Coarse entropy of metric spaces

TL;DR

This paper introduces the concept of coarse entropy in metric spaces, showing it is a coarse invariant that can only be zero or infinity, and explores its implications for space properties and embeddings.

Contribution

It establishes that coarse entropy is a coarse invariant, characterizes its dichotomy in specific spaces, and demonstrates its use in obstructing coarse embeddings.

Findings

Coarse entropy is invariant under isometries.

It can only be zero or infinity for any metric space.

Coarse entropy can obstruct coarse embeddings despite volume growth considerations.

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.