Level-2 IFS thermodynamic formalism: Gibbs probabilities in the space of probabilities and the push-forward map

Abstract

We will denote by M the space of Borel probabilities on the symbolic space Ω = { 1 , 2 ⋯ , m } N . M is equipped Monge–Kantorovich metric. We consider here the push-forward map T : M → M as a dynamical system. The space of Borel probabilities on M is denoted by M . Given a continuous function A : M → R , an a priori probability Π 0 on M , and a certain convolution operation acting on pairs of probabilities on M , we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability Π ^ ∈ M for such an operator. Under a normalization condition for A, we show the existence of some T -invariant probabilities Π ^ ∈ M . We are able to define the variational entropy of such Π ^ and a related maximization pressure problem associated to A. In some particular examples, we show how to get eigenprobabilities solutions on M for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on M for the classical (Level-1) Thermodynamic Formalism; this shows that our approach is a natural generalization of the classic case.