On 4-Reflective Complex Analytic Planar Billiards

Abstract

The famous conjecture of Ivrii (Funct Anal Appl 14(2):98–106, 1980) 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 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 author’s previous result Glutsyuk (Moscow Math J 14(2):239–289, 2014) 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.