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Working paper

On 4-relfective complex analytic planar billiards

arxiv.org. math. Cornell University, 2014. No. 1405.5990.
The famous conjecture of V.Ya.Ivrii  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 analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the   author's result  classifying 4-reflective planar algebraic counterexamples. We provide applications to real billiards: classification of 4-reflective real planar analytic pseudo-billiards; solution of the piecewise-analytic case of Tabachnikov's commuting planar billiard problem; solution of a particular case of Plakhov's Invisibility Conjecture. In particular, we retrieve the solution of Ivrii's Conjecture for quadrilateral orbits in planar billiards  in piecewise-analytic case previously obtained in a joint paper of the author with Yu.Kudryashov.