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On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

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Alexey Glutsyuk

Abstract

A planar projective billiard is a planar curve $C$ equipped with a transversal line field. It defines reflection of lines from $C$. Its projective dual is a dual billiard: a curve $\gamma\subset\mathbb{RP}^2$ equipped with a family of non-trivial projective involutions acting on its projective tangent lines and fixing the tangency points. Projective and dual billiards were introduced by S.Tabachnikov. He stated the following conjecture generalizing the famous Birkhoff Conjecture on integrable billiards to dual and projective billiards. Let a dual billiard $\gamma$ be strictly convex and closed, and let its outer neighborhood admit a foliation by closed curves (including $\gamma$) such that the involution of each tangent line to $\gamma$ permutes its intersection points with every leaf. Then $\gamma$ and the leaves are conics forming a pencil. In a recent paper the author proved this conjecture under the rational integrability assumption: existence of a non-constant rational function (integral) whose restriction to tangent lines is invariant under their involutions. He has also shown that if $\gamma$ is not closed, then it is still a conic, but the dual billiard structure needs not be defined by a pencil. He classified all the rationally integrable dual billiard structures (with singularities) on conic. In the present paper we give classification of rationally integrable dual multibilliards: collections of dual billiards and points $Q_j$ (called vertices) equipped with a family of projective involutions acting on lines through $Q_j$ from an open subset in $\mathbb{RP}^1$. As an application, we get classification of piecewise smooth projective billiards whose billiard flow has a non-constant first integral that is a rational $0$-homogeneous function of the velocity.
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ensl-03925877 , version 1 (05-01-2023)

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Alexey Glutsyuk. On rationally integrable planar dual multibilliards and piecewise smooth projective billiards. 2023. ⟨ensl-03925877⟩
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