then a further necessary condition for $S>0$ in a strip is $L_{1}+H+A<0$. If
$|L_{1}|/2H\leq 1$ then a necessary condition is $L_{1}^{2}-4AH>0$. Examples
here are Figs. 41 and 42. Note that here there are 4 different trajectories
corresponding to a single choice of constants of the motion.

Figure 41: Case: $A=1$, $L_{1}=-5$, $L_{2}=3$, $H=2$. Figure 42: Case: $A=1$,
$L_{1}=-10$, $L_{2}=-3$, $H=10$.

Case when $R^{2}=0$. When $L_{1}=-9$, $L_{2}=0$, $A=1$, $H=8$, we have
$R^{2}=0$ such that $S=0$. The trajectory in this case is a straight line on
the x-axis. The two boundary points where $p_{x}=0$ for $x>0$ are $0$ and
$0.9354$. The velocity of the particle is
$\dot{x}=\\{H,x\\}=\frac{p_{x}(1-x^{2})}{x^{2}+z^{2}}$. Substituting $z=0$ and
the expression for $p_{x}$ in terms of $x$ into the equation
gives,$\dot{x}=\sqrt{\frac{L_{1}+2H}{x^{2}}-H}$ Thus $\displaystyle\lim_{x\to
0}\dot{x}=\infty$, so the particle will go through the origin instead of
bouncing back.

##  Appendix A Review of some basic concepts in Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical Newtonian mechanics
alternate to Lagrangian mechanics. In the Hamiltonian formalism a physical
system describing the motion of a particle at time $t$ involves $n$
generalized coordinates $q_{j}(t)$, and $n$ generalized momenta $p_{j}(t)$.
The phase space of the system is described by points
$(p_{j},q_{j})\in{\mathbb{R}}^{2n}$. The generalized coordinates
$q_{j}^{\prime}s$ and generalized momenta $p_{j}^{\prime}s$ are derived from
the Lagrangian formulation. The dynamics of the system are given by Hamilton’s
equations [1, 12]

$\displaystyle\frac{dq_{j}}{dt}=+\frac{\partial{\cal H}}{\partial
p_{j}},\qquad\frac{dp_{j}}{dt}=-\frac{\partial{\cal H}}{\partial q_{j}},$
(A.1)

where ${\cal H}={\cal H}(q_{1},\dots,q_{n},p_{1},\dots,p_{n},t)$ is the
Hamiltonian, which in this paper corresponds to the total (time-independent)
energy of the system: ${\cal H}=T+V$, where $T$ and $V$ are kinetic and
potential energy, respectively. Solutions of these equations give the
trajectories of the system. Here $T$ is a function of ${\bf q}$ alone while
$V$ is a function of ${\bf q}$ alone. Explicitly,

$\displaystyle{\cal H}=\frac{1}{2m}\sum_{j,k}g^{jk}({\bf q})p_{j}p_{k}+V({\bf
q}),$ (A.2)

where $g^{jk}$ is a contravariant metric tensor on some real or complex
Riemannian manifold. That is $g^{-1}=\det(g^{jk})\neq 0$, $g^{jk}=g^{kj}$ and
the metric on the manifold is given by
$ds^{2}=\sum\limits_{j,k=1}^{n}g_{jk}dq^{j}dq^{k}$, where $(g_{jk})$ is the
covariant metric tensor, the matrix inverse to $(g^{jk})$. Under a local
transformation $q^{\prime}_{j}=f_{j}({\bf q})$ the contravariant tensor and
momenta transform according to

$\displaystyle(g^{\prime})^{\ell h}=\sum_{j,k}\frac{\partial
q^{\prime}_{\ell}}{\partial q_{j}}\frac{\partial q^{\prime}_{h}}{\partial
q_{k}}g^{jk},\qquad p^{\prime}_{\ell}=\sum_{j=1}^{n}\frac{\partial
q_{j}}{\partial q^{\prime}_{\ell}}p_{j},$

so ${\cal H}$ is coordinate independent. Here $m$ is a scaling parameter that
can often be interpreted as the mass of the particle. For the Hamiltonian
(A.2) the relation between momenta and the velocities is
$p_{j}=m\sum\limits_{\ell=1}^{n}g_{j\ell}\dot{q}_{\ell}$, so that
$T=\frac{1}{2m}\sum\limits_{j,k}g^{jk}({\bf
q})p_{j}p_{k}=\frac{m}{2}\sum\limits_{\ell,h=1}^{n}g_{\ell h}({\bf
q})\dot{q}_{\ell}\dot{q}_{h}$. Once the velocities are given, the momenta are
scaled linearly in $m$. In mechanics the exact value of $m$ may be important,
but for our purposes can be scaled to any nonzero value using the above
formulas. To make direct contact with mechanics we will often set $m=1$; for
structure calculations we will usually set $m=1/2$.

The Poisson bracket of two arbitrary functions ${\cal A}({\bf p},{\bf q})$,
${\cal B}({\bf p},{\bf q})$ on the phase space is the function

$\displaystyle\\{{\cal A},{\cal B}\\}({\bf p},{\bf
q})=\sum_{j=1}^{n}\left(\frac{\partial{\cal A}}{\partial
p_{j}}\frac{\partial{\cal B}}{\partial q_{j}}-\frac{\partial{\cal A}}{\partial
q_{j}}\frac{\partial{\cal B}}{\partial p_{j}}\right).$

The Poisson bracket obeys the following properties, for ${\cal A}$, ${\cal
B}$, ${\cal C}$ functions on the phase space and $a$, $b$ constants,

$\displaystyle\text{anti-symmetry:}\quad\\{{\cal A},{\cal B}\\}=-\\{{\cal
B},{\cal A}\\},$ $\displaystyle\text{Bilinearity:}\quad\\{{\cal A},m{\cal
B}+n{\cal C}\\}=m\\{{\cal A},{\cal B}\\}+n\\{{\cal A},{\cal C}\\},$
$\displaystyle\text{Jacobi identity:}\quad\\{{\cal A},\\{{\cal B},{\cal
C}\\}\\}+\\{{\cal B},\\{{\cal C},{\cal A}\\}\\}+\\{{\cal C},\\{{\cal A},{\cal
B}\\}\\}=0,$ $\displaystyle\text{Leibniz rule:}\quad\\{{\cal A},{\cal
BC}\\}=\\{{\cal A},{\cal B}\\}{\cal C}+{\cal B}\\{{\cal A},{\cal C}\\},$
$\displaystyle\text{chain\ rule:}\quad\\{f({\cal A}),{\cal
B}\\}=f^{\prime}({\cal A})\\{{\cal A},{\cal B}\\}.$

In terms of the Kronecker delta $\delta_{jk}$, coordinates $({\bf q},{\bf p})$
satisfy canonical relations $\\{p_{j},p_{k}\\}=\\{q_{j},q_{k}\\}=0$, and
$\\{p_{j},q_{k}\\}=\delta_{jk}$. Using the Poisson bracket, we can rewrite
Hamilton’s equations (A.1) as

$\displaystyle\\{{\cal H},q_{j}\\}=\frac{dq_{j}}{dt}=\frac{\partial{\cal
H}}{\partial p_{j}},\qquad\\{{\cal
H},p_{j}\\}=\frac{dp_{j}}{dt}=\frac{\partial{\cal H}}{\partial q_{j}}.$

For any function ${\cal F}({\bf q},{\bf p})$, its dynamics along a trajectory
${\bf q}(t)$, ${\bf p}(t)$ is $\frac{d{\cal F}}{dt}=\\{{\cal H},{\cal F}\\}$.
Thus ${\cal F}({\bf q},{\bf p})$ will be constant along a trajectory if and
only $\\{{\cal H},{\cal F}\\}=0$. If $\\{{\cal H},{\cal F}\\}=0$, we say that
$\cal F$ is a constant of the motion.

###  A.1 Classical integrability

A system with Hamiltonian ${\cal H}$ is integrable if it admits $n$ constants
of the motion ${\cal L}_{1}={\cal H}$, ${\cal L}_{2},\dots,{\cal L}_{n}$ that
are in involution:

$\displaystyle\\{{\cal L}_{j},{\cal L}_{k}\\}=0,\qquad 1\leq j,k\leq n,$ (A.3)

and are functionally independent in the sense that
$\det\big{(}\frac{\partial{\cal L}_{j}}{\partial p_{k}}\big{)}\neq 0$. Suppose
$\cal H$ is integrable with associated constants of the motion ${\cal L}_{j}$.
Then by the inverse function theorem we can solve the $n$ equations ${\cal
L}_{j}({\bf q},{\bf p})=c_{j}$ for the momenta to obtain $p_{k}=p_{k}({\bf
q},{\bf c})$, $k=1,\dots,n$, where ${\bf c}=(c_{1},\dots,c_{n})$ is a vector
of constants. For an integrable system, if a particle with position $\bf q$
lies on the common intersection of the hypersurfaces ${\cal L}_{j}=c_{j}$ for
constants $c_{j}$, then its momentum $\bf p$ is completely determined. Also,
if a particle following a trajectory of an integrable system lies on the
common intersection of the hypersurfaces ${\cal L}_{j}=c_{j}$ at time $t_{0}$,
where the ${\cal L}_{j}$ are constants of the motion, then it lies on the same
common intersection for all $t$ near $t_{0}$. Considering $p_{j}({\bf q},{\bf
c})$ and using the conditions (A.3) and the chain rule it is straightforward
to verify $\frac{\partial p_{j}}{\partial q_{k}}=\frac{\partial
p_{k}}{\partial q_{j}}$. Therefore, there exists a function $u({\bf q},{\bf
c})$ such that $p_{\ell}=\frac{\partial u}{\partial q_{\ell}}$,
$\ell=1,\dots,n$. Note that ${\cal P}_{j}({\bf q},\frac{\partial
u}{\partial{\bf q}})=c_{j}$, $j=1,\dots,n$, and, in particular, $u$ satisfies
the Hamilton–Jacobi equation

$\displaystyle{\cal H}\left({\bf q},\frac{\partial u}{\partial{\bf
q}}\right)=E,$ (A.4)

where $E=c_{1}$. By construction $\det(\frac{\partial u}{\partial
q_{j}\partial c_{k}})\neq 0,$ and such a solution of the Hamilton–Jacobi
equation depending nontrivially on $n$ parameters $\bf c$ is called a complete
integral. This argument is reversible: a complete integral of (A.4) determines
$n$ constants of the motion in involution, ${\cal P}_{1},\dots,{\cal P}_{n}$.

######  Theorem A.1.

A system is integrable if and only if (A.4) admits a complete integral.

A powerful method for demonstrating that a system is integrable is to exhibit
a complete integral by using additive separation of variables.

It is a standard result in classical mechanics that for an integrable system
one can integrate Hamilton’s equations and obtain the trajectories [1, 12].
The Hamiltonian formalism is is well suited to exploiting symmetries of the
system and an important tool in laying the framework for quantum mechanics.

###  A.2 Classical superintegrability

Let ${\cal F}=(f_{1}({\bf q},{\bf p}),\dots,f_{N}({\bf q},{\bf p}))$ be a set
of $N$ functions defined and locally analytic in some region of a
$2n$-dimensional phase space. We say ${\cal F}$ is functionally independent if
the $N\times 2n$ matrix $\left(\frac{\partial f_{\ell}}{\partial
q_{j}},\frac{\partial f_{\ell}}{\partial p_{k}}\right)$ has rank $N$
throughout the region. Necessarily $N\leq 2n$. The set is functionally
dependent if the rank is strictly less than $N$ on the region. In this case
there is a nonzero analytic function $F$ of $N$ variables such that
$F(f_{1},\dots,f_{n})=0$ identically on the region. Conversely, if $F$ exists
then the rank of the matrix is $<N$. We say that the Hamiltonian system ${\cal
H}$ is (polynomially) integrable if there exists a set of $N=n$ constants of
the motion ${\cal L}_{1}={\cal H},\dots,{\cal L}_{n}$, each polynomial in the
momenta globally defined (except possibly for isolated singularities), that is
functionally independent and in involution: $\\{{\cal L}_{j},{\cal
L}_{k}\\}$=0 for $1\leq j,k\leq n$. A classical Hamiltonian system in $n$
dimensions is maximally $($polynomially$)$ superintegrable if it admits $2n-1$
functionally independent, globally defined constants, polynomial in the
momenta (the maximum number possible [33]). At most $n$ functionally
independent constants of the motion can be in mutual involution [1]. However,
several distinct $n$-subsets of the $2n-1$ polynomial constants of the motion
for a superintegrable system could be in involution. In that case the system
is multi-integrable. An important feature of superintegrable systems is that
the orbits traced out by the trajectories can be determined algebraically,
without the need for integration. Along any trajectory each of the symmetries
is constant: ${\cal L}_{s}=c_{s}$, $s=1,\dots,2n-1$. Each equation ${\cal
L}_{s}({\bf q},{\bf p})=c_{s}$ determines a $(2n-1)$-dimensional hypersurface
in the $2n$-dimensional phase space, and the trajectory must lie in that
hypersurface. Thus the trajectory lies in the common intersection of $2n-1$
independent hypersurfaces; hence it must be a curve. Another important feature
of the trajectories is that all bounded orbits are periodic [34].

