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  1. Home > Articles & Issues >
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  3. Tilings, Quasicrysta ...
Conference paper

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

Pierre Arnoux (1), Valerie Berthe ORCID (1), Hiromi Ei (2), Shunji Ito (2)
(1) Institut de mathématiques de Luminy
(2) Tsuda College
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Imported on
November 21, 2016
Published on
December 31, 2000
Last modified on
March 31, 2025
Volume 0
DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001)
Proceedings
DOI
10.46298/dmtcs.2291
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https://about.hal.science/hal-authorisation-v1
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Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

Pierre Arnoux (1), Valerie Berthe ORCID (1), Hiromi Ei (2), Shunji Ito (2)
(1) Institut de mathématiques de Luminy
(2) Tsuda College
Abstract
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.
Keywords
  • [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]
  • Substitutions
  • translations on compact groups
  • tilings
  • atomic surface
  • fractal sets
  • Markov partitions
  • numeration systems
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