Pure point diffraction and entropy beyond the Euclidean space

TL;DR

This paper extends the understanding of entropy in pure point diffractive Delone sets beyond Euclidean spaces, showing that topological entropy remains zero in a broader setting but patch counting entropy can vary.

Contribution

It generalizes entropy results for Delone sets to sigma-compact locally compact Abelian groups and introduces a variational principle for topological entropy.

Findings

Topological entropy of Delone dynamical systems is zero in the generalized setting.

Patch counting entropy can be non-zero, infinite, or not well-defined along limits in this context.

Counterexamples demonstrate the complexity of entropy behavior beyond Euclidean spaces.

Abstract

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of sigma-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set can not be computed along a limit and even be infinite in this setting.