The polynomial constants of the motion for a system with Hamiltonian ${\cal
H}$ form the (polynomial) symmetry algebra $S_{\cal H}$ of the system, closed
under scalar multiplication and addition, multiplication and the Poisson
bracket: Indeed if $\cal H$ is a Hamiltonian with constants of the motion
${\cal L}$, ${\cal K}$. Then $\alpha{\cal L}+\beta{\cal K}$, ${\cal L}{\cal
K}$ and $\\{{\cal L},{\cal K}\\}$ are also constants of the motion. The $n$
defining constants of the motion of a polynomially integrable system do not
generate a very interesting symmetry algebra, because all Poisson brackets
vanish. However, for the $2n-1$ generators of a polynomial superintegrable
system the brackets cannot all vanish and the symmetry algebra has nontrivial
structure. The degree (or order) $O({\cal L})$ of a polynomial constant of the
motion $\cal L$ is its degree as a polynomial in the momenta. Here ${\cal H}$
has degree 2. The degree $O(F_{k})$ of a set of generators $F_{k}=\\{{\cal
L}_{1},\dots,{\cal L}_{k}\\}$, is the maximum degree of the generators. Let
$S=S_{F_{k}}$ be a symmetry algebra of a Hamiltonian system generated by the
set $F_{k}$. Clearly, many different sets $F^{\prime}_{k^{\prime}}$ can
generate the same symmetry algebra. Among all these there will be a set of
generators $F^{0}_{k_{0}}$ for which $\ell=O(F^{0}_{k_{0}})$ is a minimum.
Here $\ell$ is unique, although $F^{0}_{k_{0}}$ is not. We define the degree
(or order) of $S$ to be $\ell$. The theory of 2nd degree superintegrable
systems has been developed in papers such as [8, 16, 17, 18, 19, 22, 28, 37];
all such systems are known as are the structures of the symmetry algebras, all
of which close at degree 6.

### Acknowledgement

This work was partially supported by a grant from the Simons Foundation (#
208754 to Willard Miller, Jr.). We thank Galliano Valent for correcting an
error in an earlier draft.

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# Thermodynamic formalism for general iterated function systems with measures

Jader E. Brasil<EMAIL_ADDRESS>Elismar R. Oliveira  Rafael Rigão
Souza

###### Abstract

This paper introduces a theory of Thermodynamic Formalism for Iterated
Function Systems with Measures (IFSm). We study the spectral properties of the
Transfer and Markov operators associated to a IFSm. We introduce variational
formulations for the topological entropy of holonomic measures and the
topological pressure of IFSm given by a potential. A definition of equilibrium
state is then natural and we prove its existence for any continuous potential.
We show, in this setting, a uniqueness result for the equilibrium state
requiring only the Gâteaux differentiability of the pressure functional.

MSC2010: 37D35, 37B40, 28Dxx.
Keywords: Iterated Function System, Thermodynamic Formalism, Ergodic Theory,
Transfer Operator, Entropy, Pressure, Equilibrium States.

##  1 Introduction

The modern study of Iterated Function Systems (IFS for short) come back to the
early 80’s with the works of J. Hutchinson [20] and M. Barnsley [3]. In these
papers, the theory was unified both in the geometric and the analytical point
of view, generating what we call today the Hutchinson-Barnsley theory for IFS.
An IFS is a family of maps acting from a set to itself. Under good contraction
hypothesis there exist an invariant compact set called the fractal attractor.
Moreover, if we add weights having good continuity hypothesis, the IFS acts on
probabilities having an invariant probability whose support is the fractal
attractor set. We observe that several works on geometric features of fractals
were done in the previous decades by Mandelbrot and others, but after the 80’s
the IFS assumed the central role in the generation and study of fractals and
its applications.

For a dynamical system given by a map $T:X\to X$, an initial point $x_{0}\in
X$ is iterated by $T$ producing the orbit
$\\{x_{0},T(x_{0}),T^{2}(x_{0}),...\\}$, whose limit or the cluster points are
the objects of main interest, from a dynamical point of view. On the other
hand, for an IFS $(X,\tau_{\theta})_{\theta\in\Theta}$, we iterate the initial
point by choosing at each step a possibly different map $\tau_{\theta}:X\to
X$, indexed by the generally finite set $\Theta$, producing multiple orbits
$\\{Z_{j},j\geq
0\\}=\\{Z_{0}=x_{0},Z_{1}=\tau_{\theta_{0}}(x_{0}),Z_{2}=\tau_{\theta_{1}}(\tau_{\theta_{0}}(x_{0})),...\\}$.
We notice that the orbit is now a set of orbits controlled by the sequence
$(\theta_{0},\theta_{1},...)\in\Theta^{\mathbb{N}}$. To avoid this
complication Hutchinson defined the fractal operator $F:K(X)\to K(X)$ by

$F(B)=\bigcup_{\theta\in\Theta}\tau_{\theta}(B),$

for $B\in K(X)$, where $K(X)$ is the family of nonempty compact sets of $X$.
This operator is called the Hutchinson-Barnsley operator and a compact set is
invariant or fractal if $F(\Omega)=\Omega$. Additionally $\Omega$ is a fractal
attractor if the orbit of $B$ by $F$, given by $\\{B,F(B),F^{2}(B),...\\}$
converge, w.r.t. the Hausdorff-Pompeiu metric to $\Omega$, for any $B\in K(X)$
(see [5] for details on the Hausdorff-Pompeiu metric).

Other possible point of view to understand the dynamics of an IFS is the
probabilistic one. Suppose $X$ is now a metric space and let $\mathcal{P}(X)$
be the set of Borel probability measures defined on $X$. In this case we
consider that, in each step the function to be iterated is chosen according to
some probability, which produces a stochastic process
$\\{X_{0},X_{1},X_{2},...\\}$ where each $X_{j+1}\in X$ is randomly chosen
according to a distribution which is obtained from the previous $X_{j}$ by a
transition kernel using the IFS law. To illustrate that, we consider the
classical case of IFS with constant probabilities studied by Hutchinson,
Barnsley and many others in the beginnings of the 80’s: we consider
$\Theta=\\{1,2,...,n\\}$, meaning that we have a finite number of maps, and
each one is chosen according to a probability $p_{j}>0$ where
$p_{1}+\cdots+p_{n}=1$, constituting an IFS with probabilities (IFSp for
short). We denote by $C(X,\mathbb{R})$ the Banach space of all real continuous
functions equipped with supremum norm $\|\;\|_{\infty}$. Under this conditions
the classic transfer operator (also called Ruelle operator, see Ruelle [29,
30], Walters [33] and Fan [14]) is given by

$B_{q}(f)(x)=\int_{\Theta}f(\tau_{\theta}(x))dq_{x}(\theta)=\sum_{j=1}^{n}p_{j}\,f(\tau_{j}(x)),$
(1)

for any $f\in C(X,\mathbb{R})$, where the measure $q_{x}$ is given by
$dq_{x}(\theta)=\sum_{j=1}^{n}p_{j}\delta_{j}(\theta),\;\forall x\in X$. The
dual of $B_{q}$ is a Markov operator (see [11], [22] and[3], for details on
Markov operators and its connection with IFS) acting on $\mu$. The operator
$\mathcal{L}_{q}(\mu):=B_{q}^{*}(\mu)$, is implicitly defined by the property

$\int_{X}fd\mathcal{L}_{q}(\mu)=\int_{X}B_{q}(f)(x)d\mu,$ (2)

for any $f\in C(X,\mathbb{R})$. Given an initial distribution $\mu_{0}$, we
iterate it by the Markov operator $M:\mathcal{P}(X)\to\mathcal{P}(X)$
obtaining the distributions
$\mu_{0},\mu_{1}=\mathcal{L}_{q}(\mu_{0}),\mu_{2}=\mathcal{L}_{q}^{2}(\mu_{0}),...\in\mathcal{P}(X)$.
We have $X_{n}\sim\mu_{n}$ for all $n\geq 0$. Analogously to the fractal
attractor, we say that $\mu\in\mathcal{P}(X)$ is an invariant measure if
$\mathcal{L}_{q}(\mu)=\mu$ and that $\mu\in\mathcal{P}(X)$ is an attracting
invariant measure (or Hutchinson-Barnsley measure) if
$\mathcal{L}_{q}^{j}(\mu_{0})$ converge to $\mu$ w.r.t. the Monge-Kantorovich
metric(see [20]), for any $\mu_{0}\in\mathcal{P}(X)$. It is possible to prove
that the support of the invariant attracting measure is the fractal attractor
(see [20]).

The final feature of an IFS dynamics we need to understand is the connections
between IFS orbits and invariant measures. The first connection is given by a
celebrated result due to M. Barnsley, known as the Chaos Game Theorem (CGT for
short), which claims that, from the initial probabilities $p_{j}$’s, we can
build a probability $\mathbb{P}$ over the space $\Theta^{\mathbb{N}}$ such
that for $\mathbb{P}$-a.e.
$(\theta_{0},\theta_{1},...)\subset\Theta^{\mathbb{N}}$ the corresponding
orbit
$\\{x_{0},\tau_{\theta_{0}}(x_{0}),\tau_{\theta_{1}}(\tau_{\theta_{0}}(x_{0})),...\\}$
approximate the fractal attractor $\Omega$, for any initial point $x_{0}$. The
second connection is given by the Elton’s Ergodic Theorem (EET for short) [13]
claiming that, from the initial probabilities $p_{j}$’s, we can build a
probability $\mathbb{P}$ over the space $\Theta^{\mathbb{N}}$ such that
$\mathbb{P}$-a.e. $(\theta_{0},\theta_{1},...)\subset\Theta^{\mathbb{N}}$, the
corresponding asymptotic average of visits of the orbit
$\\{x_{0},\tau_{\theta_{0}}(x_{0}),\tau_{\theta_{1}}(\tau_{\theta_{0}}(x_{0})),...\\}$
to a measurable set $B\subset X$ is equal to $\mu(B)$, if
$\mu(\partial(B))=0$, analogously to the usual Birkhoff ergodic theorem for a
single map, where $\mu$ is the invariant measure of the IFS in consideration.
For continuous functions it means that

$\frac{1}{N}\left(f(x_{0})+f(\tau_{\theta_{0}}(x_{0}))+\cdots+f(\tau_{\theta_{N-1}}(\cdots\tau_{\theta_{0}}(x_{0})))\right)\to\int_{X}fd\mu,$

for any $f\in C(X,\mathbb{R})$, as $N\to\infty$. In other words

$\frac{1}{N}\left(\delta_{x_{0}}+\delta_{\tau_{\theta_{0}}(x_{0})}+\cdots+\delta_{\tau_{\theta_{N-1}}(\cdots\tau_{\theta_{0}}(x_{0}))}\right)\to\mu,$

in distribution. In synthesis, the CGT and the EET are random procedures to
approximate the fractal attractor and the invariant measure, respectively.

The study of the conditions under which we have, for a given IFS, a fractal
attractor which is the support of an invariant measure is called the
Hutchinson-Barnsley theory. Such conditions have been extremely relaxed and
generalized in several ways in the past forty years. A first generalization,
yet for $\Theta=\\{1,2,...,n\\}$, was for IFSp where the constant
probabilities $p_{j}>0$ such that $p_{1}+\cdots+p_{n}=1$ were replaced by
variable probabilities $p_{j}(x)>0$ where $p_{1}(x)+\cdots+p_{n}(x)=1$ for all
$x\in X$. Now, the transfer operator is defined by

$B_{q}(f)(x)=\int_{\Theta}f(\tau_{\theta}(x))dq_{x}(\theta)=\sum_{j=1}^{n}p_{j}(x)\,f(\tau_{j}(x)),$
(3)

where the measure $q_{x}$ is given by
$dq_{x}(\theta)=\sum_{j=1}^{n}p_{j}(x)\delta_{j}(\theta),\;\forall x\in X$,
for any $f\in C(X,\mathbb{R})$, Very general conditions for the existence of
the invariant measure for such IFS are given in [4]. We point out that the EET
was also proved for variable probabilities and finite functions in [13].

In Fan [14], 1999, the condition $p_{1}(x)+\cdots+p_{n}(x)=1$ is finally
dropped assuming only that each $p_{\theta}(x)\geq 0$ for
$\theta\in\\{1,...,n\\}$. In this work Fan study a contractive system which is
a triplet $(X,\tau_{\theta},p_{\theta})_{\theta\in\\{1,...,n\\}}$, where each
$\tau_{\theta}$ is a contractive map and each $p_{\theta}(x)\geq 0$ for
$\theta=1,...,n$, generalizing the notion of IFS with probabilities. In this
setting, Fan proves a Ruelle-Perron-Frobenius theorem (RPF theorem, for
short), meaning the existence of a positive eigenfunction for the operator
$B_{q}$ and an eigenmeasure for the dual operator $B_{q}^{*}$ with the same
eigenvalue which is the spectral radius of $B_{q}$.

The next key improvement was given by Stenflo [32], where random iterations
are used to represent the iterations of a so called IFS with probabilities,
$(X,\tau_{\theta},\mu)_{\theta\in\Theta}$ for an arbitrary measurable space
$\Theta$. The approach here is slightly different from the previous works on
IFS with probabilities, instead considering weights, the iterations from
$Z_{0}\in X$ are $Z_{j+1}=\tau_{I_{j}}(Z_{j})$ governed by a sequence of i.i.d
variables $\\{I_{j}\in\Theta\\}_{j\in\mathbb{N}}$, with distribution $\mu$,
generating a Markov chain $\\{Z_{j},\;j\geq 0\\}$ with transfer operator given
by

$B_{q}(f)(x)=\int_{\Theta}f(\tau_{\theta}(x))dq_{x}(\theta)=\sum_{j=1}^{n}p_{j}(x)\,f(\tau_{j}(x)),$
(4)

where the measure $q_{x}$ is given by $dq_{x}(\theta)=d\mu(\theta),\;\forall
x\in X$, the for any $f\in C(X,\mathbb{R})$, The main goal of Stenflo [32] is
to establish, when $B_{q}$ is Feller, the existence of an unique attracting
invariant measure $\pi$, for this Markov chain.

The approach in our paper is a generalization of Stenflo [32], (4), and all
classical settings (1) and (3). We take a family
$q_{x}(\cdot)\in\mathcal{M}(\Theta)$, indexed by $x\in X$, generating a Markov
chain with transfer operator given by

$B_{q}(f)(x)=\int_{\Theta}f(\tau_{\theta}(x))dq_{x}(\theta),$

for any $f\in C(X,\mathbb{R})$. The meaning of the distribution
$q_{x}(\cdot)\in\mathcal{M}(\Theta)$ is such that, the position $x$ of a
previous iteration of the IFS determine the distribution $q_{x}(\cdot)$ of
$\theta$ used to choose the function $\tau_{\theta}$ and produce the new point
$\tau_{\theta}(x)$. When $q_{x}=\mu$, for any $x\in X$, is a constant
distribution we recover the setting from Stenflo [32].

In our setting the IFSm $(X,\tau_{\theta},q)_{\theta\in\Theta}$ can be studied
as the sample paths of the Markov process $\\{Z_{j},\;j\geq 0\\}$ with initial
distribution $\mu_{0}=\mu\in\mathcal{M}(X)$ and
$\mu_{j+1}=\mathcal{L}_{q}(\mu_{j})$, where for any $\nu\in\mathcal{M}(X)$,

$\int_{X}f(x)d\mathcal{L}_{q}(\nu)(x)=\int_{X}B_{q}(f)(x)d\nu(x),$

for any $f\in C(X,\mathbb{R})$. Such degree of generality is necessary to
enlarge the range of application for the IFS theory, specially the
thermodynamic formalism. In Section 7 we present a situation where we believe
the tools developed in the previous sections can be applied when analyzing an
interesting problem in economics.

Our goal is to present a complete theory of thermodynamical formalism for
these IFS with measures, that is, good definitions for transfer operators,
invariant measures, entropy, pressure, equilibrium measures and a variational
principle. Finally, we want to use these tools to characterize the solutions
of the ergodic optimization problem.

For sake of completeness we would like to point out that we do not prove a RPF
theorem for those systems, only the existence of positive eigenfunctions, but
we establish all the results that can be derived if we have assumed such a
property. To the best of our knowledge the RPF theorem for IFSm has not been
established an it is a very hard problem. There are several works on the
matter of finding IFS for which the RPF theorem holds. Those IFS are said to
have the RPF property. In 2009 Lopes and Oliveira [24] studied those systems
renaming it as weighted systems or IFS with weights, having the RPF property,
producing a self contained notion of entropy and topological pressure through
a variational principle for holonomic measures allowing to establish a
thermodynamical formalism for IFS. Other approaches for IFS thermodynamic
formalism were developed by Urbański [31, 25, 18] and many others.

It’s worth to mention that, in Urbański et al. [18] a thermodynamic formalism
for conformal infinite (countable) iterated function systems is presented
using the conformal structure via partition functions. Also in Käenmäki [21] a
thermodynamical formalism for IFS is studied with the help of cylinder
functions, where general IFS means that $(X,\tau_{\theta})_{\theta\in\Theta}$
and $\Theta$ is the increasing union of finite alphabets. In Lopes et al. [23]
a thermodynamic formalism for shift spaces, taking values on a compact metric
space is presented, although this problem is closely related to thermodynamic
formalism for IFS when we associate the pre images of the shift map with a
respective maps producing an infinite IFS. Also in [1] a variational principle
for the specific entropy on the context of symbolic dynamics of compact metric
space alphabets was developed generalizing somehow the results in [24].

In our work we will extend the variational results in Lopes and Oliveira [24]
and, more recently, the preprint Cioletti and Oliveira [8], to a general IFS
called IFS with measures (IFSm), $(X,\tau_{\theta},q)_{\theta\in\Theta}$ for
an arbitrary compact space $\Theta$ (see Dumitru [12] for the Hutchinson-
Barnsley theory for such infinite systems or Lukawska [17] for infinite
countable ones. Other related articles are [15, 19, 27, 28, 26, 7]). We point
out that we do not use partitions to construct our entropy, only an integral
formulation, and the variational principle is taken over the holonomic
measures which enclose the invariant ones.

The structure of the paper is the following: in Section 2, we present the
basic definitions on IFS with measures (IFSm) and a fundamental result about
the eigenspace associated to the maximal eigenvalue of the transfer operator.
We also prove the existence of a positive eigenfunction for the transfer
operator associated to the spectral radius and give a constructive proof of
the existence of equilibrium states. In Section 3 we define the Markov
operator, which in the case of a normalized IFSm gives the evolution of the
distribution of the associated Markov process, and show that the set of
eigenmeasures for it is non-empty. In Section 4, we introduce holonomic
measures, which play the role of invariant measures in the IFS setting. In
Section 5 we define entropy for a IFSm, the topological pressure of a given
potential function, as well as the concept of equilibrium states. In a remark
in the end of this section we show how the classical thermodynamical formalism
for a dynamical system is a particular case of the IFSm Thermodynamic
Formalism. In Section 6 a uniqueness result for the equilibrium states is
obtained. Finally, in Section 7 we present a possible application in economic
theory of the theory developed in the previous sections.

##  2 IFSm and Transfer Operator

In this section we set up the basic notation and present a fundamental result
about the eigenspace associated to the maximal eigenvalue (or spectral radius)
of the transfer operator.

In this paper $\textbf{X},\Theta$ are compact metric spaces, equipped with
$\mathscr{B}(\textbf{X})$ and $\mathscr{B}(\Theta)$ respectively the Borel
$\sigma$-algebra for X and $\Theta$.

The Banach space of all real continuous functions equipped with supremum norm
is denoted by $C(\textbf{X},\mathbb{R})$. Its topological dual, as usual, is
identified with $\mathscr{M}_{s}(\textbf{X})$, the space of all finite Borel
signed measures endowed with total variation norm. We use the notation
$\mathscr{M}_{1}(X)$ for the set of all Borel probability measures over $X$
supplied with the weak-$*$ topology. Since we are assuming that $X$ is a
compact metric space then we have that the topological space
$\mathscr{M}_{1}(X)$ is compact and metrizable.

Take $\text{q}\hskip 0.40002pt={(\text{q}\hskip
0.40002pt_{x})}_{x\in\textbf{X}}$ a collection of measures on
$\mathscr{B}(\Theta)$, such that

  1. (q 1)

$\displaystyle\sup\text{q}\hskip 0.40002pt\equiv\sup_{x\in X}\text{q}\hskip
0.40002pt_{x}(\Theta)<\infty$,

  2. (q 2)

$\displaystyle\inf\text{q}\hskip
0.40002pt\equiv\inf_{x\in\textbf{X}}\text{q}\hskip 0.40002pt_{x}(\Theta)>0$,

  3. (q 3)

$\displaystyle x\mapsto\text{q}\hskip 0.40002pt_{x}(A)$ is a Borel map, i.e,
is $\mathscr{B}(\textbf{X})$-measurable for all fixed
$A\in\mathscr{B}(\Theta)$,

  4. (q 4)

$\displaystyle x\mapsto\text{q}\hskip 0.40002pt_{x}$ is weak∗-continuous.

An Iterated Function System with measures $q$, IFSm for short, is a triple
$\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$, where $\tau={(\tau_{\theta})}_{\theta\in\Theta}$ is a collection
of functions from X to itself with the following property

  1. ($\tau$1)

$\tau:(\Theta,\textbf{X})\mapsto\textbf{X}$, where
$\tau(\theta,x)=\tau_{\theta}(x)$ is continuous.

The $\mathcal{R}_{\text{q}\hskip 0.28001pt}$ is said to be normalized if for
all $x\in\textbf{X}$, $\text{q}\hskip 0.40002pt_{x}$ is a probability measure.

######  Definition 2.1.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$ be an IFSm. The Transfer Operator $\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}:C(\textbf{X},\mathbb{R})\righttoleftarrow$ associated to
$\mathcal{R}_{\text{q}\hskip 0.28001pt}$ is defined by:

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)(x)=\int_{\Theta}f(\tau_{\theta}(x))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta),\qquad\forall x\in X.$

$\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}$ is well defined. In
fact, $\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}$ is continuous once
that

$\left\lVert\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)\right\rVert_{\infty}=\sup_{x}\left\lvert\int\,f(\tau_{\theta}(x))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\right\rvert\leq\sup\text{q}\hskip 0.40002pt\left\lVert
f\right\rVert_{\infty}<\infty.$

Futhermore, for a fixed $f\in C(\textbf{X},\mathbb{R})$ and $x\in\textbf{X}$,
given $\varepsilon>0$, take $\delta>0$ s.t.

$\sup_{\theta\in\Theta}d(f(\tau_{\theta}(x)),f(\tau_{\theta}(y)))<\frac{\varepsilon}{2\sup\text{q}\hskip
0.40002pt},$

and

$\left\lvert\int_{\Theta}f(\tau_{\theta}(x))d\text{q}\hskip
0.40002pt_{x}(\theta)-\int_{\Theta}f(\tau_{\theta}(x))d\text{q}\hskip
0.40002pt_{y}(\theta)\right\rvert<\frac{\varepsilon}{2},$

for all $y\in\textbf{X}$ with $d(x,y)<\delta$. Then,

$\displaystyle|$ $\displaystyle\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)(x)-\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)(y)|=\left\lvert\int_{\Theta}f(\tau_{\theta}(x))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)-\int_{\Theta}f(\tau_{\theta}(y))\,\text{d}\text{q}\hskip
0.40002pt_{y}(\theta)\right\rvert$ $\displaystyle\leq\int_{\Theta}\left\lvert
f(\tau_{\theta}(x))-f(\tau_{\theta}(y))\right\rvert\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)+\left\lvert\int_{\Theta}f(\tau_{\theta}(y))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)-\int_{\Theta}f(\tau_{\theta}(y))\,\text{d}\text{q}\hskip
0.40002pt_{y}(\theta)\right\rvert$
$\displaystyle<\frac{\varepsilon}{2\sup\text{q}\hskip
0.40002pt}\int_{\Theta}\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)+\frac{\varepsilon}{2}=\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$

This shows that, for $f\in C(\textbf{X},\mathbb{R})$ and $x\in\textbf{X}$,
given $\varepsilon>0$, there is $\delta>0$ such that for every
$d(x,y)<\delta$, $\left\lvert\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)(x)-\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)(y)\right\rvert<\epsilon$, therefore $\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(f)(x)$ is continuous.

