Uniform (projective) hyperbolicity or no hyperbolicity: A dichotomy for generic conservative maps

We show that the Lyapunov exponents of volume preserving C¹ diffeomorphisms of a compact manifold are continuous at a given diffeomorphism if and only if the Oseledets splitting is either dominated or trivial. It follows that for a C¹-residual subset of volume preserving diffeomorphisms the Oseledets splitting is either dominated or trivial. We obtain analogous results in the setting of symplectic diffeomorphisms, where the conclusion is actually stronger: dominated splitting is replaced by partial hyperbolicity. We also obtain versions of these results for continuous cocycles with values in some matrix groups. In the text we give the precise statements of these results and the ideas of the proofs. The complete proofs will appear in [4].