Equilibrium States for Partially Hyperbolic Horseshoes

Renaud Leplaideur (1) , Krerley Oliveira , Isabel Rios

1 LM - Laboratoire de mathématiques de Brest

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \cite{DHRS07}. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of $C^\infty$ potentials with phase transition.

Domaines: Systèmes dynamiques [math.DS]

Renaud Leplaideur, Krerley Oliveira, Isabel Rios. Equilibrium States for Partially Hyperbolic Horseshoes. 2007. ⟨hal-00202571⟩