Action on the circle at infinity of foliations of \mathbb R²

2023/01/11 by Christian Bonatti, Bonatti, Christian
Mathematics · #37C86 #37D20 #37E10 #37E35 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2301.04530

openalex publication_date 2023/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper provides a canonical compactification of the plane \mathbb R² by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \citeMa. Moreover any homeomorphism of \mathbb R² preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on \mathbb R² preserving one foliation or two transverse foliations. In particular the action on the circle at infinity associated to an Anosov flow X on a closed 3-manifold is minimal if and only if X is non-\mathbb R-covered.

Related

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• On transverse R-covered minimal foliations
• Every noncompact surface is a leaf of a minimal foliation
• Minimality of the action on the universal circle of uniform foliations
• Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations