https://hal-ens-lyon.archives-ouvertes.fr/ensl-02443723Glutsyuk, AlexeyAlexeyGlutsyukUMPA-ENSL - Unité de Mathématiques Pures et Appliquées - ENS Lyon - École normale supérieure - Lyon - CNRS - Centre National de la Recherche ScientifiqueOn odd-periodic orbits in complex planar billiardsHAL CCSD2014[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Glutsyuk, Alexey2020-01-17 12:54:182020-01-23 01:40:242020-01-22 18:56:49enJournal articleshttps://hal-ens-lyon.archives-ouvertes.fr/ensl-02443723/document10.1007/s10883-014-9236-5application/pdf1The famous conjecture of V.Ya.Ivrii (1978) says that in every bil-liard 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 partial classification of k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invis-ibility conjecture.