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Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms


Authors: Artur Avila and Jairo Bochi
Journal: Trans. Amer. Math. Soc. 364 (2012), 2883-2907
MSC (2010): Primary 37D25, 37D30, 37C20
DOI: https://doi.org/10.1090/S0002-9947-2012-05423-7
Published electronically: February 14, 2012
MathSciNet review: 2888232
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Abstract: We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $ C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure, then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold $ C^r$-generically for any $ r \ge 1$; namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the $ L^1$-continuity of Lyapunov exponents.


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Additional Information

Artur Avila
Affiliation: Laboratoire de Probabilités et Modèles aléatoires, CNRS UMR 7599, Université de Paris VI, Paris Cedex 05, France
Address at time of publication: IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil
Email: artur@math.sunysb.edu

Jairo Bochi
Affiliation: Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Mq. S. Vicente, 225, 22453-900, Rio de Janeiro, RJ, Brazil
Email: jairo@mat.puc-rio.br

DOI: https://doi.org/10.1090/S0002-9947-2012-05423-7
Received by editor(s): January 18, 2010
Received by editor(s) in revised form: May 3, 2010
Published electronically: February 14, 2012
Additional Notes: Both authors were partially supported by CNPq–Brazil. This research was partially conducted during the period that the first author served as a Clay Research Fellow.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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