Abstract
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 are 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 set $S\subset\mathrm{GL}(d)$ satisfying an accessibility condition, for a residual subset of all continuous $S$-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on $S$ is satisfied for most common matrix groups and also for matrices that arise from discrete Schrödinger operators.
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