# Large deviation for Gibbs probabilities at zero temperature and invariant
idempotent probabilities for iterated function systems

Jairo K. Mengue E-mail<EMAIL_ADDRESS>Elismar R. Oliveira E-mail:
<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

We consider two compact metric spaces $J$ and $X$ and a uniform contractible
iterated function system $\\{\phi_{j}:X\to X\,|\,j\in J\\}$. For a Lipschitz
continuous function $A$ on $J\times X$ and for each $\beta>0$ we consider the
Gibbs probability $\rho_{{}_{\beta A}}$. Our goal is to study a large
deviation principle for such family of probabilities as $\beta\to+\infty$ and
its connections with idempotent probabilities. In the non-place dependent case
($A(j,x)=A_{j},\,\forall x\in X$) we will prove that $(\rho_{{}_{\beta A}})$
satisfy a LDP and $-I$ (where $I$ is the deviation function) is the density of
the unique invariant idempotent probability for a mpIFS associated to $A$. In
the place dependent case, we prove that, if $(\rho_{{}_{\beta A}})$ satisfy a
LDP, then $-I$ is the density of an invariant idempotent probability. Such
idempotent probabilities were recently characterized through the Mañé
potential and Aubry set, therefore we will obtain an identical
characterization for $-I$.

_Key words and phrases: iterated function systems, large deviation, idempotent
measures, Maslov measures_

_2020 Mathematics Subject Classification: Primary 37A30, 37A50, 28A33, 46E27,
60B10,60F10; Secondary 15A80, 37C30._

##  1 Introduction

We consider two compact metric spaces $(X,d_{X})$ and $(J,d_{J})$ and denote
by $C(X,\mathbb{R})$ and $C(J\times X,\mathbb{R})$, the set of continuous
functions from $X$ to $\mathbb{R}$ and $J\times X$ to $\mathbb{R}$,
respectively.

######  Definition 1.1.

Let $(X,d_{X})$ and $(J,d_{J})$ be compact metric spaces. A _uniformly
contractible iterated function system_ $(X,(\phi_{j})_{j\in J})$ is a family
of maps $\\{\phi_{j}:X\to X\,|\,j\in J\\}$ satisfying: there exists
$0<\gamma<1$ such that

$d_{X}(\phi_{j_{1}}(x_{1}),\phi_{j_{2}}(x_{2}))\leq\gamma\cdot[d_{J}(j_{1},j_{2})+d_{X}(x_{1},x_{2})],\,\,\forall
j_{1},j_{2}\in J,\forall x_{1},\,x_{2}\in X.$ (1)

We also consider the continuous map $\phi:J\times X\to X,$ defined by
$\phi(j,x)=\phi_{j}(x)$.

By one hand, in [27] it is described a thermodynamic formalism for uniformly
contractible IFS $(X,(\phi_{j})_{j\in J})$. On the other hand, in [28] it is
characterized the invariant idempotent probabilities for the so called max-
plus IFS. In the present work we exhibit a connection between these two
settings. Our objective is to study a large deviation principle (LDP) for
Gibbs probabilities in thermodynamic formalism for IFS, when the temperature
goes to zero, as an application of results in [28]. As we will see, the Mañé
potential and Aubry set will play an important role in the description of the
deviation function.

From now on we always consider the Borel sigma algebra in each metric space.
Let $B:J\times X\to(0,+\infty)$ be a continuous function and $\nu$ be a fixed
Borel probability on $J$ satisfying $\mathop{\rm supp}\nolimits(\nu)=J$. We
define the transfer operator $\mathcal{L}_{\phi,B,\nu}:C(X,\mathbb{R})\to
C(X,\mathbb{R})$ by

$\mathcal{L}_{\phi,B,\nu}(f)(x)=\int_{J}B(j,x)\cdot f(\phi_{j}(x))\,d\nu(j),$

for any $f\in C(X,\mathbb{R})$.

For each $\beta>0$ (which is interpreted as the inverse of temperature in
thermodynamic formalism) and a Lipschitz continuous function $A:J\times
X\to\mathbb{R}$ we also consider the operator $\mathcal{L}_{\phi,e^{\beta
A},\nu}$. It admits a unique pair $(\lambda_{\beta A},h_{\beta A})$ where
$\lambda_{\beta A}$ is a positive number and $h_{\beta A}:X\to\mathbb{R}$ is a
positive and continuous function, such that $\sup_{x\in X}h_{\beta A}(x)=1$
and $\mathcal{L}_{\phi,e^{\beta A},\nu}(h_{\beta A})=\lambda_{\beta A}\cdot
h_{\beta A}$. The function $q^{\beta}:J\times X\to\mathbb{R}$, defined by
$q^{\beta}(j,x):=\frac{e^{\beta A(j,x)}\cdot h_{\beta
A}(\phi_{j}(x))}{\lambda_{\beta A}\cdot h_{\beta A}(x)}$, satisfies
$\mathcal{L}_{\phi,q^{\beta},\nu}(1)=1$ and the dual operator of
$\mathcal{L}_{\phi,q^{\beta},\nu}$ has a unique invariant probability
$\rho_{{}_{\beta A}}$ on $X$, that is,

$\int_{X}f(x)d\rho_{{}_{\beta A}}(x)=\int_{X}\int_{J}{q^{\beta}(j,x)}\cdot
f(\phi_{j}(x))\,d\nu(j)\,d\rho_{{}_{\beta A}}(x),\,\,\forall f\in
C(X,\mathbb{R}).$ (2)

We call $\rho_{{}_{\beta A}}$ a Gibbs probability on $X$. Our objective is to
consider a large deviation principle for $(\rho_{{}_{\beta A}})$ as
$\beta\to+\infty$ and to present its connections with invariant idempotent
probabilities.

The Ruelle (transfer) operator was introduced by D. Ruelle [31] in the context
of equilibrium states - Gibbs measures - of infinite one-dimensional lattice
gas. It was extensively studied for IFS in the last decades and a remarkable
result was Fan’s Theorem (see [13], Theorem 1.1) which shows the existence of
a positive eigenfunction and a Gibbs probability for a contractive IFS with
Dini continuous place dependent positive weights. Several extensions of this
result appeared studying the Ruelle operator, its rate of convergence and the
existence of eigenfunctions and eigenmeasures. In [18], for example, it is
studied the conditions for the existence of such eigenfunctions together with
the Perron-Frobenius property for non-expansive finite IFSs. In [33] it is
studied the speed of convergence of the Ruelle operator for a finite IFS,
where the weights are positive Lipschitz continuous and the system satisfies
an average contractive condition. In [34], it is studied the existence of
eigenfunctions and eigenmeasures of the Ruelle operator for a finite weakly
contractive IFS.

A key generalization was given by Stenflo [32], where random iterations are
used to represent the iterations of a so called IFS with probabilities, for an
arbitrary measurable set of maps. Their approach is slightly different from
the previous works on IFS with probabilities. Its main goal was to establish,
when the transfer operator is Feller, that there exists a unique attracting
invariant measure for the Markov chain generated by the IFS. It is worth to
mention that this approach was previously introduced by [5], for an average-
contractive system, to show the existence of invariant measures. In [5], it is
assumed that $J$ is finite and the maps are Lipschitz continuous, with place
dependent and Dini continuous weights.

One can also introduce a thermodynamic formalism based on holonomic measures
according to [23], [8], [27] and [7], where several results are obtained for
fairly general IFSs.

There is a special class of IFS where the thermodynamic formalism is well
known: if $J:=\\{1,...,d\\}$ is a finite set, $X:=J^{\mathbb{N}}$ and
$\phi_{j}(x)=\phi_{j}(x_{1},x_{2},x_{3},...)=(j,x_{1},x_{2},x_{3},...)$ where
$x=(x_{1},x_{2},x_{3},...)\in X$, then $(\phi_{j})_{j\in J}$ are the inverse
branches of the shift map $\sigma$ acting in $J^{\mathbb{N}}$. We call this
model by full shift for finite alphabet. In this case, the thermodynamic
formalism for zero temperature is also widely studied as well as its
connection with ergodic optimization and max-plus algebra [2]. Considering
this case, results concerning large deviations, when the temperature goes to
zero, can be founded in [3] and [26].

In the setting of full shift for finite alphabet it is proved in [3] (more
generally for sub-shifts of finite type) that $(\rho_{{}_{\beta A}})$
satisfies a LDP, since $A$ has a unique maximizing measure. In this case the
deviation function $I$ evaluated at any point $x\in X$, is given as by
$I(x)=R(x)+R(\sigma(x))+R(\sigma^{2}(x))+\ldots$, where the function $R$ is
also presented. This is problematic for a general IFS, because we have at
hands just the maps $\phi_{j}$ which in the full shift correspond to the
inverse branches of $\sigma$.

In [26] the main idea is to use Prop. 3.2 of [30]. We believe that such
strategy can not be replicated from the full shift for finite alphabet to a
general IFS. In [26] it is also proved that the deviation function presented
in [3] fails without the hypothesis of uniqueness of maximizing measure and
another characterization of $I$ is presented. The characterization of the
deviation function for IFS in the present work is similar to one which appears
in [26].

In order to study LDP for Gibbs probabilities in a uniformly contractible IFS
we will apply recent results from [28]. We will consider the function
$q=\lim_{\beta\to\infty}q^{\beta}$, when it exists. In the non-place dependent
case ($A(j,x)=A_{j},\,\forall x\in X$) is easy to show that $q$ exists. In
this case, it is proved in [28] that there exists a unique invariant
idempotent probability for the mpIFS $(X,\phi,q)$ and it is presented its
density $\lambda$. In this case we will prove that $(\rho_{{}_{\beta A}})$
satisfy a LDP with deviation function $I=-\lambda$. In the place dependent
case, the situation is more complicated. As far as we know it is not proved a
LDP or the existence of the limit $q$, even in the full shift for finite
alphabet. In the present work we prove that, if $(\rho_{{}_{\beta A}})$
satisfy a LDP and the limit $q$ there exists, then $I=-\lambda$ where
$\lambda$ is the density of an invariant idempotent probability for the mpIFS
$(X,\phi,q)$. As such idempotent probabilities were recently characterized
through the Mañé potential and Aubry set in [28], we will obtain a
characterization for $I$ using the Mañé potential and Aubry set too.

We would like to point out that despite the fact that the theory of IFS with
compact space of maps is fairly old (see [19], [25], among others), only few
works are known for such more general setting. Furthermore, as far as we know,
the LDP theory for IFS, at zero temperature, is absent in the literature.
Although, the thermodynamic formalism and multifractal analysis of Gibbs
probabilities has shown notable advances in the last few years with many
generalizations (for instance, [14], [15], [29] and [7], among others) thus it
seems to be of paramount importance to find further connections and
developments in this areas.

We propose, initially, in Section 2, to present a discussion concerning
thermodynamic formalism for IFS, following mainly [27]. In the sequence, we
extend to the setting of IFS some well known results, in thermodynamic
formalism, for the zero temperature case. Right away, in Section 3 we present
the main definitions and results contained in [28], concerning invariant
idempotent probabilities for max-plus IFS. After the introduction of these
elements we then present their connection by considering large deviation
principles in Section 4.

