Communications in Mathematical Physics

, Volume 174, Issue 1, pp 43–56 | Cite as

Complexity of trajectories in rectangular billiards

  • Yu. Baryshnikov


To a trajectory of the billiard in a cube we assign its symbolic trajectory-the sequence of numbers of coordinate planes, to which the faces met by the trajectory are parallel. The complexity of the trajectory is the number of different words of lengthn occurring in it. We prove that for generic trajectories the complexity is well defined and calculate it, confirming the conjecture of Arnoux, Mauduit, Shiokawa and Tamura [AMST].


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AMST] Arnoux, P., Mauduit, C., Shiokawa, I., Tamura, J.-I.: Complexity of sequences defined by billiard in the cube. Bull. Soc. Math. France122, 1–12 (1994)Google Scholar
  2. [B] Bruckstein, A. M.: Self-similarity properties of digitized straight lines. In: Vision geometry, Proc. AMS Spec. Sess., 851st Meet., Hoboken/NJ (USA) 1989, Contemp. Math.119, 1–20 (1991)Google Scholar
  3. [FF] Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton, NJ: Princeton Univ. Press, 1962.Google Scholar
  4. [LP] Lunnon, W.F., Pleasants, P.A.B.: Characterization of two-distance sequences. J. Aust. Math. Soc., Ser. A53, No. 2, 198–218 (1992)Google Scholar
  5. [MH] Morse, M., Hedlund, G.A.: Symbolic dynamics II. Sturmian trajectories. Am. J. Math.62, 1–42 (1940)Google Scholar
  6. [R] Rauzy, G.: Mots infinis et arithmetique. In: Automata on Infinite Words, Lect. Notes in Comp. Sciences192, Berlin, Heidelberg, New York: Springer (1985) pp. 165–171Google Scholar
  7. [S] Stolarsky, K.B.: Beatty sequences, continuous fractions and certain shift operators. Can. Math. Bull.19, 473–482 (1976)Google Scholar
  8. [T] Tabachnikov, S.: Billiards. Preprint (1994)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Yu. Baryshnikov
    • 1
  1. 1.Department of MathematicsUniversity of OsnabrückOsnabrückGermany

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