Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps

Varandas, Paulo; Viana, Marcelo

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 555-593.

Nous prouvons l'existence d'un nombre fini d'états d' équilibre ergodiques pour une classe assez grande d'homéomorphismes locaux non-uniformément dilatants sur des espaces métriques compacts et pour les potentiels de Hölder continus à oscillation pas trop grande. Aucune structure de Markov n'est supposée. Si la transformation est topologiquement mélangeante alors il existe un unique état d' équilibre, il est une mesure exacte et vérifie une propriété de Gibbs non-uniforme. Avec quelques hypothèses supplémentaires, nous prouvons aussi que les états d' équilibre varient de façon continue avec la dynamique et le potentiel (stabilité statistique) et sont également stables sous des perturbations stochastiques de la transformation.

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact metric spaces and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.

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