Stable Ergodicity and Accessibility for certain Partially Hyperbolic Diffeomorphisms with Bidimensional Center Leaves

TL;DR

This paper proves that certain classes of partially hyperbolic diffeomorphisms with two-dimensional center leaves are stably ergodic and accessible, including perturbations of product systems and symplectomorphisms, for generic parameter choices.

Contribution

It establishes stable ergodicity and accessibility for a broad class of partially hyperbolic systems with bidimensional center leaves, extending previous results to new settings.

Findings

Accessibility holds in open and dense subsets.

Systems are stably ergodic under perturbations.

Results apply to various classes including skew products and symplectomorphisms.

Abstract

We consider classes of partially hyperbolic diffeomorphism $f:M\to M$ with splitting $TM=E^s\oplus E^c\oplus E^u$ and $\dim E^c=2$. These classes include for instance (perturbations of) the product of Anosov and conservative surface diffeomorphisms, skew products of surface diffeomorphisms over Anosov, partially hyperbolic symplectomorphisms on manifolds of dimension four with bidimensional center foliation whose center leaves are all compact. We prove that accessibility holds in these classes for $C^1$ open and $C^r$ dense subsets and moreover they are stably ergodic.