The calculus of thermodynamical formalism

• Autor(a) principal: Giulietti, Paolo
• Data de Publicação: 2018
• Outros Autores: Kloeckner, Benoît R., Lopes, Artur Oscar, Farias, Diego Marcon
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFRGS
• Texto Completo: http://hdl.handle.net/10183/182431
• topic: Estados de equilibrio; Entropia; Regularização entrópica; Transfer operators; Equilibrium states; Entropy; Regularity; Wasserstein space

Given an onto map T acting on a metric space  and an appropriate Banach space of functions X./, one classically constructs for each potential A 2 X a transfer operator LA acting on X./. Under suitable hypotheses, it is well-known that LA has a maximal eigenvalue A, has a spectral gap and defines a unique Gibbs measure A. Moreover there is a unique normalized potential of the form B D ACf f T Cc 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 7! B, and of the Gibbs map A 7! A. 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.