We consider a transformation $T:X\to X$ which can be either the shift acting on the Bernoulli space $X=\{1,2...,d\}^\mathbb{N}$ or an expanding transformation acting on the circle $X=S^1$. For a given $\alpha$-Holder potential $A: X \to \mathbb{R}$ one can consider the equilibrium probability for the pressure $P(A)$.

Denote by $\mathcal{G}$ the set of equilibrium probabilities for $\alpha$-Holder potentials.

In the Banach manifold $\mathcal{G}$ we introduce a natural Riemannian structure. For $v$ in the tangent space $T_\mu \mathcal{G}$ the value $|v|^2$ is the asymptotic variance of $v$ for $\mu=\mu_A$, where $\mu_A$ is the equilibrium probability for the normalized potential $A$.

We investigate the gradient flow of the entropy function $\mu \in \mathcal{G} \to h (\mu)$ and for a fixed Holder potential $B:X \to \mathbb{R}$ the gradient flow of the function $\mu \in \mathcal{G} \to G(\mu)= h(\mu) + \int B d \mu$. We also solve some interesting problems of maximization of entropy with constrains.

We show that the Riemannian structure on $\mathcal{G}$ is not compatible with the Wasserstein Riemannian structure.

The set of potentials depending of two coordinates is a subset of $\mathcal{G}$. It can be parametrized by the square
$\Gamma=[0,1]\times[0,1]\subset \mathbb{R}^2$. The Riemannian metric restricted to the set $\Gamma$ defines a manifold of positive curvature.

This is a joint work with P. Giulietti, B. Kloeckner and D. Marcon