# The Lyapunov exponents of generic volume-preserving and symplectic maps

### 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.

## Authors

Jairo Bochi

Instituto de Matemática, UFRGS, 91509-900 Porto Alegre, Brazil

Marcelo Viana

IMPA, 22460-320 Rio de Janeiro, Brazil