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Squirals and beyond: substitution tilings with singular continuous spectrum

Published online by Cambridge University Press:  20 March 2013

MICHAEL BAAKE  and
UWE GRIMM
MICHAEL BAAKE
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany email mbaake@math.uni-bielefeld.de
UWE GRIMM
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK email uwe.grimm@open.ac.uk
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Abstract

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} $.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1077 - 1102
Copyright
Copyright ©2013 Cambridge University Press 

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