Equilibrium measures for some partially hyperbolic systems

Vaughn Climenhaga; Yakov Pesin; Agnieszka Zelerowicz

doi:10.3934/jmd.2020006

Article Contents

2020, Volume 16: 155-205. Doi: 10.3934/jmd.2020006

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Equilibrium measures for some partially hyperbolic systems

1.
Department of Mathematics, University of Houston, Houston, TX 77204, USA URL: https://www.math.uh.edu/~climenha/
2.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA URL: http://www.math.psu.edu/pesin/
3.
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received: July 25, 2019

Revised: March 20, 2020

Published: June 2020

Partially supported by NSF grant DMS-1554794. Partially supported by NSF grant DMS-1400027. Partially supported by NSF grant DMS-1400027

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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.

Keywords:

Mathematics Subject Classification: Primary: 37D35, 37C45; Secondary: 37C40, 37D30.

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Figure 6.1. Proving Lemma 6.2

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Figure 8.1. Birkhoff averages are essentially constant

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Figure 8.2. The partitions $ {\mathcal{R}} $ and $ {\mathcal{R}}_n $, and the choice of $ \epsilon_R $

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