Toeplitz flows and model sets

TL;DR

This paper demonstrates that binary Toeplitz flows can be viewed as Delone dynamical systems from model sets, revealing new insights into their ergodic properties and entropy, with implications for the theory of aperiodic order.

Contribution

It establishes a connection between Toeplitz flows and model sets, providing new results on their ergodic behavior and entropy, answering open questions in the field.

Findings

Irregular proper model sets can be uniquely ergodic

Such model sets do not necessarily have positive entropy

The paper links Toeplitz flows with the theory of model sets

Abstract

We show that binary Toeplitz flows can be interpreted as Delone dynamical systems induced by model sets and analyse the quantitative relations between the respective system parameters. This has a number of immediate consequences for the theory of model sets. In particular, we use our results in combination with special examples of irregular Toeplitz flows from the literature to demonstrate that irregular proper model sets may be uniquely ergodic and do not need to have positive entropy. This answers questions by Schlottmann and Moody.