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C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  26 March 2009

Jairo Bochi
Jairo Bochi
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente, 225, Rio de Janeiro, CEP 22453-900, Brazil ([email protected])
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Abstract

We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

Keywords

symplectic diffeomorphismspartial hyperbolicityLyapunov exponentsgeneric properties

MSC classification

Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 49 - 93
Copyright
Copyright © Cambridge University Press 2010

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