Equilibrium states for partially hyperbolic maps with one-dimensional center

TL;DR

This paper establishes the existence and uniqueness of equilibrium states for certain partially hyperbolic maps with one-dimensional center, including measures of maximal entropy, under specific conditions.

Contribution

It proves the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center and demonstrates uniqueness for a class of maps on the 2-torus.

Findings

Existence of equilibrium states for the class of maps studied.

Uniqueness of the measure of maximal entropy.

Conditions ensuring uniqueness on the n-torus.

Abstract

We prove the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness of such measures for endomorphisms defined on the 2-torus that: have a linear model as a factor; and with the condition that this measure gives zero weight to the set where the conjugacy with the linear model fails to be invertible. In particular, we obtain the uniqueness of the measure of maximal entropy. For the n-torus, the uniqueness in the case with one-dimensional center holds for absolutely partially hyperbolic maps with additional hypotheses on the invariant leaves, namely, dynamical coherence and quasi-isometry.