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A Smörgåsbord of (bi)contact structures, Reeb flows and pseudo-Anosov flows

Thomas Barthelmé

TL;DR

The paper investigates how dynamics of two Reeb flows sharing the same contact structure can be related, introducing free homotopy data and pseudo-Anosov models as two strong lenses. It combines Barbot–Fenley theory of Anosov/pseudo-Anosov flows, cylindrical contact homology, and the theory of Anosov-supporting bicontact structures to show that, within the same Anosov contact structure, Reeb–Anosov flows have the same free homotopy data and are orbit-equivalent under suitable conditions. It also explores when Reeb flows admit pseudo-Anosov models, using Birkhoff sections to transfer the Nielsen–Thurston classification from surface dynamics to 3-manifold flows, and discusses conjectures about the rarity and rigidity of Anosov-supporting contacts, as well as the existence and uniqueness of true pseudo-Anosov models. Together, these results provide a framework for relating distinct Reeb flows via topological invariants, foliation structures, and model flows, while outlining many open questions at the intersection of contact geometry and hyperbolic dynamics.

Abstract

This note was written for the proceedings of the conference "Symplectic Geometry and Anosov Flows" held in Heideleberg in July 2024. It is meant as an invitation to the study of certain families of contact structures, centering around the following question: ``How much can one relate the dynamics of two distinct Reeb flows of the same contact structure?'' We gather some results as well as state many more questions and conjectures around that theme.

A Smörgåsbord of (bi)contact structures, Reeb flows and pseudo-Anosov flows

TL;DR

The paper investigates how dynamics of two Reeb flows sharing the same contact structure can be related, introducing free homotopy data and pseudo-Anosov models as two strong lenses. It combines Barbot–Fenley theory of Anosov/pseudo-Anosov flows, cylindrical contact homology, and the theory of Anosov-supporting bicontact structures to show that, within the same Anosov contact structure, Reeb–Anosov flows have the same free homotopy data and are orbit-equivalent under suitable conditions. It also explores when Reeb flows admit pseudo-Anosov models, using Birkhoff sections to transfer the Nielsen–Thurston classification from surface dynamics to 3-manifold flows, and discusses conjectures about the rarity and rigidity of Anosov-supporting contacts, as well as the existence and uniqueness of true pseudo-Anosov models. Together, these results provide a framework for relating distinct Reeb flows via topological invariants, foliation structures, and model flows, while outlining many open questions at the intersection of contact geometry and hyperbolic dynamics.

Abstract

This note was written for the proceedings of the conference "Symplectic Geometry and Anosov Flows" held in Heideleberg in July 2024. It is meant as an invitation to the study of certain families of contact structures, centering around the following question: ``How much can one relate the dynamics of two distinct Reeb flows of the same contact structure?'' We gather some results as well as state many more questions and conjectures around that theme.

Paper Structure

This paper contains 6 sections, 23 theorems, 1 equation, 5 figures.

Table of Contents

  1. Introduction
  2. Contact Anosov structures and $\mathbb R$-covered flows
  3. More background
  4. The free homotopy data and orbit equivalences
  5. Anosov-supporting bicontact structures
  6. Birkhoff sections and pseudo-Anosov models

Key Result

Theorem 1.1

Let $R_1$, $R_2$ be two Anosov Reeb flows of two contact structures $\xi_1,\xi_2$. Then the Anosov flows $R_1, R_2$ are orbit equivalent if and only if $\xi_1$ and $\xi_2$ are contactomorphic.

Figures (5)

  • Figure 1: The local picture of stable and unstable foliations near a $3$-prong singularity
  • Figure 2: The contact structures $\xi^\pm$ in their respective quadrants
  • Figure 3: A bitransverse Reeb vector field $R^+$
  • Figure 4: The vector fields $X$, $Y$ and the three contact structures
  • Figure 5: A lozenge and its corners

Theorems & Definitions (51)

  • Theorem 1.1: BMB
  • Definition 1.2
  • Definition 1.3
  • Definition 2.2
  • Proposition 2.3: Anosov Ano67
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 41 more