On rationally integrable planar dual and projective billiards

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 present paper we study its following generalized dual-projective version stated by S.Tabachnikov. Consider a closed smooth strictly convex planar curve γ equipped with a dual billiard structure: a family of 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 leaf. Tabachnikov's Conjecture states that then the curve γ is an ellipse and the leaves are ellipses forming a pencil. We prove Tabachnikov's Conjecture in the case, when the curve is C 4 -smooth and the foliation admits a rational first integral. To this end, we show that each C 4 -smooth germ γ of planar curve carrying a rationally integrable dual billiard structure is a conic and classify all the rationally integrable dual billiards on conics. They include the dual billiards induced by pencils of conics, two infinite series of exotic dual billiards and five more exotic ones.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

“…We prove bijectivity of the correspondence (given in remark 1.12) between rational integrals of a multibilliard and rational 0-homogeneous integrals of the flow of its dual projective billiard. This together with the results from [30] on duality between exotic dual billiards from theorem 1.17 and exotic projective billiards from theorem 1.20 and the results of the present paper on dual multibilliards will imply the main results on projective billiards.…”

“…The generalization of the Birkhoff Conjecture to dual billiards was stated by Tabachnikov in [40]. Its rationally integrable version was solved by the author of the present paper in [30]. Its polynomially integrable version for outer billiards was stated and partially studied in [40] and solved completely in [26].…”

“…It was shown there that -the underlying curve is always a conic; -besides example 1.6 of dual pencil type projective billiard, which has a quadratic integral, there are infinitely many other rationally 0-homogeneously integrable projective billiard structures on a conic, called exotic structures, with minimal degree of integral being arbitrary even number greater than 3. These results of [30] are recalled in section 1.3.…”

Complex Caustics of the Elliptic Billiard

“…Billiards in pseudo-Euclidean spaces and Minkowski metrics have been intensively studied by Dragović and Radnović [1,5,6,7]. More generally, projective billiards have been studied in any dimensions [12,9,11,10,18,19,28,26]. The results which can be found here extend previously known results [12] found by the author.…”

“…The conjecture is still open, although some partial positive answers have been given, see for example [3,16,17,18,19,21,23].…”