\(C^r\)-prevalence of stable ergodicity for a class of partially hyperbolic systems (Q2143211)

Summary: 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 neighborhood of \(f\) in \(\operatorname{Diff}^r(X,\text{Vol})\), 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).