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

Arnoux, Pierre; Berthé, Valérie; Ito, Shunji

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

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 ${\mathbb{Z}}^2$ 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 ${\mathbb{Z}}^2$ 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 ${\mathbb{Z}}^2$-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 ${\mathbb{Z}}^2$-actions by rotations.

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