On Complex Algebraic Caustics in Planar and Projective Billiards

Abstract

A caustic of a billiard is a curve whose tangent lines are reflected to its own tangent lines. A billiard is called Birkhoff caustic-integrable if there exists a topological annulus adjacent to its boundary from inside that is foliated by closed caustics. The famous Birkhoff Conjecture, studied by many mathematicians, states that the only Birkhoff caustic-integrable billiards are ellipses. The conjecture is open even for billiards whose boundaries are ovals of algebraic curves. In this case the billiard is known to have a dense family of so-called rational caustics that are also ovals of algebraic curves. We introduce the notion of a complex caustic: a complex algebraic curve whose complex tangent lines are sent by complexified reflection to its own complex tangent lines. We show that the usual billiard on a real planar curve \(\gamma\) has a complex caustic if and only if \(\gamma\) is a conic. We prove an analogous result for billiards on all the surfaces of constant curvature. These results are corollaries of the solution of S. Bolotin’s polynomial integrability conjecture: a joint result by M. Bialy, A. Mironov and the author. We extend them to the projective billiards introduced by S. Tabachnikov, which are a common generalization of billiards on surfaces of constant curvature. We also deal with a well-known class of projective billiards on conics that are defined to have caustics forming a dual conical pencil. We show that, up to restriction to a finite union of arcs, each of them is equivalent to a billiard on an appropriate surface of constant curvature.