The Lyapunov exponents of generic volume preserving and symplectic systems

We show that the integrated Lyapunov exponents of C volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all C symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2. Finally, given any closed group G ⊂ GL(d) that acts transitively on the projective space, for a residual subset of all continuous G-valued cocycles over any measure preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial.