The Lyapunov exponents of generic volume preserving and symplectic systems

• Autores: Marcelo Viana , Jairo Bochi
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 161, Nº 3, 2005, págs. 1423-1485
• Idioma: inglés
• DOI: 10.4007/annals.2005.161.1423
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• Resumen

  • We show that the integrated Lyapunov exponents of C1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents are equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume-preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all C1 symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2. Finally, given any set S ¿¿ GL(d) satisfying an accessibility condition, for a residual subset of all continuous S-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on S is satisfied for most common matrix groups and also for matrices that arise from discrete Schr¿Nodinger operators.