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.

Two conjectures on coarse conjugacy by Geller and Misiurewicz

“…Since s ′ and k ′ are fixed, and k n is bounded by K, inequality (8) implies that s n grows at least as fast as the exponential function with base F, which yields a contradiction with (7) because F > G.…”

“…In fact, in Examples 2.1 and 2.4 we arrange X to equal Y and the conjugating maps ϕ and ψ to be the identity map. This shows that imposing any extra conditions on the conjugating maps, for example bijectivity like in [7], does not make Conjecture A true. While this is not yet true in Example 2.1, we construct the transformations f and g in Example 2.4 to be bijections.…”

“…The considerations of coarse-geometric notions of dynamics are therefore timely. Indeed, while coarse entropy of [2] is inspired by topological entropy, another notion referred to as coarse entropy and inspired by algebraic entropy of group endomorphisms has also been introduced recently [7]. The introduction of coarse conjugacy is a fundamental development within this emerging direction, and our aim in the present note is to analyse two conjectures on coarse conjugacy by Geller and Misiurewicz.…”

Coarse entropy

“…We note that Zava [6] has independently defined a different notion also referred to as coarse entropy. Zava's coarse entropy is inspired by the algebraic entropy of a group endomorphism, and in the case of a surjective endomorphism, they coincide.…”