##  2 Thermodynamic Formalism for IFS

In this section we describe the setting presented mainly in [27], concerning
the thermodynamic formalism for IFS. The characterization of the entropy and
its relation with the Kullback-Leibler divergence is presented in [21].

For each $\beta>0$ and a Lipschitz continuous function $A:J\times
X\to\mathbb{R}$ we consider the operator

$\mathcal{L}_{\phi,e^{\beta A},\nu}(f)(x)=\int_{J}e^{\beta A(j,x)}\cdot
f(\phi_{j}(x))\,d\nu(j),$

where $\nu$ is a fixed probability on $J$ with $\mathop{\rm
supp}\nolimits(\nu)=J$.

######  Proposition 2.1.

Suppose that $A:J\times X\to\mathbb{R}$ has Lipschitz constant $\mathop{\rm
Lip}\nolimits(A)$. Then $\mathcal{L}_{\phi,e^{\beta A},\nu}$ admits a unique
pair $(\lambda_{\beta A},h_{\beta A})$ where $\lambda_{\beta A}$ is a positive
number and $h_{\beta A}:X\to\mathbb{R}$ is a positive and continuous function,
such that $\sup_{x\in X}h_{\beta A}(x)=1$ and $\mathcal{L}_{\phi,e^{\beta
A},\nu}(h_{\beta A})=\lambda_{\beta A}\cdot h_{\beta A}$. Furthermore,
$h_{\beta A}$ is Lipschitz continuous and $\mathop{\rm Lip}\nolimits(h_{\beta
A})\leq e^{\beta\cdot\frac{\mathop{\rm Lip}\nolimits(A)}{1-\gamma}}$, where
$\gamma$ satisfies (1).

The proof of uniqueness can be found in [27], but details concerning the
Lipschitz constant of $h_{\beta A}$ are missing. Let us then present such
details following ideas of [6] and [4].

Proof. First, note that we can drop $\beta$ in the computations. For each
$0<s<1$, we define the operator $T_{s}:C(X,\mathbb{R})\to C(X,\mathbb{R})$ by

$T_{s}(u)(x)=\log\int_{J}e^{A(j,x)+su(\phi_{j}(x))}\,d\nu(j).$

We have $|T_{s}(u)-T_{s}(v)|_{\infty}\leq s|u-v|_{\infty},$ meaning that, $T$
is a uniform contraction map on $C(X,\mathbb{R})$ with respect to the supremum
norm ($|g|_{\infty}=\sup_{x\in X}|g(x)|$). Let $u_{s}\in C(X,\mathbb{R})$ be
the unique fixed point for $T_{s}$.

We claim that for any $s\in(0,1)$, the function $u_{s}$ is Lipschitz
continuous, with $\mathop{\rm Lip}\nolimits(u_{s})\leq\frac{\mathop{\rm
Lip}\nolimits(A)}{1-\gamma}$. Indeed, for any $x_{1},x_{2}\in X$ we have

$u_{s}(x_{1})-u_{s}(x_{2})\leq\max_{j}\\{A(j,x_{1})-A(j,x_{2})+s[u_{s}(\phi_{j}(x_{1}))-u_{s}(\phi_{j}(x_{2}))]\\}.$

By iterating this inequality we get

$u_{s}(x_{1})-u_{s}(x_{2})\leq\max_{(j_{0},j_{1},...)}\left(\sum_{n=0}^{\infty}s^{n}[A(j_{n},\phi_{j_{n-1}}\circ...\circ\phi_{j_{0}}(x_{1}))-A(j_{n},\phi_{j_{n-1}}\circ...\circ\phi_{j_{0}}(x_{2}))]\right)$
$\leq\sum_{n=0}^{\infty}s^{n}Lip(A)\gamma^{n}{d(x_{1},x_{2})}=\frac{\mathop{\rm
Lip}\nolimits(A)}{1-s\gamma}{d(x_{1},x_{2})}\leq\frac{\mathop{\rm
Lip}\nolimits(A)}{1-\gamma}{d(x_{1},x_{2})}.$

Particularly
$-\frac{\operatorname{Lip}(A)}{1-\gamma}\operatorname{diam}(X)\leq u_{s}-\max
u_{s}\leq 0$ and then the family $\\{u^{*}_{s}=u_{s}-\max u_{s}\\}_{0<s<1}$ is
equicontinuous and uniformly bounded. Note also that, as $T_{s}(u_{s})=u_{s}$,
we obtain

$-|A|+s\min u_{s}\leq u_{s}(x)\leq|A|+s\max u_{s},\,\forall x\in X.$

Consequently, $-|A|\leq(1-s)\min u_{s}\leq(1-s)\max u_{s}\leq|A|$, for any
$0<s<1$. Then, there exists a subsequence $s_{n}\to 1$ and a number
$k\in\mathbb{R}$ such that $[\,(1-s_{n})\,\max u_{s_{n}}\,]\,\to k$. By
applying Arzela-Ascoli theorem we get a subsequence, also denoted by $s_{n}$,
such that $\\{u_{s_{n}}^{*}\\}_{n\geq 1}$ converges uniformly to a function
$u\in C(X,\mathbb{R})$. As $\mathop{\rm Lip}\nolimits(u_{s}^{*})=\mathop{\rm
Lip}\nolimits(u_{s})\leq\frac{\mathop{\rm Lip}\nolimits(A)}{1-\gamma}$ we get
$\mathop{\rm Lip}\nolimits(u)\leq\frac{\mathop{\rm
Lip}\nolimits(A)}{1-\gamma}$. In order to finish the proof, we will show that
$h=e^{u}$ is an eigenfunction for $\mathcal{L}_{\phi,e^{A},\nu}$ with
associated eigenvalue $e^{k}$. Indeed, for any $s\in(0,1)$ we have

$\displaystyle e^{u^{*}_{s}(x)}$ $\displaystyle=$ $\displaystyle
e^{u_{s}(x)-\max u_{s}}=e^{-(1-s)\max u_{s}+u_{s}(x)-s\max u_{s}}$
$\displaystyle=$ $\displaystyle e^{-(1-s)\max
u_{s}}\,\int_{J}e^{A(j,x)+su_{s}(\phi_{j}(x))-s\max u_{s}}\,d\nu(j).$

Taking the limit when $s\to 1$ we conclude that $u$ satisfies

$e^{u(x)}=e^{-k}\,\int_{J}e^{A(j,x)+u(\phi_{j}(x))}\,d\nu(j)=e^{-k}\mathcal{L}_{\phi,e^{A},\nu}(e^{u})(x).$

$\blacksquare$

Consider the function $q^{\beta}:J\times X\to\mathbb{R}$ which is defined by
$q^{\beta}(j,x):=\frac{e^{\beta A(j,x)}\cdot h_{\beta
A}(\phi_{j}(x))}{\lambda_{\beta A}\cdot h_{\beta A}(x)}$ and for each $j\in J$
consider also the function $q_{j}^{\beta}:X\to\mathbb{R},$ defined by
$q_{j}^{\beta}(x)=q^{\beta}(j,x)$.

Replacing $e^{\beta A}$ by the function $q^{\beta}$ we get the operator
$\mathcal{L}_{\phi,q^{\beta},\nu}$ which satisfies

$\mathcal{L}_{\phi,q^{\beta},\nu}(1)=1,$

that is, it has the eigenvalue $\lambda=1$ associated to the eigenfuntion
$h=1$. We say that this operator $\mathcal{L}_{\phi,q^{\beta},\nu}$ is
normalized.

The dual of the normalized operator $\mathcal{L}_{\phi,q^{\beta},\nu}$, which
we will denote by $\mathcal{M}_{\phi,q^{\beta},\nu}$, acting in Borel
probabilities on $X$ is given by

$\mathcal{M}_{\phi,q^{\beta},\nu}(\mu)(f)=\int_{X}\mathcal{L}_{\phi,q^{\beta},\nu}(f)(x)\,d\mu(x)=\int_{X}\int_{J}{q^{\beta}(j,x)}\cdot
f(\phi_{j}(x))\,d\nu(j)\,d\mu(x),$

for any $f\in C(X,\mathbb{R}).$ There exists a unique probability
$\rho_{{}_{\beta A}}$ (Gibbs probability) which is invariant for
$\mathcal{M}_{\phi,q^{\beta},\nu}$, that is, it satisfies (2).

We say that a probability $\pi$ on $J\times X$ is holonomic if

$\int g(x)\,d\pi(j,x)=\int g(\phi_{j}(x))\,d\pi(j,x),\,\forall g\in
C(X,\mathbb{R}).$

Let us denote by $\mathcal{H}$ the set of holonomic probabilities on $J\times
X$.

We have a natural holonomic probability $\pi_{\beta A}\in\mathcal{H}$
associated to $\beta A,$ defined by

$\int g(j,x)\,d\pi_{\beta A}(j,x):=\int_{X}\int_{J}{q^{\beta}(j,x)}\cdot
g(j,x)\,d\nu(j)\,d\rho_{{}_{\beta A}}(x),\,\,\forall g\in C(J\times
X,\mathbb{R}).$

This probability $\pi_{\beta A}$ is an equilibrium probability to $\beta A$ in
the following sense:

$P_{\nu}(\beta A):=\sup_{\pi\in\mathcal{H}}\int\beta
A\,d\pi+H_{\nu}(\pi)=\int\beta A\,d\pi_{\beta A}+H_{\nu}(\pi_{\beta A}),$ (3)

where $H_{\nu}(\pi)$ is the relative entropy of $\pi$ with respect to $\nu$.
More precisely, if the $x-$marginal of $\pi$ is a probability $\rho$ on $X$
then

$H_{\nu}(\pi)=\left\\{\begin{array}[]{cc}-\int\log(\operatorname{Jac})\,d\pi&\text{if}\,d\pi=\operatorname{Jac}(j,x)d\nu(j)d\rho(x)\\\
\\\ -\infty&\begin{array}[]{c}\text{if}\,\pi\,\text{is not absolutely
continuous}\\\ \text{with respect
to}\,\nu\times\rho\end{array}\end{array}\right..$

The above defined entropy satisfies
$H_{\nu}(\pi)=-D_{KL}(\pi\,|\,\nu\times\rho)$, where $D_{KL}$ is the Kullback-
Leibler divergence (see [21] for additional details).

We say that a measurable function $\operatorname{Jac}:J\times X\to[0,+\infty)$
is a $\nu$-Jacobian, if $\int_{J}\operatorname{Jac}(j,x)\,d\nu(j)=1,\,\forall
x\in X$. The functions $1$ and $q^{\beta}$ are $\nu$-Jacobians. In [21] it is
proved that

$H_{\nu}(\pi)=-\sup\left\\{\int\log(\operatorname{Jac})\,d\pi\,|\,\operatorname{Jac}\,\,\text{is
a Lipschitz}\,\nu\text{-Jacobian}\right\\}.$

Consequently

$H_{\nu}(\pi)\leq\int\log(1)\,d\pi=0.$

Furthermore,

$H_{\nu}(\pi_{\beta A})=-\int\log(q^{\beta})\,d\pi_{\beta A}.$

This definition of entropy does not depend of the IFS, while the operator
$\mathcal{L}$ depends of $\phi,$ as can be observed from its definition. An
interesting fact is that $P_{\nu}(\beta A)=\log(\lambda_{\beta A})$ (see
[27]).

