we call the set
$\\{(\theta,\psi(\theta)):\theta\in D\\}$ a multi-graph in
$\Theta\times\mathbb{I}$.

Jäger proved that [20, Theorem 1.14] for skew products with $C^{1}$ interval
fiber maps having a minimal homeomorphism as a base, strict negativity of the
Lyapunov exponents on a compact invariant set implies that this set is a multi
graph. Then this extended to a more general case [22, Theorem 1.2].

Let $F\in\mathcal{F}$ be a skew product over a base map $S$ as Definition
2.1.1. Then the tuple $(\Theta,\mathcal{B},m,S)$ is a measure-preserving
dynamical system.

We say [22] $K\subset\Theta\times\mathbb{I}$ is a _random compact set_ if

  1. 1.

$K_{\theta}=\\{x\in\mathbb{I}:(\theta,x)\in K\\}$ is compact for all
$\theta\in\Theta$;

  2. 2.

the functions $\theta\mapsto d(x,K_{\theta})$ are measurable for all
$x\in\mathbb{I}$.

The set of all $f$-invariant probability measures $\mu$ which project to $m$
supported on a compact $F$-invariant set $K$ is denoted by
$\mathcal{M}_{m}^{K}(F)$.

######  Theorem 2.3.1.

[22, Theorem 1.2] Let $F\in\mathcal{F}$ be a skew product with base
$(\Theta,\mathcal{B},m,S)$, the family
$(x\mapsto\log\|Df_{\theta}^{k}(x)\|)_{\theta\in\Theta}$ is equicontinuous for
all $k\in\mathbb{N}$ and $K\subset\Theta\times\mathbb{I}$ is a random compact
set such that the maximal Lyapunov exponent $\lambda(\mu,F)$ given by (2.2.4)
is negative for all $\mu\in\mathcal{M}_{m}^{K}(F)$. Then there exists an
integer $n$ such that $\\#K_{\theta}=n$ for $m$-a.e. $\theta\in\Theta$.

If the base map $S$ is a homeomorphism on a compact metric space $\Theta$,
since fiber maps are $C^{2}$ and $F$ is continuous, the next corollary can be
followed immediately.

######  Corollary 2.3.1.

Suppose $\Theta$ is a compact metric space, $m$ an invariant ergodic measure
and $S:\Theta\to\Theta$ is an ergodic measure-preserving homeomorphism.
Assume, $F\in\mathcal{F}$ is a skew product as Definition 2.1.1 over the base
map $S$ and $K$ is a compact $F$-invariant set. Further, assume that for all
$\mu\in\mathcal{M}_{m}^{K}(F)$ we have $\lambda(\mu,F)<0$. Then there exists
an integer $n$ such that $\\#K_{\theta}=n$ for $m$-a.e. $\theta\in\Theta$. In
particular, if $n>1$ then $K$ is a multi-graph.

##  3 Bony multi-graphs for skew products with interval fiber maps

Our major goal here is to describe the structure of invariant graphs and study
the geometry of attractors, mainly, in the case that the basis dynamical
system is a solenoid map. In particular, we construct robust bony multi-
graphs. In our construction, first we provide a single skew product
$\widetilde{F}$ over an expanding circle map. Then, we consider its extension
$F$ which is a skew product over the solenoid map and show that every skew
product $G$ which is close enough to $F$ admits a bony multi-graph which
carries finitely many ergodic SRB measures.

We recall the family $\mathcal{F}$ of skew products given by Definition 2.1.1.
The next theorem is the main result of this section.

######  Theorem 3.0.1.

Given any $n\in\mathbb{N}$, there exists an open set
$\mathcal{U}\subset\mathcal{F}$ of skew products of the form (1.0.1) with
interval fiber maps having a solenoid map as the base map such that each skew
product system $G$ belonging to $\mathcal{U}$ admits an attracting invariant
multi-graph or bony multi-graph having exactly $n$ SRB measures supported in
$K(G)$.

The rest of this section is devoted to proving this theorem.

###  3.1 Skew products forced by expanding circle maps

If $X$ is a metric space and $f:X\longrightarrow X$ is a continuous map, then
we say that $f$ is _weakly contractive_ whenever for each $x,y\in X$ with
$x\neq y$, $d(f(x),f(y))<d(x,y)$.

######  Remark 3.1.1.

It is a well-known fact [21, Coro. 3] that if $f$ is weakly contractive and
$X$ is compact then there exists a unique fixed point $x\in X$ of the map $f$.
Furthermore, for every $y\in X$, $\lim_{k\to\infty}f^{k}(y)=x$ uniformly. Then
we say that $x$ is a _weak attracting fixed point_. Clearly if $f$ is a weakly
contractive map then

$d(f^{n}(y),f^{n}(z))\to 0,\ as,\ n\to\infty,$

for each $y,z\in X$.

######  Definition 3.1.1.

We say that the interval map $f$ is $s$-weakly contractive if it satisfies the
following conditions:

  1. $(1)$

$f$ is a $C^{1}$ weakly contractive map.

  2. $(2)$

Let $p$ be the unique fixed point of $f$. Then, $Df(p)=1$ and for all $x\neq
0$, one has $Df(x)<1$.

The following example shows that the set of $s$-weakly contractive maps is
nonempty.

######  Example 3.1.1.

Define $f:[0.1,0.7]\to[0.1,0.7]$ by $f(x)=3.098x^{1.83}-2.5x^{2.4}+0.1$. Then
$f$ is a weakly contractive map with the unique fixed point $0.466$. In
particular, $Df(0.466)=1$ and for all $x\neq 0.466$, we have $Df(x)<1$.

In [11], the author gave another examples of $s$-weakly contractive maps.

######  Definition 3.1.2 (Weak-pair).

Take two $C^{2}$ increasing interval maps $f_{i}$, $i=0,1,$ defined on an
interval $[a,b]$, $C^{2}$-close to the identity so that they fulfill the
following conditions:

  1. $(1)$

Each $f_{i}$, $i=0,1,$ is a $s$-weakly contractive map having a (unique) weak
attracting fixed point $p_{i}$.

  2. $(2)$

For every $x\in[a,b]$, the following “contraction on average property" hold:

$\sum_{i=0}^{1}\log Df_{i}(x)<0.$ (3.1.1)

  3. $(3)$

The weak attracting fixed points $p_{i}$, $i=0,1$, are pairwise disjoint.
Moreover, $p_{i}\neq a,b$ and $g_{i}(p_{j})\neq p_{i}$ for each $j\neq i$.

Let $p_{0}<p_{1}$ and $J=[p_{0},p_{1}]\subset[a,b]$. Then we say that
$(f_{0}|_{J},f_{1}|_{J})$ is a weak-pair for $J$.

######  Remark 3.1.2.

By conditions (1) and (3) the following “covering property" holds: there
exists the points $x_{0}$ and $x_{1}$ with $p_{0}<x_{0}<x_{1}<p_{1}$ such that
for the interval $B=(x_{0},x_{1})\subset\text{int}(J)$ one has:

$\forall x\in[x_{1},x_{2}],\ Df_{i}(x)<1\ \text{and}\ \text{Cl}(B)\subset
f_{0}(B)\cup f_{1}(B).$

Note that, covering property introduced in [29]. In Figure 1 below, the pairs
$(f_{0}|_{I_{1}},f_{1}|_{I_{1}})$ and $(f_{0}|_{I_{2}},f_{1}|_{I_{2}})$ are
weak-pairs.

Figure 1: Two increasing $C^{2}$ maps $f_{0}$ and $f_{1}$ with two weak-pairs

Fix $n\geq 1$ and consider $n$ disjoint subintervals of $\mathbb{I}=[0,1]$,
say $I_{1}=[a_{1},b_{1}],\ldots,I_{n}=[a_{n},b_{n}]$, with

$0<a_{1}<b_{1}<a_{2}<\ldots<a_{n}<b_{n}<1$

and two increasing $C^{2}$ maps $f_{0},f_{1}:\mathbb{I}\to\mathbb{I}$ such
that

  1. ($i$)

$f_{0}(I_{i})\subset I_{i},f_{1}(I_{i})\subset I_{i}\ \text{and}\
(f_{0}|_{I_{i}},f_{1}|_{I_{i}})\ \text{is \ a \ weak-pair \ for \ each}\
I_{i},\ i=1,\ldots,n$, hence there exist weak attracting fixed points
$p^{0}_{i}$ and $p^{1}_{i}$ of $f_{0}$ and $f_{1}$, respectively;

  2. ($ii$)

$f_{0}$ and $f_{1}$ have repelling fixed points $c_{i}$ and $d_{i}$,
$i=1,\ldots,n-1$, respectively, such that

$0<b_{1}<c_{1}<d_{1}<a_{2}<\ldots b_{n-1}<c_{n-1}<d_{n-1}<a_{n}<b_{n}<1.$

We assume that $a_{i}<p^{0}_{i}<p^{1}_{i}<b_{i}$. By Remark 3.1.2, there exist
points $x_{i}$ and $y_{i}$, $i=1,\ldots,n$, such that
$a_{i}<x_{i}<y_{i}<b_{i}$ and intervals
$B_{i}=(x_{i},y_{i})\subset\text{int}(I_{i})$ satisfies the following covering
property:

$\forall x\in[x_{i},y_{i}],Df_{i}(x)<1,\ \text{and}\ \text{Cl}(B_{i})\subset
f_{0}(B_{i})\cup f_{1}(B_{i}).$ (3.1.2)

Consider the circle expanding map $\omega:\mathbb{T}^{1}\to\mathbb{T}^{1}$,
$\omega(t)=4t$ $(mod\ 1)$. Let

$L_{i}\subset\mathbb{T}^{1},\ i=0,1,$

be disjoint closed arcs, with the length of each arc equal to $1/4$. Then for
$0\leq i\leq 1$ we have $\omega(L_{i})=\mathbb{T}^{1}$. Take a smooth map
$\ell:\mathbb{T}^{1}\to[0,1]$ such that $\ell|L_{0}\equiv 0$,
$\ell|L_{1}\equiv 1$ and outside the $\delta$-neighborhood of $L_{0}\cup
L_{1}$, for sufficiently small $\delta>0$, $\ell(t)\in(0,1)$.

