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On simultaneous conjugacies of pairs of transverse foliations of the torus

Martin Mion-Mouton

TL;DR

This work analyzes minimal transverse bi-foliations of the torus and shows they are completely determined up to simultaneous conjugacy by their asymptotic cycles. By interpreting foliations as suspensions of circle dynamics and leveraging Schwartzman asymptotic cycles, the author constructs a global homeomorphism isotopic to the identity that simultaneously conjugates a bi-foliation to its linear model defined by $A(\mathcal{F}_\alpha)$ and $A(\mathcal{F}_\beta)$. The main result yields a dynamical Teichmüller-type classification and implies rigidity for certain automorphisms, bridging toral foliation dynamics with PSL$_2(\mathbb{Z})$ actions on the asymptotic-cycle data. These insights have implications for geometric structures associated with dynamical systems and de-Sitter metrics, and link the mapping class group dynamics to explicit linear models on the torus. $$\mathbf{T}^2$$-based bi-foliations thus admit a precise, invariant-driven description via their asymptotic directions.

Abstract

We prove in this note that two pairs of transverse minimal topological foliations of the torus are individually conjugated if, and only if they are simultaneously conjugated.

On simultaneous conjugacies of pairs of transverse foliations of the torus

TL;DR

This work analyzes minimal transverse bi-foliations of the torus and shows they are completely determined up to simultaneous conjugacy by their asymptotic cycles. By interpreting foliations as suspensions of circle dynamics and leveraging Schwartzman asymptotic cycles, the author constructs a global homeomorphism isotopic to the identity that simultaneously conjugates a bi-foliation to its linear model defined by and . The main result yields a dynamical Teichmüller-type classification and implies rigidity for certain automorphisms, bridging toral foliation dynamics with PSL actions on the asymptotic-cycle data. These insights have implications for geometric structures associated with dynamical systems and de-Sitter metrics, and link the mapping class group dynamics to explicit linear models on the torus. -based bi-foliations thus admit a precise, invariant-driven description via their asymptotic directions.

Abstract

We prove in this note that two pairs of transverse minimal topological foliations of the torus are individually conjugated if, and only if they are simultaneously conjugated.
Paper Structure (7 sections, 5 theorems, 11 equations)

This paper contains 7 sections, 5 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Foliations of the torus and suspensions
  4. Circle homeomorphisms and rotation numbers
  5. Asymptotic cycles
  6. Proof of Theorem \ref{['theoremintro-bifoliations']}
  7. Proof of Corollary \ref{['corollaryintro-automorphismesbifoliations']}

Key Result

Theorem 1

Let $(\mathcal{F}_\alpha,\mathcal{F}_\beta)$ be a minimal topological bi-foliation of the torus and $p\in\mathbf{T}^{2}$. Then $(\mathcal{F}_\alpha,\mathcal{F}_\beta)$ is simultaneously conjugated to the linear bi-foliation $(\mathcal{F}_{A(\mathcal{F}_\alpha)},\mathcal{F}_{A(\mathcal{F}_\beta)})$ d

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 2
  • Definition 2.1
  • Proposition 2.3: yano_asymptotic_1985
  • Lemma 2.4
  • Proposition 2.5
  • Definition 3.1