On the Stable Ergodicity of Berger-Carrasco's example

TL;DR

This paper proves the stable ergodicity of a specific volume-preserving, partially hyperbolic diffeomorphism that exhibits robust non-uniform hyperbolicity without relying on accessibility, advancing understanding of ergodic behavior.

Contribution

It establishes stable ergodicity for Berger-Carrasco's example without using accessibility, highlighting new techniques in non-uniform hyperbolic dynamics.

Findings

Proves stable ergodicity of Berger-Carrasco's example

Shows robustness of non-uniform hyperbolicity in the example

Demonstrates ergodic behavior without dominated splitting

Abstract

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Pierre Berger and Pablo Carrasco. This example is robustly non-uniformly hyperbolic, with two dimensional center, almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.