Denseness of ergodicity for a class of volume-preserving flows

Abstract

We consider the class of $C^1$ partially hyperbolic volume-preserving flows with one-dimensional central direction endowed with the $C^1$-Whitney topology. We prove that, within this class, any flow can be approximated by an ergodic $C^2$ volume-preserving flow and so, as a consequence, ergodicity is dense.