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 , 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.
Highlights
A tiling is a cover of Rd by its countably many subsets T with the property that T = T ◦
The diffraction measures defined for these tilings correspond to physical diffraction patterns
It is important to study the nature of diffraction measures for tilings
Summary
Concerning the spectral properties of self-affine tilings, a key conjecture is the Pisot substitution conjecture, which states that self-affine tilings obtained from substitution rules of Pisot type are pure point diffractive, that is, their diffraction measures are pure point. This is still an open problem, but there are several partial positive answers. For μX -almost all (in)n ∈ X, the S-adic tilings belonging to (in)n for ρ1, ρ2 have zero absolutely continuous diffraction spectrum Note that this is not included in Theorem 1.1 because μ(X) might be zero.
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