Constructing equilibrium states for some partially hyperbolic attractors via densities

TL;DR

This paper introduces a new method for constructing equilibrium states in certain partially hyperbolic systems, extending previous Gibbs measure constructions to more general settings with additional assumptions.

Contribution

It generalizes equilibrium state construction techniques from uniformly hyperbolic to partially hyperbolic systems under new conditions.

Findings

Addresses new issues in equilibrium state construction for partially hyperbolic systems.

Introduces conditions on center-stable and center-unstable manifolds for equilibrium states.

Builds on prior work by Pesin-Sinai, Dolgopyat, Climenhaga, and others.

Abstract

We shall describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalises our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin-Sinai and Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.