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Strongly adapted contact geometry of Anosov 3-flows

Surena Hozoori

TL;DR

The paper develops a purely contact-geometric framework for 3-dimensional Anosov flows by encoding them as strongly adapted bi-contact structures and associated Reeb dynamics. It proves a set of equivalent formulations showing that Anosovity is captured by the existence of strongly adapted contact data and yields a deformation theory via asymptotic synchronization of adapted norms. It also analyzes the global structure of the space of strong adaptations, proving an acyclic fibration over the space of Anosov flows and establishing homotopy equivalence, which enables transfer of contact-geometric invariants to dynamical classifications. The results provide a local contact model for Anosov 3-flows, unify prior foliations-based approaches with Reeb dynamics, and open avenues for using Floer-type techniques and surgery in this dynamical setting. Overall, the work strengthens the bridge between hyperbolic dynamics and low-dimensional contact topology, with potential implications for invariants and deformation theories in both fields.

Abstract

We provide a 3 dimensional contact geometric characterization of Anosov 3-flows based on interactions with Reeb dynamics. We investigate basic properties of the space of the resulting geometries and in particular show that such space is homotopy equivalent to the space of Anosov 3-flows. A technical theorem on the asymptotic synchronization of adapted norms is proved, which can be of broader interest.

Strongly adapted contact geometry of Anosov 3-flows

TL;DR

The paper develops a purely contact-geometric framework for 3-dimensional Anosov flows by encoding them as strongly adapted bi-contact structures and associated Reeb dynamics. It proves a set of equivalent formulations showing that Anosovity is captured by the existence of strongly adapted contact data and yields a deformation theory via asymptotic synchronization of adapted norms. It also analyzes the global structure of the space of strong adaptations, proving an acyclic fibration over the space of Anosov flows and establishing homotopy equivalence, which enables transfer of contact-geometric invariants to dynamical classifications. The results provide a local contact model for Anosov 3-flows, unify prior foliations-based approaches with Reeb dynamics, and open avenues for using Floer-type techniques and surgery in this dynamical setting. Overall, the work strengthens the bridge between hyperbolic dynamics and low-dimensional contact topology, with potential implications for invariants and deformation theories in both fields.

Abstract

We provide a 3 dimensional contact geometric characterization of Anosov 3-flows based on interactions with Reeb dynamics. We investigate basic properties of the space of the resulting geometries and in particular show that such space is homotopy equivalent to the space of Anosov 3-flows. A technical theorem on the asymptotic synchronization of adapted norms is proved, which can be of broader interest.

Paper Structure

This paper contains 5 sections, 32 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Bi-contact geometry and projective Anosovity
  3. Reeb dynamics of bi-contact structures and strong adaptations
  4. Asymptotic synchronization process
  5. The space of adaptations

Key Result

Theorem A

Let $X$ be a $C^\infty$ non-vanishing vector field on a closed oriented 3-manifold $M$. TFAE: (1) $X$ generates a transversally oriented Anosov flow; (2) there exists a positive contact form $\alpha_+$, such that $\alpha_+(X)=0$ and $\mathcal{L}_X\alpha_+$ is a negative contact form; (3) there exist

Figures (6)

  • Figure 1: Local behavior of flow lines in in the strongly adapted contact geometry of Anosov flows
  • Figure 2: Foliation theoretic local picture
  • Figure 3: Anosov flow action on $TM/\langle X\rangle$ (left) vs. projective Anosovity (right)
  • Figure 4: Bi-contact geometric approach to Anosov 3-flows
  • Figure 5: Projective geometry of Reeb flows in bi-contact geometry (left) Strong adaptation (right)
  • ...and 1 more figures

Theorems & Definitions (56)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Remark 1.2
  • Corollary 1.3
  • Proposition 2.2
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.7
  • Lemma 3.1
  • ...and 46 more