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We prove the following dichotomy for vector fields in a C1-residual subset ofvolume-preserving flows: for Lebesgue almost every point all Lyapunov exponentsequal to zero or its orbit has a dominated splitting. As a consequence if wehave a vector field in this residual that cannot be C1-approximated by a vectorfield having elliptic periodic orbits, then, there exists a full measure setsuch that every orbit of this set admits a dominated splitting for the linearPoincare flow. Moreover, we prove that a volume-preserving and C1-stablyergodic flow can be C1-approximated by another volume-preserving flow which isnon-uniformly hyperbolic.

Contributions to the Geometric and Ergodic Theory of Conservative Flows