On Equilibrium States for Partially Hyperbolic Horseshoes

TL;DR

This paper establishes the existence and uniqueness of equilibrium states for a class of partially hyperbolic systems derived from horseshoes, focusing on potentials with small variation and their behavior in the center-unstable direction.

Contribution

It introduces new results on equilibrium states for partially hyperbolic horseshoes with small variation potentials, extending understanding of their statistical properties.

Findings

Proves existence of equilibrium states for the systems.

Shows uniqueness of these equilibrium states.

Focuses on potentials constant on local stable manifolds.

Abstract

We prove existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Holder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced in [D\'iaz,L., Horita,V., Rios, I., Sambarino, M., Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 2009]. For the original three dimensional system, we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.