Genericity of non-uniform hyperbolicity in dimension 3

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 Mañé-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.

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