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

Thomas Barthelmé, Kathryn Mann

Abstract

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 spectral rigidity theorem for $\mathbb{R}$-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 $\mathbb{R}$-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--Vaugon, and a virtual description of the group of contact transformations of a contact Anosov structure, generalizing a result of Giroux and Massot.

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

Abstract

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 spectral 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--Vaugon, and a virtual description of the group of contact transformations of a contact Anosov structure, generalizing a result of Giroux and Massot.

Paper Structure

This paper contains 16 sections, 27 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic-like actions
  3. Orbit equivalence of Anosov flows
  4. Application 1: Classifying self orbit equivalences
  5. Application 2: Contact Anosov flows
  6. Outline of the article
  7. Proofs of Theorems \ref{['thm:hyp_action']} and \ref{['thm_main']}
  8. Skew-Anosov flows
  9. Hyperbolic-like homeomorphisms: Proof of Theorem \ref{['thm:hyp_action']}
  10. Conclusion of the proof of Theorem \ref{['thm_main']}
  11. Classifying self orbit-equivalences
  12. Proof of Theorem \ref{['thm_criterion']}
  13. Application: classification of self-orbit equivalences in special cases
  14. Application to contact flows
  15. Further applications to contact topology With Jonathan Bowden
  16. ...and 1 more sections

Key Result

Theorem 1.1

A minimal, hyperbolic-like action of a nonabelian group $G$ on $\mathbb R$ is determined up to conjugacy by the set of elements of $G$ that act with fixed points.

Figures (4)

  • Figure 1: The orbit space $\mathcal{O}$
  • Figure 2: The orbit space $\mathcal{O}$ with (part of) a string of lozenges
  • Figure 3: The configuration on the right is impossible, if the shaded lozenge is $L'_{j}$
  • Figure 4: The action of the lift of a Dehn twist on a ray, shown in $\widetilde{M}$ on the left and schematically indicating intersections with lifted neighborhoods of tori on the right.

Theorems & Definitions (68)

  • Theorem 1.1: Rigidity of hyperbolic-like actions
  • Theorem 1.2: Classification of $\mathbb R$-covered Anosov flows
  • Corollary 1.3
  • Theorem 1.4: Criterion for self-orbit equivalence
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 58 more