• Corpus ID: 118576112

Stable accessibility is C1 dense

@inproceedings{Dolgopyat2001StableAI,
  title={Stable accessibility is C1 dense},
  author={Dmitry Dolgopyat and Amie Wilkinson},
  year={2001}
}

• D. Dolgopyat, A. Wilkinson
• Published 2001
• Mathematics

— We prove that in the space of ail Cr (r ^ 1) partially hyperbolic diffeomorphisms, there is a C1 open and dense set of accessible diffeomorphisms. This settles the C1 case of a conjecture of Pugh and Shub. The same resuit holds in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms. Combining this theorem with results in [Br], [Ar] and [PugSh3], we obtain several corollaries. The first states that in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms, topological transitivity holds on an open and dense set. Further, on a symplectic n-manifold (n $C 4) the C1-closure of the stably transitive symplectomorphisms is precisely the closure of the partially hyperbolic symplectomorphisms. Finally, stable ergodicity is C1 open and dense among the volume preserving, partially hyperbolic diffeomorphisms satisfying the additional technical hypothèses of [PugSh3]. Introduction This paper is about the accessibility property of partially hyperbolic diffeomorphisms. We show that accessibility holds for a C1 open and dense set in the space of ail partially hyperbolic diffeomorphisms, thus settling the C1 version of a conjecture of Pugh and Shub [PugShl]. Partially hyperbolic diffeomorphisms are similar to Anosov diffeomorphisms, in that they possess invariant hyperbolic directions, but unlike Anosov diffeomorphisms, they can also possess invariant directions of nonhyper bolic behavior. Accessibility means that the hyperbolic directions fill up the manifold on a macroscopic scale. Accessibility often provides enough hyperbolicity for a variety of chaotic properties, such as topological transitivity [Br] and ergodicity 2000 Mathematics Subject Classification. — Primary 37D30, 37C20.