From pseudo-Anosov flows on graph manifolds to totally periodic flows

TL;DR

The paper demonstrates that pseudo-Anosov flows on graph manifolds are closely related to totally periodic or suspension Anosov flows, via a construction involving partial Birkhoff sections.

Contribution

It provides a new method to relate pseudo-Anosov flows on graph manifolds to well-understood flows using partial Birkhoff sections.

Findings

Every pseudo-Anosov flow on a graph manifold is orbit equivalent to a totally periodic or suspension Anosov flow outside finitely many orbits.

Constructs a partial Birkhoff section with genus one components that misses finitely many closed orbits.

Shows that transitive Anosov flows with orientable foliations are almost equivalent to suspension flows.

Abstract

We show that every pseudo-Anosov flow on a graph manifold is almost equivalent, i.e. orbit equivalent in the complement of a finite collection of closed orbits, to a totally periodic pseudo-Anosov flow or a suspension Anosov flow. The proof is via a hands-on construction of a partial Birkhoff section with genus one components that misses finitely many closed orbits. When combined with previous work of the author, this implies that every transitive Anosov flow on a graph manifold with orientable stable and unstable foliations is almost equivalent to a suspension Anosov flow.