Non-transitive pseudo-Anosov flows

TL;DR

This paper explores the structure of non-transitive pseudo-Anosov flows in 3-manifolds through group actions on orbit spaces, showing they are determined by boundary actions and establishing conditions for transitivity.

Contribution

It extends the concept of Anosov-like actions to non-transitive flows and demonstrates that such flows are uniquely determined by their fundamental group's boundary action.

Findings

Pseudo-Anosov flows are determined by boundary group actions.

Non-transitive flows on atoroidal 3-manifolds are necessarily transitive.

Density of periodic orbits implies transitivity in topological 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.