On the stable ergodicity of diffeomorphisms with dominated splitting

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 chainhyperbolicity. 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.

Minimality and stable Bernoulliness in dimension 3

“…We also point out that there is another theorem in [Oba19], which does not require the existence of an expanding bundle. It requires a dominated bundle…”

New examples of stably ergodic diffeomorphisms in dimension 3

“…In 2004, Tahzibi obtained in [Tah04] the first example of a stably ergodic system having no hyperbolic direction. There are some recent works by the authors in [NnRH20,Oba19], where they obtain results on stable ergodicity for non partially hyperbolic systems with some assumptions such as minimality of the strong unstable foliation, or chain hyperbolicity.…”