GENERICITY OF NONUNIFORM HYPERBOLICITY IN DIMENSION 3

For a generic conservative diffeomorphism of a closed connected 3-manifold M, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is nonuniformly hyperbolic and ergodic. This is the 3-dimensional version of the well-known result by Mane-Bochi [14, 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result was inspired by and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].