###  2.1 Zero temperature limits and ergodic optimization

This section follows ideas present in [22] and [2] which were adapted to the
present setting. It extends to IFS some results concerning zero temperature
limits. The zero temperature case corresponds to an analysis of the limits, as
$\beta\to+\infty,$ of the concepts introduced in above subsection (for
instance: $\rho_{{}_{\beta A}},\,h_{\beta A},\,\pi_{\beta A},P_{\nu}(\beta
A)$).

The following result generalizes one from [22] and will be useful in the
present work.

######  Lemma 2.2.

Let $Y,Z$ be compact metric spaces. Let $W_{\beta}:Y\times Z\to\mathbb{R}$ be
a family of measurable functions converging uniformly to a continuous function
$W:Y\times Z\to\mathbb{R}$, as $\beta\to+\infty,$ and let $\mu$ be a finite
measure on $Y$ with $\mathop{\rm supp}\nolimits(\mu)=Y$. Then

$\frac{1}{\beta}\log\int_{Y}e^{\beta W_{\beta}(y,z)}d\mu(y)\to\sup_{y\in
Y}\,W(y,z)$

uniformly on $Z,$ as $\beta\to+\infty$. The same is true if we replace $\beta$
by a sequence $\beta_{i}$ which converges to $+\infty$.

Proof. As $W$ is continuous and $Y$ is compact we have that
$\tilde{W}(z):=\sup_{y\in Y}\,W(y,z)$ defines a continuous function too. Fix
an $\epsilon>0$. As $W,\tilde{W}$ are also uniformly continuous, there exists
$\delta>0$ such that

$d(y_{1},y_{2})+d(z_{1},z_{2})<\delta\Rightarrow|W(y_{1},z_{1})-W(y_{2},z_{2})|<\epsilon$

and

$d(z_{1},z_{2})<\delta\Rightarrow|\tilde{W}(z_{1})-\tilde{W}(z_{2})|<\epsilon.$

As $W_{\beta}\to W$ uniformly we can suppose also there exists $\beta_{0}>0$
such that

$d(y_{1},y_{2})+d(z_{1},z_{2})<\delta\Rightarrow|W_{\beta}(y_{1},z_{1})-W_{\beta}(y_{2},z_{2})|<2\epsilon,\,\text{for}\,\beta>\beta_{0}.$

Let $Z_{0}\subseteq Z$ be a finite set such that $\cup_{z\in
Z_{0}}B(z,\delta)=Z$. For each $z\in Z$, let $y_{z}\in Y$ be such that
$W(y_{z},z)>\tilde{W}(z)-\epsilon$. Then, for any $z\in Z$ and $y\in
B(y_{z},\delta)$, we have $W(y,z)>\tilde{W}(z)-2\epsilon$. As $W_{\beta}\to W$
uniformly, there exists $\beta_{1}>\beta_{0}$ such that
$W_{\beta}(y,z)>\tilde{W}(z)-3\epsilon$, for any $z\in Z$, $y\in
B(y_{z},\delta)$ and $\beta>\beta_{1}$.

As $Z_{0}$ is a finite set and $\mathop{\rm supp}\nolimits(\mu)=Y,$ there
exists $\beta_{2}>\beta_{1}$ such that

$\frac{1}{\beta}\log\big{(}\mu\big{(}B(y_{z_{0}},\delta)\big{)}\big{)}>-\epsilon,\,\forall
z_{0}\in Z_{0},\,\beta>\beta_{2}.$

Given any $z\in Z$, there exists $z_{0}\in Z_{0}$ satisfying
$d(z,z_{0})<\delta$. If $\beta>\beta_{2}$ we have that

$\frac{1}{\beta}\log\int_{Y}e^{\beta
W_{\beta}(y,z)}d\mu(y)>\frac{1}{\beta}\log\int_{Y}e^{\beta(W_{\beta}(y,z_{0})-2\epsilon)}d\mu(y)\geq\frac{1}{\beta}\log\int_{B(y_{z_{0}},\delta)}e^{\beta(W_{\beta}(y,z_{0})-2\epsilon)}d\mu(y)$
$>\frac{1}{\beta}\log\left(\mu\big{(}B(y_{z_{0}},\delta)\big{)}e^{\beta(\tilde{W}(z_{0})-5\epsilon)}\right)>\tilde{W}(z_{0})-6\epsilon>\tilde{W}(z)-7\epsilon.$

On the other hand, there exists $\beta_{4}>\beta_{3}$ such that,
$W_{\beta}(y,z)<\tilde{W}(z)+\epsilon$, for any $\beta>\beta_{4}$ and $y\in
Y$. Then, for $\beta>\beta_{4}$ we have

$\frac{1}{\beta}\log\int_{Y}e^{\beta
W_{\beta}(y,z)}d\mu(y)<\tilde{W}(z)+\epsilon.$

$\blacksquare$

Remember that $\log(\lambda_{\beta A})=P_{\nu}(\beta A)$.

######  Definition 2.3.

Let us define the number $\displaystyle{m(A):=\sup_{\pi\in\mathcal{H}}\int
A\,d\pi}$.

A continuous function $V:X\to\mathbb{R}$ is a calibrated subaction of $A$ if

$\sup_{j\in J}[A(j,x)+V(\phi_{j}(x))-V(x)-m(A)]=0,\,\forall x\in X.$

A holonomic probability $\pi\in\mathcal{H}$ is maximizing with respect to $A$
if $\int A\,d\pi=m(A)$.

######  Proposition 2.4.

Consider a Lipschitz continuous function $A:J\times X\to\mathbb{R}$. Under the
above framework, we have:

  1. 1.

$\displaystyle{\lim_{\beta\to+\infty}\frac{1}{\beta}\log(\lambda_{\beta
A})=\lim_{\beta\to+\infty}\frac{P_{\nu}(\beta A)}{\beta}=m(A).}$

  2. 2.

Suppose that $\frac{1}{\beta_{n}}\log(h_{\beta_{n}A})\to V$ uniformly, as
$\beta_{n}\to+\infty$. Then $V$ is a Lipschitz calibrated subaction of $A$.

  3. 3.

Suppose that $\pi_{\beta_{n}A}\to\pi_{\infty}$ in the weak∗ topology, as
$\beta_{n}\to+\infty$. Then the probability $\pi_{\infty}$ is holonomic and
maximizing to $A$.

Proof. By definition of $h_{\beta A}$ and $\lambda_{\beta A}$ we have

$\lambda_{\beta A}=\int_{J}e^{\beta A(j,x)+\log(h_{\beta
A}(\phi_{j}(x)))-\log(h_{\beta A}(x))}\,d\nu(j),\,\,\forall x\in X.$ (4)

If $h_{\beta A}$ attains its minimum and maximum in $x_{1}$ and $x_{2}$,
respectively, we get:

$\lambda_{\beta A}=\int_{J}e^{\beta A(j,x_{2})+\log(h_{\beta
A}(\phi_{j}(x_{2})))-\log(h_{\beta A}(x_{2}))}\,d\nu(j)\leq\int_{J}e^{\beta
A(j,x_{2})}\,d\nu(j)\leq e^{\beta\cdot\max_{j,x}A(j,x)}$

and similarly

$\lambda_{\beta A}\geq\int_{J}e^{\beta A(j,x_{1})}\,d\nu(j)\geq
e^{\beta\cdot\min_{j,x}A(j,x)}.$

Then

$\min_{j,x}A(j,x)\leq\frac{1}{\beta}\log(\lambda_{\beta
A})\leq\max_{j,x}A(j,x).$

Let $k$ be an accumulation point of $\frac{1}{\beta}\log(\lambda_{\beta A})$
as $\beta\to+\infty$ and suppose that
$\frac{1}{\beta_{n}}\log(\lambda_{\beta_{n}A})\to k$. By Proposition 2.1 the
family $\frac{1}{\beta}\log(h_{\beta A})$ is equicontinuous and uniformly
bounded. By applying Arzela-Ascoli theorem, we get a subsequence of
$(\beta_{n})$, also denoted by $(\beta_{n}),$ and a Lipschitz continuous
function $V$ such that $\frac{1}{\beta_{n}}\log(h_{\beta_{n}A})\to V$
uniformly. By taking $\lim_{\beta_{n}\to+\infty}\frac{1}{\beta_{n}}\log$ in
both sides of (4) and applying Lemma 2.2 we obtain

$k=\sup_{j}[A(j,x)+V(\phi_{j}(x))-V(x)],\,\,\forall x\in X.$

We claim that $k=m(A)$. Indeed, by one hand, as $\pi_{\beta A}$ is holonomic,
we have

$\frac{1}{\beta}\log(\lambda_{\beta A})=\frac{P_{\nu}(\beta
A)}{\beta}=\frac{\int\beta A\,d\pi_{\beta A}+H_{\nu}(\pi_{\beta
A})}{\beta}\stackrel{{\scriptstyle H_{\nu}\leq 0}}{{\leq}}\frac{\int\beta
A\,d\pi_{\beta A}}{\beta}=\int A\,d\pi_{\beta A}\leq m(A).$

Therefore $k\leq m(A)$. On the other hand, if $\pi$ is any holonomic
probability, as $k\geq A(j,x)+V(\phi_{j}(x))-V(x)$ we get $k\geq\int A\,d\pi$.
Therefore, $k\geq m(A)$.

We conclude that $k=m(A)$ for any possible convergent sequence and then

$\lim_{\beta\to+\infty}\frac{1}{\beta}\log(\lambda_{\beta A})=m(A).$

Clearly, we also get

$m(A)=\sup_{j}[A(j,x)+V(\phi_{j}(x))-V(x)],\,\,\forall x\in X,$

(and then $V$ is a calibrated subaction) for any uniform limit of
$\frac{1}{\beta}\log(h_{\beta A})$, as $\beta\to\infty$.

