On 4-Reflective Complex Analytic Planar Billiards

Abstract

The famous conjecture of V.Ya.Ivrii [14] 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 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 [7] classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C 4 -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 C 4 -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

On the Circumcenters of Triangular Orbits in Elliptic Billiard

“…Complex reflection law and complex planar billiards were introduced and studied by A. Glutsyuk in [6] and [7]. See also [8] where they were applied to solve the two-dimensional Tabachnikov's Commuting Billiard conjecture and a particular case of two-dimensional Plakhov's Invisibility conjecture with four reflections.…”

Billiards in confocal quadrics as a Pluri-Lagrangian system

“…A conjecture byTabachnikov [6] that the commutation of billiard maps characterizes confocal quadrics has been settled, under certain assumptions, in[1].…”

Open problems, questions and challenges in finite- dimensional integrable systems

“…These transformations commute; see [91]. The converse was recently proved by Glutsyuk [113], as a consequence of his classification of 4-reflective complex planar billiards. For outer billiards, an analogous theorem is that if the outer billiard transformations on two nested curves commute, then the curves are concentric homothetic ellipses [114].…”