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

TL;DR

This paper investigates large deviation principles for Gibbs probabilities at high inverse temperature and their connection to invariant idempotent probabilities in iterated function systems, revealing new links with dynamical potentials.

Contribution

It establishes a large deviation principle for Gibbs probabilities in both place-dependent and non-place-dependent cases and characterizes associated invariant idempotent probabilities.

Findings

Proves LDP for Gibbs probabilities as temperature approaches zero.

Identifies the deviation function as the density of invariant idempotent probabilities.

Connects the deviation function with Mañé potential and Aubry set characterizations.

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 Ma\~{n}\'{e} potential and Aubry set, therefore we will obtain an identical characterization for $-I$.