Approximate controllability on the group of volume-preserving diffeomorphisms

Abstract: We study controlability issues for the group of volume-preserving diffeomorphisms of the torus $\mathbb T^d$ for system $\dot x=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on $\mathbb T^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.