On 4-reflective complex analytic planar billiards
Abstract
The famous conjecture of V.Ya.Ivrii 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 its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar -smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.
Cite
@article{arxiv.1405.5990,
title = {On 4-reflective complex analytic planar billiards},
author = {Alexey Glutsyuk},
journal= {arXiv preprint arXiv:1405.5990},
year = {2015}
}
Comments
To appear in Journal of Geometric Analysis. 69 pages, 14 figures. New changes: simplifying and strengthening the last proposition in Subsection 4.2; generalizing and simplifying definitions of billiard combinations in Section 6; updating list of known k-reflective billiards in Section 6; new bibliographic references; minor polishing of some parts