Invariance principle and non-compact center foliations

TL;DR

This paper generalizes the invariance principle for partially hyperbolic diffeomorphisms to include non-compact center foliations, enabling classification of measures and analysis of physical measures in broader dynamical systems.

Contribution

It extends the invariance principle to systems with non-compact center leaves, including discretized Anosov flows, and applies this to classify measures and study physical measures.

Findings

Invariance principle holds for non-compact center foliations.

Classification of measures of maximal entropy achieved.

Analysis of physical measures for perturbations of Anosov flows.

Abstract

We prove a generalization of a so called "invariance principle" for partially hyperbolic diffeomorphisms: if an invariant probability measure has all its center Lyapunov exponents equal to zero then the measure admits a center disintegration that is invariant by stable and unstable holonomies. This was known for systems admitting a foliation by compact center leaves, and we extend it to a larger class which contains discretized Anosov flows. We use our result to classify measures of maximal entropy and study physical measures for perturbations of the time-one map of Anosov flows.