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 uniformly contractible iterated function system \(\{\phi _j: X \rightarrow X \, | \, j \in J \}\). 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 \(\beta \rightarrow +\infty \) 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 rate 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 \(-I\).