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Moscow Mathematical Journal

Volume 14, Issue 2, April–June 2014  pp. 239–289.

On Quadrilateral Orbits in Complex Algebraic Planar Billiards

Authors:  Alexey Glutsyuk
Author institution:Permanent address: CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon 07, France,
Laboratoire J.-V. Poncelet (UMI 2615 du CNRS and the Independent University of Moscow)
National Research University Higher School of Economics, Russia


The famous conjecture of V. Ya. Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii’s conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we provide classification of the so-called real algebraic pseudo-billiards with open set of quadrilateral orbits: billiards formed by four real algebraic curves; the reflections allow to change the side with respect to the reflecting tangent line.

2010 Mathematics Subject Classification. 37C25, 37F05, 51N15, 14E15.

Keywords: Billiard, periodic orbit, complex algebraic curve, complex reflection law, complex Euclidean metric, isotropic line, complex confocal conics, birational transformation.

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