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 $\{\phi_j: X \to X \, | \, j \in J \}$. 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 $-I$.