Non-transitive pseudo-Anosov flows

Abstract

We study (topological) pseudo-Anosov flows from the perspective of the associated group actions on their orbit spaces and boundary at infinity. We extend the definition of Anosov-like action from [BFM22] from the transitive to the general non-transitive context and show that one can recover the basic sets of a flow, the Smale order on basic sets, and their essential features, from such general group actions. Using these tools, we prove that a pseudo-Anosov flow in a $3$ manifold is entirely determined by the associated action of the fundamental group on the boundary at infinity of its orbit space. We also give a proof that any topological pseudo-Anosov flow on an atoroidal 3-manifold is necessarily transitive, and prove that density of periodic orbits implies transitivity, in the topological rather than smooth case.

v2: a few additions and modifications to deal with isolated prong singularities (Def 3.3)