Equilibrium States for Partially Hyperbolic Horseshoes

TL;DR

This paper investigates the ergodic properties of invariant measures in a specific type of partially hyperbolic horseshoe, establishing hyperbolicity of measures and existence of equilibrium states, including phase transitions.

Contribution

It proves all ergodic measures are hyperbolic and demonstrates the existence of equilibrium measures for continuous potentials in this setting.

Findings

All recurrent points have non-zero Lyapunov exponents.

Existence of equilibrium measures for any continuous potential.

An example of phase transition in a family of smooth potentials.

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 \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.