Billiard complexity in the hypercube

Bedaride, Nicolas; Hubert, Pascal

doi:10.5802/aif.2274

Billiard complexity in the hypercube
[Complexité du billard cubique multi-dimensionnel]

Bedaride, Nicolas ¹ ; Hubert, Pascal ²

¹ Fédération de recherches des unités de mathématiques de Marseille UMR 6632 Laboratoire d’Analyse Topologie et Probabilités Av. Escadrille Normandie-Niemen 13397 Marseille Cedex 20 (France)
² Fédération de recherches des unités de mathématiques de Marseille UMR 6632 Laboratoire d’Analyse Topologie et Probabilités av. Escadrille Normandie-Niemen 13397 Marseille Cedex 20 (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 719-738.

Résumé
Abstract

On considère l’application du billard dans le cube de $ℝ^{d}$ . On code cette application par les faces du cube. On obtient un langage, dont on cherche à évaluer la complexité. On montre que l’ordre de grandeur de cette fonction est $n^{3 d - 3}$ .

We consider the billiard map in the hypercube of $ℝ^{d}$ . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3 d - 3}$ is the order of magnitude of the complexity.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2274

Classification : 37A35, 37C35, 05A16, 11N37, 28D
Keywords: Symbolic dynamic, billiard, words, complexity function
Mot clés : Dynamique symbolique, billard, mots, complexité

Affiliations des auteurs :

Bedaride, Nicolas ¹ ; Hubert, Pascal ²

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     volume = {57},
     number = {3},
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     zbl = {1138.37017},
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Bedaride, Nicolas; Hubert, Pascal. Billiard complexity in the hypercube. Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 719-738. doi : 10.5802/aif.2274. https://www.numdam.org/articles/10.5802/aif.2274/

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