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

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 Π0 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.