On quadrilateral orbits in complex algebraic planar billiards

2013/09/07 by Glutsyuk, Alexey · 1 citation
#14E15 #37C25 #37F05 #51N15 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.1843

Abstract

The famous conjecture of V.Ya.Ivrii (1978) says that \it 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.

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