On odd-periodic orbits in complex planar billiards
Résumé
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.
Domaines
Systèmes dynamiques [math.DS]Origine | Fichiers produits par l'(les) auteur(s) |
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