We study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.
"Coarse dynamics and fixed-point theorem." Nagoya Math. J. 202 1 - 13, June 2011. https://doi.org/10.1215/00277630-1260414