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Anosov flows in dimension 3: an outside look

Rafael Potrie

TL;DR

This survey collects and organizes the 3D theory of Anosov flows, emphasizing the intricate link between dynamics, foliation theory, and 3-manifold topology. It centers on Barbot–Fenley leaf-space theory, the dichotomy between $\mathbb{R}$-covered and non-$\mathbb{R}$-covered flows, and the geometry of leaves, including Candel hyperbolization and quasi-geodesic flowlines. A core thread is the use of orbit and bifoliated planes, lozenges, and universal circles to translate 3D dynamics into 2D actions and to connect with contact geometry via the Barbot–Marty correspondence. The work highlights both established results (e.g., Regulating flows and universal circles) and open questions, pointing to rich interactions with taut foliations, left-orderability of fundamental groups, and modern analytic approaches to hyperbolic dynamics. Overall, the notes provide a comprehensive scaffold linking topological obstructions, leaf-space geometry, and concrete constructions (suspensions, geodesic flows, surgeries) to a broader landscape of geometric structures in 3-manifolds.

Abstract

These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agustín Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results from the theory of Anosov flows. The point of view of the course is to provide an overview of the main results and questions in the subject, with emphasis on the interaction with topology, geometry, specially symplectic geometry and contact aspects of the theory. Some detail is given in the presentation of the Barbot-Fenley theory of leaf spaces. Hopefully the notes will contribute in gaining a working knowledge of the theory and its many beautiful connections.

Anosov flows in dimension 3: an outside look

TL;DR

This survey collects and organizes the 3D theory of Anosov flows, emphasizing the intricate link between dynamics, foliation theory, and 3-manifold topology. It centers on Barbot–Fenley leaf-space theory, the dichotomy between -covered and non--covered flows, and the geometry of leaves, including Candel hyperbolization and quasi-geodesic flowlines. A core thread is the use of orbit and bifoliated planes, lozenges, and universal circles to translate 3D dynamics into 2D actions and to connect with contact geometry via the Barbot–Marty correspondence. The work highlights both established results (e.g., Regulating flows and universal circles) and open questions, pointing to rich interactions with taut foliations, left-orderability of fundamental groups, and modern analytic approaches to hyperbolic dynamics. Overall, the notes provide a comprehensive scaffold linking topological obstructions, leaf-space geometry, and concrete constructions (suspensions, geodesic flows, surgeries) to a broader landscape of geometric structures in 3-manifolds.

Abstract

These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agustín Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results from the theory of Anosov flows. The point of view of the course is to provide an overview of the main results and questions in the subject, with emphasis on the interaction with topology, geometry, specially symplectic geometry and contact aspects of the theory. Some detail is given in the presentation of the Barbot-Fenley theory of leaf spaces. Hopefully the notes will contribute in gaining a working knowledge of the theory and its many beautiful connections.

Paper Structure

This paper contains 31 sections, 32 theorems, 2 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. Classical definition
  4. Topological definition
  5. Some topological obstructions
  6. Foliations and leaf spaces
  7. Individual leaf spaces
  8. ${\mathbb R}$-covered Anosov flows and transitivity
  9. Orbit space and the bifoliated plane
  10. Dichotomy for ${\mathbb R}$-covered Anosov flows
  11. Lozenges
  12. Some properties of non-${\mathbb R}$-covered Anosov flows
  13. One step up map of skewed-${\mathbb R}$-covered Anosov flows
  14. Geometry of leaves
  15. Candel's theorem
  16. ...and 16 more sections

Key Result

Theorem 2.1

The set of Anosov vector fields is open in the $C^1$-topology and any pair of $C^1$-close Anosov vector fields are orbit equivalent.

Figures (13)

  • Figure 1: Local picture of an Anosov flow. (Figure by Elena Gomes)
  • Figure 2: Foliation charts. (Figure by Elena Gomes)
  • Figure 3: If a pair of leaves intersects in more than one component, one can see the non-Hausdorffness inside each of the leaves.
  • Figure 4: The bifoliated plane for some examples. (Figure by Christian Bonatti.)
  • Figure 5: Depiction of the argument in case 2
  • ...and 8 more figures

Theorems & Definitions (62)

  • Theorem 2.1: Structural stability of Anosov flows
  • Theorem 2.2: Stable manifold theorem
  • Definition 2.3
  • Theorem 3.1: Margulis
  • Theorem 3.2: Ghys Ghys
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Definition 4.3
  • ...and 52 more