Large Deviation for Gibbs Probabilities at Zero Temperature and Invariant Idempotent Probabilities for Iterated Function Systems

We consider two compact metric spaces J and X and a uniformly contractible iterated function system {phi(j ): X -> X | j is an element of J}. For a Lipschitz continuous function A on J x 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 -> +infinity and its connections with idempotent probabilities. In the non-place dependent case (A (j, x) = A (j), for all(x )is an element of 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 Ma & ntilde;& eacute; potential and Aubry set, therefore we will obtain an identical characterization for -I.