## Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001

### Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.)

### DMTCS Conference Volume AA (2001), pp. 59-78

author: | Pierre Arnoux, Valérie Berthé, Hiromi Ei and Shunji Ito |
---|---|

title: | Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions |

keywords: | Substitutions, translations on compact groups, tilings, atomic surface, fractal sets, Markov partitions, numeration systems |

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 irrational examples 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. |

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reference: | Pierre Arnoux and
Valérie Berthé and Hiromi Ei and Shunji Ito
(2001), Tilings, Quasicrystals, Discrete Planes, Generalized
Substitutions, and Multidimensional Continued Fractions, in
Discrete Models: Combinatorics, Computation, and
Geometry, DM-CCG 2001, Robert Cori and Jacques Mazoyer
and Michel Morvan and Rémy Mosseri (eds.),
Discrete Mathematics and Theoretical Computer Science
Proceedings AA, pp. 59-78 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

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