The Lyapunov exponents of generic zero divergence three-dimensional vector fields

Abstract

Abstract We prove that for a C 1 -generic (dense G δ ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p ∈ M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C 1 -dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p ∈ M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.