Billiard in a triangle and quadratic homogeneous foliations on $ {\mathbf{C}}^{2}$

Ferrán Valdez

doi:10.1016/j.crma.2008.01.013

Dynamical Systems

Billiard in a triangle and quadratic homogeneous foliations on

C^{2}

Presented by: Étienne Ghys

Ferrán Valdez ¹

¹ IRMAR, Université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 317-322.

Abstract
Résumé

This Note is partly an announcement. We describe the geodesic foliation of the translation surface associated to a triangular billiard in terms of real and holomorphic foliations defined by quadratic homogeneous vector fields on $C^{2}$ . Our technique allows, in particular, to state the existence of real foliations by curves on $C^{2}$ arising from homogeneous quadratic vector fields and having a dense set of periodic orbits.

Cette Note est partiellement une annonce. Nous décrivons le feuilletage géodésique de la surface de translation associée au billiard sur un triangle en termes de feuilletages réels et holomorphes définis par des champs de vecteurs quadratiques homogènes de $C^{2}$ . Nos techniques nous permettent, entre autres, de constater l'existence des feuilletages réels en courbes de $C^{2}$ définis par des champs de vecteurs homogènes, quadratiques et pour lesquels les orbites périodiques sont denses.

Received: 2007-07-12
Accepted: 2008-01-07
Published online: 2008-03-01

DOI: 10.1016/j.crma.2008.01.013

Author's affiliations:

Ferrán Valdez ¹

¹ IRMAR, Université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France

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     title = {Billiard in a triangle and quadratic homogeneous foliations on $ {\mathbf{C}}^{2}$},
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Ferrán Valdez. Billiard in a triangle and quadratic homogeneous foliations on $ {\mathbf{C}}^{2}$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 317-322. doi : 10.1016/j.crma.2008.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.013/

References
Cited by

[1] M. Boshernitzan; G. Galperin; T. Krüger; S. Troubetzkoy Periodic billiard orbits are dense in rational polygons, Trans. Amer. Math. Soc., Volume 350 (1998) no. 9, pp. 3523-3535

[2] T.A. Driscoll; L.N. Trefethen Schwarz–Christoffel Mapping, Cambridge Monographs on Applied and Computational Mathematics, vol. 8, Cambridge University Press, Cambridge, 2002

[3] H. Masur; S. Tabashnikov Rational billiards and flat structures, Handbook of Dynamical Systems, vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089

[4] S. Troubetzkoy Periodic billiard orbits in right triangles http://hal.ccsd.cnrs.fr/ccsd-00005014 (preprint)

[5] A.N. Zemljakov; A.B. Katok Topological transitivity of billiards in polygons, Mat. Zametki, Volume 18 (1975) no. 2, pp. 291-300

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