# The calculus of thermodynamical formalism

### Paolo Giulietti

Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil### Benoît R. Kloeckner

Université Paris-Est - Créteil Val-de-Marne, France### Artur O. Lopes

Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil### Diego Marcon

Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

## 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 $\mathscr L_A$ acting on $\mathcal X(\Omega)$. Under suitable hypotheses, it is well-known that $\mathscr L_A$ 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.

## Cite this article

Paolo Giulietti, Benoît R. Kloeckner, Artur O. Lopes, Diego Marcon, The calculus of thermodynamical formalism. J. Eur. Math. Soc. 20 (2018), no. 10, pp. 2357–2412

DOI 10.4171/JEMS/814