On the Transitivity of Invariant Manifolds of Conservative Flows

The main result of this work is the following: for volume preserving flows on compact manifolds with the $C^r$ topology, $1\leqq r\leqq \mathrm{\infty }$ , the closure of every invariant manifold of periodic orbits and singularities is a chain transitive set. We also develop to new local constructions, which surprise by the simplicity of the arguments. One, a local perturbation to change an orbit to a nearby without altering its past. The other is a flow box theorem in the context of volume preserving flows, a result that is well known for Hamiltonians or general flows.