Contributions to the Geometric and Ergodic Theory of Conservative Flows

We prove the following dichotomy for vector fields in a C-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincaré flow. Moreover, we prove that a volumepreserving and C-stably ergodic flow can be C-approximated by another volume-preserving flow which is non-uniformly hyperbolic. MSC 2000: primary 37D30, 37D25; secondary 37A99, 37C10. keywords: Volume-preserving flows; Lyapunov exponents; Dominated splitting; Stable ergodicity.