On odd-periodic orbits in complex planar billiards

TL;DR

This paper investigates the complex analogue of Ivrii's conjecture for odd-periodic billiard orbits, proving results for triangular orbits and certain algebraic curves, with implications for real billiards and invisibility theory.

Contribution

It establishes new results for odd-periodic orbits in complex planar billiards, especially for triangular orbits and algebraic curves avoiding isotropic points.

Findings

Proved positive results for triangular orbits.

Established conditions under which odd-periodic orbits exist.

Applied findings to real billiard and invisibility problems.

Abstract

The famous conjecture of V.Ya.Ivrii (1978) 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 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 real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory.