Abundance of stable ergodicity

Abstract

We consider the set ${\mathcal{P}\mathcal{H}}_{\omega }(M)$ 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 ${\mathcal{P}\mathcal{H}}_{\omega }(M)$. This solves a conjecture of Pugh and Shub, in this setting.