Squirals and beyond: Substitution tilings with singular continuous spectrum

TL;DR

This paper investigates substitution tilings, specifically squiral patterns, revealing their singular continuous diffraction and mixed dynamical spectrum, and generalizes the findings to higher dimensions.

Contribution

It provides a constructive proof for the spectral properties of squiral tilings and extends the analysis to bijective block substitutions in higher dimensions.

Findings

Squiral tilings have purely singular continuous diffraction.

The dynamical spectrum of squiral tilings is mixed, with pure point and singular continuous parts.

The methods generalize to bijective block substitutions in any dimension.

Abstract

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of $\mathbb{Z}^{2}$. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalisation to bijective block substitutions of trivial height on $\mathbb{Z}^{d}$.