Finally, suppose that $\pi_{\beta_{n}A}\to\pi_{\infty}$ in the weak∗ topology,
as $\beta_{n}\to+\infty$. Then $\pi_{\infty}$ is holonomic and $\int
A\,d\pi_{\infty}\leq m(A)$. On the other hand,

$m(A)=\lim_{\beta_{n}\to+\infty}\frac{P_{\nu}(\beta_{n}A)}{\beta}\stackrel{{\scriptstyle
H_{\nu}\leq 0}}{{\leq}}\lim_{\beta_{n}\to+\infty}\int
A\,d\pi_{\beta_{n}A}=\int A\,d\pi_{\infty}.$

$\blacksquare$

Given $x\in X$, $n\in\mathbb{N}$ and a finite sequence
$\omega=(j_{1},j_{2},...,j_{n})\in J^{n}$ we denote

$\phi_{\omega}(x)=\phi_{(j_{1},...,j_{n})}(x):=\phi_{j_{1}}\circ\dots\circ\phi_{j_{n}}(x)$

and

${\rm
Sum}(A,\omega,x):=A({j_{1}},\phi_{(j_{2},...j_{n})}(x))+A(j_{2},\phi_{(j_{3},...,j_{n})}(x))+\dots+A(j_{n},x)-n\cdot
m(A).$

Given $x,y\in X$ and $\varepsilon>0$ we define

$S_{A,\varepsilon}(x,y)=\sup_{n\in\mathbb{N}}\left[\sup_{\omega\in
J^{n}\,|\,d(x,\phi_{\omega}(y))<\varepsilon}{\rm Sum}(A,\omega,y)\right],$ (5)

which can be $-\infty$ if the set $\\{\omega\in
J^{n}\,|\,d(x,\phi_{\omega}(y))<\varepsilon\\}$ is empty for any $n$. Clearly
$0<\epsilon<\epsilon^{\prime}\Rightarrow S_{q,\varepsilon}\leq
S_{q,\varepsilon^{\prime}}$, so we can define the Mañé potential
$S_{A}:X\times X\to\mathbb{R}\cup\\{-\infty\\},$ by

$S_{A}(x,y)=\lim_{\varepsilon\to 0}S_{A,\varepsilon}(x,y).$ (6)

The Aubry set is defined as

$\Omega_{A}=\\{x\in X|S_{A}(x,x)=0\\}.$ (7)

Let $V$ be a calibrated subaction to $A$ and define
$q(j,x):=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$. Then, by definition of calibrated
subaction, we have $\sup_{j\in J}q(j,x)=0,\,\forall\,x\in X$. For any
holonomic probability $\pi$ on $J\times X$ we have $\int q\,d\pi=\int
A\,d\pi-m(A)$. Then $m(q)=0$ and a holonomic probability $\pi$ is maximizing
to $A$ if and only if it is maximizing to $q$.

######  Theorem 2.5.

Let $A:J\times X\to\mathbb{R}$ be a Lipschitz function and $V$ be any
continuous calibrated subaction to $A$. Let us define the function $q:J\times
X\to\mathbb{R}$ by $q(j,x):=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$. Then
$S_{q}(x,y)=S_{A}(x,y)+V(x)-V(y)$ and $\Omega_{q}=\Omega_{A},$ which is a non
empty set. Furthermore,

$V(x)=\sup_{z\in\Omega_{A}}[S_{A}(z,x)+V(z)],\,\,\forall x\in X.$ (8)

Proof. We have $m(q)=0$, then, for $\omega=(j_{1},...,j_{n})$,

${\rm
Sum}(q,\omega,y):=q(j_{1},\phi_{(j_{2},...j_{n})}(y))+q({j_{2}},\phi_{(j_{3},...,j_{n})}(y))+\dots+q(j_{n},y).$

As $q(j,x)=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$ we have

${\rm Sum}(q,\omega,y)={\rm Sum}(A,\omega,y)+V(\phi_{\omega}(y))-V(y).$

Then

$S_{q}(x,y)=\lim_{\varepsilon\to 0}\sup_{n\in\mathbb{N}}\left[\sup_{\omega\in
J^{n}\,|\,d(x,\phi_{\omega}(y))<\varepsilon}({\rm
Sum}(A,\omega,y)+V(\phi_{\omega}(y))-V(y))\right]$ $=S_{A}(x,y)+V(x)-V(y).$

The Aubry sets satisfy $\Omega_{A}=\\{x\in X\,|\,S_{A}(x,x)=0\\}$ and
$\Omega_{q}=\\{x\in X\,|\,S_{q}(x,x)=0\\}$. As $S_{A}(x,x)=S_{q}(x,x)$ we
obtain $\Omega_{A}=\Omega_{q}.$

Let us denote also $q_{j}(x)=q(j,x)$. We still need to prove that
$\Omega_{A}\neq\emptyset$ and $\sup_{z\in\Omega_{A}}S_{q}(z,x)=0,\,\forall
x\in X.$ As $V$ is a calibrated subaction we have $\sup_{j\in
J}q(j,x)=0,\forall x\in X$ and so $S_{q}\leq 0$. From now on we fix $x\in X$
and prove that there exists $\tilde{x}\in\Omega_{A}$ such that
$S_{q}(\tilde{x},x)=0$.

Let $(j_{n})$ be a sequence of points of $J$ satisfying $q_{j_{1}}(x)=0$ and
$q_{j_{n+1}}(\phi_{j_{n}}\circ...\circ\phi_{j_{1}}(x))=0$ for all $n\geq 1$.
Let
$x_{n}:=\phi_{(j_{n},...,j_{1})}(x)=\phi_{j_{n}}\circ...\circ\phi_{j_{1}}(x)$.

As $X$ is compact there exists a subsequence $n_{i}$ and a point $\tilde{x}$
such that $x_{n_{i}}\to\tilde{x}$. It follows from the definition of
$S_{q}(\cdot,\cdot)$ that $S_{q}(\tilde{x},x)\geq\lim_{n_{i}\to+\infty}{\rm
Sum}(q,(j_{n_{i}},...,j_{1}),x)=0$ and then $S_{q}(\tilde{x},x)=0$. In what
follows we will prove that $\tilde{x}\in\Omega$.

Let $\omega_{k,l}:=(j_{k},j_{k-1},...,j_{l})$ for $k\geq l$. By construction
of $(j_{n})$ we have $q_{j_{n+1}}(x_{n})=0$ and

${\rm Sum}(q,\omega_{k,n+1},x_{n})=0,\,\,\,\forall k>n.$ (9)

Given $\varepsilon>0$ there exist $n,m\in\mathbb{N}$ such that $n<m$ and
$x_{n},x_{m}\in B(\tilde{x},\varepsilon/2)$. Observe that
$x_{m}=\phi_{\omega_{m,n+1}}(x_{n})$ and as $\phi$ satisfies (1) we get

$d(\tilde{x},\phi_{\omega_{m,n+1}}(\tilde{x}))\leq
d(\tilde{x},x_{m})+d(x_{m},\phi_{\omega_{m,n+1}}(\tilde{x}))$
$=d(\tilde{x},x_{m})+d(\phi_{\omega_{m,n+1}}(x_{n}),\phi_{\omega_{m,n+1}}(\tilde{x}))<\varepsilon.$

It follows that

$S_{q,\varepsilon}(\tilde{x},\tilde{x})\geq{\rm
Sum}(q,\omega_{m,n+1},\tilde{x}).$

As $A$ is Lipschitz continuous and $\phi$ satisfies (1), there exists $C>0$
such that

${\rm Sum}(q,\omega_{m,n+1},\tilde{x})={\rm
Sum}(A,\omega_{m,n+1},\tilde{x})+V(\phi_{\omega_{m,n+1}}(\tilde{x}))-V(\tilde{x})$
$\geq{\rm
Sum}(A,\omega_{m,n+1},x_{n})-\frac{C}{1-\gamma}\frac{\varepsilon}{2}+V(\phi_{\omega_{m,n+1}}(\tilde{x}))-V(\tilde{x})$
$={\rm
Sum}(q,\omega_{m,n+1},x_{n})-\frac{C}{1-\gamma}\frac{\varepsilon}{2}+[V(\phi_{\omega_{m,n+1}}(\tilde{x}))-V(\phi_{\omega_{m,n+1}}(x_{n}))]-[V(\tilde{x})-V(x_{n})].$

Therefore, applying also equation (9) and using that $V$ is continuous, we
have $S_{q}(\tilde{x},\tilde{x})=\displaystyle\lim_{\varepsilon\to
0}S_{q,\varepsilon}(\tilde{x},\tilde{x})\geq 0.$ As we always have $S_{q}\leq
0$, we complete the proof.

$\blacksquare$

##  3 On invariant idempotent probabilities

In this section we recall definitions and results containing in [28]. The
theory of idempotent probabilities was introduced by Maslov (also called
Maslov measures) to model problems of optimization as integrals with respect
to a max-plus structure, see [16], [17], [11] and [20] for classical results
on Maslov measures, also [24], [35], [36], [10] and [9] for the existence of a
unique invariant idempotent probability for non-place dependent IFSs.

Let us consider $\mathbb{R}_{\max}:=\mathbb{R}\cup\\{-\infty\\}$ endowed with
the operations

  1. 1.

$\oplus:\mathbb{R}_{\max}\times\mathbb{R}_{\max}\to\mathbb{R}_{\max}$, where
$a\oplus b:=\max(a,b)$ assuming $a\oplus-\infty:=a$.

  2. 2.

$\odot:\mathbb{R}_{\max}\times\mathbb{R}_{\max}\to\mathbb{R}_{\max}$, where
$a\odot b:=a+b$ assuming $a\odot-\infty:=-\infty$.

We consider the following conventions and notations: for any function
$f:A\to\mathbb{R}_{\max}$,

  1. 1.

$\bigoplus_{a\in A}:=\sup_{a\in A}$;

  2. 2.

$\bigoplus_{a\in A}f(a)=-\infty$, if $A=\emptyset$;

  3. 3.

$\bigoplus_{a\in A}f(a)=-\infty$, if $f(a)=-\infty,\,\forall a\in A$ .

  4. 4.

$\bigoplus_{a\in A}f(a)=+\infty$, if $A\neq\emptyset$ and $\\{f(a)\,|\,a\in
A\\}$ is not bounded above.

######  Definition 3.1.

A functional $m:C(X,\mathbb{R})\to\mathbb{R}$ is an idempotent probability on
$X$ if

  1. 1.

$m(a\odot f)=a\odot m(f)$, $\forall a\in\mathbb{R}$ and $\forall f\in
C(X,\mathbb{R})$;

  2. 2.

$m(f\oplus g)=m(f)\oplus m(g)$, $\forall f,g\in C(X,\mathbb{R})$;

  3. 3.

$m(0)=0$.

We denote by $I(X)$ the set of idempotent probabilities on $X$.

The idempotent probabilities are closely related with upper semi-continuous
functions (u.s.c.). We denote by $U(X,\mathbb{R}_{\max})$ the set of u.s.c.
functions $\lambda:X\to\mathbb{R}_{\max}$ satisfying $\\{x\in
X|\lambda(x)\neq-\infty\\}\neq\varnothing.$ The next result is well known in
the literature (mainly from [16]) and a proof for the present setting can be
founded in [28].

######  Theorem 3.2.

A functional $\mu:C(X,\mathbb{R})\to\mathbb{R}$ is an idempotent probability
if and only if there exists $\lambda\in U(X,\mathbb{R}_{\max})$ satisfying

$\mu(\psi)=\bigoplus_{x\in X}\lambda(x)\odot\psi(x),\,\,\,\,\forall\,\psi\in
C(X,\mathbb{R})\,\,\,\text{and}\,\,\,\bigoplus_{x\in X}\lambda(x)=0.$

If $\mu\in I(X)$ there is a unique such function $\lambda$ in
$U(X,\mathbb{R}_{\max})$, which is called its density.

######  Definition 3.3.

Let $(X,d_{X})$ and $(J,d_{J})$ be compact metric spaces. A max-plus IFS
(mpIFS for short) $\mathcal{S}=(X,(\phi_{j})_{j\in J},(q_{j})_{j\in J})$ is a
uniformly contractive IFS $(X,(\phi_{j})_{j\in J})$ endowed with a normalized
family of weights $(q_{j})_{j\in J}$, which is a family of functions
$\\{q_{j}:X\to\mathbb{R}\,|\,j\in J\\}$ satisfying:
i. there exists a constant $C>0$ such that

$|q_{j}(x_{1})-q_{j}(x_{2})|\leq C\cdot d_{X}(x_{1},x_{2}),\,\,\forall j\in
J,\;\forall x_{1},\,x_{2}\in X;$ (10)

ii.

