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

Abstract

We prove that for $r\in {\mathbb{N}}_{\ge 2}\cup \{\infty \}$, for any dynamically coherent, center bunched and strongly pinched volume preserving $C^r$ partially hyperbolic diffeomorphism $f:X\to X$, 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 neighborhood of $f$ in ${\mathrm{Diff}}^r(X,\mathrm{Vol})$, 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 (1997); 2. a generic version of an open question of Hirsch–Pugh–Shub (1977); and 3. a generic version of an open question of Pugh–Shub (1997).