Transitive Anosov flows on non-compact manifolds

Abstract

In this article we study topological transitivity of Anosov flows on non-compact 3-manifolds. We provide homological conditions under which the lifts of a transitive Anosov flow to certain infinite covers of the manifold remain transitive. With some deep results in 3-manifold topology, we then deduce that such cover can be obtained for any non-graph manifold admitting a transitive Anosov flow. Moreover, for a large class of Anosov flows known as $\mathbb{R}$-covered, which are always transitive, we show that their lifts to any regular covers are either transitive or consist exclusively of wandering orbits. Finally, we construct a family of transitive Anosov flows on non-compact manifolds that satisfy a homotopical characterization of suspension flows, answering the flow version of a question concerning the existence of transitive Anosov diffeomorphisms on non-compact manifolds.