On rationally integrable planar dual and projective billiards

Inventiones Mathematicae. 2026. Vol. 245. P. 977–1058.

Glutsyuk A.

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study its following generalized dual version stated by S. Tabachnikov. Consider a closed smooth strictly convex curve γ ⊂ ℝℙ2 equipped with a dual billiard structure: a family of non-trivial projective involutions acting on its projective tangent lines and fixing the tangency points. Suppose that its outer neighborhood admits a foliation by closed curves including γ such that the involution of each tangent line permutes its intersection points with every curve. Then the curves are conics forming a pencil. We prove positive answer in the case, when the curve γ is C4-smooth and the foliation admits a rational first integral. In this case the dual billiard is rationally integrable, which means that there exists a non-constant rational function, namely the integral, whose restriction to each tangent line is invariant under its involution. To this end, we show that each C4-smooth germ of curve carrying a rationally integrable dual billiard structure is a conic, and the dual billiard structure extends holomorphically with isolated singularities to the complexified conic. We classify rationally integrable dual billiards on conic with isolated singularities. They include the dual billiards induced by pencils of conics and infinitely many exotic examples. We deduce analogous results for projective billiards via duality.

Keywords: caustic, каустика, rational integral, рациональный интеграл, pencil of conics, projective billiard, проективный бильярд, пучок коник, двойственный бильярд, dual billiard