On the stable ergodicity of diffeomorphisms with dominated splitting

TL;DR

This paper develops new criteria for stable ergodicity in diffeomorphisms with dominated splitting using chain-hyperbolicity, expanding understanding beyond the partially hyperbolic case and demonstrating density results.

Contribution

It introduces two novel criteria for stable ergodicity outside the partially hyperbolic setting, one for diffeomorphisms with dominated splitting and another for weakly partially hyperbolic systems.

Findings

Established criteria for stable ergodicity using chain-hyperbolicity.

Proved $C^1$-density of stable ergodicity in certain weakly partially hyperbolic systems.

Extended ergodic theory beyond classical hyperbolic scenarios.

Abstract

In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the $C^1$-density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms.