On 4-reflective complex analytic planar billiards

Abstract

The famous conjecture of V.Ya.Ivrii 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 its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar $C^4$-smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise $C^4$-smooth). We provide a survey and a small technical result concerning higher number of complex reflections.