Periodic billiard orbits in right triangles

Serge Troubetzkoy[1]

  • [1] Institut de mathématiques de Luminy, Centre de physique théorique, Case 907, 13288 Marseille cedex 9 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 29-46
  • ISSN: 0373-0956

Abstract

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There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

How to cite

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Troubetzkoy, Serge. "Periodic billiard orbits in right triangles." Annales de l’institut Fourier 55.1 (2005): 29-46. <http://eudml.org/doc/116190>.

@article{Troubetzkoy2005,
abstract = {There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.},
affiliation = {Institut de mathématiques de Luminy, Centre de physique théorique, Case 907, 13288 Marseille cedex 9 (France)},
author = {Troubetzkoy, Serge},
journal = {Annales de l’institut Fourier},
keywords = {Polygonal billiard; periodic orbits; symmetries},
language = {eng},
number = {1},
pages = {29-46},
publisher = {Association des Annales de l'Institut Fourier},
title = {Periodic billiard orbits in right triangles},
url = {http://eudml.org/doc/116190},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Troubetzkoy, Serge
TI - Periodic billiard orbits in right triangles
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 29
EP - 46
AB - There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.
LA - eng
KW - Polygonal billiard; periodic orbits; symmetries
UR - http://eudml.org/doc/116190
ER -

References

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  1. M. Boshernitzan, Billiards and rational periodic directions in polygons, Amer. Math. Monthly 99 (1992), 522-529 Zbl0776.58034MR1166001
  2. M. Boshernitzan, G. Galperin, T. Krüger, S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons, Trans. AMS 350 (1998), 3523-3535 Zbl0910.58013MR1458298
  3. B. Cipra, R. Hanson, A. Kolan, Periodic trajectories in right triangle billiards, Phys. Rev. E52 (1995), 2066-2071 MR1388476
  4. G. Galperin, Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons, Comm. Math. Phys. 91 (1983), 187-211 Zbl0529.70001MR723547
  5. G. Galperin, A. Stepin, Ya. Vorobets, Periodic billiard trajectories in polygons: generating mechanisms, Russian Math. Surveys 47 (1992), 5-80 Zbl0777.58031MR1185299
  6. G. Galperin, D. Zvonkine, Periodic billiard trajectories in right triangles and right-angled tetrahedra, Regular and Chaotic Dynamics 8 (2003), 29-44 Zbl1023.37021MR1963966
  7. E. Gutkin, Billiard in polygons, Physica D19 (1986), 311-333 Zbl0593.58016MR844706
  8. E. Gutkin, Billiard in polygons: survey of recent results, J. Stat. Phys. 83 (1996), 7-26 Zbl1081.37525MR1382759
  9. E. Gutkin, S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards, Proceedings of the International Congress of Dynamical Systems (1996), 21-45, F. Ledrappier et al. eds., Montevideo, Uruguay Zbl0904.58036
  10. A. Katok, B. Hasselblatt, Encyclopedia of Mathematics and its Applications, 54 (1995), Cambridge University Press Zbl0878.58020MR1326374
  11. H. Masur, Closed trajectories of a quadratic differential with an application to billiards, Duke Math. J. 53 (1986), 307-313 Zbl0616.30044MR850537
  12. H. Masur, S. Tabachnikov, Rational billiards and flat structures, 1A (2002), 1015-1089, North-Holland Zbl1057.37034
  13. T. Ruijgrok, Periodic orbits in triangular billiards, Acta Physica Polonica B22 (1991), 955-981 
  14. J. Schmeling, S. Troubetzkoy, Inhomogeneous Diophantine approximation and angular recurrence for polygonal billiards, Math. Sb. 194 (2003), 295-309 Zbl1043.37028MR1992153
  15. S. Tabachnikov, Billiards, (1995), Soc. Math. France Zbl0833.58001MR1328336
  16. S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regul. Chaotic Dyn. 9 (2004), 1-12 Zbl1049.37024MR2058893

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