Abundance of stable ergodicity

Abstract

We consider the set PHω(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 C2 diffeomorphism in an open and dense subset of PH ω(M). This solves a conjecture of Pugh and Shub, in this setting.