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Orbit equivalences of $\mathbb{R}$–covered Anosov flows and hyperbolic-like actions on the line

Thomas Barthelmé and Kathryn Mann

Appendix: Thomas Barthelmé, Kathryn Mann and Jonathan Bowden

Geometry & Topology 28 (2024) 867–899

We prove a rigidity result for group actions on the line whose elements have what we call “hyperbolic-like” dynamics. Using this, we give a rigidity theorem for –covered Anosov flows on 3–manifolds, characterizing orbit equivalent flows in terms of the elements of the fundamental group represented by periodic orbits. As consequences of this, we give an efficient criterion to determine the isotopy classes of self-orbit equivalences of –covered Anosov flows, and prove finiteness of contact Anosov flows on any given manifold.

In the appendix, with Jonathan Bowden, we prove that orbit equivalences of contact Anosov flows correspond exactly to isomorphisms of the associated contact structures. This gives a powerful tool to translate results on Anosov flows to contact geometry and vice versa. We illustrate its use by giving two new results in contact geometry: the existence of manifolds with arbitrarily many distinct Anosov contact structures, answering a question of Foulon, Hasselblatt and Vaugon, and a virtual description of the group of contact transformations of a Anosov contact structure, generalizing a result of Giroux and Massot.

Anosov flows, actions on real line
Mathematical Subject Classification
Primary: 37D20, 57M60
Received: 13 January 2022
Revised: 20 September 2022
Accepted: 18 October 2022
Published: 13 March 2024
Proposed: David Fisher
Seconded: Mladen Bestvina, Leonid Polterovich
Thomas Barthelmé
Department of Mathematics and Statistics
Queen’s University
Kingston ON
Kathryn Mann
Department of Mathematics
Cornell University
Ithaca, NY
United States
Jonathan Bowden
Fakultät für Mathematik
Universität Regensburg

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