The calculus of thermodynamical formalism

Given an onto map T acting on a metric space Ω and an appropriate Banach space of functions X(Ω), one classically constructs for each potential A ∈ X a transfer operator LA acting on X(Ω). Under suitable hypotheses, it is well-known that LA has a maximal eigenvalue λA, has a spectral gap and defines a unique Gibbs measure μA. Moreover there is a unique normalized potential of the form B = A + f - fT + 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 N of normalized potentials, of the normalization map A ↦ B, and of the Gibbs map A ↦ μA. We give an easy proof of the fact that N is an analytic submanifold of X and that the normalization map is analytic; we compute the derivative of the Gibbs map; and we endow 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.