  We will denote by $\mathcal{M}$ the space of Borel probabilities on the
symbolic space $\Omega=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ 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 $\Pi_0$ 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 $\hat{\Pi}\in\mathfrak{M}$ for such an operator. Under a
normalization condition for $A$, we show the existence of some
$\mathfrak{T}$-invariant probabilities $\hat{\Pi}\in\mathfrak{M}.$ We are able
to define the variational entropy of such $\hat{\Pi}$ 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.
