Most Invariant Manifolds of Conservative Systems have Transitive Closure

The main result of this work in the following: for generic volume preserving flows on compact manifolds with the Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document} topology, 1≤r≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le r\le \infty $$\end{document}, the closure of every invariant manifold of periodic orbits and singularities is a chain transitive set. This was already known for generic symplectic and volume preserving diffeomorphisms.