Equilibrium measures for some partially hyperbolic systems

TL;DR

This paper establishes the existence and uniqueness of equilibrium measures for certain partially hyperbolic systems using geometric measure theory and Bowen property conditions.

Contribution

It introduces a novel approach combining geometric measure theory with thermodynamic formalism to prove uniqueness of equilibrium measures in partially hyperbolic systems.

Findings

Unique equilibrium measure for systems with bounded expansion in the center-stable bundle

Construction of reference measures on unstable leaves analogous to Hausdorff measure

Convergence of averaged pushforwards to a Gibbs measure

Abstract

We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.