Characterization of Entropy for Spacing shifts

TL;DR

This paper investigates the entropy of spacing shifts, establishing conditions under which the entropy is zero or positive, and linking these to properties of the set P, including proximality and intersective sets.

Contribution

It characterizes when the entropy of a spacing shift is zero or positive, connecting entropy to set properties like intersectivity and elta-star sets, and addresses an open question.

Findings

Zero entropy implies proximality of the system.

Zero entropy occurs iff P is the complement of an intersective set.

Positive entropy implies P is a elta-star set, but not always.

Abstract

Suppose $P\subseteq \mathbb{N}$ and let $(\Sigma_P,\,\sigma_P)$ be the space of a spacing shift. We show that if entropy $h_{\sigma_P}=0$ then $(\Sigma_P,\,\sigma_P)$ is proximal. Also $h_{\sigma_P}=0$ if and only if $P=\mathbb N\setminus E$ where $E$ is an intersective set. Moreover, we show that $h_{\sigma_P}>0$ implies that $P$ is a $\Delta^*$ set; and by giving a class of examples, we show that this is not a sufficient condition. Then there is enough results to solve question 5 given in [J. Banks et al., \textit{Dynamics of Spacing Shifts}, Discrete Contin. Dyn. Syst., to appear.].