$\bigoplus_{j\in J}q_{j}(x)=0,\,\,\forall x\in X;$ (11)

iii. the function $q:J\times X\to\mathbb{R}$, defined by $q(j,x)=q_{j}(x)$, is
continuous.

We also use the compact notation $\mathcal{S}=(X,\phi,q)$ to denote a mpIFS
(assuming that the set of indices $J$ is known). Considering the setting of
above section, if $A:J\times X\to\mathbb{R}$ is Lipschitz continuous and
$V:X\to\mathbb{R}$ is a Lipschitz calibrated subaction of $A$, then defining
$q(j,x)=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$ we get a normalized family of
weights. This is the case of $\displaystyle
q(j,x)=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(q^{\beta}(j,x))$, where
$q^{\beta}(j,x)=\frac{e^{\beta A(j,x)}\cdot h_{\beta A}(\phi_{j}(x))}{h_{\beta
A}(x)\cdot\lambda_{\beta A}}$, when such limit exists.

######  Definition 3.4.

To each mpIFS $\mathcal{S}=(X,\phi,q)$ we assign the following operators:
1\. $\mathcal{L}_{\phi,q}^{0}:C(X,\mathbb{R})\to C(X,\mathbb{R})$, defined by

$\mathcal{L}_{\phi,q}^{0}(f)(x):=\bigoplus_{j\in J}q_{j}(x)\odot
f(\phi_{j}(x)).$ (12)

2\. $\mathcal{M}_{\phi,q}^{0}:I(X)\to I(X),$ defined by

$\mathcal{M}_{\phi,q}^{0}(\mu):=\bigoplus_{j\in
J}\mu(q_{j}\odot(f\circ\phi_{j})).$ (13)

3\. $L_{\phi,q}^{0}:U(X,\mathbb{R}_{\max})\to U(X,\mathbb{R}_{\max}),$ defined
by

$L_{\phi,q}^{0}(\lambda)(x):=\bigoplus_{(j,y)\in\phi^{-1}(x)}q_{j}(y)\odot\lambda(y).$
(14)

The operators $\mathcal{L}_{\phi,q}^{0}$ and $\mathcal{M}_{\phi,q}^{0}$ play
the hole of the operators $\mathcal{L}_{\phi,q^{\beta},\nu}$ and
$\mathcal{M}_{\phi,q^{\beta},\nu}$, which were introduced in section 2, when
considered the zero temperature case and the scale $\frac{1}{\beta}\log$ (see
also Lemma 2.2).

Next theorem establishes the relation between the three operators in
Definition 3.4.

######  Theorem 3.5.

Given a function $\lambda\in U(X,\mathbb{R}_{\max})$ satisfying
$\oplus_{x}\lambda(x)=0$ and the associated idempotent probability
$\mu=\bigoplus_{x\in X}\lambda(x)\odot\delta_{x}\in I(X)$ we have that
$\mathcal{M}_{\phi,q}^{0}(\mu)=\bigoplus_{x\in
X}L_{\phi,q}(\lambda)(x)\odot\delta_{x}$, that is,
$\mathcal{M}_{\phi,q}^{0}(\mu)$ has density $L_{\phi,q}(\lambda)$ where
$\lambda$ is the density of $\mu$. Furthermore

$\mathcal{M}_{\phi,q}^{0}(\mu)(f)=\mu(\mathcal{L}_{\phi,q}^{0}(f)),$

for any $f\in C(X,\mathbb{R})$, that is, $\mathcal{M}_{\phi,q}^{0}$ is the
max-plus dual of $\mathcal{L}_{\phi,q}^{0}$.

######  Definition 3.6.

An idempotent probability $\mu\in I(X)$ with density $\lambda\in
U(X,\mathbb{R}_{\max})$ is called invariant (with respect to the mpIFS) if it
satisfies any of the following equivalent conditions:
1\. $\mathcal{M}_{\phi,q}^{0}(\mu)=\mu$;
2\. $L_{\phi,q}(\lambda)=\lambda$;
3\. $\mu(\mathcal{L}_{\phi,q}^{0}(f))=\mu(f)$, for any $f\in C(X,\mathbb{R})$.

In [28] the idempotent invariant probabilities were characterized using the
Mañé potential and the Aubry set. Let us present these results below.

######  Theorem 3.7.

Consider any function $\overline{\lambda}:X\to[-\infty,0]$ satisfying
$\bigoplus_{x\in X}\bar{\lambda}(x)=0$ and

$\bar{\lambda}(x)=\bigoplus_{z\in\Omega_{q}}[S_{q}(x,z)\odot\bar{\lambda}(z)].$

Then, $\overline{\mu}(f):=\bigoplus_{x\in X}(\overline{\lambda}(x)\odot f(x))$
is an idempotent probability which is invariant for the operator
$\mathcal{M}_{\phi,q}^{0}$. Reciprocally, if $\mu\in I(X)$ is invariant for
$\mathcal{M}_{\phi,q}^{0}$ and $\lambda\in U(X,\mathbb{R}_{\max})$ is its
(unique u.s.c.) density, then $\lambda$ satisfies the equation.

$\lambda(x)=\bigoplus_{z\in\Omega_{q}}[S_{q}(x,z)\odot\lambda(z)].$ (15)

We remark that we have a symmetric correspondence between the equation which
characterizes the calibrated subactions in Theorem 2.5 and the equation which
characterizes densities of invariant idempotent probabilities in Theorem 3.7.
The main differences are the use of $S_{A}$ instead $S_{q}$ and the order of
variables.

In [28] it is presented an example proving that a max-plus IFS can have
infinitely many possible idempotent invariant probabilities. On the other
hand, if $\phi_{j}(x)=\phi_{j}$ does not depend of $x,$ the idempotent
invariant probability is unique. This will play an important hole in Section 4
and we present this result below.

Given any $\omega=(j_{1},j_{2},...)\in J^{\mathbb{N}}$, there exists a unique
point, denoted by $\pi(\omega)\in X$ satisfying

$\pi(\omega)=\lim_{n\to+\infty}\phi_{j_{1}}\circ...\circ\phi_{j_{n}}(x)$

for any point $x$ of $X$.

######  Theorem 3.8.

Given a mpIFS, $S=(X,\phi,q)$, if $q_{j}(x)=q_{j}$ does not depend on $x$,
then there exists a unique invariant idempotent probability for
$\mathcal{M}_{\phi,q}^{0}$. Let $\lambda\in U(X,\mathbb{R}_{\max})$ be its
density. Then for any $z\in\Omega$ we have

$\lambda(x)=S_{q}(x,z)=\bigoplus_{(j_{1},j_{2},j_{3},...)\in\pi^{-1}(x)}[q_{j_{1}}+q_{j_{2}}+q_{j_{3}}+...].$

##  4 Zero temperature limits and Large Deviations

In [1] it is presented a necessary and sufficient condition for a family of
probabilities to satisfy a large deviation principle through the notion of
idempotent measures introduced by Maslov. In the present section we will prove
that the idempotent probabilities invariant for $\mathcal{M}_{\phi,q}$ are
related with large deviation principles for the measures $\rho_{{}_{\beta A}}$
at zero temperature. This will provide a connection between Sections 2 and 3.

######  Definition 4.1.

Let $(\mu_{\beta})_{\beta>0}$ be a family of probabilities on $X$. We will say
that $(\mu_{\beta})$ satisfy a large deviation principle (LDP) if there exists
a lower semi-continuous (l.s.c.) rate function $I:X\rightarrow[0,+\infty]$
such that
1\. for each closed set $C\subset X$

$\limsup_{\beta\to+\infty}\frac{1}{\beta}\log\mu_{\beta}(C)\leq-\inf_{\omega\in
C}I(\omega);$

2\. for each open set $U\subset X$

$\liminf_{\beta\to+\infty}\frac{1}{\beta}\log\mu_{\beta}(U)\geq-\inf_{\omega\in
U}I(\omega).$

Remark: As $X$ is open and closed and $\mu_{\beta}(X)=1$, we obtain
$\inf_{w\in X}I(w)=0.$ Furthermore, we assume $\log(0)=-\infty,$
$\displaystyle{\lim_{\beta\to+\infty}\frac{1}{\beta}\log(0)=-\infty}$ and
$\displaystyle{-\inf_{x\in\emptyset}I(x)=-\infty}$.

Next two results are very useful in the present work.

######  Lemma 4.2.

[Varadhan’s Lemma] Let $(\mu_{\beta})_{\beta>0}$ be a family of probabilities
on the compact metric space $X$ which satisfies a LDP with rate function
$I:X\rightarrow[0,+\infty]$. Then, for any function $f\in C(X,\mathbb{R})$ we
have

$\lim_{\beta\to\infty}\frac{1}{\beta}\log(\rho_{{}_{\beta
A}}(e^{\beta{f}}))=\sup_{x\in X}[f(x)-I(x)].$

Proof. See Theorem 4.3.1 in [12]. $\blacksquare$

######  Lemma 4.3.

[Bric’s Inverse Varadhan’s Lemma] Let $(\mu_{\beta})_{\beta>0}$ be a family of
probabilities on the compact metric space $X$. Suppose that for any function
$f\in C(X,\mathbb{R})$ there exists the limit

$\Gamma(f):=\lim_{\beta\to\infty}\frac{1}{\beta}\log(\rho_{{}_{\beta
A}}(e^{\beta{f}})).$

Then $(\mu_{\beta})$ satisfies a LDP. Furthermore, denoting by
$I:X\rightarrow[0,+\infty]$ its deviation function, we have

$\Gamma(f)=\sup_{x\in X}[f(x)-I(x)],\,\forall\,f\in C(X,\mathbb{R}).$

Proof. See Theorem 4.4.2 in [12]. $\blacksquare$

###  4.1 Irreducible Aubry set and a large deviation principle

In this subsection we investigate the uniqueness of idempotent invariant
probabilities through the Aubry set and proves a LDP.

######  Definition 4.4.

We will say that $\Omega_{A}$ is irreducible if $S_{A}(x,y)=0$ for any
$x,y\in\Omega_{A}$.

As an example, in [28] it is proved that $\Omega_{q}$ is irreducible if
$q_{j}(x)=q_{j}$ does not depend of $x$.

######  Corollary 4.5.

If $\Omega_{A}$ is irreducible then any calibrated subaction is constant on
$\Omega_{A}$ and any two calibrated subactions are equal, up to the addition
of a constant.

