Discrete planes, ℤ 2 -actions, Jacobi-Perron algorithm and substitutions
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, p. 305-349

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.

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.

DOI : https://doi.org/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 
@article{AIF_2002__52_2_305_0,
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     pages = {305-349},
     doi = {10.5802/aif.1889},
     zbl = {1017.11006},
     mrnumber = {1906478},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2002__52_2_305_0}
}

Arnoux, Pierre; Berthé, Valérie; Ito, Shunji. Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 305-349. doi : 10.5802/aif.1889. http://www.numdam.org/item/AIF_2002__52_2_305_0/

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