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Non-transitive pseudo-Anosov flows

Thomas Barthelmé, Christian Bonatti, Kathryn Mann

TL;DR

This work extends the Anosov-like action framework to nontransitive bifoliated planes and leverages the orbit-space perspective to study topological pseudo-Anosov flows. The authors show that the fundamental group action on the boundary at infinity $\partial P$ determines the flow up to orbit equivalence, extending known transitive results to the nontransitive setting via a Smale-class/Smale-chain calculus and boundary dynamics. Key contributions include transitivity of topological pseudo-Anosov flows on atoroidal 3-manifolds, density implications for transitivity from periodic orbits, and a complete boundary-determined classification of flows through circle actions on $S^1$; the framework also provides a toolkit for constructing and obstructing flows using fatgraphs and blow-up/ping-pong techniques. The results unify the orbit-space and boundary viewpoints, clarify the roles of wandering sets and boundary leaves, and yield practical criteria for when flows are determined by boundary data, with explicit examples and a toolbox for building pseudo-Anosov flows.

Abstract

We study (topological) pseudo-Anosov flows from the perspective of the associated group actions on their orbit spaces and boundary at infinity. We extend the definition of Anosov-like action from [BFM22] from the transitive to the general non-transitive context and show that one can recover the basic sets of a flow, the Smale order on basic sets, and their essential features, from such general group actions. Using these tools, we prove that a pseudo-Anosov flow in a $3$ manifold is entirely determined by the associated action of the fundamental group on the boundary at infinity of its orbit space. We also give a proof that any topological pseudo-Anosov flow on an atoroidal 3-manifold is necessarily transitive, and prove that density of periodic orbits implies transitivity, in the topological rather than smooth case.

Non-transitive pseudo-Anosov flows

TL;DR

This work extends the Anosov-like action framework to nontransitive bifoliated planes and leverages the orbit-space perspective to study topological pseudo-Anosov flows. The authors show that the fundamental group action on the boundary at infinity determines the flow up to orbit equivalence, extending known transitive results to the nontransitive setting via a Smale-class/Smale-chain calculus and boundary dynamics. Key contributions include transitivity of topological pseudo-Anosov flows on atoroidal 3-manifolds, density implications for transitivity from periodic orbits, and a complete boundary-determined classification of flows through circle actions on ; the framework also provides a toolkit for constructing and obstructing flows using fatgraphs and blow-up/ping-pong techniques. The results unify the orbit-space and boundary viewpoints, clarify the roles of wandering sets and boundary leaves, and yield practical criteria for when flows are determined by boundary data, with explicit examples and a toolbox for building pseudo-Anosov flows.

Abstract

We study (topological) pseudo-Anosov flows from the perspective of the associated group actions on their orbit spaces and boundary at infinity. We extend the definition of Anosov-like action from [BFM22] from the transitive to the general non-transitive context and show that one can recover the basic sets of a flow, the Smale order on basic sets, and their essential features, from such general group actions. Using these tools, we prove that a pseudo-Anosov flow in a manifold is entirely determined by the associated action of the fundamental group on the boundary at infinity of its orbit space. We also give a proof that any topological pseudo-Anosov flow on an atoroidal 3-manifold is necessarily transitive, and prove that density of periodic orbits implies transitivity, in the topological rather than smooth case.

Paper Structure

This paper contains 22 sections, 71 theorems, 4 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Anosov-like actions and bifoliated planes
  3. Structure of the bifoliated planes
  4. Axiom \ref{['Axiom_non-separated']} and non-separated leaves
  5. Product and scalloped regions
  6. Finite branching
  7. Smale classes and topological transitivity
  8. Smale chains and the structure of wandering sets
  9. Structure of the regular and "wandering" sets
  10. Boundary leaves
  11. Smale Chains
  12. Virtually cyclic point stabilizers and $\mathbb{Z}^2$ invariant Smale chains
  13. $\mathbb{Z}^2$ invariant Smale chains
  14. Smale bounded actions and density of non-corner leaves
  15. Smale-bounded actions
  16. ...and 7 more sections

Key Result

Theorem 1.2

If $P$ is a bifoliated plane with either prongs or nonseparated leaves and $G$ has an Anosov-like action, then point stabilizers for the action are trivial or virtually (index at most 4) isomorphic to $\mathbb Z$.

Figures (15)

  • Figure 1:
  • Figure 2: A scalloped region, with its two infinite lines of lozenges
  • Figure 3: Proof of local product structure
  • Figure 4: If $g_1(L_1) = L_2$ then $g$ has a fixed point in $L_1 \cap L_2$
  • Figure 5: Two TL substrips
  • ...and 10 more figures

Theorems & Definitions (170)

  • Definition 1.1: Anosov-like action
  • Theorem 1.2: Point stabilizers
  • Theorem 1.3: Atoroidal implies transitive
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10: Circle actions classify flows
  • ...and 160 more