Orbit equivalences of pseudo-Anosov flows

TL;DR

This paper classifies transitive Anosov and pseudo-Anosov flows on closed 3-manifolds up to orbit equivalence, revealing that most flows are determined by their periodic orbits and introducing invariants for special cases.

Contribution

It provides a classification theorem for these flows, introduces a framework based on Anosov-like actions on bifoliated planes, and shows how flows are determined by their actions on ideal boundaries.

Findings

Flows are classified by free homotopy classes of periodic orbits.

Exceptional flows with 'tree of scalloped regions' require additional invariants.

Anosov-like actions are determined by their boundary actions.

Abstract

We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $\pi_1(M)$-orbit of tree gives a complete invariant of orbit equivalence classes of flows. The framework for the proof is a more general result about \emph{Anosov-like actions} of abstract groups on bifoliated planes, showing that the homeomorphism type of the bifoliation and the conjugacy class of the action can be recovered from knowledge of which elements of the group act with fixed points. As a consequence, we show that Anosov flows are determined up to orbit equivalence by the action on the ideal boundary of their orbit spaces, and more generally that transitive Anosov-like actions on bifoliated planes are determined up to conjugacy by their actions on the plane's ideal boundary: any conjugacy between two such actions on their ideal circles can be extended uniquely to a conjugacy on the interior of the plane.