$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents
Dynamical Systems
2010-05-03 v5 Probability
Symplectic Geometry
Abstract
We prove that if is a -generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if 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.
Cite
@article{arxiv.0801.2960,
title = {$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents},
author = {Jairo Bochi},
journal= {arXiv preprint arXiv:0801.2960},
year = {2010}
}
Comments
Final version. To appear in Journal of the Institute of Mathematics of Jussieu