On 4-reflective complex analytic planar billiards
Résumé
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 C4-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
C4-smooth).We provide a survey and a small technical result concerning higher
number of complex reflections.
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