Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions
[Plans discrets, actions de ℤ 2 , algorithme de Jacobi-Perron et substitutions]
Arnoux, Pierre ; Berthé, Valérie ; Ito, Shunji
Annales de l'Institut Fourier, Tome 52 (2002), p. 305-349 / Harvested from Numdam
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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.

Publié le : 2002-01-01
DOI : https://doi.org/10.5802/aif.1889
Classification:  11A55,  11J70,  40A15,  68R15
Mots clés: substitutions, fractions continues généralisées, plans discrets, pavages, algorithme de Jacobi-Perron, induction, actions de ℤ 2 , suites doubles 
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     author = {Arnoux, Pierre and Berth\'e, Val\'erie and Ito, Shunji},
     title = {Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions},
     journal = {Annales de l'Institut Fourier},
     volume = {52},
     year = {2002},
     pages = {305-349},
     doi = {10.5802/aif.1889},
     mrnumber = {1906478},
     zbl = {1017.11006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2002__52_2_305_0}
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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) pp. 305-349. doi : 10.5802/aif.1889. http://gdmltest.u-ga.fr/item/AIF_2002__52_2_305_0/

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