Billiard complexity in the hypercube

Nicolas Bedaride[1]; Pascal Hubert[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 (2007)

  • Volume: 57, Issue: 3, page 719-738
  • ISSN: 0373-0956

Abstract

top
We consider the billiard map in the hypercube of . 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 is the order of magnitude of the complexity.

How to cite

top

Bedaride, Nicolas, and Hubert, Pascal. "Billiard complexity in the hypercube." Annales de l’institut Fourier 57.3 (2007): 719-738. <http://eudml.org/doc/10239>.

@article{Bedaride2007,
abstract = {We consider the billiard map in the hypercube of $\mathbb\{R\}^d$. 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 $n^\{3d-3\}$ is the order of magnitude of the complexity.},
affiliation = {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); 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)},
author = {Bedaride, Nicolas, Hubert, Pascal},
journal = {Annales de l’institut Fourier},
keywords = {Symbolic dynamic; billiard; words; complexity function; symbolic dynamic},
language = {eng},
number = {3},
pages = {719-738},
publisher = {Association des Annales de l’institut Fourier},
title = {Billiard complexity in the hypercube},
url = {http://eudml.org/doc/10239},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Bedaride, Nicolas
AU - Hubert, Pascal
TI - Billiard complexity in the hypercube
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 719
EP - 738
AB - We consider the billiard map in the hypercube of $\mathbb{R}^d$. 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 $n^{3d-3}$ is the order of magnitude of the complexity.
LA - eng
KW - Symbolic dynamic; billiard; words; complexity function; symbolic dynamic
UR - http://eudml.org/doc/10239
ER -

References

top
  1. P. Arnoux, C. Mauduit, I. Shiokawa, J. Tamura, Complexity of sequences defined by billiard in the cube, Bull. Soc. Math. France 122 (1994), 1-12 Zbl0791.58034MR1259106
  2. Yu. Baryshnikov, Complexity of trajectories in rectangular billiards, Comm. Math. Phys. 174 (1995), 43-56 Zbl0839.11006MR1372799
  3. N. Bedaride, Billiard complexity in rational polyhedra, Regul. Chaotic Dyn. 8 (2003), 97-104 Zbl1023.37024MR1963971
  4. N. Bedaride, Entropy of polyhedral billiard, (2005) Zbl1200.37034
  5. N. Bedaride, A generalization of Baryshnikov’s formula., (2006) 
  6. J. Berstel, M. Pocchiola, A geometric proof of the enumeration formula for Sturmian words, Internat. J. Algebra Comput. 3 (1993), 349-355 Zbl0802.68099MR1240390
  7. J. Cassaigne, Complexité et facteurs spéciaux, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 67-88 Zbl0921.68065MR1440670
  8. J. Cassaigne, P. Hubert, S. Troubetzkoy, Complexity and growth for polygonal billiards, Ann. Inst. Fourier 52 (2002), 835-847 Zbl1115.37312MR1907389
  9. William Fulton, Intersection theory, Springer-Verlag 2 (1998) Zbl0885.14002MR1644323
  10. G. Galʼperin, T. Krüger, S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys. 169 (1995), 463-473 Zbl0924.58043MR1328732
  11. G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, (1979), The Clarendon Press Oxford University Press, New York Zbl0020.29201MR568909
  12. P. Hubert, Complexité de suites définies par des billards rationnels, Bull. Soc. Math. France 123 (1995), 257-270 Zbl0836.58013MR1340290
  13. A. Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard, Comm. Math. Phys. 111 (1987), 151-160 Zbl0631.58020MR896765
  14. H. Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems 10 (1990), 151-176 Zbl0706.30035MR1053805
  15. F. Mignosi, On the number of factors of Sturmian words, Theoret. Comput. Sci. 82 (1991), 71-84 Zbl0728.68093MR1112109
  16. M. Morse, G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42 Zbl0022.34003MR745

NotesEmbed ?

top

You must be logged in to post comments.