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

TL;DR

This paper proves that for generic symplectic diffeomorphisms, the Oseledets splitting is either trivial or partially hyperbolic, with all center exponents vanishing if the system is not Anosov, confirming a long-standing conjecture.

Contribution

It establishes a full proof of Mañé's 1983 conjecture using a novel probabilistic perturbation method involving random walks.

Findings

Oseledets splitting is trivial or partially hyperbolic for generic systems.

All center Lyapunov exponents vanish if the system is not Anosov.

Introduces a new probabilistic technique for constructing perturbations.

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. Ma\~{n}\'{e} in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.