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Article

On quadrilateral orbits in complex algebraic planar billiards

Moscow Mathematical Journal. 2014. Vol. 14. No. 2. P. 239-289.

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 counterexam- ples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we pro- vide 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 re- flecting tangent line.