Proof. For any $x\in\Omega_{A}$ and any calibrated subaction $V$, applying
Theorem 2.5, we have

$V(x)=\sup_{z\in\Omega_{A}}S_{A}(z,x)+V(z)=\sup_{z\in\Omega_{A}}V(z).$

This proves that $V$ is constant in $\Omega_{A}$. Let us call
$V_{\Omega}:=V(z),\,\forall z\in\Omega_{A}$. Then, for any $x\in X$ and any
calibrated subaction $V$, we have

$V(x)=\left(\sup_{z\in\Omega_{A}}S_{A}(z,x)\right)+V_{\Omega}.$

In this way if $V$ and $U$ are any two calibrated subactions, we have
$V(x)-U(x)=V_{\Omega}-U_{\Omega},\;\forall\,x\in X$. $\blacksquare$

######  Corollary 4.6.

If $\Omega_{A}$ is irreducible then there exists the uniform limit

$\displaystyle{V=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(h_{\beta A})}$

and consequently the uniform limit
$\displaystyle{q=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(q^{\beta})}$, which
satisfies $q(j,x)=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$.

Proof. By construction of $h_{\beta A}$ we have $\sup_{x\in X}h_{\beta
A}(x)=1$ and the family $\frac{1}{\beta}\log(h_{\beta A})$ is equicontinuous
(see Proposition 2.1). Let $V$ be the unique calibrated subaction satisfying
$\sup_{x\in X}V(x)=0$ (any two of they differ just by a constant). By applying
Arzela-Ascoli theorem we get $\frac{1}{\beta}\log(h_{\beta A})\to V$
uniformly. $\blacksquare$

######  Proposition 4.7.

It suppose that $\Omega_{A}$ is irreducible. Let
$q=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(q^{\beta})=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$.
There exists a unique invariant idempotent probability for the mpIFS
$(X,\phi,q)$. Its density $\lambda$ satisfies
$\lambda(x)=S_{q}(x,z),\,\,\forall z\in\Omega_{A}$.

Proof. From Theorem 2.5 we have $\Omega:=\Omega_{A}=\Omega_{q}$. From Theorem
3.7, if $\rho$ is an invariant idempotent probability with density $\lambda$
we have

$\lambda(x)=\sup_{z\in\Omega}S_{q}(x,z)+\lambda(z),\,\forall x\in X.$

If $x\in\Omega$ then we obtain
$\lambda(x)=\sup_{z\in\Omega}\lambda(z)=:\lambda_{\Omega}$ is a constant. If
$x\in X$ we have then

$\lambda(x)=\lambda_{\Omega}+\sup_{z\in\Omega}S_{q}(x,z).$

As $\sup_{x\in X}\lambda(x)=0$ and $\sup_{x\in
X}\sup_{z\in\Omega}S_{q}(x,z)=0$ we get $\lambda_{\Omega}=0$ and then

$\lambda(x)=\sup_{z\in\Omega}S_{q}(x,z).$

In [28] it is proved that $S_{q}(x,z)\geq
S_{q}(x,y)+S_{q}(y,z),\,\forall\,x,y,z\in X$. Then, if $z_{1}\in\Omega$, we
have $\lambda(x)=S_{q}(x,z_{1})$, because

$\lambda(x)\geq
S_{q}(x,z_{1})\geq\sup_{z\in\Omega}[S_{q}(x,z)+S_{q}(z,z_{1})]=\sup_{z\in\Omega}S_{q}(x,z)=\lambda(x).$

This proves that the invariant idempotent probability for the mpIFS
$(X,\phi,q)$ is unique and its density is $\lambda(x)=S_{q}(x,z_{1}),\,\forall
z_{1}\in\Omega$. $\blacksquare$

######  Theorem 4.8.

Suppose that $\Omega_{A}$ is irreducible. Let

$q=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(q^{\beta})=A(j,x)+V(\phi_{j}(x))-V(x)-m(A).$

Then $(\rho_{{}_{\beta A}})$ satisfy a LDP with rate function

$I(x)=-S_{q}(x,z)=-S_{A}(x,z)-V(x)+V(z),\,\forall z\in\Omega_{A}.$

Furthermore,

$\rho(f):=\sup_{x\in X}[f(x)-I(x)],$

defines an idempotent probability with density $-I$, which is the unique
invariant to the mpIFS $(X,\phi,q).$ For any continuous function $f\in
C(X,\mathbb{R})$ we have

$\lim_{\beta\to\infty}\frac{1}{\beta}\log(\rho_{{}_{\beta A}}(e^{\beta
f}))=\rho(f).$

Before we present the proof of Theorem 4.8, we need two lemmas.

######  Lemma 4.9.

Let $g:J\times X\to\mathbb{R}$ be a continuous function and for each
$\beta>0$, let $g_{\beta}:J\times X\to\mathbb{R}$ be a continuous function.
Suppose that $g_{\beta}\to g$ uniformly and that
$\frac{1}{\beta}\log(\int_{X}e^{\beta\cdot g(j,x)}\,d\rho_{\beta A}(x))$
converges to a function $\Gamma^{*}(j)$ uniformly on $J$. Then

$\frac{1}{\beta}\log(\int_{X}e^{\beta\cdot g_{\beta}(j,x)}\,d\rho_{\beta
A}(x))\to\Gamma^{*}(j)$

uniformly on $J$. The same is true if we replace $\beta$ by an increasing
sequence $\beta_{i}$ which converges to $+\infty$.

Proof. For any $\epsilon>0$ there exists $\beta_{0}$ such that
$g-\epsilon<g_{\beta}<g+\epsilon$, $\beta>\beta_{0}$ and

$\Gamma^{*}(j)-\epsilon<\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot
g(j,x)}\,d\rho_{\beta
A}(x)\right)<\Gamma^{*}(j)+\epsilon,\,\,\forall\beta>\beta_{0}.$

Then we have, for $\beta>\beta_{0}$,

$\Gamma^{*}(j)-2\epsilon<\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot
g(j,x)}\,d\rho_{\beta
A}(x)\right)-\epsilon=\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot
g(j,x)-\epsilon}\,d\rho_{\beta A}(x)\right)$
$\leq\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot
g_{\beta}(j,x)}\,d\rho_{\beta
A}(x)\right)<\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot
g(j,x)+\epsilon}\,d\rho_{\beta A}(x)\right)$
$=\frac{1}{\beta}\log\left(\int_{X}e^{\beta\cdot g(j,x)}\,d\rho_{\beta
A}(x)\right)+\epsilon<\Gamma^{*}(j)+2\epsilon.$

$\blacksquare$

######  Lemma 4.10.

Suppose for any $f:X\to\mathbb{R}$ there exists the limit

$\Gamma(f):=\lim_{\beta\to+\infty}\frac{1}{\beta}\log\left(\int_{X}e^{\beta
f(x)}\,d\rho_{\beta A}(x)\right).$

Let $q:J\times X\to\mathbb{R}$ be a continuous function and consider the
function $\Gamma^{*}:J\to\mathbb{R}$, defined by
$\Gamma^{*}(j)=\Gamma(q_{j}+f\circ\phi_{j})$, where $q_{j}(x)=q(j,x)$. Then
$\frac{1}{\beta}\log(\int_{X}(e^{\beta(q+f\circ\phi)})d\rho_{\beta A})$
converges uniformly to $\Gamma^{*}$ on $J$. The same is true if we replace
$\beta$ by an increasing sequence $\beta_{i}$ which converges to $+\infty$.

Proof. Fix any $\epsilon>0$. As $q$ and $f$ are continuous, $\phi$ satisfies
(1) and $J$ is compact, there is $\delta>0$ such that
$d(j,l)<\delta\Rightarrow|q_{j}(x)-q_{l}(x)+f(\phi_{j}(x))-f(\phi_{l}(x))|<\epsilon,\,\forall\,x\in
X.$ As $J$ is compact and the balls $B(j,\delta)$, $j\in J,$ form an open
covering of $J$ we can obtain a finite set $J_{0}\subset J$ such that the
balls $B(j,\delta)$, $j\in J_{0}$, form an open covering of $J$ too. As this
set $J_{0}$ is finite, there is $n_{0}\in\mathbb{N}$ such that

$\Gamma(q_{j}+f\circ\phi_{j})-\epsilon<\frac{1}{\beta}\log(\int_{X}e^{\beta(q_{j}+f\circ\phi_{j})}d\rho_{\beta
A})<\Gamma(q_{j}+f\circ\phi_{j})+\epsilon,\,\,\forall j\in
J_{0},\forall\beta>n_{0}.$

For any $l\in J$ there is $j\in J_{0}$ such that $d(l,j)<\delta$, then,
$q_{j}+f\circ\phi_{j}-\epsilon<q_{l}+f\circ\phi_{l}<q_{j}+f\circ\phi_{j}+\epsilon$.
From definition of $\Gamma$ we immediately get $\Gamma(f+a)=\Gamma(f)+a$ for
any $a\in\mathbb{R}$ and $[f\leq g\Rightarrow\Gamma(f)\leq\Gamma(g)]$. Then,
for $\beta>n_{0}$,

$\Gamma(q_{l}+f\circ\phi_{l})-3\epsilon\leq\Gamma(q_{j}+f\circ\phi_{j}+\epsilon)-3\epsilon=\Gamma(q_{j}+f\circ\phi_{j})-2\epsilon$
$<\frac{1}{\beta}\log\left(\int_{X}e^{\beta(q_{j}+f\circ\phi_{j})}\,d\rho_{\beta
A}\right)-\epsilon=\frac{1}{\beta}\log(\int_{X}e^{\beta(q_{j}+f\circ\phi_{j}-\epsilon)}\,d\rho_{\beta
A})$
$\leq\frac{1}{\beta}\log\left(\int_{X}e^{\beta(q_{l}+f\circ\phi_{l})}\,d\rho_{\beta
A}\right)<\frac{1}{\beta}\log\left(\int_{X}e^{\beta(q_{j}+f\circ\phi_{j}+\epsilon)}\,d\rho_{\beta
A}\right)$
$<\Gamma(q_{j}+f\circ\phi_{j})+2\epsilon=\Gamma(q_{j}+f\circ\phi_{j}-\epsilon)+3\epsilon\leq\Gamma(q_{l}+f\circ\phi_{l})+3\epsilon.$

$\blacksquare$

Proof of Theorem 4.8.  From Proposition 4.7 there exists a unique invariant
idempotent probability $\rho$ for the mpIFS $(X,\phi,q)$. Fix $\tilde{f}\in
C(X,\mathbb{R})$ and suppose by contradiction there exists an increasing
sequence of positive numbers $(\beta_{i})$ such that

$\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}\tilde{f}}))\neq\rho(\tilde{f}).$

As $C(X,\mathbb{R})$ is separable, there exists a enumerable set $F$ of
continuous functions which is dense in $C(X,\mathbb{R})$ with the uniform
norm. By applying a Cantor’s diagonal argument we can construct a subsequence
(also denoted by $(\beta_{i})$) such that the limit
$\lim_{\beta_{i}\to+\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}f}))$
there exists for any $f\in F$. We denote by

$\Gamma(f):=\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}f}))$

for any $f\in F$.

Claim 1. The limit
$\Gamma(g):=\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}g}))$
there exists for any $g\in C(X,\mathbb{R})$.

Proof of Claim 1. Given $\epsilon>0$ there exists $f\in F$ such that
$f(x)-\epsilon\leq g(x)\leq f(x)+\epsilon$ for any $x\in X$. Therefore

$\Gamma(f)-\epsilon=\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}(f-\epsilon)}))\leq\liminf_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}g}))$
$\leq\limsup_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}g}))\leq\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}(f+\epsilon)}))\leq\Gamma(f)+\epsilon.$

Then

$\limsup_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}g}))-\liminf_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\rho_{\beta_{i}A}(e^{\beta_{i}g}))\leq
2\epsilon.$

Making $\epsilon\to 0$ we get the existence of the limit $\Gamma(g)$. This
finish the proof of claim.

Claim 2. The map $\Gamma:C(X,\mathbb{R})\to\mathbb{R}$, given by
$\Gamma(f)=\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}f}))$
defines an idempotent probability.

Proof of Claim 2.

  1. 1.

$\Gamma(0)=\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(1))=0$;

  2. 2.

$\Gamma(\lambda\odot\psi)=\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}\lambda}\cdot
e^{{\beta_{i}}\psi}))=\lambda\odot\Gamma(\psi)$, for all
$\lambda\in\mathbb{R}$ and $\psi\in C(X,\mathbb{R})$;

  3. 3.

$\Gamma(\varphi\oplus\psi)=\Gamma(\varphi)\oplus\Gamma(\psi)$, for all
$\varphi,\psi\in C(X,\mathbb{R})$.
To check this last equation just consider the sequence of inequalities:

$\Gamma(\varphi\oplus\psi)=\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}(\varphi\oplus\psi)})\geq\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}\varphi})=\Gamma(\varphi).$

Similarly we get such inequality for $\psi$ and so

$\Gamma(\varphi\oplus\psi)\geq\Gamma(\varphi)\oplus\Gamma(\psi).$

On the other hand

$\Gamma(\varphi\oplus\psi)=\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}(\varphi\oplus\psi)}))$
$\leq\lim_{{\beta_{i}}\to\infty}\frac{1}{{\beta_{i}}}\log(\rho_{\beta_{i}A}(e^{{\beta_{i}}\varphi})+\rho_{\beta_{i}A}(e^{{\beta_{i}}\psi}))=\Gamma(\varphi)\oplus\Gamma(\psi).$

Claim 3. The idempotent probability $\Gamma$ is invariant for the operator
$\mathcal{M}_{\phi,q}^{0}$.

Proof of Claim 3. Replace $f$ by $e^{\beta f}$ in equation (2) and then take
$\lim_{\beta_{i}\to\infty}\frac{1}{\beta_{i}}\log(\cdot)$ in both sides of
this equation. In this way we get

$\frac{1}{\beta_{i}}\log\left(\int_{X}e^{\beta_{i}f}\,d\rho_{\beta_{i}A}(x)\right)=\frac{1}{\beta_{i}}\log\left(\int_{X}\int_{J}q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)})\,d\nu(j)d\rho_{\beta_{i}A}(x)\right).$

Applying Fubini’s theorem we have

$\frac{1}{\beta_{i}}\log\left(\int_{X}e^{\beta_{i}f}\,d\rho_{\beta_{i}A}(x)\right)=\frac{1}{\beta_{i}}\log\left(\int_{J}\int_{X}q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)})\,d\rho_{\beta_{i}A}(x)d\nu(j)\right)$
$=\frac{1}{\beta_{i}}\log\left(\int_{J}\int_{X}e^{\beta_{i}[\frac{1}{\beta_{i}}\log\left(q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)})\right)]}\,d\rho_{\beta_{i}A}(x)d\nu(j)\right).$

As
$\frac{1}{\beta_{i}}\log(q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)}))\to
q(j,x)+f\circ\phi_{j}(x)$ uniformly, applying Lemma 4.9 and Lemma 4.10 we get
that
$\frac{1}{\beta_{i}}\log\int_{X}e^{\beta_{i}[\frac{1}{\beta_{i}}\log(q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)}))]}\,d\rho_{\beta_{i}A}(x)$
converges to $\Gamma(q_{j}+f\circ\phi_{j})$ uniformly on $J$. Then, applying
Lemma 2.2, we get

$\frac{1}{\beta_{i}}\log\left(\int_{J}\int_{X}e^{\beta_{i}[\frac{1}{\beta_{i}}\log(q_{j}^{\beta_{i}}(x)\cdot(e^{\beta_{i}f\circ\phi_{j}(x)}))]}\,d\rho_{\beta_{i}A}(x)d\nu(j)\right)\to\sup_{j\in
J}[\Gamma(q_{j}+f\circ\phi_{j})].$

Therefore,

$\Gamma(f)=\sup_{j}[\Gamma(q_{j}+f\circ\phi_{j})]\,\,\,\forall\,f\in
C(X,\mathbb{R}).$

This shows that $\Gamma$ is invariant for the operator
$\mathcal{M}_{\phi,q}^{0}$ acting on idempotent probabilities. This proves
Claim 3.

From Proposition 4.7, there exists a unique invariant idempotent probability
for $\mathcal{M}_{\phi,q}^{0}$ which we denote by $\rho$. Then $\Gamma=\rho$
and the initial hypothesis of this proof is false. We conclude that for any
continuous function $f$,
$\rho(f)=\lim_{\beta\to\infty}\frac{1}{\beta}\log(\rho_{{}_{\beta A}}(e^{\beta
f}))$. Let us call by $\lambda$ the density of $\rho$. Then $\lambda\in
U(X,\mathbb{R}_{\max})$ and from Proposition 4.7 it satisfies

$\lambda(x)=S_{q}(x,z),\,\,\forall z\in\Omega_{A}.$

We still need to prove that the Gibbs measures $\rho_{{}_{\beta A}}$ satisfies
a LDP with rate function $I(x)=-\lambda(x)$. But this is just a consequence of
Theorem 3.2 and Lemma 4.3.

$\blacksquare$

###  4.2 Example: the non-place dependent case

In this subsection we suppose $X$ and $J$ are compact metric spaces. We
consider a uniformly contractible IFS $\mathcal{S}=(X,(\phi_{j})_{j\in J})$
and also fixed numbers $A_{j},j\in{J}$, such that the function
$A:J\to\mathbb{R}$, given by $A(j)=A_{j}$ is Lipschitz continuous.

For each $\beta>0$, the operator $\mathcal{L}_{\phi,e^{\beta A},\nu}$ has the
eigenfunction $h_{\beta A}=1$ associated to the eigenvalue $\lambda_{\beta
A}=\int_{J}e^{\beta A_{j}}\,d\nu(j)$. Then, we consider the numbers given by
$q_{j}^{\beta}=\frac{e^{\beta A_{j}}}{\int_{J}e^{\beta A_{i}}d\nu(i)},\,j\in
J$. Observe that $\int_{J}q_{j}^{\beta}d\nu(j)=1$. We will call
$(q_{j}^{\beta})_{j\in J}$ a Gibbs density on $J$ for temperature
$\frac{1}{\beta}$. There exists a unique Borel probability $\rho_{{}_{\beta
A}}$ on $X$ (Gibbs probability) which is invariant for the dual operator
$\mathcal{M}_{\phi,q^{\beta},\nu}$. It satisfies

$\rho_{{}_{\beta A}}(f)=\int_{J}q_{j}^{\beta}\cdot\rho_{{}_{\beta
A}}(f\circ\phi_{j})\,d\nu(j)\,\,\forall f\in C(X,\mathbb{R}).$ (16)

As $q_{j}^{\beta}=\frac{e^{\beta A_{j}}}{\int_{J}e^{\beta A_{i}}d\nu(i)}$ we
obtain from Lemma 2.2

$q_{j}:=\lim_{\beta{\to\infty}}\frac{1}{\beta}\log(q_{j}^{\beta})=A_{j}-\max_{i\in
J}A_{i}.$

Then $q_{j}\leq 0$ and $\bigoplus_{j\in J}q_{j}=0$. As $A$ is also continuous,
we conclude that $\mathcal{S}=(X,(\phi_{j})_{j\in J},(q_{j})_{j\in J})$ is a
max-plus IFS.

From Theorem 3.8 there exists a unique invariant idempotent probability $\rho$
for $\mathcal{M}_{\phi,q}^{0}$. This probability satisfies

$\rho(f)=\bigoplus_{j\in J}q_{j}\odot\rho(f\circ\phi_{j})\,\,\forall f\in
C(X,\mathbb{R})$ (17)

and its density $\lambda\in U(X,\mathbb{R}_{\max})$ satisfies

$\lambda(x)=\bigoplus_{(j_{1},j_{2},j_{3},...)\in\pi^{-1}(x)}[q_{j_{1}}+q_{j_{2}}+q_{j_{3}}+...].$

######  Theorem 4.11.

Consider the above framework. Then the family of Gibbs probabilities
$(\rho_{{}_{\beta A}})$ satisfy a LDP and its rate function is given by

$I(x)=-\bigoplus_{(j_{1},j_{2},j_{3},...)\in\pi^{-1}(x)}[q_{j_{1}}+q_{j_{2}}+q_{j_{3}}+...].$

Proof. From Theorem 3.8, if $x,z\in\Omega_{A}$ we have
$S_{q}(x,z)=S_{q}(x,x)=0$. Then $\Omega_{A}$ is irreducible and therefore the
proof is consequence of Theorem 4.8. $\blacksquare$

###  4.3 The place dependent case

In this section we will prove the following result.

######  Theorem 4.12.

Suppose that the family of invariant probabilities $(\rho_{{}_{\beta
A}})_{\beta>0}$ satisfy a LDP and denote by $I$ its deviation function.
Suppose that there exists the uniform limit
$V=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(h_{\beta A})$. Then, the uniform
limit $q=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(q^{\beta})$ there exists
and it satisfies $q(j,x)=A(j,x)+V(\phi_{j}(x))-V(x)-m(A),$ which defines a
normalized family of weights. Furthermore, $-I$ is the density of an
idempotent invariant probability $\rho$ for the mpIFS
$\mathcal{S}=(X,\phi,q)$, that is

$-I(x)=\sup_{z\in\Omega}[S_{q}(x,z)-I(z)],\,\,\forall x\in X.$ (18)

Proof. Suppose that the family of invariant probabilities $(\rho_{{}_{\beta
A}})_{\beta>0}$ satisfy a LDP and denote by $I$ its deviation function. Then
$I$ is l.s.c and $\inf_{x\in X}I(x)=0$. By applying Lemma 4.2 we get that, for
any $f\in C(X,\mathbb{R})$, there exists the limit
$\Gamma(f):=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(\rho_{{}_{\beta
A}}(e^{\beta f}))$ and it satisfies

$\Gamma(f)=\sup_{x\in X}[f(x)-I(x)].$

Therefore, $\Gamma$ defines an idempotent probability with density $-I$
(observe that $-I$ is u.s.c and $\sup_{x\in X}-I(x)=0$).

