  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.
