On Odd-periodic Orbits in Complex Planar Billiards

被引:5
|
作者
Glutsyuk, Alexey [1 ,2 ,3 ,4 ]
机构
[1] CNRS, F-75700 Paris, France
[2] ENS Lyon, UMPA, UMR 5669, Lyon, France
[3] Lab JV Poncelet, UMI 2615, Lyon, France
[4] Natl Res Univ Higher Sch Econ HSE, Moscow, Russia
关键词
Real (complex) planar analytic billiard; Periodic orbit; Complex Euclidean metric; Isotropic lines; Complex reflections; Real planar analytic pseudo-billiard; Invisibility; POINTS; BODIES; SET;
D O I
10.1007/s10883-014-9236-5
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
The famous conjecture of V. Ya. Ivrii (1978) 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 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 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 invisibility conjecture.
引用
收藏
页码:293 / 306
页数:14
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