Preprint A147/2002
The Lyapunov exponents of generic volume preserving and symplectic systems

Marcelo Viana | Bochi, Jairo

**Keywords: **
Volume-preserving maps | symplectic maps | Lyapunov exponents

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 $G \subset GL(d,\mathbb{R})$ 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.