Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

Pierre Arnoux; Valerie Berthe; Hiromi Ei; Shunji Ito

doi:10.46298/dmtcs.2291

Pierre Arnoux ; Valerie Berthe ; Hiromi Ei ; Shunji Ito - Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

dmtcs:2291 - Discrete Mathematics & Theoretical Computer Science, January 1, 2001, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) - https://doi.org/10.46298/dmtcs.2291

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued FractionsArticle

Authors: Pierre Arnoux ¹; Valerie Berthe ¹; Hiromi Ei ²; Shunji Ito ²

1 Institut de mathématiques de Luminy
2 Tsuda College

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.

https://doi.org/10.46298/dmtcs.2291

Source: HAL:hal-01182971v1

Volume: DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001)

Section: Proceedings

Published on: January 1, 2001

Imported on: November 21, 2016

Keywords: Substitutions,translations on compact groups,tilings,atomic surface,fractal sets,Markov partitions,numeration systems,[INFO] Computer Science [cs],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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Sébastien Labbé, 2021, Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations, Journal of Modern Dynamics, 17, 0, pp. 481, 10.3934/jmd.2021017, https://doi.org/10.3934/jmd.2021017.

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V. V. Krasil’shchikov;A. V. Shutov, 2012, Distribution of points of one-dimensional quasilattices with respect to a variable module, Russian Mathematics, 56, 3, pp. 14-19, 10.3103/s1066369x12030036.

M.-A. Jacob-Da Col;P. Tellier, 2009, Quasi-linear transformations and discrete tilings, Theoretical Computer Science, 410, 21-23, pp. 2126-2134, 10.1016/j.tcs.2009.01.032.

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E.O. Harriss;J.S.W. Lamb, 2004, Canonical substitutions tilings of Ammann–Beenker type, Theoretical Computer Science, 319, 1-3, pp. 241-279, 10.1016/j.tcs.2004.02.014.

Jean-Pierre Gazeau;Jean-Louis Verger-Gaugry, 2004, Geometric study of the beta-integers for a Perron number and mathematical quasicrystals, Journal de Théorie des Nombres de Bordeaux, 16, 1, pp. 125-149, 10.5802/jtnb.437, https://doi.org/10.5802/jtnb.437.

Valentin E. Brimkov;Reneta P. Barneva, 2004, Plane digitization and related combinatorial problems, Discrete Applied Mathematics, 147, 2-3, pp. 169-186, 10.1016/j.dam.2004.09.010, https://doi.org/10.1016/j.dam.2004.09.010.

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Tan Bo;Wen Zhixiong;Zhang Yiping, 2003, The structure of invertible substitutions on a three-letter alphabet, Analysis in Theory and Applications, 19, 4, pp. 365-382, 10.1007/bf02835535.

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