Anosov flows with the same periodic orbits

Thomas Barthelmé; Sergio Fenley; Kathryn Mann

doi:10.3934/jmd.2025006

Article Contents

2025, Volume 21: 361-400. Doi: 10.3934/jmd.2025006

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Anosov flows with the same periodic orbits

1.
Department of Math & Stats, Queen's University, 48 University Ave., Kingston, ON K7L 3N6, Canada
2.
Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL 32306, USA
3.
Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA

Received: December 27, 2024

Published online: April 11, 2025

TB: Partially supported by the NSERC (funding reference number RGPIN-2017-04592).
SF: Partially supported by NSF DMS-2054909.
KM: Partially supported by NSF CAREER grant DMS-1933598 and a Sloan fellowship.

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Abstract

In [8], 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.

Keywords:

Mathematics Subject Classification: Primary: 37C15, 37D20, 37C27; Secondary: 37C10, 37C86, 57R30.

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Figure 1. A line of adjacent lozenges and a scalloped region. The leaves $ f_i^{j, u} $ are vertical (blue), $ f_i^{j, s} $ are horizontal (red)

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Figure 2. The intersection of two scalloped regions

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Figure 3. The model block with flow $ \psi^+ $; The shaded region is a Birkhoff annulus

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Figure 4. Left: two model blocks glued along a half-face, the rear/right block is flipped vertically. Orbits $ \alpha_i $ are labelled according to the block $ N_1 $. Right: lifting these blocks to $ \widetilde{{M}} $ and projecting to $ \mathscr O_\phi $ gives two adjacent lozenges with corners fixed by the element representing the fiber. In this image, the flow direction is out of the page, towards the reader

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Figure 5. Local structure of the 4-regular tree $ \mathscr{T}' $ (grey) and surface $ \Sigma' $, indicating how to glue blocks. The product of each 2-cell with $ \mathbb R $ can be thought of as $ \widetilde{N} $, where $ \mathscr{T}' \times \mathbb R $ is the lift of the Birkhoff annulus in the cell

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Figure 6. An admissible fat graph $ X_1 $—one identifies the top and bottom as well as the right and left sides by translation. (For clarity, only some of the boundary components of the associated surface are shown). The highlighted vertex is the unique vertex that is not adjacent to any "quadrilateral" boundary component (a boundary component following exactly 4 edges), thus invariant under all automorphisms. To produce a family of examples, take $ X_n $ to be the result of replacing the 2 bottom rows of squares with $ 2n $ rows

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Figure 7. A fat graph with two vertices and four edges (with indicated gluings), seen also as a quotient of a fat graph in $ \mathbb R^2 $

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References

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