2013/09/07 by Alexey Glutsyuk, Glutsyuk, Alexey · 1 citation
Mathematics · #14N05 #37C25 #37F05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1309.1849
openalex publication_date 2013/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory.