Complexity and growth for polygonal billiards
[Complexité et diagonales généralisées dans les billards polygonaux]
Cassaigne, J.1 ; Hubert, Pascal1 ; Troubetzkoy, Serge2
1 Institut de Mathématiques de Luminy, Case 907, 13288 Marseille Cedex 9 (France)
2 Centre de Physique Théorique, Institut de Mathématiques de Luminy, Case 907, 13288 Marseille Cedex 9 (France)
Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 835-847.

Nous établissons un lien entre la fonction de complexité et le nombre de diagonales généralisées pour un billard polygonal. Dans le cas où le billard est rationnel, la fonction de complexité est comprise entre deux polynômes cubiques; elle a une asymptotique cubique lorsque le polygone pave le plan.

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.

DOI : 10.5802/aif.1903
Classification : 37C35
Keywords: complexity, polygonal billiards, generalized diagonals, bispecial words
Mot clés : complexité, billards polygonaux, diagonales généralisées, mots bispéciaux
Cassaigne, J. 1 ; Hubert, Pascal 1 ; Troubetzkoy, Serge 2

1 Institut de Mathématiques de Luminy, Case 907, 13288 Marseille Cedex 9 (France)
2 Centre de Physique Théorique, Institut de Mathématiques de Luminy, Case 907, 13288 Marseille Cedex 9 (France) 
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Cassaigne, J.; Hubert, Pascal; Troubetzkoy, Serge. Complexity and growth for polygonal billiards. Annales de l'Institut Fourier, Tome 52 (2002) no. 3, pp. 835-847. doi : 10.5802/aif.1903. http://archive.numdam.org/articles/10.5802/aif.1903/

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