Robust ergodic properties in partially hyperbolic dynamics

TL;DR

This paper investigates the ergodic behavior of partially hyperbolic systems with mostly contracting central directions, extending previous results to local diffeomorphisms and analyzing stability under perturbations.

Contribution

It extends Bonatti and Viana's work to local diffeomorphisms and demonstrates that such systems form a C^2-open set with dense statistical stability.

Findings

Existence and finiteness of physical measures established for local diffeomorphisms.

Such systems form a C^2-open set with dense statistical stability.

All mostly contracting systems are stable under small random perturbations.

Abstract

We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti, Viana about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C^2-open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations.