The Lyapunov exponents of generic volume preserving and symplectic   systems
  We show that the integrated Lyapunov exponents of $C^1$ 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 $C^1$ 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 $G \subset GL(d,\mathrm{R})$ 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. 
Jairo Bochi and Marcelo Viana