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

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 (1982). 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.