The structure of branching in Anosov flows of 3-manifolds

Abstract

In this article we study the topological structure of the lifts to the universal of the stable and unstable foliations of $3$-dimensional Anosov flows. In particular we consider the case when these foliations do not have Hausdorff leaf space. We completely determine the structure of the set of non separated leaves from a given leaf in one of these foliations. As a consequence of this suspensions are characterized, up to topological conjugacy, as the only $3$-dimensional Anosov flows without freely homotopic closed orbits. Furthermore the structure of branching is related to the topology of the manifold: if there are infinitely many leaves not separated from each other, then there is an incompressible torus transverse to the flow. Transitivity is not assumed for these results. Finally, if the manifold has negatively curved fundamental group we derive some important properties of the limit sets of leaves in the universal cover.