TAILIEUCHUNG - Đề tài "The Lyapunov exponents of generic volume-preserving and symplectic maps "

We show that the integrated Lyapunov exponents of C 1 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: | Annals of Mathematics The Lyapunov exponents of generic volume-preserving and symplectic maps By Jairo Bochi and Marcelo Viana Annals of Mathematics 161 2005 1423 1485 The Lyapunov exponents of generic volume-preserving and symplectic maps By Jairo Bochi and Marcelo Viana To Jacob Palis on his 60th birthday with friendship and admiration. Abstract 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 diffeomor-phisms 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 c 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 Schrodinger operators. 1. Introduction Lyapunov exponents describe the asymptotic evolution of a linear cocycle over a transformation positive or negative exponents correspond to exponential growth or decay of the norm respectively whereas vanishing exponents mean lack of exponential behavior. Partially supported by CNPq Profix and Faperj Brazil. . thanks the Royal Institute of Technology for its hospitality. . is grateful for the hospitality of College de France Universite de Paris-Orsay and Institut de Mathematiques de Jussieu. 1424 JAIRO BOCHI .

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