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On curves with Poritsky property

Alexey Glutsyuk 1
1 UMPA
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
Abstract : For a given closed convex planar curve γ with smooth boundary and a given p > 0, the string construction is obtained by putting a string surrounding γ of length p + |γ| to the plane. Then we pull some point of the string "outwards from γ" until its final position A, when the string becomes stretched completely. The set of all the points A thus obtained is a planar convex curve Γ p. The billiard reflection T p from the curve Γ p acts on oriented lines, and γ is a caustic for Γ p : that is, the family of lines tangent to γ is T p-invariant. The action of the reflection T p on the tangent lines to γ ≃ S 1 induces its action on the tangency points: a circle diffeomorphism T p : γ → γ. We say that γ has string Poritsky property, if it admits a parameter t (called Poritsky-Lazutkin string length) in which all the transformations T p are translations t → t + c p. These definitions also make sense for germs of curves γ. Poritsky property is closely related to the famous Birkhoff Conjecture. It is classically known that each conic has string Poritsky property. In 1950 H.Poritsky proved the converse: each germ of planar curve with Poritsky property is a conic. In the present paper we extend this Poritsky's result to germs of curves to all the simply connected complete Riemannian surfaces of constant curvature and to outer billiards on all these surfaces. We also consider the general case of curves with Poritsky property on any two-dimensional surface with Riemannian metric and prove a formula for the derivative of the Poritsky-Lazutkin length as a function of the natural length parameter. In this general setting we also prove the following uniqueness result: a germ of curve with Poritsky property is uniquely determined by its 4-th jet. In the Euclidean case this statement follows from the above-mentioned Poritsky’s result.
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Submitted on : Thursday, November 21, 2019 - 5:56:08 PM
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  • HAL Id : ensl-02374972, version 1
  • ARXIV : 1901.01881

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Alexey Glutsyuk. On curves with Poritsky property. 2019. ⟨ensl-02374972⟩

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