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 uniform contractible iterated function system \φⱼ: X → X \, | \, j ∈ J \. For a Lipschitz continuous function A on J × X and for each β>0 we consider the Gibbs probability ρ_β A. Our goal is to study a large deviation principle for such family of probabilities as β → +∞ and its connections with idempotent probabilities. In the non-place dependent case (A(j,x)=Aⱼ,\,∀ x∈ X) we will prove that (ρ_β 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 (ρ_β 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 $-I$.