Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

TL;DR

This paper classifies totally periodic pseudo-Anosov flows in graph manifolds, showing they are topologically equivalent to constructed models and characterizing their isotopic equivalence through topological and dynamical data.

Contribution

It establishes a classification of these flows by constructing model flows and providing criteria for their isotopic equivalence based on topological and dynamical invariants.

Findings

Each flow is topologically equivalent to a model flow.

Two model flows are isotopically equivalent iff they share the same topological and dynamical data.

Classification relies on glueing neighborhoods of Birkhoff annuli and Dehn surgeries.

Abstract

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we constructed in a previous article. A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (ie. there is a isotopy of the manifold mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.