Minimality and stable Bernoulliness in dimension 3

Abstract

In 3-dimensional manifolds, we prove that generically in Diff 1 m (M 3 ), the existence of a minimal expanding invariant foliation implies stable Bernoulliness.

New examples of stably ergodic diffeomorphisms in dimension 3

“…One possible strategy to prove conjecture 1.1 in three-manifolds could be to prove the following: Conjecture 1.2. [NnRH20] Generically in Diff 1 m (M 3 ) if f admits a dominated splitting, then there is a uniformly expanding or contracting f-invariant minimal foliation.…”

“…Conjecture 1.1 would then follow from the following fact: Theorem 1.3. [ NnRH20] generically in Diff 1 m (M 3 ) if f admits a uniformly expanding or contracting invariant minimal foliation then f is stably ergodic.…”

“…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.…”

“…Conjecture 1.1. [NnRH20] Generically in Diff 1 m (M), either all Lyapunov exponents are zero, or f is stably ergodic and stably non-uniformly hyperbolic. An equivalent statement is: generically in Diff 1 m (M) dominated splitting implies stable ergodicity.…”