Now, consider an isotopy

$\displaystyle f_{t}(x)=(1-(\ell(t)^{2}))f_{0}(x)+(\ell(t)^{2})f_{1}(x)$
(3.1.3)

between $f_{0}$ and $f_{1}$. Then, for each $t\in L_{0}$, $f_{t}=f_{0}$ and
for each $t\in L_{1}$, $f_{t}=f_{1}$. It is easy to see that

$\displaystyle\forall t\in\mathbb{T}^{1}\setminus B_{\delta}(L_{0}\cup L_{1})\
\text{and}\ \forall x\in I_{i},\ i=1,\cdots,n,\ \text{one has}\ Df_{t}(x)<1.$
(3.1.4)

We now define a skew product over the expanding circle map $\omega$,
corresponding to the fiber maps $f_{0}$ , $f_{1}$ by

$\displaystyle\widetilde{F}:\mathbb{T}^{1}\times I\to\mathbb{T}^{1}\times
I,\quad\widetilde{F}(t,x)=(\omega(t),f(t,x)),$ (3.1.5)

where $f(t,x):=f_{t}(x)$ given by (3.1.3). In the rest of this article we fix
the skew products $\widetilde{F}$ given by (3.1.5).

Clearly, by construction, for each $1\leq i\leq n$, the compact set
$D_{i}:=\mathbb{T}^{1}\times I_{i}$ is a trapping region for $\widetilde{F}$.
Let us take

$A_{max}(\widetilde{F}):=\bigcap_{n\geq
0}\widetilde{F}^{n}(\mathbb{T}^{1}\times\mathbb{I}),\ \text{and}\
\Lambda_{i}:=\bigcap_{n\geq 0}\widetilde{F}^{n}(\mathbb{T}^{1}\times I_{i}),\
1\leq i\leq n.$ (3.1.6)

Then $\Lambda_{i}$, $1\leq i\leq n$, are maximal attractors of $\widetilde{F}$
corresponding to the trapping regions $D_{i}$ and the union
$\bigcup_{i=1}^{n}\Lambda_{i}$ is a compact invariant set contained in the
global attractor $A_{max}(\widetilde{F})$.

###  3.2 Extension skew products and invariant graphs

Since the base map $\omega:\mathbb{T}^{1}\to\mathbb{T}^{1}$ is non-invertible,
hence we can always find an extension $S:\Omega\to\Omega$ that is invertible.
By extension we mean that there exists a surjective map
$\textbf{p}:\Omega\to\mathbb{T}^{1}$ such that, $\textbf{p}\circ
S=\omega\circ\textbf{p}$. To begin with, take $\Omega$ to be the set of all
pre-orbits of $\omega$, that is, all sequences $(t_{-n})_{n\leq 0}$ satisfying
$\omega(t_{-n})=t_{-n+1}$ for every $n>0$. Consider the map
$\textbf{p}:\Omega\to\mathbb{T}^{1}$ sending each sequence $(t_{-n})_{n\geq
0}$ to its term $t_{0}$ of order zero. Observe that
$\mathbf{p}(\Omega)=\mathbb{T}^{1}$. Finally, we define $S:\Omega\to\Omega$ by

$S(\dots,t_{-n},\dots,t_{0})=(\dots,t_{-n},\dots,t_{0},\omega(t_{0})).$

It is clear that $S$ is well defined and satisfies $\mathbf{p}\circ
S=\omega\circ\mathbf{p}$. The inverse limit space $\Omega$ is endowed with the
product topology, then it is easy to see that $S$ is a homeomorphism on
$\Omega$.

######  Remark 3.2.1.

On ergodic point of view, given an ergodic measure $\mu^{+}$ defined on Borel
subsets of $\mathbb{T}^{1}$ there exists a unique measure $\mu$ defined on
Borel subsets of $\Omega$ such that $\mathbf{p}_{\ast}\mu=\mu^{+}$. We mention
that since $\omega:\mathbb{T}^{1}\longrightarrow\mathbb{T}^{1}$ is a
$C^{2}$-expanding endomorphism then $\omega$ possesses an absolutely
continuous invariant ergodic measure $\nu^{+}$ whose density is bounded and
bounded away from zero. [27]. Moreover, $\nu^{+}$ is equivalent to Lebesgue.
Thus, $(\Omega,S)$ has an invariant ergodic measure $\nu$ inherited from the
invariant measure $\nu^{+}$ for $\omega$ on $\mathbb{T}^{1}$, i.e.
$\mathbf{p}_{\ast}\nu=\nu^{+}$.

For the skew products $\widetilde{F}$ given by (3.1.5), we define an extension
skew product map $F=F(\widetilde{F})$ on $\Omega\times\mathbb{I}$ by

$F(\textbf{t},x)=(S(\textbf{t}),f(\textbf{t},x))=(S(\textbf{t}),f_{t_{0}}(x)),$
(3.2.1)

for each $\textbf{t}=(\dots,t_{-n},\dots,t_{0})\in\Omega$, where $f_{t_{0}}$
is given by (3.1.3). Note that the inverse map $F^{-1}$ is given by

$F^{-1}(\mathbf{t},x)=(\dots,t_{-2},t_{-1},(f_{t_{-1}})^{-1}(x)).$

Then $\mathbf{p}\times id$ is a semi conjugacy between $\widetilde{F}$ and
$F$.

######  Remark 3.2.2.

By construction, the fiber maps of $\widetilde{F}$ and its extension $F$ are
the same.

In the rest of this paper, we fix the skew product $F$ with the fiber maps
$f_{\textbf{t}}$, $\textbf{t}\in\Omega$. By construction, for each $1\leq
i\leq n$, the compact set $\Omega\times I_{i}$ is a trapping region for $F$.
Let us take

$A_{max}(F):=\bigcap_{n\geq 0}F^{n}(\Omega\times\mathbb{I}),\ \text{and}\
\Delta_{i}:=\bigcap_{n\geq 0}F^{n}(\Omega\times I_{i}),\ 1\leq i\leq n.$
(3.2.2)

Then $\Delta_{i}$, $1\leq i\leq n$, are maximal attractors of $F$
corresponding to the trapping regions $\Omega\times I_{i}$ and the union
$K:=\bigcup_{i=1}^{n}\Delta_{i}$ is a compact invariant set contained in the
global attractor $A_{max}(F)$.

Let us take

$F_{i}:=F|_{\Omega\times I_{i}},\ i=1,\dots,n.$ (3.2.3)

######  Lemma 3.2.1.

The skew product $F_{i}$ satisfies the non-uniformly contraction condition
given by Definition 2.2.7 for each, $i=1,\dots,n$.

###### Proof.

By construction, for each $\textbf{t}\in\Omega$, the fiber map
$f_{\textbf{t}}$ of $F$ is a $C^{2}$ diffeomorphism. Hence, there exists a
$\varsigma\in L^{1}(\Omega,\nu)$ such that
$d(f_{\textbf{t}}(x),f_{\textbf{t}}(x^{\prime}))<\varsigma(\textbf{t})d(x,x^{\prime})$,
for $\nu$-almost every t. Then, by [38, Lem. 5.1], for all $m\geq 0$

$\varsigma_{m,i}(\textbf{t})=\sup_{x,x^{\prime}\in
I_{i}}\frac{d(f_{\textbf{t}}^{m}(x),f_{\textbf{t}}^{m}(x^{\prime}))}{d(x,x^{\prime})},\
i=1,\cdots,n$ (3.2.4)

exists for almost every t and it is measurable, where
$f^{m}_{\textbf{t}}=f_{S^{m-1}(\textbf{t})}\circ\ldots\circ
f_{S(\textbf{t})}\circ f_{\textbf{t}}$. Also, by [38, Lem. 5.3], the sequence
$\varsigma_{m,i}$ is submultiplicative. It immediately follows that
$\log\varsigma_{m,i}$ is subadditive, that is
$\log\varsigma_{m+k,i}(\textbf{t})\leq\log\varsigma_{m,i}(\textbf{t})+\log\varsigma_{k,i}(S^{m}(\textbf{t}))$.
We can thus apply the Kingman’s subadditive ergodic theorem [24, Thm. 5.4] to
$\log\varsigma_{m,i}$ to deduce that the limit

$\lambda_{i}(\textbf{t})=\lim_{m\to+\infty}\frac{1}{m}\log\varsigma_{m,i}(\textbf{t}),$
(3.2.5)

exists for $\nu$-almost every t and is $S$ invariant. Moreover,

$\lim_{m\to\infty}\frac{1}{m}\textnormal{log}\varsigma_{m,i}(\textbf{t})=\inf_{m}\frac{1}{m}\textnormal{log}\varsigma_{m,i}(\textbf{t})$
(3.2.6)

and it is constant by the ergodicity of $\nu$, denoted by $\lambda_{i}(\nu)$.
Note that, by definition, for each $t=(\dots,t_{-1},t_{0})\in\Omega$,

$f^{m}_{\textbf{t}}=f_{S^{m-1}(\textbf{t})}\circ\dots\circ
f_{S(\textbf{t})}\circ f_{\textbf{t}}=f_{\omega^{m-1}(t_{0})}\circ\dots\circ
f_{\omega(t_{0})}\circ f_{t_{0}}.$

On the other hand, by (3.1.4), for each
$\textbf{t}=(\ldots,t_{-1},t_{0})\in\Omega$ with
$t_{0}\in\mathbb{T}^{1}\setminus B_{\delta}(L_{0}\cup L_{1})$ and every $x\in
I_{i}$, we have $\|Df_{\textbf{t}}(x)\|<1$. It is known that [6] for Lebesgue
almost all $t_{0}$ the set $\\{\omega^{n}(t_{0}):n\in\mathbb{N}\\}$ is dense
in $\mathbb{T}^{1}$. Since, $\nu^{+}$ is equivalent to Lebesgue, the same
holds for $\nu^{+}$-almost all $t_{0}$. Hence, for $\nu^{+}$-almost all
$t_{0}$, there exists $n\in\mathbb{N}$ such that
$\omega^{n}(t_{0})\in\mathbb{T}^{1}\setminus B_{\delta}(L_{0}\cup L_{1})$.
This implies that $f_{\omega^{n}(t_{0})}$ is a uniformly contracting map. By
these facts, since $\nu$ inherited from $\nu^{+}$, there exists an upper bound
$\lambda<0$ such that $\lambda_{i}(\nu)\leq\lambda<0$ (note that for each $t$,
by definition, the fiber map $f_{t}$ is either weakly contractive or uniformly
contracting). Hence, $F_{i}$ satisfies the non-uniformly contraction condition
for each $i=1,\dots,n$. ∎

Given a small $\varepsilon>0$, take a $\varepsilon$-neighborhood $\mathcal{U}$
of $F$ with respect to the metric defined by (2.2.3). Taking $\varepsilon$
small enough such that for each $G\in\mathcal{U}$ and for all $i=1,\dots,n$,
the region $\Omega\times I_{i}$ is a trapping region for $G|_{\Omega\times
I_{i}}$. For each $G\in\mathcal{G}$, we take

$G_{i}:=G|_{\Omega\times I_{i}},\ i=1,\dots,n.$ (3.2.7)

######  Lemma 3.2.2.

Let the $\varepsilon$-neighborhood $\mathcal{U}$ of $F$ defined above be small
enough. Then for every perturbed skew product $G\in\mathcal{U}$, the
restricted skew products $G_{i}$, $i=1,\dots,n$, given by (3.2.7) satisfy the
non-uniformly contraction condition given by Definition 2.2.7.

###### Proof.

By Lemma 3.2.1, for each $i=1,\dots,n$, the skew product $F_{i}$ satisfies the
non-uniformly contraction condition. So, by (3.2.6), for $\nu$-almost t, the
limit
$\lim_{m\to\infty}\frac{1}{m}\textnormal{log}\varsigma_{m,i}(\textbf{t})=\inf_{m}\frac{1}{m}\textnormal{log}\varsigma_{m,i}(\textbf{t})$
exists and it is constant, by the ergodicity of $\nu$, denoted by
$\lambda_{i}(\nu)$; in particular, $\lambda_{i}(\nu)<0$. Take

$\lambda=\max\\{\lambda_{i}(\nu):i=1,\dots,n\\}.$ (3.2.8)

Clearly, $\lambda<0$. Take a skew product $G\in\mathcal{U}$. Let $G$ defined
by $G(\textbf{t},x)=(S(\textbf{t}),\textbf{g}(\textbf{t},x))$. Then, by
definition of the metric given by $(\ref{m})$,
$\sup_{\textbf{t}}dist_{C^{2}}(f_{\textbf{t}}^{\pm},g_{\textbf{t}}^{\pm})<\varepsilon$.
Take

$\zeta_{m,i}(\textbf{t}):=\sup_{x,x^{\prime}\in
I_{i}}\frac{d(g_{\textbf{t}}^{m}(x),g_{\textbf{t}}^{m}(x^{\prime}))}{d(x,x^{\prime})},$
(3.2.9)

for each $i=1,\dots,n$. Since, for each t, the fiber map $g_{\textbf{t}}$ is a
$C^{2}$ diffeomorphism, there exists a $\varsigma\in L^{1}(\Omega,\nu)$ such
that
$d(g_{\textbf{t}}(x),g_{\textbf{t}}(x^{\prime}))<\varsigma(\textbf{t})d(x,x^{\prime})$,
for $\nu$-almost every t. Then, by [38, Lem. 5.1], for all $m\geq 0$,
$\zeta_{m,i}(\textbf{t})$ exists for almost every t and it is measurable.
Also, by [38, Lem. 5.3], the sequence $\zeta_{m,i}$ is submultiplicative and
$\log\zeta_{m,i}$ is subadditive. By the Kingman’s subadditive ergodic theorem
[24, Thm. 5.4], the limit

$\lambda_{i}(\textbf{t},G)=\lim_{m\to+\infty}\frac{1}{m}\log\zeta_{m,i}(\textbf{t}),$
(3.2.10)

exists for $\nu$-almost every t and is $S$ invariant. Moreover,

$\lim_{m\to\infty}\frac{1}{m}\textnormal{log}\zeta_{m,i}(\textbf{t})=\inf_{m}\frac{1}{m}\textnormal{log}\zeta_{m,i}(\textbf{t})$
(3.2.11)

and it is constant by the ergodicity of $\nu$, denoted by
$\lambda_{i}(\nu,G)$. We need to see that this number is negative.

Fix small $\delta$ with $\lambda+\delta<0$. By (3.2.6) and (3.2.11),
$\lambda_{i}(\nu)=\inf_{n}\frac{1}{m}\textnormal{log}\varsigma_{m,i}(\textbf{t})$
and
$\lambda_{i}(\nu,G)=\inf_{m}\frac{1}{m}\textnormal{log}\zeta_{m,i}(\textbf{t})$,
for $\nu$-a.e. t. Thus, for a typical point t for $\lambda_{i}(\nu)$ and
$\lambda_{i}(\nu,G)$, there exists $m_{0}=m_{0}(\textbf{t})\in\mathbb{N}$ such
that
$\frac{1}{m_{0}}\textnormal{log}\varsigma_{m_{0},i}(\textbf{t})<\lambda+\delta$,
where $\lambda$ is given by (3.2.8). Taking $\varepsilon$ small enough, we get
$\frac{1}{m_{0}}\textnormal{log}\zeta_{m_{0},i}(\textbf{t})<\lambda+\delta<0$.
Consequently,

$\lambda_{i}(\nu,G)=\inf_{m}\frac{1}{m}\textnormal{log}\zeta_{m,i}(\textbf{t})<\frac{1}{m_{0}}\textnormal{log}\zeta_{m_{0},i}(\textbf{t})<\lambda+\delta<0.$

Thus $G_{i}$ satisfies the non-uniformly contraction condition. ∎

Note that, for each $1\leq i\leq n$, the compact set $\Omega\times I_{i}$ is a
trapping region for each $G\in\mathcal{U}$. Let us take

$A_{max}(G):=\bigcap_{n\geq 0}G^{n}(\Omega\times I),\ \text{and}\
\Delta_{i}(G):=\bigcap_{n\geq 0}G^{n}(\Omega\times I_{i}),\ 1\leq i\leq n.$
(3.2.12)

Then $\Delta_{i}(G)$, $1\leq i\leq n$, are maximal attractors corresponding to
the trapping regions $\Omega\times I_{i}$ and the union
$K(G):=\bigcup_{i=1}^{n}\Delta_{i}(G)$ is a compact invariant set contained in
the global attractor $A_{max}(G)$.

Applying Theorem 2.2.2 we get the next result (see also[7, Thm. 5], [38, Thm.
1.4] or [5, Pro. 2.3]).

######  Theorem 3.2.1.

Let $G\in\mathcal{U}$, where the open set $\mathcal{U}$ is given by Lemma
3.2.2. Then there exist $S$-invariant sets $\Omega_{i}(G)\subseteq\Omega$,
$i=1,\dots,n$, such that $\nu(\Omega_{i}(G))=1$ and measurable functions
$\gamma_{G,i}:\Omega_{i}(G)\to I_{i}$ such that $\Gamma_{G,i}$, the graphs of
$\gamma_{G,i}$, are invariant under $G$. Furthermore $\Gamma_{G,i}$,
$i=1,\dots,n$, are attracting $(G,\nu)$ invariant graphs.

