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 -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 -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 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 par rotations.
Keywords: substitutions, generalized continued fractions, discrete plans, tilings, Jacobi-Perron algorithm, induction, ${\mathbb {Z}}^2$-actions, two-dimensional sequences
Mot clés : substitutions, fractions continues généralisées, plans discrets, pavages, algorithme de Jacobi-Perron, induction, actions de ${\mathbb {Z}}^2$, suites doubles
@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}, pages = {305--349}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {2}, year = {2002}, doi = {10.5802/aif.1889}, mrnumber = {1906478}, zbl = {1017.11006}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1889/} }
TY - JOUR AU - Arnoux, Pierre AU - Berthé, Valérie AU - Ito, Shunji TI - Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions JO - Annales de l'Institut Fourier PY - 2002 SP - 305 EP - 349 VL - 52 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1889/ DO - 10.5802/aif.1889 LA - en ID - AIF_2002__52_2_305_0 ER -
%0 Journal Article %A Arnoux, Pierre %A Berthé, Valérie %A Ito, Shunji %T Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions %J Annales de l'Institut Fourier %D 2002 %P 305-349 %V 52 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1889/ %R 10.5802/aif.1889 %G en %F 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/articles/10.5802/aif.1889/
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