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

Abstract

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. Mañé in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

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

References in corpus (3)

• Genericity of zero Lyapunov exponents
• Generic dynamics of 4-dimensional C2 Hamiltonian systems
• Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles

Cited by in corpus (4)

• Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle
• $C^1$-openness of non-uniform hyperbolic diffeomorphisms with bounded $C^2$ norm
• Generic area-preserving reversible diffeomorphisms
• Cocycles with Quasi-Conformality I: Stability and abundance