For the skew product $F$ and any small perturbation $G\in\mathcal{U}$, we
define measures $\nu_{\gamma_{F,i}}$ and $\nu_{\gamma_{G,i}}$ on $\Omega\times
I_{i}$ by

$\nu_{\gamma_{F,i}}:=\nu\circ(id\times\gamma_{F,i})^{-1}|_{\Omega\times
I_{i}},\ \
\nu_{\gamma_{G,i}}:=\nu\circ(id\times\gamma_{G,i})^{-1}|_{\Omega\times
I_{i}},$ (3.2.13)

for each $i=1,\dots,n$. Since $\nu$ is ergodic, so they are also ergodic. In
particular, they are supported on the maximal attractors $\Delta_{i}(F)$ and
$\Delta_{i}(G)$, respectively.

Note that for the case $n=1$, i.e. we have only one invariant graph, you can
see more details of construction and the proofs in [44].

######  Definition 3.2.1 (Bony attracting graph and bony multi-graph).

Let $G$ be a skew product with interval fiber maps and an invertible base map
$S$.

  1. $(1)$

An attracting bony graph is an attracting invariant graph that is the union of
the graph of a continuous function $\gamma$ defined on some set of full
measure and a set of vertical closed intervals (bones) contained in the
closure of the graph, see [23, 44]. If the graph function $\gamma$ defined on
the whole space $\Omega$, then we call the graph of $\gamma$ a continuous
attracting invariant graph.

  2. $(2)$

A compact invariant set $K$ is a bony multi-graph for $G$ if $K$ is a multi-
graph composed of the finite union of attracting invariant graphs and at least
one of the invariant graphs contained in $K$ is an attracting bony graph.

In [44, Theorem 1], we investigated a certain class of skew products whose
fiber maps are increasing $C^{2}$-interval maps over the base map
$(\Omega,S)$. We proved the existence of an open set of skew products such
that any skew product $G$ belonging to this set admits a unique attracting
invariant graph for which the following dichotomy is ascertained. This
invariant graph is either a continuous attracting graph or an attracting bony
graph. In both cases they carry an SRB measure. Here, we extend the result to
the case that $G$ admits more than one attracting invariant graph.

The next result can be followed by applying [44, Theorem 1] to our setting by
any modification (see also [44, Lemma 2.6]).

######  Theorem 3.2.2.

Given a skew product system $G$ belonging to $\mathcal{U}$, the maximal
attractors $\Delta_{i}(G)$, $i=1,\dots,n$, defined by (3.2.12), satisfy the
following properties:

  1. $(1)$

The maximal attractor $\Delta_{i}(G)$ is either a continuous attracting
invariant graph or an attracting bony graph. In the case $\Delta_{i}(G)$ is an
attracting bony graph, the graph function $\gamma_{G,i}$ defined on a subset
$\Omega_{i}(G)\subseteq\Omega$ with total measure and there exists a family of
vertical closed intervals (bones), one bone in each fiber $\textbf{t}\times
I_{i}$, with $\textbf{t}\in\Omega\setminus\Omega_{i}(G)$; in particular, the
bones are contained in the closure of the graph $\Gamma_{G,i}$ and
$\text{Cl}(\Gamma_{G,i})=\Delta_{i}(G)$.

  2. $(2)$

The invariant ergodic measures $\nu_{\gamma_{G,i}}$, $i=1,\dots,n$, given by
(3.2.13), supported on the closure of $\Gamma_{G,i}$, are SRB measures.

However, weakly contractive fiber maps prevent the existence of bones in the
skew product, but bones appear in small perturbations of the original system.
By [44, Lemma 2.5], we get the next lemma. It shows that the set of
$G\in\mathcal{U}$ having an attracting bony graph is nonempty.

######  Lemma 3.2.3.

There exists a skew product $G\in\mathcal{U}$ and an index $1\leq i\leq n$
such that $\Delta_{i}(G)$ is the closure of an attracting bony graph. In
particular, the subset of bones has the cardinality of the continuum and is
dense in the attractor.

######  Lemma 3.2.4.

There exists an upper semicontinuous extension of the graph function
$\gamma_{\mathbb{G},i}:\Omega_{i}(G)\to I_{i}$ to the whole space $\Omega$.

###### Proof.

By Theorem 3.2.2, there exist $S$-invariant sets $\Omega_{i}(G)$ such that
$\nu(\Omega_{i}(G))=1$ and measurable function $\gamma_{G,i}:\Omega_{i}(G)\to
I_{i}$ such that $\Gamma_{G,i}$, the graph of $\gamma_{G,i}$, is invariant
under $G$.

Let $\textbf{t}\in\Omega$ and $\varepsilon>0$ be given. By Theorem 3.2.2,
$\text{Cl}(\Gamma_{G,i})=\Delta_{i}(G)$. Take
$I_{i,\textbf{t}}:=\\{\textbf{t}\\}\times I_{i}$ and
$\Delta_{G,i,\textbf{t}}:=\Delta_{i}(G)\cap I_{i,\textbf{t}}$. Then

$\Delta_{G,i,\textbf{t}}=\bigcap_{n\geq 0}g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ
g_{S^{-n}(\textbf{t})}(I_{i})=\lim_{n\to\infty}g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(I_{i}),\
i=1,\dots,n.$

Note that $g_{\textbf{t}}\circ g_{S^{-1}(\textbf{t})}\circ\dots\circ
g_{S^{-n}(\textbf{t})}(I_{i})$ is a sequence of nested intervals, and thus
$\Delta_{G,i,\textbf{t}}$ is either an interval or a single point. In
particular, if $\textbf{t}\in\Omega_{i}(G)$ then $\Delta_{G,i,\textbf{t}}$ is
a single point.

If $n$ is big enough then

$g_{\textbf{t}}\circ g_{S^{-1}(\textbf{t})}\circ\dots\circ
g_{S^{-n}(\textbf{t})}(I_{i})\subset
U_{\frac{\varepsilon}{2}}(\Delta_{G,i,\textbf{t}}),$

where $U_{\frac{\varepsilon}{2}}(\Delta_{G,i,\textbf{t}})$ is
$\frac{\varepsilon}{2}$-neighborhood of $\Delta_{G,i,\textbf{t}}$. Let
$\textbf{t}^{\prime}$ be sufficiently close to t. Then
$g_{\textbf{t}^{\prime}}\circ g_{S^{-1}(\textbf{t}^{\prime})}\circ\dots\circ
g_{S^{-n}(\textbf{t}^{\prime})}$ is $C^{2}$ close to $g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}$ and hence

$g_{\textbf{t}^{\prime}}\circ g_{S^{-1}(\textbf{t}^{\prime})}\circ\dots\circ
g_{S^{-n}(\textbf{t}^{\prime})}(I_{i})\subset
U_{\frac{\varepsilon}{2}}(\Delta_{G,i,\textbf{t}}).$

Then

$\Delta_{G,i,\textbf{t}^{\prime}}\subset g_{\textbf{t}^{\prime}}\circ
g_{S^{-1}(\textbf{t}^{\prime})}\circ\dots\circ
g_{S^{-n}(\textbf{t}^{\prime})}(I_{i})\subset
U_{\frac{\varepsilon}{2}}(g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(I_{i}))\subset
U_{\varepsilon}(\Delta_{G,i,\textbf{t}}).$ (3.2.14)

This implies the upper-semicontinuity of $\Delta_{G,i,\textbf{t}}$. This
semicontinuity will immediately imply the continuity of its graph part.

Indeed, by (3.2.14), we obtain that

$\text{diam}(\Delta_{G,i,\textbf{t}^{\prime}})\leq\text{diam}(\Delta_{G,i,\textbf{t}})+2\varepsilon.$

If $\Delta_{G,i,\textbf{t}}$ is a single point, then
$\text{diam}(\Delta_{G,i,\textbf{t}})=0$. Then, by this fact and (3.2.14),

$|\text{diam}(\Delta_{G,i,\textbf{t}})-\text{diam}(\Delta_{G,i,\textbf{t}^{\prime}})|=\text{diam}(\Delta_{G,i,\textbf{t}^{\prime}})\leq\text{diam}(\Delta_{G,i,\textbf{t}})+2\varepsilon\leq
2\varepsilon.$

This implies continuity at t.

Consider an extension of $\gamma_{G,i}:\Omega_{i}(G)\to I_{i}$ to the whole
space $\Omega$ as the following: for each $\textbf{t}\in\Omega$, take

$\gamma_{i}(\textbf{t})=\lim_{n\to\infty}g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(a_{i}),$ (3.2.15)

where $a_{i}$ is the left endpoint of $I_{i}$. By (3.2.14), if
$\textbf{t}\in\Omega_{i}(G)$, then $\lim_{n\to\infty},g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ
g_{S^{-n}(\textbf{t})}(I_{i})=\lim_{n\to\infty},g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(a_{i})$. This
fact implies that the map $\gamma_{i}$ as defined in (3.2.15) is an extension
of $\gamma_{G,i}$. We claim that $\gamma_{i}$ is upper semicontinuous.

