A Note on Entropy of Delone Sets

Abstract

In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non-compact) locally compact Abelian groups that contain Meyer sets.

arXiv admin note: text overlap with arXiv:1901.08985 This Article is a revised and renamed version of 'Entropy theory for Delone sets in locally compact abelian groups'