Abstract
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].
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[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)
[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)
[FF] Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton, NJ: Princeton Univ. Press, 1962.
[LP] Lunnon, W.F., Pleasants, P.A.B.: Characterization of two-distance sequences. J. Aust. Math. Soc., Ser. A53, No. 2, 198–218 (1992)
[MH] Morse, M., Hedlund, G.A.: Symbolic dynamics II. Sturmian trajectories. Am. J. Math.62, 1–42 (1940)
[R] Rauzy, G.: Mots infinis et arithmetique. In: Automata on Infinite Words, Lect. Notes in Comp. Sciences192, Berlin, Heidelberg, New York: Springer (1985) pp. 165–171
[S] Stolarsky, K.B.: Beatty sequences, continuous fractions and certain shift operators. Can. Math. Bull.19, 473–482 (1976)
[T] Tabachnikov, S.: Billiards. Preprint (1994)
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Communicated by Ya. G. Sinai
The author was supported by DFG.
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Yu. Baryshnikov Complexity of trajectories in rectangular billiards. Commun.Math. Phys. 174, 43–56 (1995). https://doi.org/10.1007/BF02099463
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DOI: https://doi.org/10.1007/BF02099463