C r-prevalence of stable ergodicity for a class of partially hyperbolic systems - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of the European Mathematical Society Année : 2022

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

(1, 2) ,
1
2

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).

Dates et versions

hal-03696976 , version 1 (16-06-2022)

Identifiants

Citer

Martin Leguil, Zhiyuan Zhang. C r-prevalence of stable ergodicity for a class of partially hyperbolic systems. Journal of the European Mathematical Society, 2022, 24 (9), pp.3379-3438. ⟨10.4171/JEMS/1163⟩. ⟨hal-03696976⟩
9 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook Twitter LinkedIn More