C r-prevalence of stable ergodicity for a class of partially hyperbolic systems
Résumé
We prove that for r E N>2 U \oo\, for any dynamically coherent, center bunched and strongly pinched volume preserving Cr partially hyperbolic diffeomorphism f : X ! X, if either (1) its center foliation is uniformly compact, or (2) its center-stable and center-unstable foliations are of class C1, then there exists a C1-open neighborhood off in Diffr .X; Vol/, in which stable ergodicity is Cr-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).