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There is no minimal action of \(\mathbb Z^2\) on the plane. (English) Zbl 1256.37015

Summary: We prove that there is no minimal action of \(\mathbb Z^{2}\) by homeomorphisms on the plane. This may be seen as a generalization of the Le Calvez-Yoccoz theorem: there exists no minimal homeomorphism of the infinite annulus.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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