When dealing with the spectral radius of a positive operator, via Gelfand’s
formula, we will need to consider the norm of iterates of such operator, which
is defined by a supremum which is attained in the constant function, as we
will see in the proof of Theorem 2.5. For this reason we will need the
following proposition:

######  Proposition 2.2.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$ be a IFSm. Then for the $N$-th iteration of $\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}$ we have

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)(x)=\int_{\Theta^{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})$

where,

$\displaystyle\,\text{dP}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\equiv\prod_{j=1}^{N}\,\text{d}\text{q}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})$, $x_{0}=x$ and
$x_{j+1}=\tau_{\theta_{j}}x_{j}$.

Proof. This expression can be obtained by proceeding a formal induction on
$N$. For $N=2$ and $x=x_{0}$, we have

$\displaystyle\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{2}(1)(x_{0})$ $\displaystyle=\int_{\Theta}\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}(1)(\tau_{\theta_{0}}(x_{0}))\,\text{d}\text{q}\hskip
0.40002pt_{x_{0}}(\theta_{0})$
$\displaystyle=\int_{\Theta}\int_{\Theta}\,\text{d}\text{q}\hskip
0.40002pt_{x_{1}}(\theta_{1})\,\text{d}\text{q}\hskip
0.40002pt_{x_{0}}(\theta_{0})$
$\displaystyle=\int_{\Theta^{2}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip 0.07996pt(\theta_{0},\theta_{1}).$

And, if

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)(x)=\int_{\Theta^{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1}),$

then

$\displaystyle\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}^{N+1}(1)(x)$
$\displaystyle=\int_{\Theta}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)(x_{1})\,\text{d}\text{q}\hskip 0.40002pt(\theta_{0})$
$\displaystyle=\int_{\Theta}\int_{\Theta^{N}}\,\text{dP}_{x_{1}}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{1},\ldots,\theta_{N})\,\text{d}\text{q}\hskip
0.40002pt(\theta_{0})$
$\displaystyle=\int_{\Theta}\cdots\int_{\Theta}\left(\prod_{j=0}^{N-1}\,\text{d}\text{q}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})\right)\,\text{d}\text{q}\hskip
0.40002pt(\theta_{0})$
$\displaystyle=\int_{\Theta}\cdots\int_{\Theta}\left(\prod_{j=1}^{N+1}\,\text{d}\text{q}\hskip
0.40002pt_{x_{N+1-j}}(\theta_{N+1-j})\right)$
$\displaystyle=\int_{\Theta^{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots\,\theta_{N}).$

$\blacksquare$

######  Remark 2.3.

The formal notation used for $\,\text{P}_{x}^{\hskip 0.85355pt\text{q}\hskip
0.28001pt}\hskip 0.40002pt$, in fact, means that $\,\text{P}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip 0.40002pt$ is a measure in
$\Theta^{N}$ defined by,

$\,\text{P}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.40002pt(\Theta_{0}\times\cdots\times\Theta_{N-1})=\int_{\Theta_{0}}\cdots\int_{\Theta_{N-1}}\,\text{d}\text{q}\hskip
0.40002pt_{x_{N-1}}(\theta_{N-1})\cdots\,\text{d}\text{q}\hskip
0.40002pt_{x_{0}}(\theta_{0}).$

In the case $N=2$ for instance,

$\displaystyle\,\text{P}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.40002pt(\Theta_{0}\times\Theta_{1})$
$\displaystyle=\int_{\Theta_{0}}\int_{\Theta_{1}}\,\text{d}\text{q}\hskip
0.40002pt_{\tau_{\theta_{0}}x}(\theta_{1})\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta_{0})$ $\displaystyle=\int_{\Theta_{0}}\text{q}\hskip
0.40002pt_{\tau_{\theta_{0}}x}(\Theta_{1})\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta_{0}).$

Note that $\text{q}\hskip 0.40002pt_{\tau_{\theta_{0}}(x_{0})}(\Theta_{1})$,
with fixed $\Theta_{1}$ and $x_{0}$, is a function of $\theta_{0}$ that is
measurable: indeed, if $A\in\mathscr{B}(\Theta)$,
$f_{A}:\textbf{X}\to\mathbb{R}$ defined by $f_{A}(x)=q_{x}(A)$ is measurable
by (q 3) and by ($\tau$1) implies $\tau$ is measurable. Thus,
$F_{A}:=f_{A}\circ\tau$ is measurable.

######  Proposition 2.4.

If $f:\textbf{X}\to\mathbb{R}$ is a measurable nonnegative function, then

$H(x):=\int_{\Theta_{0}}f\circ\tau(\theta,x)\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta),$

is measurable.

Using Proposition 2.4, it is a simple induction to prove that

$x\mapsto\,\text{P}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.40002pt(\Theta_{0}\times\cdots\times\Theta_{N-1})=\int_{\Theta_{0}}\,\text{P}_{\tau_{\theta_{0}}x_{0}}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.40002pt(\Theta_{1}\times\cdots\times\Theta_{N-1})\,\text{d}\text{q}\hskip
0.40002pt_{x_{0}}(\theta_{0})$

is measurable for any $\Theta_{i}\in\mathcal{B}(\Theta)$.

In this way we conclude that $\,\text{P}_{x}^{\hskip 0.85355pt\text{q}\hskip
0.28001pt}\hskip 0.40002pt$ is well defined for each space $\Theta^{N}$.

######  Theorem 2.5.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$ be a IFSm and suppose that there are a positive number $\rho$ and
a strictly positive continuous function $h:\textbf{X}\to\mathbb{R}$ such that
$\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}(h)=\rho h$. Then the
following limit exits

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\ln\left(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}^{N}(1)(x)\right)=\ln\rho$ (5)

the convergence is uniform in $x$ and $\rho=\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt})$ is the spectral radius of
$\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}$ acting on
$C(X,\mathbb{R})$.

Proof. From the hypothesis we can build a normalized IFSm
$\mathcal{R}_{\text{p}\hskip 0.28001pt}=(\textbf{X},\tau,\text{p}\hskip
0.40002pt)$ where

$\,\text{d}\text{p}\hskip
0.40002pt_{x}(\theta)=\frac{h(\tau_{\theta}(x))}{\rho
h(x)}\,\text{d}\text{q}\hskip 0.40002pt_{x}(\theta).$

Note that $\,\text{dP}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt$ and $\,\text{dP}_{x}^{\hskip 0.85355pt\text{p}\hskip
0.28001pt}\hskip 0.07996pt$ are related in the following way

$\displaystyle\,\text{dP}_{x}^{\hskip 0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})$
$\displaystyle=\prod_{j=1}^{N}\,\text{d}\text{q}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})$ $\displaystyle=\prod_{j=1}^{N}\frac{\rho
h(x_{N-j})}{h(\tau_{\theta_{N-j}}(x_{N-j}))}\,\text{d}\text{p}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})$
$\displaystyle=\rho^{N}\prod_{j=1}^{N}\frac{h(x_{N-j})}{h(x_{N-j+1})}\,\text{d}\text{p}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})$
$\displaystyle=\rho^{N}\frac{h(x_{0})}{h(x_{N})}\prod_{j=1}^{N}\,\text{d}\text{p}\hskip
0.40002pt_{x_{N-j}}(\theta_{N-j})$
$\displaystyle=\rho^{N}\frac{h(x_{0})}{h(x_{N})}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{n-1})$

where $x_{0}=x$ and $x_{j+1}=\tau_{\theta_{j}}x_{j}$.

Since X is compact and $h$ is a strictly positive continuous function, we have
constants $0<a<1<b$, independent of $x$, such that

$a\leq h(x_{0})/h(x_{N})\leq b.$

Using the Proposition 2.2 and the above inequalities, we obtain for any fixed
$N\in\mathbb{N}$ the following expression

$\displaystyle\frac{1}{N}\ln(\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)(x))$
$\displaystyle=\frac{1}{N}\ln\left(\int_{\Theta^{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{q}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle=\frac{1}{N}\ln\left(\int_{\Theta^{N}}\rho^{N}\frac{h(x_{0})}{h(x_{N})}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle=\ln\rho+\frac{1}{N}\ln\left(\int_{\Theta^{N}}\frac{h(x_{0})}{h(x_{N})}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right).$

Furthermore,

$\displaystyle\frac{1}{N}\ln\left(\int_{\Theta^{N}}\frac{h(x_{0})}{h(x_{N})}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle\geq\frac{1}{N}\ln\left(\int_{\Theta^{N}}a\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle=\frac{1}{N}\ln
a+\frac{1}{N}\ln\int_{\Theta_{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip 0.07996pt$
$\displaystyle=\frac{1}{N}\ln a\xrightarrow{\,N\to\infty\,}0$

and

$\displaystyle\frac{1}{N}\ln\left(\int_{\Theta^{N}}\frac{h(x_{0})}{h(x_{N})}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle\leq\frac{1}{N}\ln\left(\int_{\Theta^{N}}b\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip
0.07996pt(\theta_{0},\ldots,\theta_{N-1})\right)$
$\displaystyle=\frac{1}{N}\ln
b+\frac{1}{N}\ln\int_{\Theta_{N}}\,\text{dP}_{x}^{\hskip
0.85355pt\text{p}\hskip 0.28001pt}\hskip 0.07996pt$
$\displaystyle=\frac{1}{N}\ln b\xrightarrow{\,N\to\infty\,}0.$

Therefore, for every $N\geq 1$ we have

$\sup_{x\in\textbf{X}}\left\lvert\frac{1}{N}\ln\left(\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)(x)\right)-\ln\rho\right\rvert\leq\frac{\ln b-\ln a}{N},$

which proves (5). Now using Gelfand’s formula for the spectral radius and the
fact that, for a positive operator $T$ defined on $C(X,\mathbb{R})$ we have
$\left\lVert T\right\rVert=\left\|T(1)\right\|_{\infty}$, where $\left\lVert
T\right\rVert$ denotes the usual norm operator $\left\lVert
T\right\rVert=\sup_{\\{\|f\|_{\infty}\leq 1\\}}\|T(f)\|_{\infty}$, we have

$\displaystyle\left\lvert\ln\rho(\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt})-\ln\rho\right\rvert$
$\displaystyle=\left\lvert\ln\left(\limsup_{N\to\infty}\left\lVert\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}^{N}\right\rVert^{\frac{1}{N}}\right)-\ln\rho\right\rvert=\limsup_{N\to\infty}\left\lvert\frac{1}{N}\ln\left\lVert\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}^{N}\right\rVert-\ln\rho\right\rvert$
$\displaystyle=\limsup_{N\to\infty}\left\lvert\frac{1}{N}\ln\left(\left\|\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)\right\|_{\infty}\right)-\ln\rho\right\rvert$
$\displaystyle\leq\limsup_{N\to\infty}\ \
\sup_{x\in\textbf{X}}\left\lvert\frac{1}{N}\ln\left(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}^{N}(1)(x)\right)-\ln\rho\right\rvert$
$\displaystyle\leq\limsup_{N\to\infty}\frac{\ln b-\ln a}{N}=0.$

$\blacksquare$

We will now address the question of existence of positive eigenfunctions for
the transfer operator associated to the spectral radius, and give a
constructive proof of the existence of equilibrium states.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$ assuming that there is $\mu$ a probability on $\Theta$ s.t.
$\forall x\in\textbf{X}$, $\text{q}\hskip 0.40002pt_{x}\ll\mu$ and
$J:\Theta\times\textbf{X}\to\mathbb{R}$, defined by
$J(\theta,x):=\frac{\,\text{d}\text{q}\hskip
0.28001pt_{x}}{\,\text{d}\mu}(\theta)$, a continuous function. Define
$u(x,\theta)=\log J(\theta,x)$, and consider a parametric family of variable
discount functions $\delta_{n}:[0,+\infty)\to\mathbb{R}$, where
$\delta_{n}(t)\to I(t)=t$, when $n\to\infty$, pointwise and the normalized
limits $\lim_{n}w_{n}(x)-\max w_{n}$ of the fixed points

$w_{n}(x):=\log\int_{\Theta}e^{u(\theta,x)+\delta_{n}(w_{n}(\tau(\theta,x)))}\,\text{d}\mu=\log\int_{\Theta}e^{\delta_{n}(w_{n}(\tau(\theta,x)))}\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)$

for a discounted transfer operator (see[9], Definition 3.2 for the operator
definition and Theorem 3.6 for the existence of the fixed points) associated
to a variable discount decision-making process. We now check that the
requirements in [9] are fulfilled in our setting. Consider
$S_{n}:=(\textbf{X},\Theta,\Psi,\tau,u,\delta_{n})$ where $\Psi(x)=\Theta$ for
all $x\in\textbf{X}$ and the sequence $(\delta_{n})$ satisfies the
admissibility conditions:

  1. 1.

the contraction modulus $\gamma_{n}$ of $\delta_{n}$ is also a variable
discount function;

  2. 2.

