Coarse entropy of metric spaces
William Geller, Michał Misiurewicz, Damian Sawicki
TL;DR
The paper defines the coarse entropy $h_\infty$ for metric spaces via $\delta$-pseudoorbits and $R$-separated pseudo-orbits, and proves that it is the same for all isometric embeddings and is a coarse invariant. It establishes a sharp zero–infinity dichotomy, showing $h_\infty(X)$ is either $0$ or $\infty$, and completely characterizes this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. It also demonstrates that coarse entropy can obstruct coarse embeddings and that the dichotomy interacts nontrivially with volume growth. The results provide a robust, measureless large-scale invariant that distinguishes coarse-geometric types and informs embedding obstructions with potential applications in geometric group theory and coarse topology.
Abstract
Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential--exponential growth dichotomy, 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. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.