$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents

Details $C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents $C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents

2008-01-18

arxiv.org/abs/0801.2960v5

J. Inst. Math. Jussieu, 9 (2010), 49-93

Mathematics

Dynamical Systems

Final version. To appear in Journal of the Institute of Mathematics of Jussieu

Scientific paper

10.1017/S1474748009000061

We prove that if $f$ is a $C^1$-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 center bundle vanish. This establishes in full a result announced by R. Ma\~{n}\'{e} in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

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