On a notion of entropy in coarse geometry

Abstract

The notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.

Keywords

• coarse space
• quasi-coarse space
• coarse entropy
• algebraic entropy
• growth of metric spaces
• coarse equivalence
• 20k30
• 37b40
• 54h20
• 54c70
• 54a99
• 54e15
• 54e99