Absence of absolutely continuous diffraction spectrum for certain S-adic tilings. (21st October 2021)

• Record Type: Journal Article
• Authors: Nagai, Yasushi
• Is Part Of: Nonlinearity. Volume 34:Number 11(2021)
• Page Start: 7963
• Page End: 7990
• Publication Date: 2021-10-21
• Subjects: tiling -- diffraction -- S-adic sequence -- Lyapunov exponent -- renormalisation; 52C23 -- 52C22; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac2a51
• Languages: English
• ISSNs: 0951-7715

Abstract

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 S-adic tilings from a family of binary block substitutions.