Coarse entropy of metric spaces

William Geller, Michał Misiurewicz and Damian Sawicki

Abstract

Recently, for a self-map of a metric space, the two first-named authors defined its coarse entropy. The definition uses pseudoorbits instead of true orbits of the map. As a result, there are spaces in which the coarse entropy of the identity, which we call the coarse entropy of the space, is positive. Unlike some related notions like volume growth, it is defined for an arbitrary metric space. We investigate its connections with other properties of the space. We show that it can only be either zero or infinity, and although for sufficiently nice spaces the dichotomy zero-infinite coarse entropy coincides with the dichotomy subexponential-exponential growth, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces.