Squirals and beyond: substitution tilings with singular continuous spectrum (Q2925256)

Translation-bounded measures on \(\mathbb{R}^d\) determine associated dynamical systems, and in the case of pure point discrete spectrum these are well understood. Pure point spectra arise naturally in periodic and almost periodic systems in (quasi-)crystallography, and in that setting there is a good relationship between the spectrum of pure point dynamical systems and pure point diffraction spectra -- one can discern the dynamical spectrum from diffraction data. For mixed spectra the situation is much more intricate, and as usual the presence of singular continuous spectra causes particular difficulties. Motivated in part by mixed spectra being identifiable in diffraction spectra of real materials, and progress in understanding the relationship between the dynamical spectrum and the diffraction spectrum in various specific classes of substitution systems, examples where specific obstacles to progress can be examined are of great interest. The system studied here -- the squiral aperiodic tiling, a tiling of the plane produced by one prototile with infinitely many edges (and its mirror image) arising from a simple geometric inflation rule -- is attractive in this way as it is genuinely higher-dimensional (that is, not built up in some simple way from one-dimensional systems) and contains singular continuous spectrum. Here an explicit and constructive proof is given that the dynamical spectrum is of mixed type (both pure point and singular continuous) and the diffraction is purely singular continuous. The method used will also apply to certain bijective block substitutions on \(\mathbb{Z}^d\).