On curves with the Poritsky property

Journal of Fixed Point Theory and Applications. 2022. Vol. 24. No. 2. Article 35.

Glutsyuk A.

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism Tp:γ→γ. We say that γ has string Poritsky property, if it admits a parameter t (called Poritsky--Lazutkin string length parameter) in which all the transformations Tp are translations t↦t+cp. 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 surfaces with Riemannian metric 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.

Keywords: integrability, интегрируемость, бильярды, billiards, Poritsky property, Риманова метрика, symplectomorphisms, Riemannian metric, свойство Порицкого, симплектоморфизмы