Indeed, let $\gamma_{i}(\textbf{t})<c$, for some $c\in(0,1)$. Then, by
definition, $\lim_{n\to\infty}g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(a_{i})<c$. Hence,
there exists $n_{0}\in\mathbb{N}$ such that $g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}(a_{i})<c$, for
each $n\geq n_{0}$. Note that the fiber maps are increasing interval maps.
Take $\varepsilon>0$ small enough such that if an interval map $f$ is
$C^{2}$-$\varepsilon$-close to $g_{\textbf{t}}\circ
g_{S^{-1}(\textbf{t})}\circ\dots\circ g_{S^{-n}(\textbf{t})}$ then
$f(a_{0})<c$. Now take $\textbf{t}^{\prime}$ sufficiently close to t such that
$g_{\textbf{t}^{\prime}}\circ g_{S^{-1}(\textbf{t}^{\prime})}\circ\dots\circ
g_{S^{-n}(\textbf{t}^{\prime})}$ is $C^{2}$-$\varepsilon$-close to
$g_{\textbf{t}}\circ g_{S^{-1}(\textbf{t})}\circ\dots\circ
g_{S^{-n}(\textbf{t})}$. Then $g_{\textbf{t}^{\prime}}\circ
g_{S^{-1}(\textbf{t}^{\prime})}\circ\dots\circ
g_{S^{-n}(\textbf{t}^{\prime})}(a_{0})<c$. This fact implies the upper
semicontinuity of $\gamma_{i}$. ∎

By Corollary 2.3.1, Lemma 3.2.2 and Theorem 3.2.2, we get the next result.

######  Corollary 3.2.1.

Let $G\in\mathcal{U}$ and consider the compact invariant set
$K(G):=\bigcup_{i=1}^{n}\Delta_{i}(G)$, where $\Delta_{i}(G)$, $i=1,\dots,n$,
are given by (3.2.12). Then $K(G)$ is an invariant multi-graph or a bony
multi-graph. In particular, $K(G)$ carries $n$ ergodic SRB measures
$\nu_{\gamma_{G,i}}$, $i=1,\dots,n$, given by (3.2.13).

##  4 Thermodynamic properties of invariant graphs

Take an skew product $G\in\mathcal{U}$ given by Theorem 3.2.1. Here, we
discuss thermodynamic properties of invariant graphs $\Gamma_{G,i}$,
$i=1,\dots,n$, given by Theorem 3.2.2, and hence the bony multi-graph $K(G)$.

Let $(X,d)$ be a non-empty, compact metric space and $T:X\to X$ be a
continuous transformation. For each $n\in\mathbb{N}$ define for $x,y\in X$ the
metric

$d_{n}(x,y):=\max\\{d(T^{i}(x),T^{i}(y)):0\leq i<n\\}.$

Given some $\epsilon>0$, a subset $\emptyset\neq E\subseteq K\subseteq X$ is
called $(\epsilon,n)$-_separated_ in $K$, if

$\inf\\{d_{n}(x,y):x\neq y\in E\\}\geq\epsilon.$

In addition, $E\subseteq K$ is called _maximally_ $(\epsilon,n)$-_separated_
in $K$, if for all $z\in K$ the set $E\cup\\{z\\}$ is not
$(\epsilon,n)$-separated anymore. Using Zorn’s lemma, for every non-empty
subset $K\subset X$ there exists a maximally $(\epsilon,n)$-separated set
$E\subset K$.

In what follows, the set of all $T$-invariant probability measures is denoted
by $\mathcal{M}_{T}(X)$. Moreover, we denote by
$\mathcal{E}_{T}(X)\subseteq\mathcal{M}_{T}(X)$ the set of all $T$-invariant,
ergodic probability measures on $X$. Recall for $\mu\in\mathcal{M}_{T}(X)$, we
denote by $h_{\mu}(T)$ the measure-theoretic entropy of $T$ with respect to
$\mu$. For dynamical systems $(X,T)$ the quantity $h_{top}(T)$ denotes the
topological entropy of $(X,T)$, and one has

$h_{top}(T)=\sup\\{h_{\mu}(T):\mu\in\mathcal{M}_{T}(X)\\}.$

Let $(X,T)$ be a dynamical system and $\phi:X\to\mathbb{R}$ be an arbitrary
function. For every subset $K\subseteq X$, define

$P_{K}(T,\phi):=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup_{E}\sum_{x\in
E}\exp\sum_{i=0}^{n-1}\phi(T^{i}(x)),$

where the supremum is taken over all $(\epsilon,n)$-separated sets in $K$.

The variational principle for the topological pressure of continuous functions
was proven in [43]: One has

$P_{X}(T,\phi)=\sup\\{h_{\mu}(X)+\int_{X}\phi d\mu\\},$

where the supremum is taken over all ergodic $T$-invariant Borel probability
measures $\mu$ on $X$. Here $h_{\mu}(T)$ denotes the measure theoretic entropy
of $T$ with respect to $\mu$.

Take an skew product $G\in\mathcal{U}$ given by Theorem 3.2.1. We recall that
the base map $S$ is an extension of expanding circle map
$\omega:\mathbb{T}^{1}\to\mathbb{T}^{1}$, $\omega(t)=4t\ (mod1)$. In the
setting of uniformly expanding maps, equilibrium states always exist and they
are unique SRB measures if the potential is Hölder continuous and the dynamics
is topologically exact, see [41, Theorem 12.1]. Clearly $\omega$ is a
topologically exact uniformly expanding map and hence it admits a unique SRB
equilibrium measure $\nu^{+}_{\phi}$ for each Hölder continuous potential
$\phi$. Moreover, it is supported on the whole $\mathbb{T}^{1}$.

As we have seen in Subsection 4.2, there exists a semiconjugacy
$\textbf{p}:\Omega\to\mathbb{T}^{1}$ sending each sequence
$\textbf{t}=(\ldots,t_{-n},\ldots,t_{-1},t_{0})\in\Omega$ to its term $t_{0}$
of order zero. Given an ergodic measure $\mu^{+}$ defined on Borel subsets of
$\mathbb{T}^{1}$ there exists a unique measure $\mu$ defined on Borel subsets
of $\Omega$ such that $\mu^{+}=\textbf{p}_{\ast}\mu$.

Here, we investigate the relation between entropy and topological pressure of
the expanding circle map $\omega$ and its extension $S$. First, we need the
following results.

Assume $f:M\longrightarrow M$ and
$\tilde{f}:\tilde{M}\longrightarrow\tilde{M}$ are two continuous
transformations of compact topological spaces $M$ and $\tilde{M}$,
respectively. If $f$ is a topological factor of $\tilde{f}$, then
$h_{top}(f)\leq h_{top}(\tilde{f})$, see [41]. Also, Ledrappier-Walter’s
formula states the relation between metric entropy of
$\tilde{f}:\tilde{M}\to\tilde{M}$ and its topological factor $f:M\to M$:

######  Theorem 4.0.1.

$(\mathbf{Ledrappier-Walter^{\prime}s\ formula}).$[26] Let $\tilde{M}$ and $M$
be compact metric spaces and let
$\tilde{f}:\tilde{M}\longrightarrow\tilde{M}$, $f:M\longrightarrow M$ and
$\pi:\tilde{M}\longrightarrow M$ be continuous maps such that $\pi$ is
surjective and $\pi\circ\tilde{f}=f\circ\pi$, then

$\sup_{\tilde{\mu};\pi_{*}\tilde{\mu}=\mu}h_{\tilde{\mu}}(\tilde{f})=h_{\mu}(f)+\int
h_{top}(\tilde{f},\pi^{-1}(y))d\mu(y).$

######  Lemma 4.0.1.

The following two facts hold:

  1. $(1)$

For any probability measures $\mu$ and $\mu^{+}$ invariant under $S$ and
$\omega$, respectively, with $\mu^{+}=\textbf{p}_{\ast}\mu$, one has that
$h_{\mu^{+}}(\omega)=h_{\mu}(S)$.

  2. $(2)$

Let $\phi$ be a Hölder continuous potential for the expanding circle map
$\omega$. Then $P_{\omega}(\phi)=P_{S}(\psi)$, where
$\psi=\phi\circ\textbf{p}$.

###### Proof.

(1) Since

$\textbf{p}^{-1}(t)=\\{(\ldots,t_{-n},\ldots,t_{-1},t_{0})\in\Omega:t_{0}=t\\}$

we observe that $h_{top}(\omega,\textbf{p}^{-1}(t))=0$ for every
$t\in\mathbb{T}^{1}$ because we can choose a subset of $\textbf{p}^{-1}(t)$
with finite cardinality as $n$-generator for every $n\in\mathbb{N}$. Thus, we
apply the Ledrappier-Walter’s formula to conclude that
$h_{\mu^{+}}(\omega)=h_{\mu}(S)$, see Section 3.3 of [32] for more details.

