Squirals and beyond: Substitution tilings with singular continuous spectrum

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 ${\mathbb{Z}}^d$.

23 pages, 7 figures

References in corpus (5)

• Pure Point Dynamical and Diffraction Spectra
• Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
• The singular continuous diffraction measure of the Thue-Morse chain
• Examples of substitution systems and their factors
• On the notions of symmetry and aperiodicity for Delone sets

Cited by in corpus (8)

• Aperiodic crystals and beyond
• On the Fourier Analysis of Measures with Meyer Set Support
• Singular substitutions of constant length
• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions
• A note on measures vanishing at infinity
• Why do Meyer sets diffract?
• Squiral diffraction
• New results on tilings via cup products and Chern characters on tiling spaces