Absence of absolutely continuous diffraction spectrum for certain S-adic tilings

Abstract

Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence of the absolutely continuous diffraction spectrum for most of the S-adic tilings from a family of binary block substitutions.

30 pages, two figures

References in corpus (6)

• Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction
• Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction
• Spectral analysis of a family of binary inflation rules
• Shifts of finite type and random substitutions
• Lyapunov exponents for binary substitutions of constant length
• Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions