Topological matchings and amenability

Friedrich Martin Schneider, Andreas Thom

Published 2015-02-08Version 1

We introduce a novel quantity for general dynamical systems, which we shall call the mean topological matching number. We prove that a dynamical system is amenable if its mean topological matching number equals one. Furthermore, we show that the converse is true provided that the considered dynamical system does not contain any isolated points. We conclude that a Hausdorff topological group $G$ is amenable if and only if the mean topological matching number of the associated action of $G$ on itself equals one.