Suppose there exists the uniform limit
$V=\lim_{\beta\to+\infty}\frac{1}{\beta}\log(h_{\beta A})$. Then $V$ is a
calibrated subaction to $A$. We have, by definition of $q^{\beta}$,

$\frac{1}{\beta}\log(q^{\beta}(j,x))=A(j,x)+\frac{1}{\beta}\log(h_{\beta
A}(\phi_{j}(x)))-\frac{1}{\beta}\log(h_{\beta
A}(x))-\frac{1}{\beta}\log(\lambda_{\beta A}).$

Then $\frac{1}{\beta}\log(q^{\beta}(j,x))\to q(j,x)$ uniformly, where
$q(j,x)=A(j,x)+V(\phi_{j}(x))-V(x)-m(A)$. As $V$ is a calibrated subaction,
the function $q$ defines a normalized family of weights. Let us consider the
mpIFS $\mathcal{S}=(X,\phi,q)$.

Claim: $\Gamma$ defines an idempotent probability which is invariant for the
operator $\mathcal{M}_{\phi,q}^{0}$.

Proof of Claim: It follows from the same reasoning as Claim 3 in the proof of
Theorem 4.8.

Finally, as $-I$ is the density of an invariant idempotent probability for the
mpIFS $\mathcal{S}=(X,\phi,q)$, we obtain, from Theorem 3.7 that

$-I(x)=\sup_{z\in\Omega}[S_{q}(x,z)-I(z)].$

$\blacksquare$

Is a remarkable fact that equations (18) and (8) represents a kind of duality
between calibrated subactions (associated to eigenfunction in zero
temperature) and deviation functions (associated to Gibbs probabilities in
zero temperature). On the other hand LDP provides a connection between Gibbs
probabilities in thermodynamic formalism and invariant idempotent
probabilities in mpIFS.

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#  Can quantum non-locality be connected to extra-dimensions?

Marco Genovese  Italian National Metrology Research Institute (INRIM), Strada
delle Cacce 91, I-10135 Turin, Italy and INFN, sede di Torino, via P.Giuria 1,
10125 Turin, Italy.

###### Abstract

Quantum non locality, as described by EPR paradox, represents one of the
mysteries at the very foundations of quantum mechanics. Here we suggest to
investigate if it can be understood by considering extra dimensions.

##  I Introduction

Quantum non-locality, is a fundamental argument of the debate on the
foundations of quantum mechanics (QM)bell ; mm ; cl71 ; as82 ; lf1 ; lf2 ; lf3
; int ; eli18 ; gis20 ; gra21 . Furthermore, it is also becoming an important
resource for emerging quantum technologies qt1 ; qt2 ; qt3 ; qt4 ; qt5 ; qt6 .
This prompts the need of a clear understanding of quantum non-locality and, in
particular, of its compatibility with special relativity.

Even if one can rigorously demonstrate that it can not lead to any
superluminal transmission (signaling) GRW80 , peaceful coexistence between
special relativity and QM would require more, i.e., it would be necessary to
understand a coherent description among different observers shi . For
instance, how to reconciliate two observers that see a different temporal
order in the collapse of two entangled particles? The answers to this question
span from being an unsolved problem mau ; Macc19 ; st ; st2 ; st3 ; st4 ; st5
; st6 , requiring a preferred foliation to relativistic space–time, to being
only apparent, since accounts of entangled systems undergoing collapse yielded
by different reference frames can be considered as no more than differing
accounts of the same process and events Myr ; there is a form of holism
associated to QM description of composite systems dar21 , the factorizable
state after the collapse on a certain hypersurface is merely one way of
slicing together local parts, entangled states are a superposition of such
splicings. Also a connection between quantum non-locality and wormholes,
summarized by the sentence ”ER=EPR” (Einstein Rosen wormholes are tied to
Einstein-Podolsky-Rosen paradox) has been suggested by Maldacena and Susskind
ER  or the question looses its problematic aspects in superdeterministic
theories th1 ; t1 .

##  II A solution in higher dimensional spaces

Be that as it may, rebus sic stantibus, an undoubted difficulty in
understanding the collapse in composite systems persists. This is true for
standard quantum mechanics, but it is even more important for modifications of
quantum mechanics formalism where a spontaneous collapse occurs. For instance,
Ref.s c1 ; c2  demonstrated that creating a relativistic version of
spontaneous collapse models ”à la GRW” is impossible, while Ref.s c ; g ; p
showed that certain collapse models lead to superluminal signaling.
Furthermore while Bell inequalities tests exclude local hidden variable
theories bell ; mm ; cl71 ; as82 ; lf1 ; lf2 ; lf3 ; int , non-local hidden
variable theories still remain a valid alternative to standard quantum
mechanics int ; db1 ; db2 ; q1 ; q2 ; q3 , but in this case non-locality must
be understood in their framework.

Here we suggest that one should explore the possibility that the quantum non-
locality ”problem” can be solved in higher dimensional spaces mm . In the
following we will present in little more detail this idea suggested in mm ,
discussing some motivations and detailing some difficulties to be considered
in developing it. It is not our intention to provide a precise model: probably
several different ways can be gone along leading to rather different models
and various mathematical problems must be solved on these paths. We limit
ourselves to introducing this new idea, hoping this letter can stimulate
further studies contributing significantly to the debate on quantum non-
locality.

Let us consider a $n>4$ space-time with $n-4$ more (spatial) dimensions and
let us suppose that, while the usual fields only “live” in $3+1$ dimensions,
the collapse involves also other dimensions, eventually being induced by “some
field” propagating also in these extra-dimensions. We are considering the
collapse on the position variable, since in general every measurement at the
end reduces to a position measurement.

Every n-dimensional space can be “contracted” around a single point of a
higher dimensional space. For instance, a plane can be wrapped as tied as one
wishes around a single point in three-dimensionall space: two points of a
napkin can be several centimeters apart moving on the napkin, but they can be
very next in the third dimension after squeezing the napkin in a hand. Thus,
two far points in the “ordinary” space can be as close as wanted considering
extra-dimensions. Incidentally, this “contraction” is an isometry (the
distances in the 3+1 are unchanged) and leaves invariant the Gaussian
curvature. Nonetheless, a precise characterisation of the effects on local
curvature would depend on the specific model and would represent a significant
point in elaborating specific models.

If this extra-dimensions are extremely small (for example around the Planck
scale, as supposed for instance by some quantum gravity theory qg ; m  ),
every event of the 3+1 dimensional space could be connected with any other
event in an “extremely short” effective distance through the compactified
dimensions. Therefore, the collapse of one of two remote entangled particles
could cause the collapse of the other with the propagation of a subluminal
signal through the compactified dimensions. Quantum non-locality would reduce
to the fact that only the wave function collapse would be affected by these
extra dimensions.

This kind of phenomenon could either be included in a more general theory
going beyond quantum mechanics (for example, some Planck scale theory already
predicts compactified dimensions) or to find a use in building relativistic
collapse models r ; r1 ; r2 ; r3 ; r4 ; r5 . For example a simple
generalization of Tumulka’s one tum  should eventually be able to incorporate
it.

As a simple example, one could eventually consider some general non-unitary
evolution operator

$U(t)=Exp\\{-i\int d^{4}xd^{n}y{\it O}_{I}(x,y,t)\\}$ (1)

where ${\it O}_{I}(x,y,t)={\it H}_{I}(x,t)+C_{I}(x,y,t)$, ${\it H}_{I}(x,t)$
being the usual Hermitian–Hamiltonian density of quantum fields leaving in a
3+1 dimensional space (x), while the component $C_{I}(x,y,t)$ also inclues the
extra dimensions, y. $C_{I}(x,y,t)$ is not Hermitian and eventually induces
the collapse by inducing a non-unitary evolution. When one traces over the
degrees of freedom corresponding to the not Hermitian component,
$C_{I}(x,y,t)$, the usual unitary evolution is eventually restored, as in ’t
Hooft finite degrees of freedom model th .

For instance, let’s suppose one has a scalar field with a usual Hamiltonian,
where x are usual space time coordinates and y additional compactified
dimensions

$1/2\partial\Psi^{*}(x,y)\partial\Psi(x,y)+g(x,y)(\Psi^{*}(x,y)\Psi(x,y))^{n}$

if the coupling g has a real part depending on x and an imaginary part
depending on y the total Hamiltonian is not Hermitian and originates a non-
unitary evolution. Nonetheless, when tracing over the y degrees of freedom one
recovers a Hermitian Hamiltonian and a unitary evolution. In particular one
can consider the case when the evolution operator, containing the non
Hermitian component of the Hamiltonian, reduces to a projector collapsing the
state in a precise position with a probability of collapse proportional to the
number of matter field particles ($b^{\dagger}b$). This guarantees, as in all
collapse models, the persistency of quantum superpositions for microscopic
objects and a smooth transition to localised objects for macroscopic ones:

$U=Exp[-\int d^{d}yC_{I}(x,y,t)]\approx Exp[-g(x-q)^{2}b^{\dagger}(x)b(x)]$
(2)

where $C_{I}(x,y,t)$ can eventually be evaluated by inverting Eq.2.

In alternative the collapse induced by these effects can be included in a
Lindblad equation on the line of lin .

An interesting point becomes the causal connection among events. Actually, two
points in space can be very far in the usual dimensions and therefore two
events in these space locations cannot be connected by a causal signal if not
after a long propagation. Nevertheless, they are rapidly connected when the
collapse is considered, since this propagates in the compactified extra
dimension.If the effect of the signal through extra dimensions only affects
the collapse, this would not allow faster than light signaling in usual
dimensions. For example, Ghirardi-Rimini-Weber theorem GRW80  guarantees that
one cannot use EPR correlations and the collapse induced by one observer for
superluminal signaling.

Another consideration concerns the recent theorem demonstrated by Ref. b22
that collapse dynamics must always be diffusive: the hypotheses of this
theorem (in particular the one related to translational invariance) could not
be necessary in the models discussed here.

In summary, while one can suppose that usual quantum evolution/interactions
only happen in ordinary space with a unitary evolutions, the collapse of the
wave function could be related to the non-unitary evolution in additional
dimensions, that being compactified allows a ”practically” instantaneous
collapse independently by distance in ordinary space. This somehow reproduces
t’Hooft model th  in an infinite dimensional field theory.

##  III Conclusions

In conclusion, we suggest that quantum non-locality and the problems connected
to the temporal order of wave function collapse for entangled systems can
eventually find a solution by considering extra dimensions.

We do not propose a specific mathematical model, that can be connected, for
instance, to an eventual theory describing quantum gravity, but we suggest
that this idea is worth of being further explored.

Finally, it is worth considering the question of whether such a scheme could
lead to observable physical consequences. Of course, models introducing
spontaneous collapse of the wave function though this mechanism would lead to
the same effects expected in general in collapse models ab ; ab1 ; ab2 ; 11 .

Furthermore, one could expect these extra dimensions (and eventual extra
field/fields) could provide several possible observable effects. Long-distance
effects deriving from crossing these extra dimensions can eventually appear,
for example inducing correlations otherwise forbidden by propagation in
ordinary dimensions. Just by speculating a little these correlations could
eventually even be the source of ultra-quantum technologies or affect present
quantum technologies (as a threat to QKD security proofs). Furthermore, vacuum
energy (eventually in connection to cosmologic dark energy) can be affected by
this kind of phenomenon. Nonetheless, the study of different possible models
in this general framework and their possible effects is beyond the purpose of
this paper simply introducing this idea.

##  IV Acknowledgements

This research was supported by San Paolo Fundation, project QuaFuPhy.

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