New criteria for ergodicity and nonuniform hyperbolicity

Abstract

In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. 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 nonzero 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 2-dimensional center bundle. This is motivated by a well-known conjecture of Pugh and Shub.