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

Abstract

We consider two compact metric spaces $J$ and $X$ and a uniform contractible iterated function system $\{\ddot{I}{•}_j:X\to X|j\in J\}$. For a Lipschitz continuous function $A$ on $J\times X$ and for each ${\hat{I}}^2>0$ we consider the Gibbs probability $\ddot{I}{}_{{}_{{\hat{I}}^{2}A}}$. Our goal is to study a large deviation principle for such family of probabilities as ${\hat{I}}^2\to +\infty$ 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 $(\ddot{I}{}_{{}_{{\hat{I}}^{2}A}})$ 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 $(\ddot{I}{}_{{}_{{\hat{I}}^{2}A}})$ 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 $-I$.

24 pages