$\delta_{n}(0)=0$ and $\delta_{n}(t)\leq t$ for any $t\in(0,+\infty)$;

  3. 3.

for any fixed $\alpha>0$ we have $\delta_{n}(t+\alpha)-\delta_{n}(t)\to\alpha$
when $n\to\infty$, uniformly in $t>0$.

######  Theorem 2.6.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}$ and $(\delta_{n})$ in above
conditions such that the above defined $u$ satisfies:

  1. 1.

$u$ is uniformly $\delta$-bounded for $(\delta_{n})$;

  2. 2.

$u$ is uniformly $\delta$-dominated for $(\delta_{n})$.

Then there exists a positive and continuous eigenfunction $h$ such that
$\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}(h)=\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt})h$.

Proof. Theorem 3.28 of [9] implies that there exists $k\in[0,\left\lVert
u\right\rVert_{\infty}]$ and $\varphi(x):=e^{h(x)}$ continuous and positive
function with

$e^{k}\varphi(x)=\int_{\Theta}\varphi\circ\tau(\theta,x)\,e^{u(x,\theta)}\,\text{d}\mu(\theta)=\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(\varphi)(x),$

for all $x\in\textbf{X}$. Now use the Theorem 2.5 and the theorem is proven.
$\blacksquare$

##  3 Markov Operator and its Eigenmeasures

In this section we define the Markov Operator, which in the case of a
normalized IFSm gives the evolution of the distribution of the associated
Markov Process, and show that the set of eigenmeasures for it is non-empty.

######  Definition 3.1.

The Markov Operator $\mathcal{L}_{\text{q}\hskip
0.28001pt}:\mathscr{M}_{s}(X)\righttoleftarrow$ is the unique bounded linear
operator satisfying

$\int_{\textbf{X}}f\,\text{d}[\mathcal{L}_{q}(\mu)]=\int_{\textbf{X}}\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(f)\,\text{d}\mu,$

for all $\mu\in\mathscr{M}_{s}(X)$ and $f\in C(\textbf{X},\mathbb{R})$.

In the case of a normalized IFSm, we can consider the Markov Process
$\\{Z_{j},j\geq 0\\}$ with initial distribution $Z_{0}\sim\mu_{0}$, where
$\mu_{0}\in\mathscr{M}_{1}(X)$, and $Z_{j+1}=\tau_{{\theta}_{j}}(Z_{j})$ for
$j\geq 0$, where ${{\theta}_{j}}\sim q_{Z_{j}}$. Then, if $Z_{j}\sim\mu_{j}$,
we have $\mu_{j+1}=\mathcal{L}_{\text{q}\hskip 0.28001pt}(\mu_{j})$.

######  Theorem 3.2.

Let $\mathcal{R}_{\text{q}\hskip 0.28001pt}=(\textbf{X},\tau,\text{q}\hskip
0.40002pt)$ be a IFSm. Then there exists a positive number
$\rho\leq\rho(\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt})$ such that
the set $\mathcal{G}^{*}(\text{q}\hskip
0.40002pt)=\\{\nu\in\mathscr{M}_{1}(X):\mathcal{L}_{q}\nu=\rho\nu\\}$ is not
empty.

Proof. Notice that the mapping

$\mathscr{M}_{1}(X)\ni\gamma\mapsto\frac{\mathcal{L}_{q}(\gamma)}{\mathcal{L}_{q}(\gamma)(X)}$

sends $\mathscr{M}_{1}(X)$ to itself. From its convexity and compactness, in
the weak topology which is Hausdorff when $X$ is metric and compact, it
follows from the continuity of $\mathcal{L}_{q}$ and the Tychonov-Schauder
Theorem that there is at least one probability measure $\nu$ satisfying
$\mathcal{L}_{q}(\nu)=(\mathcal{L}_{q}(\nu)(X))\,\nu$.

We claim that

$\displaystyle\inf_{x\in\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)\leq\mathcal{L}_{q}(\gamma)(\textbf{X})\leq\sup_{x\in\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)$ (6)

for every $\gamma\in\mathscr{M}_{1}(X)$.

Indeed,

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(1)(x)=\int_{\Theta}1\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=\text{q}\hskip 0.40002pt_{x}(\Theta),$
$\mathcal{L}_{\text{q}\hskip
0.28001pt}(\gamma)(\textbf{X})=\int_{\textbf{X}}1\,\text{d}[\mathcal{L}_{\text{q}\hskip
0.28001pt}\gamma]=\int_{\textbf{X}}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(1)\,\text{d}\gamma=\int_{\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)\,\text{d}\gamma(x),$
$0<\inf_{x\in\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)\leq\int_{\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)\,\text{d}\gamma(x)\leq\sup_{x\in\textbf{X}}\text{q}\hskip
0.40002pt_{x}(\Theta)<\infty.$

From the inequality $\eqref{des1-exi-auto-medida}$ it follows that

$0<\rho\equiv\sup\\{\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu)(X):\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu)=(\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu)(X))\,\nu\\}<+\infty.$

By a compactness argument one can show the existence of
$\nu\in\mathscr{M}_{1}(\textbf{X})$ so that $\mathcal{L}_{\text{q}\hskip
0.28001pt}\nu=\rho\nu$. Indeed, let ${(\nu_{n})}_{n\in\mathbb{N}}$ be a
sequence such that $\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu_{n})(\textbf{X})\uparrow\rho$, when $n$ goes to infinity. Since
$\mathscr{M}_{1}(X)$ is compact metric space in the weak topology we can
assume, up to subsequence, that $\nu_{n}\rightharpoonup\nu$. This convergence
together with the continuity of $\mathcal{L}_{\text{q}\hskip 0.28001pt}$
provides

$\mathcal{L}_{\text{q}\hskip
0.28001pt}\nu=\lim_{n\to\infty}\mathcal{L}_{\text{q}\hskip
0.28001pt}\nu_{n}=\lim_{n\to\infty}\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu_{n})(X)\nu_{n}=\rho\,\nu,$

thus showing that the set
$\mathcal{G}^{*}({q})\equiv\\{\nu\in\mathscr{M}_{1}(X):\mathcal{L}_{\text{q}\hskip
0.28001pt}\nu=\rho\,\nu\\}\neq\emptyset$.

To finish the proof we observe that by using any
$\nu\in\mathcal{G}^{*}(\text{q}\hskip 0.40002pt)$, we get the following
inequality

$\displaystyle\rho^{N}=\int_{X}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}^{N}(1)\,\text{d}\nu\leq\left\lVert\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}^{N}\right\rVert.$

From this inequality and Gelfand’s Formula, it follows that
$\rho\leq\rho(B_{{q}})$. $\blacksquare$

##  4 Holonomic Measure and Disintegrations

Now we introduce holonomic measures, which play the role of invariant measures
in the IFS setting.

An invariant measure for a classical dynamical system
$T:\textbf{X}\righttoleftarrow$ on a compact space is a measure $\mu$
satisfying for all $f\in C(X,\mathbb{R})$

$\int_{\textbf{X}}f(T(x))\,\text{d}\mu=\int_{\textbf{X}}f(x)\,\text{d}\mu,\quad\text{equivalently}\quad\int_{\textbf{X}}f(T(x))-f(x)\,\text{d}\mu=0.$

From the Ergodic Theory point of view the natural generalization of this
concept for an IFS $\mathcal{R}=(\textbf{X},\tau)$ is the concept of holonomy.

Consider the cartesian product space $\Omega\equiv\textbf{X}\times\Theta$ and
for each $f\in C(X,\mathbb{R})$ its “$\Theta$-differential”
$\text{d}f:\Omega\to\mathbb{R}$ defined by $[\text{d}_{x}f](\theta)\equiv
f(\tau_{\theta}(x))-f(x)$.

######  Definition 4.1.

A measure $\hat{\mu}$ over $\Omega$ is said holonomic, with respect to an IFS
$\mathcal{R}$ if for all $f\in C(X,\mathbb{R})$ we have

$\displaystyle\int_{\Omega}[\text{d}_{x}f](\theta)\,d\hat{\mu}(x,\theta)=0.$

Notation,

$\displaystyle\mathcal{H}\left(\mathcal{R}\right)\equiv\\{\hat{\mu}\,|\,\hat{\mu}\text{
is a holonomic probability measure with respect to}\ \mathcal{R}\\}.$

Since $\Omega$ is compact the set of all holonomic probability measures is
obviously convex and compact. It is also not empty because $\Omega$ is compact
and any average

$\hat{\mu}_{N}\equiv\frac{1}{N}\sum_{j=0}^{N-1}\delta_{(x_{j},{\theta}_{j})},$

where $x_{j+1}=\tau_{{\theta}_{j}}(x_{j})$ and $x_{0}\in\textbf{X}$ is fixed,
will have their cluster points in $\mathcal{H}\left(\mathcal{R}\right)$.
Indeed, for all $N\geq 1$ we have the following identity

$\displaystyle\int_{\Omega}[\text{d}_{x}f](\theta)\,d\hat{\mu}_{N}(x,\theta)$
$\displaystyle=\frac{1}{N}\sum_{j=0}^{N-1}[\text{d}_{x_{j}}f](\theta_{j})=\frac{1}{N}(f(\tau_{\theta_{N-1}}(x_{N-1}))-f(x_{0})).$

From the above expression is easy to see that if $\hat{\mu}$ is a cluster
point of the sequence ${(\hat{\mu}_{N})}_{N\geq 1}$, then there is a
subsequence ${(N_{k})}_{k\to\infty}$ such that

$\displaystyle\int_{\Omega}[\text{d}_{x}f](\theta)\,d\hat{\mu}(x,\theta)$
$\displaystyle=\lim_{k\to\infty}\int_{\Omega}[\text{d}_{x}f](\theta)\,d\hat{\mu}_{N_{k}}(x,\theta)$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}(f(x_{N_{k}})-f(x_{0}))=0.$

######  Theorem 4.2 (Disintegration).

Let $X$ and $Y$ be compact metric spaces, $\hat{\mu}:\mathscr{B}(Y)\to[0,1]$ a
Borel probability measure, $T:Y\to X$ a Borel mensurable function and for each
$A\in\mathscr{B}(X)$ define a probability measure
$\mu(A)\equiv\hat{\mu}(T^{-1}(A))$. Then there exists a family of Borel
probability measures ${(\mu_{x})}_{x\in X}$ on $Y$, uniquely determined
$\mu$-a.e, such that

  1. 1.

$\mu_{x}(Y\backslash T^{-1}(x))=0$, $\mu$-a.e;

  2. 2.

$\displaystyle\int_{Y}f\,d\hat{\mu}=\int_{X}\left[\int_{T^{-1}(x)}\\!\\!\\!f(y)\
d\mu_{x}(y)\right]d\mu(x)$.

This decomposition is called the disintegration of $\hat{\mu}$, with respect
to $T$.

Proof. For a proof of this theorem, see [10] p.78 or [2], Theorem 5.3.1.
$\blacksquare$

In this paper we are interested in disintegrations in cases where $Y$ is the
Cartesian product $\Omega\equiv\textbf{X}\times\Theta$ and
$T:\Omega\to\textbf{X}$ is the projection on the first coordinate. In such
cases if $\hat{\mu}$ is any Borel probability measure on $\Omega$, then
follows from the first conclusion of Theorem 4.2 that the disintegration of
$\hat{\mu}$ provides for each $x\in\textbf{X}$ a unique probability measure
$\mu_{x}$ ($\mu$-a.e.) supported on $\\{x\\}\times\Theta$. So we can write the
disintegration of $\hat{\mu}$ as
$\text{d}\hat{\mu}(x,\theta)=\text{d}\mu_{x}(\theta)\text{d}\mu(x)$, where
here we are abusing notation identifying $\mu_{x}(\\{x\\}\times A)$ with
$\mu_{x}(A)$.

