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Billiards in polygons and homogeneous foliations on C2

Published online by Cambridge University Press:  01 February 2009

FERRÁN VALDEZ
FERRÁN VALDEZ*
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany (email: [email protected])
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Abstract

In this article we construct a new framework for the study of polygonal billiards. To every polygonal billiard we associate a holomorphic foliation on C2. The dynamics of the billiard ball is linked to the directional flow of the complex vector field defining the associated holomorphic foliation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 255 - 271
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Boshernitzan, M., Galperin, G., Krüger, T. and Troubetzkoy, S.. Periodic billiard orbits are dense in rational polygons. Trans. Amer. Math. Soc. 350(9) (1998), 3523–3535.CrossRefGoogle Scholar
[2]Cerveau, D.. Densité des feuilles de certaines équations de Pfaff à 2 variables. Ann. Inst. Fourier (Grenoble) 33(1) (1983), 185–194.Google Scholar
[3]Cerveau, D. and Mattei, J.-F.. Formes Intégrables Holomorphes Singulières (Astérisque, 97). Société Mathématique de France, Paris, 1982.Google Scholar
[4]Camacho, C. and Lins Neto, A.. Geometric Theory of Foliations. Birkhäuser Boston, Boston, MA, 1985, 205 pp.Google Scholar
[5]Driscoll, T. A. and Trefethen, L. N.. Schwarz–Christoffel Mapping (Cambridge Monographs on Applied and Computational Mathematics, 8). Cambridge University Press, Cambridge, 2002.Google Scholar
[6]Ghys, E.. Topologie des feuilles génériques. Ann. of Math. (2) 141(2) (1995), 378–422.Google Scholar
[7]Gutkin, E. and Judge, C.. Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. 2003(2) (2000), 191–213.Google Scholar
[8]Gutkin, E. and Troubetzkoy, S.. Directional flows and strong recurrence for polygonal billiards. International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362). Longman, Harlow, 1996, pp. 21–45.Google Scholar
[9]Kerckhoff, S., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124(2) (1986), 293–311.CrossRefGoogle Scholar
[10]Masur, H. and Tabashnikov, S.. Rational Billiards and Flat Structures (Handbook of Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 1015–1089.Google Scholar
[11]Paul, E.. Étude topologique des formes logarithmiques fermées. Invent. Math. 95(2) (1989), 395–420.CrossRefGoogle Scholar
[12]Richards, I.. On the classification of noncompact surfaces. Trans. Amer. Math. Soc. 106 (1963), 259–269.Google Scholar
[13]Troubetzkoy, S.. Periodic billiard orbits in right triangles II, Preprint, http://hal.ccsd.cnrs.fr/ccsd-00005014.Google Scholar
[14]Vorobets, Ya. B.. Ergodicity of billiards in polygons: explicit examples. Uspekhi Mat. Nauk 51 ((4) (310)) (1996), 151–152 (Engl. Transl. Russian Math. Surveys 51 (1) (1996), 177–178).Google Scholar
[15]Zemljakov, A. N. and Katok, A. B.. Topological transitivity of billiards in polygons. Mat. Zametki 18(2) (1975), 291–300 (in Russian).Google Scholar