$C^{r}-$prevalence of stable ergodicity for a class of partially hyperbolic systems

Abstract

We prove that for $r \in \mathbb{N}_{\geq 2} \cup \{\infty\}$, for any dynamically coherent, center bunched and strongly pinched volume preserving $C^r$ partially hyperbolic diffeomorphism $f \colon X \to X$, if either (1) its center foliation is uniformly compact, or (2) its center-stable and center-unstable foliations are of class $C^1$, then there exists a $C^1$-open neighbourhood of $f$ in ${\rm Diff}^r(X,\mathrm{Vol})$, in which stable ergodicity is $C^r$-prevalent in Kolmogorov's sense. In particular, we verify Pugh-Shub's stable ergodicity conjecture in this region. This also provides the first result that verifies the prevalence of stable ergodicity in the measure-theoretical sense. Our theorem applies to a large class of algebraic systems. As applications, we give affirmative answers in the strongly pinched region to: 1. an open question of Pugh-Shub in \cite{PS}; 2. a generic version of an open question of Hirsch-Pugh-Shub in \cite{HPS}; and 3. a generic version of an open question of Pugh-Shub in \cite{HPS}.