On rationally integrable planar dual and projective billiards

TL;DR

This paper investigates a generalized dual billiard problem on convex curves in the projective plane, proving that under certain smoothness and integrability conditions, the curves are conics and classifying all such integrable structures.

Contribution

It extends the classical billiard conjecture to a dual projective setting, proving conic classification under rational integrability assumptions and identifying all such structures.

Findings

Curves with rationally integrable dual billiard structures are conics.

Classified all rationally integrable dual billiards on conics.

Identified exotic dual billiard examples beyond pencils of conics.

Abstract

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 states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others. In the paper we study its following generalized dual version stated by S.Tabachnikov. Consider a closed smooth strictly convex curve $\gamma\subset\mathbb{RP}^2$ 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 $\gamma$) such that the involution of each tangent line permutes its intersection points with every leaf. Then $\gamma$ and the leaves are conics forming a pencil. We prove positive answer in the case, when the curve $\gamma$ is $C^4$-smooth and the foliation admits a rational first integral. To this end, we show that each $C^4$-smooth germ $\gamma$ of planar curve carrying a rationally integrable dual billiard structure is a conic and classify all the rationally integrable dual billiards on (punctured) conic. They include the dual billiards induced by pencils of conics, two infinite series of exotic dual billiards and five more exotic ones.