Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux, Pierre; Berthé, Valérie; Ito, Shunji

doi:10.5802/aif.1889

Discrete planes,

ℤ^{2}

-actions, Jacobi-Perron algorithm and substitutions
[Plans discrets, actions de

ℤ^{2}

, algorithme de Jacobi-Perron et substitutions]

Arnoux, Pierre ¹ ; Berthé, Valérie ¹ ; Ito, Shunji ²

¹ Institut de Mathématiques de Luminy, Campus de Luminy, Case 901, 13288 Marseille Cedex 9 (France)
² Tsuda College, Tsuda Machi, Kodaira, Tokyo 187 (Japon)

Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 305-349.

Résumé
Abstract

Nous définissons des substitutions bi-dimensionnelles; ces substitutions engendrent des suites doubles reliées à des approximations discrètes de plans irrationnels. Elles sont obtenues au moyen de l’algorithme classique de Jacobi Perron, en définissant l’induction d’une action de $ℤ^{2}$ par rotations sur le cercle. On donne ainsi une interprétation géométrique nouvelle de l’algorithme de Jacobi-Perron, comme application opérant sur l’espace des paramètres des actions de $ℤ^{2}$ par rotations.

We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a $ℤ^{2}$ -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of $ℤ^{2}$ -actions by rotations.

MR Zbl | 6 citations dans Numdam

DOI : 10.5802/aif.1889

Classification : 11A55, 11J70, 40A15, 68R15
Keywords: substitutions, generalized continued fractions, discrete plans, tilings, Jacobi-Perron algorithm, induction,

Z^{2}

-actions, two-dimensional sequences
Mot clés : substitutions, fractions continues généralisées, plans discrets, pavages, algorithme de Jacobi-Perron, induction, actions de

Z^{2}

, suites doubles

Affiliations des auteurs :

Arnoux, Pierre ¹ ; Berthé, Valérie ¹ ; Ito, Shunji ²

¹ Institut de Mathématiques de Luminy, Campus de Luminy, Case 901, 13288 Marseille Cedex 9 (France)
² Tsuda College, Tsuda Machi, Kodaira, Tokyo 187 (Japon)

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     title = {Discrete planes, ${\mathbb {Z}}^2$-actions, {Jacobi-Perron} algorithm and substitutions},
     journal = {Annales de l'Institut Fourier},
     pages = {305--349},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     mrnumber = {1906478},
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}

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Arnoux, Pierre; Berthé, Valérie; Ito, Shunji. Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 305-349. doi : 10.5802/aif.1889. https://www.numdam.org/articles/10.5802/aif.1889/

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