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
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Given an onto map TT acting on a metric space Ω\Omega and an appropriate Banach space of functions X(Ω)\mathcal X(\Omega), one classically constructs for each potential AXA \in \mathcal X a transfer operator LA\mathscr L_A acting on X(Ω)\mathcal X(\Omega). Under suitable hypotheses, it is well-known that LA\mathscr L_A has a maximal eigenvalue λA\lambda_A, has a spectral gap and defines a unique Gibbs measure μA\mu_A. Moreover there is a unique normalized potential of the form B=A+ffT+cB=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 N\mathcal N, of the normalization map ABA\mapsto B, and of the Gibbs map AμAA\mapsto \mu_A. We give an easy proof of the fact that N\mathcal N is an analytic submanifold of X\mathcal X and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow N\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