(2) First we observe that if $\phi$ is Hölder continuous then
$\phi\circ\textbf{p}$ is also Hölder continuous. Since
$h_{top}(\omega,\textbf{p}^{-1}(t))=0$ for every $t\in\mathbb{T}^{1}$, by the
previous lemma and applying Ledrappier-Walter’s formula, we have

$\displaystyle P_{\omega}(\phi)\leq P_{S}(\psi)$ $\displaystyle=$
$\displaystyle\sup\\{h_{\mu}(S)+\int\psi d\mu\\}$ $\displaystyle=$
$\displaystyle\sup\\{h_{\mu^{+}}(\omega)+\int
h_{top}(\omega,\textbf{p}^{-1}(t))d\mu^{+}(t)+\int\phi d\mu^{+}\\}$
$\displaystyle=$ $\displaystyle P_{\omega}(\phi),$

where the supremum is taken over all probability measures $\mu$ invariant
under $S$. Note that given an invariant measure $\mu^{+}$ defined on Borel
subsets of $\mathbb{T}^{1}$ there exists a unique invariant measure $\mu$
defined on Borel subsets of $\Omega$ such that $\mu^{+}=\textbf{p}_{\ast}\mu$.
∎

By Theorem 3.2.1, there exists $S$-invariant sets
$\Omega_{i}(G)\subseteq\Omega$, $i=1,\dots,n$, such that
$\nu(\Omega_{i}(G))=1$ and measurable functions $\gamma_{G,i}:\Omega_{i}(G)\to
I_{i}$ such that $\Gamma_{G,i}$, the graphs of $\gamma_{G,i}$, are invariant
under $G$. Furthermore $\Gamma_{G,i}$, $i=1,\dots,n$, are attracting $(G,\nu)$
invariant graphs. Let us take the skew products

$G_{i}:=G|_{\Omega\times I_{i}},\
G_{i}(\textbf{t},x)=(S(\textbf{t}),g(\textbf{t},x)),\ 1\leq i\leq n.$ (4.0.1)

By Lemma 3.2.4, we extend measurable functions $\gamma_{G,i}$ to the whole
space $\Omega$ so that the extended functions are upper semicontinuous. We
also write the extended functions by the same notation $\gamma_{G,i}$.

Recently, a lot of attention has been done to extend the definition of
pressure to not necessarily continuous functions $\phi$, and to prove a
corresponding variational principle. A variational principle for sub-additive,
upper semi-continuous sequences of functions was established in [9] and [3].
This result was recently generalized in [13] for weighted topological pressure
on systems with upper semi-continuous entropy mapping. Then, Rauch [33]
extended the original definitions of pressure to discontinuous $\phi$, and
compare them to the classical ones. Furthermore he determined several classes
of functions, which admit variational inequalities and principles.

Following [33], we call $\phi$ to be _quasi-integrable with respect to_ $\mu$,
if either $\int_{X}\phi^{+}d\mu<\infty$ or $\int_{X}\phi^{-}d\mu<\infty$,
where $\phi^{+}:=\max(\phi,0)$ and $\phi^{+}:=\max(-\phi,0)$. The set of all
measurable $\phi:X\to\mathbb{R}$, which are quasi-integrable for all
$\mu\in\mathcal{M}_{T}(X)$, is defined by $Q_{T}(X)$. We call $\phi\in
Q_{T}(X)$ _quasi-integrable with respect to_ $T$. Let be $\phi\in Q_{T}(X)$.
The function $\phi$ is _upper semi-continuous with respect to_ $T$, if the
following holds: If $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ is a sequence of atomic
probability measures $\mu_{n}$ such that there exists a
$\mu\in\mathcal{M}_{T}(X)$ satisfying $\mu_{n}\to\mu$ in the weak∗-topology,
then

$\limsup_{n\to\infty}\int_{X}\phi d\mu_{n}\leq\int_{X}\phi d\mu.$

The set of all upper semi-continuous functions with respect to $T$ is denoted
by $U_{T}(X)\subseteq Q_{T}(X)$, see [33] for more details.

######  Theorem 4.0.2.

[33, Theorem C] Let $(X,T)$ be a dynamical system satisfying
$h_{top}(T)<\infty$. If $\phi:X\to\mathbb{R}$ is upper semi-continuous with
respect to $T$ (see the above definition), then one has

$P_{T}(X,\phi)=\sup\\{h_{\mu}(T)+\int_{X}\phi d\mu\\},$

where the supremum is taken over all $T$-invariant Borel probability measures
$\mu$ on $X$.

A function $\phi:X\to\mathbb{R}$ is called _upper semi-continuous_ , if
$\\{x\in X:\phi(x)<c\\}$ is an open set for every $c\in\mathbb{R}$. By
definition every upper semi-continuous function is also Borel measurable. We
denote the set of all upper semi-continuous functions $\phi:X\to\mathbb{R}$ by
$U(X)$. As $X$ is compact, every $\phi\in U(X)$ is bounded from above (see for
example [1] Theorem 2.43). This immediately yields $U(X)\subseteq Q_{T}(X)$.
In addition, by Proposition 6 of [33], one has $U(X)\subseteq U_{T}(X)$ for
every continuous mapping $T:X\to X$.

######  Corollary 4.0.1.

[33, Corollary 6] Let $h_{top}(T)<\infty$ and $\phi\in U(X)$. Then one has

$P_{T}(X,\phi)=\sup\\{h_{\mu}(T)+\int_{X}\phi d\mu\\},$

where the supremum is taken over all $T$-invariant Borel probability measures
$\mu$ on $X$.

Also, we recall the following proposition from [33].

######  Proposition 4.0.1.

Let $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ be a sequence of Borel probability
measures with limit measure $\mu$ in the weak∗ topology. Then one has for
$\varphi\in U(X)$

$\limsup_{n\to\infty}\int_{X}\varphi d\mu_{n}\leq\int_{X}\varphi d\mu.$

######  Proposition 4.0.2.

Consider the skew products $G_{i}=G|_{\Omega\times I_{i}}$, $1\leq i\leq n$,
given by (4.0.1) and let $\phi_{i}:\Omega\times I_{i}\to\mathbb{R}$ be a
Hölder continuous potential. Take $\psi_{i}:\Omega\to\mathbb{R}$ defined by
$\psi_{i}(\textbf{t})=\phi_{i}(\textbf{t},\gamma_{G,i}(\textbf{t}))$. Then
there exist some equilibrium state $\mu_{\psi_{i}}$ for $(S,\psi_{i})$.

###### Proof.

By Lemma 3.2.4, $\psi_{i}$ is upper semicontinuous. By Corollary 4.0.1, one
has that

$P_{S}(\Omega,\psi_{i})=\sup\\{h_{\mu}(S)+\int_{\Omega}\psi_{i}d\mu\\},$

where the supremum is taken over all $S$-invariant Borel probability measures
$\mu$ on $\Omega$. Since the expanding circle map $\omega$ is a $C^{2}$ local
diffeomorphism then the natural extension $S$ is locally Lipschitz continuous,
i.e., given $\textbf{t}\in\Omega$ there exists a neighborhood $V_{\textbf{t}}$
such that for every $\textbf{t}_{1},\textbf{t}_{2}\in S(V_{\textbf{t}})$ we
have

$d_{\Omega}(S^{-1}(\textbf{t}_{1}),S^{-1}(\textbf{t}_{2}))\leq\sigma(\textbf{t})d_{\Omega}(\textbf{t}_{1},\textbf{t}_{2})$

where $\sigma(\textbf{t})=\|D\omega^{-1}\|\circ\textbf{p}(\textbf{t})$. In
particular, $S$ is a Ruelle expanding map and hence it is expansive. By
Corollary 9.2.17 of [41], the entropy function of an expansive transformation
in a compact metric space is upper semi-continuous. By Proposition 4.0.1 and
upper semicontinuity of $\psi_{i}$, the function
$\mu\mapsto\int_{\Omega}\psi_{i}d\mu$ is also upper semicontinuous. By these
facts the map $\mu\mapsto h_{\mu}(S)+\int_{\Omega}\psi_{i}d\mu$ is upper
semicontinuous and hence, there exists some equilibrium state
$\mu_{\psi_{i}}$. Indeed, let $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ be a sequence
of Borel probability invariant measures such that

$h_{\mu_{n}}(S)+\int_{\Omega}\psi_{i}d\mu_{n}\quad\text{converges \ to}\quad
P_{S}(\Omega,\psi_{i}).$

Since the space invariant Borel probability measures is compact, there exists
some accumulation point $\mu_{\psi_{i}}$. By this fact and upper
semicontinuity of $\mu\mapsto h_{\mu}(S)+\int_{\Omega}\psi_{i}d\mu$, we get

$h_{\mu_{\psi_{i}}}(S)+\int_{\Omega}\psi_{i}d\mu_{\psi_{i}}\geq\liminf_{n\to\infty}h_{\mu_{n}}(S)+\int_{\Omega}\psi_{i}d\mu_{n}=P_{S}(\Omega,\psi_{i}),$

so $\mu_{\psi_{i}}$ is an equilibrium state, as stated. ∎

Let $\phi:\Omega\times I\to\mathbb{R}$ be a Hölder continuous potential that
does not depend on the fiber. This means that the function
$\phi(\textbf{t},.):\Omega\to\mathbb{R}$ is constant, for each
$\textbf{t}\in\Omega$ fixed. Take the restriction
$\phi_{i}:=\phi|_{\Omega\times I_{i}}$, $i=1,\dots,n$. Clearly, $\phi_{i}$ is
also a Hölder continuous potential. Hence, as above, the potential $\phi_{i}$
induces an upper semicontinuous potential $\psi_{i}:\Omega\to\mathbb{R}$
defined by
$\psi_{i}(\textbf{t})=\phi_{i}(\textbf{t},\gamma_{G,i}(\textbf{t}))$.

