Anosov flows with the same periodic orbits

TL;DR

This paper explores the classification of transitive pseudo-Anosov flows on closed 3-manifolds, especially focusing on cases with complex topological features called trees of scalloped regions, and constructs examples of flows sharing free homotopy data but not orbit equivalent.

Contribution

It characterizes topological features corresponding to trees of scalloped regions and classifies flows with identical free homotopy data that are not orbit equivalent.

Findings

Identifies topological features linked to trees of scalloped regions.

Classifies flows sharing free homotopy data but not orbit equivalent.

Provides explicit examples of non-orbit equivalent flows with same free homotopy data.

Abstract

In [Orbit equivalences of pseudo-Anosov flows, arXiv:2211.10505], it was proved that transitive pseudo-Anosov flows on any closed 3-manifold are determined up to orbit equivalence by the set of free homotopy classes represented by periodic orbits, provided their orbit space does not contain a feature called a "tree of scalloped regions." In this article we describe what happens in these exceptional cases: we show what topological features in the manifold correspond to trees of scalloped regions, completely classify the flows which do have the same free homotopy data, and construct explicit examples of flows with the same free homotopy data that are not orbit equivalent.