Abundance of stable ergodicity
Amie Wikinson | Bonatti, Christian | Matheus, Carlos | Viana, Marcelo
stably ergodicity | partial hyperbolicity | volume preserving
We consider the set $\cP\cH_\omega(M)$ of volume preserving partially hyperbolic diffeomorphisms on a compact manifold having $1$-dimensional center bundle. We show that the volume measure is ergodic, and even Bernoulli, for any $C^2$ diffeomorphism in an open and dense subset of $\cP\cH_\omega(M)$. This solves a conjecture of Pugh and Shub, in this setting.