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Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Surena Hozoori

Abstract

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and discuss a framework to use tools from contact and symplectic geometry and topology in the study of Anosov dynamics. We also discuss some uniqueness results regarding the underlying (bi)-contact structures for an Anosov flow and give a characterization of Anosovity based on Reeb flows.

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Abstract

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and discuss a framework to use tools from contact and symplectic geometry and topology in the study of Anosov dynamics. We also discuss some uniqueness results regarding the underlying (bi)-contact structures for an Anosov flow and give a characterization of Anosovity based on Reeb flows.

Paper Structure

This paper contains 7 sections, 36 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background On Contact And Symplectic Topology
  3. Anosovity And The Geometry Of Expansion
  4. Contact and Symplectic Geometric Characterization of Anosov Flows
  5. Uniqueness Of The Underlying (Bi-)Contact Structures
  6. A Characterization of Anosovity Based on Reeb Flows and Consequences
  7. Bi-Contact Topology And Anosov Dynamics: Remarks And Questions

Key Result

Theorem 1.1

Let $\phi^t$ be a flow on the 3-manifold $M$, generated by the $C^1$ vector field $X$. Then $\phi^t$ is Anosov, if and only if, $\langle X \rangle= \xi_+ \cap \xi_-$, where $\xi_+$ and $\xi_-$ are transverse positive and negative contact structures, respectively, and there exist contact forms $\alph

Figures (3)

  • Figure 1: The local behavior of (projectively) Anosov flows
  • Figure 2: $TM/\langle X \rangle \simeq E^s\oplus E^u$
  • Figure 3: Uniqueness of the supporting bi-contact structure

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Corollary 1.12
  • ...and 82 more