Mixing-like properties for some generic and robust dynamics

TL;DR

This paper demonstrates that for certain robustly transitive dynamical systems, the set of Bernoulli measures is dense among invariant measures, introducing the large periods property as a key concept.

Contribution

The paper introduces the large periods property and proves its robustness, showing that Bernoulli measures are dense in invariant measures for generic homoclinic classes.

Findings

Bernoulli measures are dense among invariant measures on certain classes.

The large periods property is robust for these classes.

The entire manifold can be a homoclinic class for a dense set of diffeomorphisms.

Abstract

We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robustly topologically mixing.