Preprint C48/2006
The Lyapunov exponents of conservative continuous-time dynamical systems

Mário Bessa

**Keywords: **
Lyapunov Exponents | Dominated splitting | volume-preserving flows.

We prove that for a C1-generic subset of
all the conservative vector fields on 3-dimensional compact
manifolds without singularities, we have for Lebesgue a.e. point p that either the
Lyapunov exponents at p are zero or the vector field is an Anosov vector field.
We also prove a similar version of the previous result in the setting of
conservative
non-autonomous linear differential systems in the C0-topology.
Finally we prove that for a C1-dense subset of all the conservative vector fields on
3-dimensional compact
manifolds with singularities, we have for Lebesgue a.e. point that either the
Lyapunov
exponents at p are zero or p belongs to a compact invariant set with dominated
splitting
for the linear Poincare flow.