We recall that the expanding circle map $\omega$ possesses an absolutely
continuous invariant ergodic measure $\nu^{+}$ which is equivalent to
Lebesgue. Then, $(\Omega,S)$ has an invariant ergodic measure $\nu$ inherited
from the invariant measure $\nu^{+}$, i.e. $\nu^{+}=\textbf{p}_{\ast}\nu$.

######  Proposition 4.0.3.

Let $\phi:\Omega\times I\to\mathbb{R}$ be a Hölder continuous potential and
take $\phi_{i}=\phi|_{\Omega\times I_{i}}$, $i=1,\dots,n$. Consider the skew
product $G\in\mathcal{U}$ and $G_{i}=G|_{\Omega\times I_{i}}$, $1\leq i\leq
n$, given by (4.0.1). Assume the measure $\mu_{\psi_{i}}$, given by
Proposition 4.0.2, is ergodic and it is not singular with respect to the
invariant ergodic measure $\nu$. Then
$\mu_{\phi_{i}}=\mu_{\psi_{i}}\circ(id\times\gamma_{G,i})^{-1}$ is an
equilibrium state associated to $(G_{i},\phi_{i})$ which is supported on the
maximal attractors $\Delta_{i}(G)$.

###### Proof.

Take the measure $\mu_{\psi_{i}}$, given by Proposition 4.0.2, which is
ergodic and it is not singular with respect to the invariant ergodic measure
$\nu$. As both measures are $S$-invariant and ergodic, this implies that these
measures coincide.

Since $\mu_{\psi_{i}}$ is ergodic, so $\mu_{\phi_{i}}$ is also ergodic. In
particular, since $\mu_{\psi_{i}}$ is equivalent to $\nu$, so $\mu_{\phi_{i}}$
is supported on the maximal attractors $\Delta_{i}(G)$.

By Lemma 3.2.2, $\lambda_{i}(\nu,G_{i})$ is negative and $G_{i}$ satisfies the
non-uniformly contraction condition, hence, for $\nu$\- almost every
$\textbf{t}\in\Omega$, $h_{top}(G_{i},\pi^{-1}(\textbf{t}))=0$. Since
$\mu_{\psi_{i}}$ is equivalent to $\nu$, thus, for $\mu_{\psi_{i}}$\- almost
every $\textbf{t}\in\Omega$, $h_{top}(G_{i},\pi^{-1}(\textbf{t}))=0$. Applying
Ledrappier-Walter’s formula, we have

$\displaystyle P_{S}(\psi_{i})\leq P_{G_{i}}(\phi_{i})$ $\displaystyle=$
$\displaystyle\sup\\{h_{\hat{\mu}}(G_{i})+\int\phi_{i}d\hat{\mu}\\}$
$\displaystyle=$ $\displaystyle\sup\\{h_{\mu}(S)+\int
h_{top}(G_{i},\pi^{-1}(\textbf{t}))d\mu(\textbf{t})+\int\psi_{i}d\mu\\}$
$\displaystyle=$ $\displaystyle P_{S}(\psi_{i}).$

Hence, $P_{S}(\psi_{i})=P_{G_{i}}(\phi_{i})$ and $\mu_{\phi_{i}}$ is an
equilibrium state for $(G_{i},\phi_{i})$. ∎

##  5 Skew products over the toral baker map

Here, a toral baker map on $\mathbb{T}^{2}=\mathbb{T}^{1}\times\mathbb{T}^{1}$
is defined by

$\displaystyle H:\mathbb{T}^{2}\to\mathbb{T}^{2},\
H(t,s)=(bt(mod1),\frac{(s+[bs])}{b}),$ (5.0.1)

for some positive integer $b$ and for each $\theta=(t,s)\in\mathbb{T}^{2}$.
This map is bijective but not continuous. Moreover, it preservers the Lebesgue
measure $m$. Hence, $(\mathbb{T}^{2},\mathcal{B},m,H)$ is a measure-preserving
dynamical system, in the sense of Arnold [2], where $\mathcal{B}$ is the Borel
$\sigma$-algebra on $\mathbb{T}^{2}$.

Assume $\mathcal{F}_{H}\subset\mathcal{F}$ denotes the family of all skew
product transformations $\mathbb{G}$ with the base map $H$ defined on
$\mathbb{T}^{2}\times\mathbb{I}$ of the form

$\mathbb{G}:\mathbb{T}^{2}\times\mathbb{I}\to\mathbb{T}^{2}\times\mathbb{I},\
\mathbb{G}(\theta,x)=(H(\theta),g_{\theta}(x)),$ (5.0.2)

where $(\theta,x)\in\mathbb{T}^{2}\times\mathbb{I}$,
$\theta=(t,s)\in\mathbb{T}^{2}$ and the fiber maps $g_{\theta}$ depends on
$\theta=(t,s)$ only through $t$, so we can write $g_{\theta}=g_{t}$, where
$\mathcal{F}$ is given by Definition 2.1.1. By definition,
$\mathbb{G}(\mathbb{T}^{2}\times\mathbb{I})\subset\mathbb{T}^{2}\times\text{int}(\mathbb{I})$.

Note that the baker map $H$ is bijective but not continuous. Moreover, it is a
canonical extension of the corresponding expanding circle map $\omega(t)=bt\
\text{mod 1}$, where $b$ is a positive integer. Here, we assume that $b=4$.

######  Lemma 5.0.1.

There exists a measure-theoretical isomorphism
$h:(\mathbb{T}^{2},H,m)\to(\Omega,S,\nu)$, where $m$ is the Lebesgue measure
on $\mathbb{T}^{2}$.

###### Proof.

We define $h:\mathbb{T}^{2}\to\Omega$ by $h(\theta)=(\ldots,t_{-1},t_{0})$,
for $\theta=(t,s)\in\mathbb{T}^{2}$ so that $(t_{0},s_{0})=(t,s)$ and for each
$j\in\mathbb{N}$, $(t_{-j},s_{-j})=H^{-j}(t_{0},s_{0})$. Then, it is easy to
see that $h$ is a bijective map. Indeed, the injectivity of the baker map $H$
implies the injectivity of $h$. Assume
$\textbf{t}=(\ldots,t_{-1},t_{0})\in\Omega$. Take

$t=t_{0},\ s=\sum_{i=0}^{\infty}\frac{\lfloor 4t_{-i-1}\rfloor}{4^{(i+1)}},\
\text{and}\ \theta=(t,s).$

Then $h(\theta)=\textbf{t}$ and hence, $h$ maps the two dimensional torus
$\mathbb{T}^{2}$ onto the inverse limit space $\Omega$. Also, $h\circ H=S\circ
h$. It is easy to see that, $h$ is a measure-theoretical isomorphism between
the two measure preserving dynamical systems $(\mathbb{T}^{2},H,m)$ and
$(\Omega,S,\nu)$, where $m$ is the Lebesgue measure on $\mathbb{T}^{2}$. ∎

Let us take the invariant graph $\widetilde{F}$ is given by (3.1.5). It admits
an extension $\mathbb{F}\in\mathcal{F}_{H}$ of the form (5.0.2) and an
extension $F$ given by (3.2.1) over the solenoid map $S$.

######  Theorem 5.0.1.

There exists an open set $\mathcal{U}_{H}\subset\mathcal{F}_{H}$ of skew
products over the baker map $H$ such that each skew product
$\mathbb{G}\in\mathcal{U}_{H}$ admits an attracting multi-graph or an
attracting bony multi-graph.

###### Proof.

Take any skew product $\mathbb{G}\in\mathcal{F}_{H}$ over the baker map $H$ of
the form (5.0.2) sufficiently small to $\mathbb{F}$. Note that for iterates of
$\mathbb{G}$ we denote

$\mathbb{G}^{n}(\theta,x)=(H^{n}(\theta),g_{H^{n-1}(\theta)}\circ\cdots\circ
g_{\theta}(x))=(H^{n}(\theta),g_{\theta}^{n}(x)).$

Since the fiber maps $g_{\theta}$ depend on $\theta=(t,s)$ only through $t$,
so we can write $g^{n}_{\theta}(x)=g_{\omega^{n-1}(t)}\circ\dots\circ
g_{t}(x)$. Thus $g^{n}_{\theta}(x)=g^{n}_{\textbf{t}}$, for each
$\textbf{t}=(\dots,t_{-1},t_{0})\in\Omega$ with $t_{0}=t$. By this fact, for
each skew product $G\in\mathcal{U}$ over the base map $S$ satisfying in the
conclusion of Lemma 3.2.2, we associate a skew product $\mathbb{G}$ over the
baker map $H$ defined by
$\mathbb{G}(\theta,x)=(H(\theta),g(\theta,x))=(H(\theta),g_{t}(x))$,
$\theta=(t,s)\in\mathbb{T}^{2}$, where $g_{t}$, $t\in\mathbb{T}^{1}$, are the
fiber maps of $G$. Therefore, there exist an open set
$\mathcal{U}_{H}\subset\mathcal{F}_{H}$ and a one to one correspondence
between the skew products $G\in\mathcal{U}$ over the base map $S$ and the skew
products $\mathbb{G}\in\mathcal{U}_{H}$ over the baker map $H$. Moreover,
$(h\times id)\circ\mathbb{G}=G\circ(h\times id)$.

