Squirals and beyond: substitution tilings with singular continuous spectrum

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, wit...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2014-08, Vol.34 (4), p.1077-1102
Main Authors: BAAKE, MICHAEL, GRIMM, UWE
Format: Article
Language:English
Subjects:
Balancing
Blocking
Diffraction
Dynamical systems
Equivalence
Inflation
Lattice theory
Lattices
Spectrum analysis
Tiling
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Summary: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 generalization to bijective block substitutions of trivial height on ${ \mathbb{Z} }^{d} $.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2012.191