The calculus of thermodynamical formalism

Abstract

Given an onto map $T$ acting on a metric space $\Omega$ and an appropriate Banach space of functions $\mathcal{X}(\Omega )$, one classically constructs for each potential $A\in \mathcal{X}$ a transfer operator ${\mathcal{L}}_A$ acting on $\mathcal{X}(\Omega )$. Under suitable hypotheses, it is well-known that ${\mathcal{L}}_A$ has a maximal eigenvalue ${\lambda }_A$, has a spectral gap and defines a unique Gibbs measure ${\mu }_A$. Moreover there is a unique normalized potential of the form $B=A+f-f\circ T+c$ acting as a representative of the class of all potentials defining the same Gibbs measure.

The goal of the present article is to study the geometry of the set of normalized potentials $\mathcal{N}$, of the normalization map $A\mapsto B$, and of the Gibbs map $A\mapsto {\mu }_A$. We give an easy proof of the fact that $\mathcal{N}$ is an analytic submanifold of $\mathcal{X}$ and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow $\mathcal{N}$ with a natural weak Riemannian metric (derived from the asymptotic variance) with respect to which we compute the gradient flow induced by the pressure with respect to a given potential, e.g. the metric entropy functional. We also apply these ideas to recover in a wide setting existence and uniqueness of equilibrium states, possibly under constraints.