Equilibrium states for partially hyperbolic horseshoes

Abstract

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. 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 ∞ potentials with phase transition.

The structure of the space of ergodic measures of transitive partially hyperbolic sets

“…Porcupine-like horseshoes were introduced in [12] as model for internal heterodimensional cycles in horseshoes. Later these horseshoes were generalized and studied in a series of papers from various points of view: topological ( [7,8,9], thermodynamical ( [21,9,24,25]) and fractal ([13]) 1 . This line of research is also closely related to the study of socalled bony attractors and sets (see [17] for a survey and references).…”

Equilibrium stability for non-uniformly hyperbolic systems

“…Castro and Nascimento in [12] showed uniqueness of the maximal entropy measure for partially hyperbolic attractors semiconjugated to nonuniformly expanding maps. For a family of partially hyperbolic horseshoes introduced by Díaz, Horita, Rios and Sambarino in [17] the existence of equilibrium states for any continuous potential was proved by Leplaideur, Oliveira and Rios in [19]. Later, Rios and Siqueira [24] proved uniqueness of equilibrium states for a class of Hölder continuous potentials with small variation and which do not depend on the the stable direction.…”

Equilibrium states for Mañé diffeomorphisms

“…The techniques for the results in the previous paragraph are not well suited to the study of equilibrium states for ϕ = 0, which remains largely unexplored. When the first version of the present work appeared [arXiv:1505.06371v1], the only available references for this subject were existence results for a certain class of partially hyperbolic horseshoes [25], with uniqueness results only for potentials constant on the center-stable direction [2]. An improved picture has emerged since then.…”