Model sets with positive entropy in Euclidean cut and project schemes

TL;DR

This paper constructs specific model sets in Euclidean spaces via cut and project schemes that exhibit positive topological entropy, with entropy related to the boundary measure of the window, applicable to various window types.

Contribution

It introduces a method to generate model sets with positive entropy in Euclidean spaces, expanding understanding of their dynamical complexity.

Findings

Model sets with positive entropy are constructed in Euclidean spaces.

Entropy is proportional to the boundary measure of the window in a probabilistic setting.

Applicable to both proper windows and those with empty interior.

Abstract

We construct model sets arising from cut and project schemes in Euclidean spaces whose associated Delone dynamical systems have positive toplogical entropy. The construction works both with windows that are proper and with windows that have empty interior. In a probabilistic construction, the entropy almost surely turns out to be proportional to the measure of the boundary of the window.