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Complexity and growth for polygonal billiards

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  • Law
  • Mathematics


Complexity and growth for polygonal billiards AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER J. CASSAIGNE, Pascal HUBERT & Serge TROUBETZKOY Complexity and growth for polygonal billiards Tome 52, no 3 (2002), p. 835-847. <> © Association des Annales de l’institut Fourier, 2002, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (, implique l’accord avec les conditions générales d’utilisation ( Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques , 835- COMPLEXITY AND GROWTH FOR POLYGONAL BILLIARDS by J. CASSAIGNE, P. HUBERT &#x26; S. TROUBETZKOY 1. Introduction. A billiard ball, i.e. a point mass, moves inside a polygon Q c JR2 with unit speed along a straight line until it reaches the boundary 0Q, then instantaneously changes direction according to the mirror law "the angle of incidence is equal to the angle of reflection," and continues along the new line. How complex is the game of billiards in a polygon? The first results in this direction, proven independently by Sinai [S] and Boldrighini, Keane and Marchetti [BKM] is that the metric entropy with respect to the invariant phase volume is zero. Sinai’s proof in fact shows more, the "metric complexity" grows at most polynomially. Furthermore, it is known that the topological entropy (in various senses) is zero [K], [GKT], [GH]. To prove finer results there are two natural quantities one can count, one is the number of generalized diagonals, that is (orient

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