Given a skew product transformation $G\in\mathcal{U}$, take the invariant
graphs $\gamma_{G,i}$ given by Theorem 3.2.1 defined on the subset
$\Omega_{i}(G)\subseteq\Omega$ with total measure. By Theorem 3.2.2,
$\text{Cl}(\Gamma_{G,i})=\Delta_{i}(G)$, where $\Delta_{i}(G)$ is the maximal
attractor given by (3.2.12), $i=1,\dots,n$.

Now, for each $\theta=(t,s)\in\mathbb{T}^{2}$, define
$\gamma_{\mathbb{G},i}(t,s)=\gamma_{G,i}(h(t,s))$. We denote the graph of
$\gamma_{\mathbb{G},i}$ by $\Gamma_{\mathbb{G},i}$. Notice that the fiber map
of $\mathbb{G}$ is constant along the stable leaves of the baker map $H$ given
by vertical fibers $\\{t\\}\times\mathbb{T}^{1}$, hence
$\gamma_{\mathbb{G},i}(t,s)$ is constant along the stable leaves of $H$.

Take $I_{i}(\textbf{t})=\\{\textbf{t}\\}\times I_{i}$ for
$\textbf{t}=(\cdots,t_{-1},t_{0})\in\Omega$, and

$I_{i}(\textbf{t},m,G)=g_{S^{-1}(\textbf{t})}\circ\cdots\circ
g_{S^{-m}(\textbf{t})}(I_{i})=g_{t_{-1}}\circ\cdots\circ g_{t_{-m}}(I_{i}),$

for $i=1,\dots,n$. Then

$\Delta_{i}(G)\bigcap I_{i}(\textbf{t})=\bigcap_{n\geq
0}I_{i}(\textbf{t},m,G).$

Note that, by construction, the fiber maps of the skew products $G$ and
$\mathbb{G}$ are the same. Hence, by these observations and the previous
lemma, $\text{Cl}(\Gamma_{\mathbb{G},i})$ is an attracting invariant graph or
an attracting bony graph for $\mathbb{G}$.

Let us take

$A_{max}(\mathbb{G}):=\bigcap_{n\geq 0}\mathbb{G}^{n}(\mathbb{T}^{2}\times
I),\ \text{and}\ \Delta_{i}(\mathbb{G}):=\bigcap_{n\geq
0}\mathbb{G}^{n}(\mathbb{T}^{2}\times I_{i}),\ 1\leq i\leq n.$ (5.0.3)

Then $\Delta_{i}(\mathbb{G})=\text{Cl}(\Gamma_{\mathbb{G},i})$, $1\leq i\leq
n$, and thus the union
$K(\mathbb{G}):=\bigcup_{i=1}^{n}\Delta_{i}(\mathbb{G})$ is an attracting
multi-graph or bony multi-graph. ∎

##  6 Data Availability

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Aliprantis, C., Border, K.,: Infinite Dimensional Analysis, Third edition, Springer-Verlag, Berlin (2006)
  * [2] Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics, Springer-Verlag Berlin and Heidelberg GmbH $\&$ Co. (2002)
  * [3] Barral, J., Feng, D.-J.: Weighted thermodynamic formalism on subshifts and applications, Asian J. Math., 16, 319–352 (2012)
  * [4] Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes in Math. 470, Springer-Verlag, Berlin (1975)
  * [5] Broomhead, D., Hadjiloucas D., and Nicol M.: Random and deterministic perturbation of a class of skew-product systems. Dyn. Stab. Syst. 14, 115-128 (1999).
  * [6] Bugeaud, Y.: Distribution modulo one and Diophantine approximation, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2012)
  * [7] Campbell, K.M.: Observational noise in skew product systems. Phys. D. 107, 43–56 (1997)
  * [8] Campbell, K.M., Davies, M.E.: The existence of inertial functions in skew product systems. Nonlinearity 9, 801–817 (1996)
  * [9] Cao, Y.-L., Feng D.-J., Huang, W.: The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst. 20, 639-657 (2008)
  * [10] Davies, M.E., Campbell, K.M.: Linear recursive filters and nonlinear dynamics. Nonlinearity 9, 487-499 (1996)
  * [11] Edalat, A.: Power Domains and Iterated Function Systems. Inform. and Comput. 124, 182-197 (1996)
  * [12] Fadaei, S., Keller, G., Ghane, F. H.: Invariant graphs for chaotically driven maps, Nonlinearity. 31 (11), 5329-5349 (2018)
  * [13] Feng, D.-J. Huang, W.: Variational principles for weighted topological pressure. Preprint, arXiv:1412.0078v1.
  * [14] Furstenberg, H.: Strict ergodicity and transformation of the torus. Am. J. Math., 83, 573–601 (1961)
  * [15] Gelfert, K., Oliveira, D.: Invariant multi-graphs in step skew products, Dyn. Syst. 39 (2019)
  * [16] Hirsch, M., Pugh, C.: Stable manifolds and hyperbolic sets. Bull. Am. Math. Soc. 75, 149–152 (1969)
  * [17] Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
  * [18] Homburg, A. J.: Synchronization in Minimal Iterated Function Systems on Compact Manifolds, Bulletin of the Brazilian Mathematical Society, 49 (3), 615–635 (2018)
  * [19] Hunt, B.R., Ott, E., Yorke, J.A.: Differentiable generalized synchronization of chaos. Phys. Rev. E 55, 4029 (1997)
  * [20] Jäger, T.: skew product systems with one dimensional fibres. Lecture notes. (2013)
  * [21] Jäger, T.: Quasiperiodically forced interval maps with negative Schwarzian derivative. Nonlinearity, 16(4), 1239-1255 (2003)
  * [22] Jäger, T., Keller, G.: Random minimality and continuity of invariant graphs in random dynamical systems, Trans. Amer. Math. Soc. 368, 6643-6662 (2016)
  * [23] Kleptsyn, V., Volk, D.: Physical measures for random walks on interval. Mosc. Math. J. 14(2), 339–365 (2014)
  * [24] Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter, Berlin (1985)
  * [25] Kudryashov, Y. G.: Bony attractors. Funkts. Anal. Prilozhen. 44(3), 73-76 (2010): English transl. Functional Anal. Appl. 44(3), 219-222 (2010)
  * [26] Ledrappier, F., Walters, P.: A relativised variational principle for continuous transformations. J.Land. Math. Soc. 16, 568-579 (1977)
  * [27] Mane, R.: Ergodic Theory and Differentiable Dynamics. Vol. 8 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1987)
  * [28] Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99 177–195, (1985).
  * [29] Nassiri, M., Pujals, E. R.: Robust transitivity in hamiltonian dynamics. Annales scientifiques de l’École Normale Supérieure, Série 4, Tome 45 (2), 191-239 (2012).
  * [30] Pecora, L.M., Carroll, T.L.: Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data. Chaos 6, 432–439 (1996)
  * [31] Pikovsky, A., Rosenblum, M., and Kurths, J.: Synchronization. A universal concept in nonlinear sciences. Cambridge University Press (2001)
  * [32] Ramos, V., Viana, M.: Equilibrium states for hyperbolic potentials. Nonlinearity. 30, 825-847 (2017)
  * [33] Rauch, M.: Variational principles for the topological pressure of measurable potentials. Discrete Contin. Dyn. Syst. Series S 10(2), 367-394 (2017)
  * [34] Ruelle, D., Sinai, Y.G.: From dynamical systems to statistical mechanics and back. Physica A: Statistical Mechanics and its Applications. 140(1-2), 1–8 (1986)
  * [35] Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E. 51, 980–994 (1995)
  * [36] Sinai, Y.G.: Gibbs measures in ergodic theory. Russian Math. Surveys. 27, 21–69 (1972)
  * [37] Stark, J.: Invariant graphs for forced systems. Phys. D. 109, 163-179 (1997)
  * [38] Stark, J.: Regularity of invariant graphs for forced systems. Ergod. Theory Dyn. Syst. 9, 155-199 (1999)
  * [39] Stark, J., Davies, M.E.: Recursive filters driven by chaotic signals. In: IEE Colloquium on Exploiting Chaos in Signal Processing. IEE Digest. vol. 143, 1–516 (1994)
  * [40] Stark, J., Sturman, R.: Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity. 13(1), 113-143 (2000)
  * [41] Viana, M., Oliveira, K.: Foundationa of Ergodic Theory. Cambridge Studies in Advanced Mathematics (2016)
  * [42] Viana, M., Yang, J.: Physical measures and absolute continuity for one-dimensional center direction. Ann. de Inst. Henri Poincare (C) Non Linear Anal. 30(5) (2013), 845–877.
  * [43] Walter, P.: An Introduction to Ergodic Theory. Springer-Verlag, New York, Heidelberg, Berlin (1982)
  * [44] Zaj, M., Ghane, F.H.: Non hyperbolic Solenoidal thick bony attractors. Qual. Theory Dyn. Syst. 18 (1), 35-55 (2019)