Distribution of patches in tilings and spectral properties of corresponding dynamical systems
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Publication:3196533
DOI10.1088/0951-7715/28/8/2617zbMATH Open1360.37052arXiv1409.2645OpenAlexW2963496146MaRDI QIDQ3196533
Publication date: 29 October 2015
Published in: Nonlinearity (Search for Journal in Brave)
Abstract: A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our results throw light on inverse problem of deducing information of tilings from information of diffraction measures, in a quite general setting.
Full work available at URL: https://arxiv.org/abs/1409.2645
Tilings in (n) dimensions (aspects of discrete geometry) (52C22) Quasicrystals and aperiodic tilings in discrete geometry (52C23)
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