A characterization of linearly repetitive cut and project sets

Haynes, A., Koivusalo, H. and Walton, J. (2018) A characterization of linearly repetitive cut and project sets. Nonlinearity, 31(2), pp. 515-539. (doi: 10.1088/1361-6544/aa9528)

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Abstract

For the development of a mathematical theory which can be used to rigorously investigate physical properties of quasicrystals, it is necessary to understand regularity of patterns in special classes of aperiodic point sets in Euclidean space. In one dimension, prototypical mathematical models for quasicrystals are provided by Sturmian sequences and by point sets generated by substitution rules. Regularity properties of such sets are well understood, thanks mostly to well known results by Morse and Hedlund, and physicists have used this understanding to study one dimensional random Schrödinger operators and lattice gas models. A key fact which plays an important role in these problems is the existence of a subadditive ergodic theorem, which is guaranteed when the corresponding point set is linearly repetitive. In this paper we extend the one-dimensional model to cut and project sets, which generalize Sturmian sequences in higher dimensions, and which are frequently used in mathematical and physical literature as models for higher dimensional quasicrystals. By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with cubical windows. We also prove that these are precisely the collection of such sets which satisfy subadditive ergodic theorems. The results are explicit enough to allow us to apply them to known classical models, and to construct linearly repetitive cut and project sets in all pairs of dimensions and codimensions in which they exist.

Item Type:Articles
Additional Information:Research supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540. HK also gratefully acknowledges the support of the Osk. Huttunen foundation.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Walton, Dr Jamie
Authors: Haynes, A., Koivusalo, H., and Walton, J.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Nonlinearity
Publisher:IOP Publishing
ISSN:0951-7715
ISSN (Online):1361-6544
Published Online:10 January 2018
Copyright Holders:Copyright © 2018 IOP Publishing Ltd and London Mathematical Society
First Published:First published in Nonlinearity 31(2): 515-539
Publisher Policy:Reproduced under a Creative Commons License

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