The calculus of thermodynamical formalism

Given an onto map T acting on a metric space Omega and an appropriate Banach space of functions chi(Omega), one classically constructs for each potential A is an element of chi a transfer operator L-A acting on chi(Omega). Under suitable hypotheses, it is well-known that 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 o 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 N of normalized potentials, of the normalization map A bar right arrow B, and of the Gibbs map A bar right arrow mu(A). We give an easy proof of the fact that N is an analytic submanifold of chi 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.