Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward map

We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $Ω=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $Π_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hatΠ\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hatΠ\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hatΠ$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case.