Genericity of non-uniform hyperbolicity in dimension 3

TL;DR

This paper proves that for generic conservative diffeomorphisms in three dimensions, the system is either non-uniformly hyperbolic and ergodic or has all Lyapunov exponents vanishing, extending known surface results to 3D.

Contribution

It establishes the 3D analogue of a well-known surface result, confirming a conjecture by Avila and Bochi for dimension 3.

Findings

Generic 3D conservative diffeomorphisms have globally dominated Oseledets splitting.

Such systems are either ergodic non-uniformly hyperbolic or have zero Lyapunov exponents almost everywhere.

Partially hyperbolic sets with positive measure and one-dimensional center have strong homoclinic intersections.

Abstract

For a generic conservative diffeomorphism of a 3-manifold M, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is non-uniformly hyperbolic and ergodic. This is the 3-dimensional version of a well-known result by Ma\~n\'e-Bochi, stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result inspired and answers in the positive for dimension 3 a conjecture by Avila and Bochi. We also prove that all partially hyperbolic sets with positive measure and center dimension one have a strong homoclinic intersection. This implies that Cr generically for any r, a diffeomorphism contains no proper partially hyperbolic sets with positive measure and center dimension one.