Statistical properties of the maximal entropy measure for partially hyperbolic attractors

TL;DR

This paper establishes the existence, uniqueness, and statistical properties of the maximal entropy measure for certain partially hyperbolic systems, including decay of correlations, CLT, and stability, using cone metrics.

Contribution

It introduces new techniques to prove statistical properties of maximal entropy measures for partially hyperbolic attractors, extending results to systems derived from solenoids and Anosov.

Findings

Existence and uniqueness of maximal entropy measure.

Exponential decay of correlations for Hölder observables.

Statistical stability for systems derived from solenoids.

Abstract

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove exponential decay of correlations for H\"older continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from solenoid we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov. Paper published in ETDS in Jan. 2016.