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.