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Cr−prevalence of stable ergodicity for a...
Leguil, Martin...

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Cr−prevalence of stable ergodicity for a class of partially hyperbolic systems by Leguil, Martin ( Author )
Lemonjar Open Access
N.A
13-07-2015
We prove that for r∈N≥2∪{∞}, 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 neighbourhood of f 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 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}.
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http://arxiv.org/abs/1507.03556
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