Abundance of stable ergodicity (Q1884666)

Summary: We consider the set \({\mathcal P}{\mathcal H}_\omega(M)\) of volume-preserving partially hyperbolic diffeomorphisms on a compact manifold having one-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 \textit{C. Pugh} and \textit{M. Shub} [J. Complexity 13, 125--179 (1997; Zbl 0883.58029)], in this setting.