Abstract
The Thue–Morse system is a paradigm of singular continuous diffraction in one dimension. Here, we consider a planar generalisation, constructed by a bijective block substitution rule, which is locally equivalent to the squiral inflation rule. For balanced weights, its diffraction is purely singular continuous. The diffraction measure is a two-dimensional Riesz product that can be calculated explicitly.
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Baake M, Grimm U (2008) The singular continuous diffraction measure of the Thue–Morse chain. J Phys A, Math Theor 41:422001. arXiv:0809.0580
Baake M (2002) Diffraction of weighted lattice subsets. Can Math Bull 45:483–498. arXiv:math.MG/0106111
Baake M, Gähler F, Grimm U (2012) Spectral and topological properties of a family of generalised Thue–Morse sequences. J Math Phys 53:032701. arXiv:1201.1423
Baake M, Grimm U (2011) Kinematic diffraction from a mathematical viewpoint. Z Kristallogr 226:711–725. arXiv:1105.0095
Baake M, Grimm U (2012) Mathematical diffraction of aperiodic structures. Chem Soc Rev 41:6821–6843. arXiv:1205.3633
Baake M, Grimm U (2012) Squirals and beyond: substitution tilings with singular continuous spectrum. Ergod Theory Dyn Syst, to appear. arXiv:1205.1384
Baake M, Grimm U (2013) Theory of aperiodic order: a mathematical invitation. Cambridge University Press, Cambridge, to appear
Grünbaum B, Shephard GC (1987) Tilings and patterns. Freeman, New York
Hof A (1995) On diffraction by aperiodic structures. Commun Math Phys 169:25–43
Kakutani S (1972) Strictly ergodic symbolic dynamical systems. In: LeCam LM, Neyman J, Scott EL (eds) Proceedings of the 6th Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 319–326
Frank NP (2005) Multi-dimensional constant-length substitution sequences. Topol Appl 152:44–69
Queffélec M (2010) Substitution dynamical systems—spectral analysis, 2nd edn. LNM, vol 1294 Springer, Berlin
Welberry TR, Withers RL (1991) The rôle of phase in diffuse diffraction patterns and its effect on real-space structure. J Appl Crystallogr 24:18–29
Withers RL (2005) Disorder, structured diffuse scattering and the transmission electron microscope. Z Kristallogr 220:1027–1034
Acknowledgements
We thank Tilmann Gneiting and Daniel Lenz for discussions. This work was supported by the German Research Council (DFG), within the CRC 701.
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Grimm, U., Baake, M. (2013). Squiral Diffraction. In: Schmid, S., Withers, R., Lifshitz, R. (eds) Aperiodic Crystals. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6431-6_2
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DOI: https://doi.org/10.1007/978-94-007-6431-6_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-6430-9
Online ISBN: 978-94-007-6431-6
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