On quadrilateral orbits in complex algebraic planar billiards

TL;DR

This paper investigates the complex algebraic analogue of Ivrii's conjecture for quadrilateral billiard orbits, providing a complete classification of algebraic counterexamples with open sets of such orbits.

Contribution

It offers the first full classification of 4-reflective algebraic billiards in the complex projective plane, identifying all counterexamples to the conjecture.

Findings

Complete classification of 4-reflective algebraic billiards

Identification of algebraic counterexamples with open sets of quadrilateral orbits

Advancement in understanding complex algebraic billiard dynamics

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 algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits.