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Pré-Publication, Document De Travail Année : 2022

On rationally integrable planar dual and projective billiards

Alexey Glutsyuk

Résumé

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 C4-smooth and the foliation admits a rational first integral. To this end, we consider an arbitrary C4-smooth germ γ of planar curve carrying a rationally integrable dual billiard structure: this means that there exists a rational function whose restriction to each line tangent to γ is invariant under the corresponding projective involution. We show that then the curve γ is always 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.
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Dates et versions

hal-03476328 , version 1 (12-12-2021)
hal-03476328 , version 2 (14-11-2022)

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Alexey Glutsyuk. On rationally integrable planar dual and projective billiards. 2022. ⟨hal-03476328v2⟩
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