There is no minimal action of Z^2 on the plane

TL;DR

This paper proves that the group Z^2 cannot act minimally on the plane, meaning no orbit can be dense, using advanced topological theorems related to homeomorphisms.

Contribution

It establishes a new non-existence result for minimal group actions on the plane, leveraging theorems about homeomorphisms and the annulus.

Findings

No minimal Z^2 action exists on the plane

Utilizes Le Calvez-Yoccoz's theorem on the annulus

Employs Brouwer homeomorphism theory

Abstract

In this paper it is proved that there is no minimal action (i.e. every orbit is dense) of Z^2 on the plane. The proof uses the non-existence of minimal homeomorphisms on the infinite annulus (Le Calvez-Yoccoz's theorem), and the theory of Brouwer homeomorphisms.