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A symplectic viewpoint on Anosov flows

Thomas Massoni

TL;DR

The paper addresses how $3$-dimensional Anosov dynamics can be understood through contact and symplectic geometry by developing the Mitsumatsu bicontact framework and its Liouville refinements. It surveys definitions, basic properties, and the Mitsumatsu construction that attaches Liouville structures to projectively Anosov and semi-Anosov flows, and introduces Anosov Liouville structures (AL) and a generalized framework (gAL) that yields contractible parameter spaces and invariance under orbit equivalence. A key contribution is a symplectic characterization of volume preserving Anosov flows and the development of generalized Liouville structures whose Floer-theoretic invariants depend only on the flow, not the particular filling. These results illuminate deep connections between hyperbolic dynamics and symplectic topology, suggesting that Liouville geometry encodes dynamical data and offering potential pathways to classify and distinguish Anosov flows via their associated Liouville and bicontact structures.

Abstract

This survey explores the geometry of three-dimensional Anosov flows from the perspective of contact and symplectic geometry, following the work of Mitsumatsu, Eliashberg-Thurston, Hozoori, and the author. We also present a few original results and discuss various open questions and conjectures.

A symplectic viewpoint on Anosov flows

TL;DR

The paper addresses how -dimensional Anosov dynamics can be understood through contact and symplectic geometry by developing the Mitsumatsu bicontact framework and its Liouville refinements. It surveys definitions, basic properties, and the Mitsumatsu construction that attaches Liouville structures to projectively Anosov and semi-Anosov flows, and introduces Anosov Liouville structures (AL) and a generalized framework (gAL) that yields contractible parameter spaces and invariance under orbit equivalence. A key contribution is a symplectic characterization of volume preserving Anosov flows and the development of generalized Liouville structures whose Floer-theoretic invariants depend only on the flow, not the particular filling. These results illuminate deep connections between hyperbolic dynamics and symplectic topology, suggesting that Liouville geometry encodes dynamical data and offering potential pathways to classify and distinguish Anosov flows via their associated Liouville and bicontact structures.

Abstract

This survey explores the geometry of three-dimensional Anosov flows from the perspective of contact and symplectic geometry, following the work of Mitsumatsu, Eliashberg-Thurston, Hozoori, and the author. We also present a few original results and discuss various open questions and conjectures.

Paper Structure

This paper contains 33 sections, 25 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Anosov and projectively Anosov flows
  4. Definitions
  5. Basic properties
  6. First examples.
  7. Structural stability.
  8. Invariant foliations.
  9. Regularity of Anosov bundles.
  10. Defining pairs
  11. Contact and symplectic geometry
  12. Contact structures
  13. Basic features.
  14. Tight vs. overtwisted.
  15. Symplectic and Liouville structures
  16. ...and 18 more sections

Key Result

Lemma 2.4

A projectively Anosov flow $\Phi$ is Anosov if and only if $\Phi$ and $\Phi^{-1}$ are semi-Anosov.

Figures (2)

  • Figure 1: Bicontact structures supporting a vector field/flow.
  • Figure 2: Anosov/dominated splitting and supporting bicontact structure.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem
  • Proposition 2.6
  • Definition 2.7
  • Definition 4.1
  • Theorem 4.2
  • ...and 36 more