Contributions to the geometric and ergodic theory of conservative flows

Abstract

Abstract We prove the following dichotomy for vector fields in a $C^1$ -residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$ -stably ergodic flow can be $C^1$ -approximated by another volume-preserving flow which is non-uniformly hyperbolic.