New criteria for ergodicity and non-uniform hyperbolicity

TL;DR

This paper introduces a new criterion for establishing ergodicity and non-uniform hyperbolicity in smooth measures of diffeomorphisms, enhancing understanding of ergodic components and Lyapunov exponents.

Contribution

It provides a novel criterion for ergodicity and hyperbolicity, and demonstrates the density of stably ergodic diffeomorphisms in certain partially hyperbolic systems.

Findings

New criterion for ergodicity and non-uniform hyperbolicity

Global ergodicity achieved using topological devices like blenders

Stably ergodic diffeomorphisms are dense among volume-preserving partially hyperbolic diffeomorphisms with 2D center

Abstract

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.