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On odd-periodic orbits in complex planar billiards

Abstract : The 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.
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Submitted on : Friday, January 17, 2020 - 12:54:18 PM
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Alexey Glutsyuk. On odd-periodic orbits in complex planar billiards. Journal of Dynamical and Control Systems, Springer Verlag, 2014, 20 (3), pp.293-306. ⟨10.1007/s10883-014-9236-5⟩. ⟨ensl-02443723⟩

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