Consider any IFSm $\mathcal{R}_{\text{q}\hskip
0.28001pt}=(\textbf{X},\tau,\text{q}\hskip 0.40002pt)$ and a probability
$\nu\in\mathscr{M}_{1}(\textbf{X})$ such that the Markov operator associated
to the IFSm $\mathcal{R}_{\text{q}\hskip 0.28001pt}$ satisfies

$\mathcal{L}_{\text{q}\hskip 0.28001pt}(\nu)=\nu,$

then it is possible to define a holonomic probability measure
$\hat{\mu}\in\mathcal{H}\left(\mathcal{R}\right)$ given by
$\text{d}\hat{\mu}(x,\theta)=\text{d}q_{x}(\theta)\,\text{d}\nu(x)$. Indeed,
fixed $f\in C(X,\mathbb{R})$, we obtain

$\int_{\Omega}[\text{d}_{x}f](\theta)\,\text{d}\hat{\mu}(x,\theta)=0\Longleftrightarrow\int_{\Omega}f(\tau_{\theta}(x))\,\text{d}\hat{\mu}(x,\theta)=\int_{\Omega}f(x)\,\text{d}\hat{\mu}(x,\theta).$

But,

$\int_{\textbf{X}}\int_{\Theta}f(\tau_{\theta}(x))\,\text{d}q_{x}(\theta)\text{d}\nu(x)=\int_{\textbf{X}}\int_{\Theta}f(x)\,\text{d}q_{x}(\theta)\text{d}\nu(x).$

if, and only if,

$\int_{\textbf{X}}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(f)\,\text{d}\nu=\int_{\textbf{X}}f\,d\nu,\;\forall f\in
C(X,\mathbb{R}).$

Which is equivalent to $\mathcal{L}_{\text{q}\hskip 0.28001pt}(\nu)=\nu$.
Since any disintegration $\hat{\mu}$ has a marginal $\mu_{x}=q_{x}$,
$\nu-a.e.$, this Borel probability measure $\hat{\mu}$ on $\Omega$, will be
called a holonomic lifting of $\nu$, with respect to
$\mathcal{R}_{\text{q}\hskip 0.28001pt}$.

##  5 Entropy and Pressure for IFSm

We now define two concepts of entropy, compare then, show sufficient
conditions for them to be equal and introduce the topological pressure of a
given potential, as well as the concept of equilibrium states. We show in this
section a first result on the existence of equilibrium states. In all that
follows, the a priori measure has a special role (see [23]).

As in the previous section the mapping $T:\Omega\to X$ denotes the projection
on the first coordinate. Even when not explicitly mentioned, any
disintegrations of a probability measure $\hat{\nu}$, defined over $\Omega$,
will be from now on considered with respect to $T$.

######  Definition 5.1 (Variational Entropy).

Let $\mathcal{R}$ be an IFS,
$\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$, $\mu$ a probability on
$\Theta$ and $d\hat{\nu}(x,\theta)=d\nu_{x}(\theta)d\nu(x)$ a disintegration
of $\hat{\nu}$, with respect to $T$. The variational entropy of $\hat{\nu}$
with a priori probability $\mu$ is defined by

$h_{\text{v}}^{\mu}(\hat{\nu})\equiv\inf_{\begin{subarray}{c}g\,\in\,C(\textbf{X},\mathbb{R})\\\
g>0\end{subarray}}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(g)}{g}\,\text{d}\nu\right\\}.$

######  Definition 5.2.

When $\text{q}\hskip 0.40002pt={(\text{q}\hskip
0.40002pt_{x})}_{x\in\textbf{X}}$ is a family of measures on $\Theta$ and
$\mu$ a probability on $\Theta$, and $\nu$ is a probability on X, we say that
the family q is $\nu$-almost everywhere absolutely continuous with respect to
$\mu$ when $\text{q}\hskip 0.40002pt_{x}\ll\mu$ for $\nu$-almost everywhere
$x$ on X and write $\text{q}\hskip 0.40002pt\ll_{\nu}\mu$.

If $\text{q}\hskip 0.40002pt\ll_{\nu}\mu$, we define $J_{x}(\theta)$ such that
$J_{x}=\frac{\,\text{d}\text{q}\hskip 0.28001pt_{x}}{\,\text{d}\mu}$ when
$\text{q}\hskip 0.40002pt_{x}\ll\mu$ and $J_{x}(\theta)=0$ otherwise.

######  Definition 5.3 (Average entropy).

Let $\mathcal{R}$ be an IFS,
$\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$,
$d\hat{\nu}(x,\theta)=d\nu_{x}(\theta)d\nu(x)$ a disintegration of $\hat{\nu}$
with respect to $T$ and $\mu$ a probability on $\Theta$ such that
$(\nu_{x})\ll_{\nu}\mu$. The average entropy of $\hat{\nu}$ with respect to
$\mu$ is defined by

$h_{\text{a}}^{\mu}(\hat{\nu})\equiv-\int_{\Omega}\ln
J_{x}(\theta)\,\text{d}\nu_{x}(\theta)\,\text{d}\nu(x).$

######  Definition 5.4 (Optimal Function).

Let $\mathcal{R}$ be an IFS,
$\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$,
$\text{d}\hat{\nu}(x,\theta)=\text{d}\nu_{x}(\theta)\,\text{d}\nu(x)$ a
disintegration of $\hat{\nu}$ with respect to $T$ and $\mu$ a probability on
$\Theta$ such that $(\nu_{x})\ll_{\nu}\mu$. We say that a positive function
$g\in C(\textbf{X},\mathbb{R})$ is optimal, with respect to the system of
measures associated to the desintegration
$\text{d}\hat{\nu}(x,\theta)=\text{d}\nu_{x}(\theta)\,\text{d}\nu(x)$ if we
have

$J_{x}(\theta)=\frac{g(\tau_{\theta}(x))}{\text{B}\hskip
0.40002pt_{\mu}(g)(x)}.$

######  Proposition 5.5.

Let $\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=Q_{x}(\theta)\,\text{d}\mu(\theta)$ and
$dp_{x}(\theta)=P_{x}(\theta)\,\text{d}\mu(\theta)$ be probabilities, and
suppose $P$ is continuous, positive and bounded away from zero, while $Q$ is a
positive function which is integrable with respect to
$\,\text{d}\text{q}\hskip 0.40002pt_{x}(\theta)\,\text{d}\nu(x)$. Then we have

$-\int_{\textbf{X}}\int_{\Theta}\log(Q_{x}(\theta))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)\leq-\int_{\textbf{X}}\int_{\Theta}\log(P_{x}(\theta))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x).$

Proof. Using Jensen’s Inequality on $f(x)=-x\log(x)$, which is a concave
function, we have

$\displaystyle\int_{\textbf{X}}\int_{\Theta}-\log$
$\displaystyle\left(\frac{Q_{x}(\theta)}{P_{x}(\theta)}\right)\frac{Q_{x}(\theta)}{P_{x}(\theta)}\,\text{d}\text{p}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)$
$\displaystyle=\int_{\Omega}f\left(\frac{Q_{x}(\theta)}{P_{x}(\theta)}\right)\,\text{d}\text{p}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)$ $\displaystyle\leq
f\left(\int_{\Omega}\,\text{d}\text{p}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)\right)=f(1)=0.$

Then,

$\displaystyle\int_{\textbf{X}}\int_{\Theta}$
$\displaystyle-\log\left(\frac{Q_{x}(\theta)}{P_{x}(\theta)}\right)\frac{Q_{x}(\theta)}{P_{x}(\theta)}P_{x}(\theta)\,\text{d}\mu(\theta)\,\text{d}\nu(x)$
$\displaystyle=\int_{\textbf{X}}\int_{\Theta}-\log\left(\frac{Q_{x}(\theta)}{P_{x}(\theta)}\right)\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)\leq 0.$

Therefore,

$-\int_{\textbf{X}}\int_{\Theta}\log\left(Q_{x}(\theta)\right)\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)\leq-\int_{\textbf{X}}\int_{\Theta}\log\left(P_{x}(\theta)\right)\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x).$

$\blacksquare$

######  Theorem 5.6.

Let $\mathcal{R}$ be an IFS,
$\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$,
$\text{d}\hat{\nu}(x,\theta)=\text{d}\nu_{x}(\theta)\,\text{d}\nu(x)$ a
disintegration of $\hat{\nu}$ with respect to $T$, and $\mu$ a probability on
$\Theta$ such that $(\nu_{x})\ll_{\nu}\mu$. Then

$h_{\text{a}}^{\mu}(\hat{\nu})\leq h_{\text{v}}^{\mu}(\hat{\nu})\leq 0.$

Moreover, if there exists some optimal function $\phi$, with respect to
$\mathcal{R}_{{q}}$, then

$h_{\text{a}}^{\mu}(\hat{\nu})=h_{\text{v}}^{\mu}(\hat{\nu})=\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(\phi)}{\phi}\,\text{d}\nu.$

Proof. From the definition of variational entropy we obtain

$h_{\text{v}}^{\mu}(\hat{\nu})=\inf_{\begin{subarray}{c}g\in
C(\textbf{X},\mathbb{R})\\\
g>0\end{subarray}}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(g)}{g}\,\text{d}\nu\right\\}\leq\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(1)}{1}\,\text{d}\nu=0.$

It remains to show that $h_{\text{a}}^{\mu}(\hat{\nu})\leq
h_{\text{v}}^{\mu}(\hat{\nu})$. Let $g:X\to\mathbb{R}$ be continuous positive
function and define for each $x\in X$ a probability where
$\,\text{d}\text{p}\hskip
0.40002pt_{x}(\theta)=g(\tau_{\theta}(x))/\text{B}\hskip
0.40002pt_{\mu}(g)(x)\,\text{d}\mu(\theta)$. From Proposition 5.5 and the
properties of the holonomic measures we get the following inequalities for any
continuous and positive function $g$:

$\displaystyle h_{\text{a}}^{\mu}(\hat{\nu})$
$\displaystyle=-\int_{\textbf{X}}\int_{\Theta}\ln
J_{x}(\theta)\,\text{d}\text{q}\hskip 0.40002pt_{x}(\theta)\,\text{d}\nu(x)$
$\displaystyle\leq-\int_{\textbf{X}}\int_{\Theta}\ln\left(\frac{g\circ\tau_{\theta}}{\text{B}\hskip
0.40002pt_{\mu}(g)}\right)\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\,\text{d}\nu(x)$
$\displaystyle=-\int_{\textbf{X}}\left[\int_{\Theta}\ln(g\circ\tau_{\theta})\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)-\int_{\Theta}\ln(\text{B}\hskip
0.40002pt_{\mu}(g))\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)\right]\,\text{d}\nu(x)$
$\displaystyle=-\int_{\textbf{X}}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(\ln g)\,\text{d}\nu+\int_{\textbf{X}}\ln(\text{B}\hskip
0.40002pt_{\mu}(g))\,\text{d}\nu$ $\displaystyle=-\int_{\textbf{X}}\ln
g\,\text{d}\nu+\int_{\textbf{X}}\ln(\text{B}\hskip
0.40002pt_{\mu}(g))\,\text{d}\nu$
$\displaystyle=\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(g)}{g}\,\text{d}\nu,$

Therefore, $h_{\text{a}}^{\mu}(\hat{\nu})\leq h_{\text{v}}^{\mu}(\hat{\nu})$.
Furthermore, if $J_{x}(\theta)=\phi\circ\tau_{\theta}(x)/\text{B}\hskip
0.40002pt_{\mu}(\phi)(x)$ for some $\phi>0$ continuous function, then
$h_{\text{a}}^{\mu}(\hat{\nu})=\int_{\textbf{X}}\log\frac{\text{B}\hskip
0.28001pt_{\mu}(\phi)}{\phi}=h_{\text{v}}^{\mu}(\hat{\nu})$. $\blacksquare$

######  Remark 5.7.

We would like to address a very important distinction between
$h_{\text{v}}^{\mu}(\hat{\nu})$ and $h_{\text{a}}^{\mu}(\hat{\nu})$. The first
one, the variational entropy, is to be used at the variational principle and
to define the equilibrium states having no additional requirements except that
$\mu$ is a probability. On the other hand the average entropy is only defined
for holonomic probabilities having a marginal absolutely continuous with
respect to $\mu$ and will not be used in a variational principle. The only
role of this quantity is to be a lower bound to the variational entropy, when
it does exist.

######  Definition 5.8.

Let $\psi:\textbf{X}\to\mathbb{R}$ be a positive continuous function, $\mu$ a
probability on $\Theta$, and
$\mathcal{R}_{\psi}(\textbf{X},\tau,\text{q}\hskip 0.40002pt)$ an IFSm, where
$\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=\psi\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)$. The
topological pressure of $\psi$, with respect to $\mathcal{R}_{\psi}$, is
defined by

$P(\psi)=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g\in
C(\textbf{X};\mathbb{R})\\\
g>0}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(g)}{g}\,\text{d}\nu\right\\},$ (7)

where $\nu:=\pi_{*}\hat{\nu}$ for $\pi:\Omega\to\textbf{X}$ the X projection.

######  Lemma 5.9.

Let $\psi:\textbf{X}\to\mathbb{R}$ be a positive continuous function and
$\mathcal{R}_{\psi}=(\textbf{X},\tau,\text{q}\hskip 0.40002pt)$ be the $IFSm$
defined above, where $\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=\psi\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)$. Then,
the topological pressure of $\psi$ is alternatively given by

$P(\psi)=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\left\\{h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\log\psi\,\text{d}\nu\right\\}.$

Proof. First, note that $\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(g)=\text{B}\hskip 0.40002pt_{\mu}(g\cdot\psi)$ where
$(g\cdot\psi)(x)=g(x)\psi(x)$. In fact

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(g)(x)=\int_{\Theta}g\circ\tau_{\theta}(x)\psi\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)=\int_{\Theta}(g\cdot\psi)\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)=\text{B}\hskip
0.40002pt_{\mu}(g\cdot\psi)(x).$

To finish the proof, we only need to use the pressure’s definition and some
basic properties as follows:

$\displaystyle P(\psi)$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g>0}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(g)}{g}\,\text{d}\nu\right\\},$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g>0}\left\\{\int_{\textbf{X}}\log\psi\,\text{d}\nu-\int_{\textbf{X}}\log\psi\,\text{d}\nu+\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(g)}{g}\,\text{d}\nu\right\\}$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g>0}\left\\{\int_{\textbf{X}}\log\psi\,\text{d}\nu+\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(g\cdot\psi)}{g\cdot\psi}\,\text{d}\nu\right\\}$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\left\\{\int_{\textbf{X}}\log\psi\,\text{d}\nu+\inf_{\tilde{g}>0}\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\mu}(\tilde{g})}{\tilde{g}}\,\text{d}\nu\right\\}$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\left\\{h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\log\psi\,\text{d}\nu\right\\}.$

$\blacksquare$

######  Remark 5.10.

Note that if $\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=\frac{\psi\circ\tau_{\theta}(x)}{\text{B}\hskip
0.28001pt_{\mu}(\psi)(x)}\,\text{d}\mu(\theta)$, by the Theorem 3.2 there
exists $\rho>0$ and $\nu$ s.t. $\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu)=\rho\nu$.

But,

$\displaystyle\rho$
$\displaystyle=\int_{\textbf{X}}\text{d}\mathcal{L}_{\text{q}\hskip
0.28001pt}(\nu)=\int_{\textbf{X}}\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(1)(x)\,\text{d}\nu(x)=\int_{\Omega}\frac{\psi\circ\tau_{\theta}(x)}{\text{B}\hskip
0.40002pt_{\mu}(\psi)(x)}\,\text{d}\mu(\theta)\,\text{d}\nu(x)$
$\displaystyle=\int_{\textbf{X}}{\text{B}\hskip
0.40002pt_{\mu}(\psi)(x)}^{-1}\int_{\Theta}\psi\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)\,\text{d}\nu(x)=\int_{\textbf{X}}\,\text{d}\nu=1.$

Therefore we have

$P(\psi)\geq\sup_{\\{\nu\,|\,\mathcal{L}_{\text{q}\hskip
0.20001pt}(\nu)=\nu\\}}\int_{\textbf{X}}\ln B_{\mu}(\psi)\,\text{d}\nu.$

######  Definition 5.11 (Equilibrium States).

Let $\mathcal{R}$ be an IFS, $\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$
and $\mu$ a probability on $\Theta$. Let $\psi:\textbf{X}\to\mathbb{R}$ be a
positive continuous function. We say that the holonomic measure $\hat{\nu}$ is
an equilibrium state for $(\psi,\mu)$ if

$h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\log\psi\,\text{d}\nu=P(\psi).$

######  Lemma 5.12.

Let X and Y be compact metric spaces and $T:\textbf{Y}\to\textbf{X}$ be a
continuous mapping. Then the push-forward mapping
$\Phi_{T}\equiv\Phi:\mathscr{M}_{1}(\textbf{Y})\to\mathscr{M}_{1}(\textbf{X})$
given by

$\Phi(\hat{\mu})(A)=\hat{\mu}(T^{-1}(A)),\quad\text{where}\
\hat{\mu}\in\mathscr{M}_{1}(\textbf{Y})\ \text{and}\
A\in\mathscr{B}(\textbf{X})$

is weak-$*$ to weak-$*$ continuous.

Proof. Since we are assuming that X and Y are compact metric spaces then we
can ensure that the weak-$*$ topology of both $\mathscr{M}_{1}(\textbf{Y})$
and $\mathscr{M}_{1}(\textbf{X})$ are metrizable. Therefore is enough to prove
that $\Phi$ is sequentially continuous. Let $(\hat{\mu}_{n})_{n\in\mathbb{N}}$
be a sequence in $\mathscr{M}_{1}(\textbf{Y})$ so that
$\hat{\mu}_{n}\rightharpoonup\hat{\mu}$. For any continuous real function
$f:\textbf{X}\to\mathbb{R}$ we have from change of variables theorem that

$\int_{\textbf{X}}f\,d[\Phi(\hat{\mu}_{n})]=\int_{\textbf{Y}}f\circ
T\,d\hat{\mu}_{n},$

for any $n\in\mathbb{N}$. From the definition of the weak-$*$ topology, it
follows that the right hand side above converges when $n\to\infty$, and we
have

$\lim_{n\to\infty}\int_{\textbf{X}}f\,d[\Phi(\hat{\mu}_{n})]=\lim_{n\to\infty}\int_{\textbf{Y}}f\circ
T\,d\hat{\mu}_{n}=\int_{\textbf{Y}}f\circ
T\,d\hat{\mu}=\int_{\textbf{X}}f\,d[\Phi(\hat{\mu})].$

The last equality shows that
$\Phi(\hat{\mu}_{n})\rightharpoonup\Phi(\hat{\mu})$ and consequently the
weak-$*$ to weak-$*$ continuity of $\Phi$. $\blacksquare$

For any $\hat{\nu}\in\mathcal{H}(\mathcal{R})$ it is always possible to
disintegrate it as $d\hat{\nu}(x,i)=d\nu_{x}(i)d[\Phi(\hat{\nu})](x)$, where
$\Phi(\hat{\nu})\equiv\nu$ is the probability measure on
$\mathscr{B}(\textbf{X})$, defined for any $A\in\mathscr{B}(\textbf{X})$ by

$\displaystyle\nu(A)\equiv\Phi(\hat{\nu})(A)\equiv\hat{\nu}(T^{-1}(A)),$ (8)

where $T:\Omega\to\textbf{X}$ is the canonical projection of the first
coordinate. This observation together with the previous lemma allow us to
define a continuous mapping from $\mathcal{H}(\mathcal{R})$ to
$\mathscr{M}_{1}(\textbf{X})$ given by
$\hat{\nu}\longmapsto\Phi(\hat{\nu})\equiv\nu$.

We now prove a theorem ensuring the existence of equilibrium states for any
continuous positive function $\psi$. Although this theorem has clear and
elegant proof and works in great generality it has the disadvantage of
providing no description of the set of equilibrium states.

######  Theorem 5.13 (Existence of Equilibrium States).

Let $\mathcal{R}$ be an IFS, $\psi:X\to\mathbb{R}$ a positive continuous
function and $\mu$ a probability on $\Theta$. Then the set of equilibrium
states for $(\psi,\mu)$ is not empty.

Proof. As we observed above we can define a weak-$*$ to weak-$*$ continuous
mapping

$\mathcal{H}(\mathcal{R})\ni\hat{\nu}\longmapsto\nu\in\mathscr{M}_{1}(\textbf{X}),$

where $d\hat{\nu}(x,i)=d\nu_{x}(i)d\nu(x)$ is the above constructed
disintegration of $\hat{\nu}$. From this observation, it follows that for any
fixed positive continuous $g$ the mapping
$\mathcal{H}(\mathcal{R})\ni\hat{\nu}\longmapsto\int_{\textbf{X}}\ln(B_{\mu}(g)/g)\,d\nu$
is continuous with respect to the weak-$*$ topology. Therefore the mapping

$\mathcal{H}(\mathcal{R})\ni\hat{\nu}\longmapsto\inf_{\begin{subarray}{c}g\in
C(\textbf{X},\mathbb{R})\\\
g>0\end{subarray}}\left\\{\int_{\textbf{X}}\ln\frac{B_{\mu}(g)}{g}\,d\nu\right\\}\equiv
h_{\text{v}}^{\mu}(\hat{\nu}).$

is upper semi-continuous (USC) which implies by standard results that the
following mapping is also USC

$\mathcal{H}(\mathcal{R})\ni\hat{\nu}\longmapsto
h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\ln(\psi(x))\,d\nu(x).$

Since $\mathcal{H}(\mathcal{R})$ is compact in the weak-$*$ topology and the
above mapping is USC, it follows that this mapping attains its supremum at
some $\hat{\mu}\in\mathcal{H}(\mathcal{R})$, i.e.,

$\sup_{\hat{\nu}\in\mathcal{H}(\mathcal{R})}\left\\{h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\ln\psi\,d\nu\right\\}=h_{\text{v}}^{\mu}(\hat{\mu})+\int_{\textbf{X}}\ln\psi\,d\mu$

thus proving the existence of at least one equilibrium state. $\blacksquare$

Let $\psi:\textbf{X}\to\mathbb{R}$ be a positive continuous function, $\mu$ a
probability on $\Omega$, and
$\mathcal{R}_{\psi}(\textbf{X},\tau,\text{q}\hskip 0.40002pt)$ an IFSm, where
$\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=\psi\circ\tau_{\theta}(x)\,\text{d}\mu(\theta)$. Suppose
that there is $h$ a positive continuous function such that $\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(h)=\text{B}\hskip
0.40002pt_{\mu}(h\cdot\psi)=\rho(\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt})h$. Then we can define, following Definition 5.4,
$\mathcal{R}_{\text{p}\hskip 0.28001pt}=(\textbf{X},\tau,\text{p}\hskip
0.40002pt)$ where

$\frac{\,\text{d}\text{p}\hskip
0.40002pt_{x}}{\,\text{d}\mu}(\theta):=\frac{(h\cdot\psi)\circ\tau_{\theta}(x)}{\text{B}\hskip
0.40002pt_{\mu}(h\cdot\psi)}=\frac{(h\cdot\psi)\circ\tau_{\theta}(x)}{\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt})h}=\frac{h\circ\tau_{\theta}(x)}{\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt})h}\cdot\frac{dq_{x}}{\,\text{d}\mu}(\theta).$

The IFSm $\mathcal{R}_{\text{p}\hskip 0.28001pt}$ is called the normalization
of $\mathcal{R}_{\text{q}\hskip 0.28001pt}$. Take $\mathcal{L}_{\text{p}\hskip
0.28001pt}(\nu)=\nu$ and let $\hat{\nu}$ be the holonomic lifting of $\nu$.
Then by the Theorem 5.6 we know that

$h_{\text{v}}^{\mu}(\hat{\nu})=\int_{\textbf{X}}\log\frac{\text{B}\hskip
0.40002pt_{\mu}(h\cdot\psi)}{h\cdot\psi}\,\text{d}\nu=\log\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt})-\int_{\textbf{X}}\log\psi\,\text{d}\nu.$

Then, choosing this $\hat{\nu}$ as particular in supremum given in Lema 5.9,
$P(\psi)\geq
h_{\text{v}}^{\mu}(\hat{\nu})+\int_{\textbf{X}}\log\psi\,\text{d}\nu=\log\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt})$.

But, remember that the pressure, defined in expression $(\ref{pressure sup
inf})$, is

$P(\psi)=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g\in
C(\textbf{X};\mathbb{R})\\\
g>0}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(g)}{g}\,\text{d}\nu\right\\},$

then,

$\inf_{g\in C(\textbf{X};\mathbb{R})\\\
g>0}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(g)}{g}\,\text{d}\nu\right\\}\leq\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\text{q}\hskip
0.28001pt}(h)}{h}\,\text{d}\nu=\log\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}).$

Taking the supremum over $\mathcal{H}\left(\mathcal{R}\right)$ in both sides
of above inequality, we have $P(\psi)\leq\log\rho(\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt})$. Since the reverse inequality is
already shown, we prove that

$P(\psi)=\log\rho(\text{B}\hskip 0.40002pt_{\text{q}\hskip 0.28001pt}).$

######  Remark 5.14.

We end this section by showing that the $IFSm$ Thermodynamic Formalism
generalizes the Thermodynamical Formalism for a dynamical system. Let $\Theta$
be a compact metric space and $\textbf{X}=\Theta^{\mathbb{N}}$. For each
$\theta\in\Theta$ define
$\sigma_{\theta}(x_{1},x_{2},\ldots)=(\theta,x_{1},x_{2},\ldots)$ the inverse
branch of the right shift $\sigma$. Take $\mu$ a a-priori probability on
$\Theta$. Let $\psi:\Omega\to\mathbb{R}$ be a positive potential and
$A=\log\psi$. Now we define $\,\text{d}\text{q}\hskip
0.40002pt_{x}(\theta)=e^{A\circ\sigma_{\theta}(x)}\,\text{d}\mu(\theta)$. We
have

$\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(\phi)=\int_{S}e^{A\circ\sigma_{\theta}(x)}\phi\circ\sigma_{\theta}(x)\,\text{d}\mu(\theta)=L\hskip
0.07996pt_{A}(\phi)(x),$

where $L\hskip 0.07996pt_{A}$ is the Ruelle Operator for the right shift
$\sigma$ and the potential $A$ (see [23] for more details). By Definition 5.8,

$\displaystyle P(\psi)$
$\displaystyle=\sup_{\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)}\inf_{g>0}\left\\{\int_{\textbf{X}}\ln\frac{\text{B}\hskip
0.40002pt_{\text{q}\hskip 0.28001pt}(g)}{g}\,\text{d}\nu\right\\}$
$\displaystyle=\sup_{\nu\in\mathcal{M}_{\sigma}}\inf_{g>0}\left\\{\int_{\textbf{X}}\ln\frac{L\hskip
0.07996pt_{A}(g)}{g}\,\text{d}\nu\right\\}.$

The last expression is exactly the pressure of the potential $A$ in
Thermodynamical Formalism. Suppose that there is $\phi_{A}$ a positive
continuous function such that $\text{B}\hskip 0.40002pt_{\text{q}\hskip
0.28001pt}(\phi_{A})=L\hskip
0.07996pt_{A}(\phi_{A})=\rho(\mathcal{R}_{A})\phi_{A}=\lambda_{A}\phi_{A}$.
Then $P(e^{A})=\log\lambda_{A}$. For instance, we know that if $A$ is Hölder,
then there exists such $\phi_{A}>0$ function. From this example, we can see
that the $IFSm$ Thermodynamic Formalism, in certain sense, generalizes the
Thermodynamical Formalism for a dynamical system. When we look at the family
$\\{\sigma_{\theta}\\}_{\theta\in\Theta}$ of functions, we are looking at the
inverse branches of the dynamical system.

##  6 Pressure Differentiability and Equilibrium States

We show in this section a uniqueness result for the equilibrium state
introduced in the last section. In order to do that we will need to consider
the functional $p:C(\textbf{X},\mathbb{R})\to\mathbb{R}$ given by

$\displaystyle p(\varphi)=P(\exp(\varphi)).$ (9)

It is immediate to verify that $p$ is a convex and finite valued functional.
We say that a Borel signed measure $\nu\in\mathscr{M}_{s}(X)$ is a subgradient
of $p$ at $\varphi$ if it satisfies the following subgradient inequality

$p(\eta)\geq p(\varphi)+\nu(\eta-\varphi)\;\;\mbox{for
any}\;\;\eta\in\mathscr{M}_{s}(X).$

The set of all subgradients at $\varphi$ is called subdifferential of $p$ at
$\varphi$ and denoted by $\partial p(\varphi)$. It is well-known that if $p$
is a continuous mapping then $\partial p(\varphi)\neq\emptyset$ for any
$\varphi\in C(\textbf{X},\mathbb{R})$.

We observe that for any pair $\varphi,\eta\in C(\textbf{X},\mathbb{R})$ and
$0<t<s$, it follows from the convexity of $p$ the following inequality

$s(p(\varphi+t\eta)-p(\varphi))\leq t(p(\varphi+s\eta)-p(\varphi)).$

In particular, the one-sided directional derivative
$d^{+}p(\varphi):C(\textbf{X},\mathbb{R})\to\mathbb{R}$ given by

$d^{+}p(\varphi)(\eta)=\lim_{t\downarrow
0}\frac{p(\varphi+t\eta)-p(\varphi)}{t}$

is well-defined for any $\varphi\in C(\textbf{X},\mathbb{R})$.

######  Theorem 6.1.

For any fixed $\varphi\in C(\textbf{X},\mathbb{R})$ we have

  1. 1.

the signed measure $\nu\in\partial p(\varphi)$ iff $\nu(\eta)\leq
d^{+}p(\varphi)(\eta)$ for all $\eta\in C(X,\mathbb{R})$;

  2. 2.

the set $\partial p(\varphi)$ is a singleton iff $d^{+}p(\varphi)$ is the
Gâteaux derivative of $p$ at $\varphi$.

Proof. This theorem is a consequence of Theorem 7.16 and Corollary 7.17 of the
reference [6]. $\blacksquare$

######  Corollary 6.2.

Let $\mathcal{R}$ be an IFS, $\psi:\textbf{X}\to\mathbb{R}$ a positive
continuous function, $\mu$ a probability on $\Theta$ and $\Phi(\hat{\nu})=\nu$
for $\hat{\nu}\in\mathcal{H}\left(\mathcal{R}\right)$ where $\nu$ is given by
disintegration with respect to $T$. If the functional $p$ defined on
$\eqref{def-funcional-p}$ is Gâteaux differentiable at
$\varphi\equiv\log\psi$, then

$\\#\\{\Phi(\hat{\mu}):\ \hat{\mu}\ \text{is an equilibrium state for}\
\psi\\}=1.$

Proof. Suppose that $\hat{\mu}$ is an equilibrium state for $\psi$. Then we
have from the definition of the pressure that

$\displaystyle p(\varphi+t\eta)-p(\varphi)$
$\displaystyle=P(\psi\exp(t\eta))-P(\psi)$ $\displaystyle\geq
h_{v}(\hat{\mu})+\int_{\textbf{X}}\ln\psi\ d\mu+\int_{\textbf{X}}t\eta\,d\mu-
h_{v}(\hat{\mu})-\int_{\textbf{X}}\ln\psi\ d\mu$
$\displaystyle=t\int_{\textbf{X}}\eta\,d\mu.$

Since we are assuming that $p$ is Gâteaux differentiable at $\varphi$, it
follows from the above inequality that $\mu(\eta)\leq d^{+}p(\varphi)(\eta)$
for all $\eta\in C(\textbf{X},\mathbb{R})$. From this inequality and Theorem
6.1 we can conclude that $\partial p(\varphi)=\\{\mu\\}$. Therefore for all
equilibrium state $\hat{\mu}$ for $\psi$ we have $\Phi(\hat{\mu})=\partial
p(\varphi)$, thus finishing the proof. $\blacksquare$

##  7 Possible Application in Economics

In Gupta et al. [16] a chaos game is used to represent a time series as a PC
plot and compare similarities and dissimilarities in different time frame such
as the global pandemic of COVID-19. More precisely, the author consider the
set $X=[0,1]^{2}$ as the base space and the four linear contractions

$\left\\{\begin{array}[]{l}\tau_{A}(x,y)=(0.5x,0.5y)\\\
\tau_{B}(x,y)=(0.5x+0.5,0.5y)\\\ \tau_{C}(x,y)=(0.5x,0.5y+0.5)\\\
\tau_{D}(x,y)=(0.5x+0.5,0.5y+0.5)\end{array}\right.$ (10)

so $(X,\tau_{\theta})_{\theta\in\Theta},\Theta=\\{A,B,C,D\\}$, is a classic
contractive system whose attractor is $X$ itself. Consider the identification:
A – if the market falls more than 0.01% of the previous value,
B – if the market falls less than 0.01% of the previous value,
C – if the market gains less than 0.01% of the previous value and
D – if the market gains more than 0.01% of the previous value,
in this way the time series of length $N$ associated to a certain economic
indicator is translated in to a genetic sequence

$\gamma=(DACCADCDACDC...AACCBADD)\in\Theta^{N}.$

Fixed an arbitrary initial point $Z_{0}=(x_{0},y_{0})=(0.5,0.5)$ the chaos
game consists in iterating $(x_{0},y_{0})$ by each map
$Z_{1}=(x_{1},y_{1})=\tau_{D}(x_{0},y_{0}),Z_{2}=(x_{2},y_{2})=\tau_{A}(x_{1},y_{1}),Z_{3}=(x_{3},y_{3})=\tau_{C}(x_{2},y_{2}),....$
according to $\gamma$. Considering $M\geq 2$ and the diadic partition of $X$
given by

$\bigcup_{\gamma^{\prime}\in\Theta^{M}}\tau_{\gamma^{\prime}_{M}}(\cdots(\tau_{\gamma^{\prime}_{1}}(X))),$

the PC plot $W$ is a grey scale picture where the color of the each individual
part
$\Lambda=\tau_{\gamma^{\prime}_{M}}(\cdots(\tau_{\gamma^{\prime}_{1}}(X)))$ is
the frequency of visits of the chaos game orbit $\\{Z_{j},j\geq 0\\}$ to
$\Lambda$ that is,

$W(\Lambda)=\frac{1}{N}\,\sharp\\{j=0,...,N-1\,|\,Z_{j}\in\Lambda\\}\sim\mu(\Lambda).$

Obviously,
$\nu_{N}=\sum_{\Lambda}W(\Lambda)\delta_{(x_{\Lambda},\,y_{\Lambda})}$, where
$(x_{\Lambda},\,y_{\Lambda})\in\Lambda$ is arbitrary, is a discrete
probability and, if $\mu(\partial(\Lambda))=0$ then by the EET ([13],
Corollary 2), when $N\to\infty$, $\nu_{N}$ converge in distribution to the
invariant measure $\mu$ for the IFS with probabilities
$(X,\tau_{\theta},p_{\theta})_{\theta\in\Theta}$, where
$p_{A},p_{B},p_{C},p_{D}$ are the relative frequency of each symbol $A,B,C,D$
in $\gamma$, respectively.
For instance, if $N=100$ and if a certain time series produces the genetic
sequence

$\gamma=(A,A,D,D,A,...,A,D,B,C,C,B,A,D)\in\\{A,B,C,D\\}^{100},$

we obtain the frequencies
$[p_{A},p_{B},p_{C},p_{C}]=[0.39,\,0.17,\,0.15,\,0.29]$, and considering $M=4$
we obtain the following PC plot

Figure 1: PC plot where each square represents one element $\Lambda$ of the
diadic partition and grey scale value
$0\leq\frac{1}{N}\,\sharp\\{j=0,...,N-1\,|\,Z_{j}\in\Lambda\\}\leq 1$.

which is an approximation for the invariant measure $\mu$ of the associated
IFS with probabilities $[0.39,\,0.17,\,0.15,\,0.29]$.

In order to generalize this idea we need to consider an infinite compact
continuous range of values of the economic indicator, such as
$\Theta=[0\%,100\%]$, instead of taking only four values
$\Theta=\\{A,B,C,D\\}$. Also, it is not reasonable to suppose that the
probability of a change of $\theta\%$ in the indicator is independent of the
current state of the indicator: the distribution of the occurrence of
$\theta\in[0\%,100\%]$, given the current state $Z\in X$ must be a measure of
probability $q_{Z}(\cdot)$ over $[0\%,100\%]$. Therefore, we believe the
theory developed in the previous sections should be used when making this
generalization.

Aknowledgement: _We would like to thanks Prof. Leandro Cioletti whose
contribution on the related preprint[8] was fundamental to establish the tools
and ideas that we now generalize in our work._

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