Action on the circle at infinity of foliations of ${\mathbb R}^2 $

TL;DR

This paper constructs a canonical circle at infinity for ${f R}^2$ based on singular foliations, extending homeomorphisms and analyzing the minimality of the induced action, especially in the context of Anosov flows.

Contribution

It introduces a new compactification of ${f R}^2$ via a circle at infinity linked to foliations, generalizing Mather's idea, and studies the minimality of the action on this circle.

Findings

The circle at infinity is canonical and invariant under foliation-preserving homeomorphisms.

Conditions are provided for the minimality of the action on the circle at infinity.

The action is minimal if and only if the Anosov flow is non-${f R}$-covered.

Abstract

This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ 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 R}^2 $ 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.