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Four equivalent properties of integrable billiards

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Abstract

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net.

Let us say that a strictly geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses.

In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke’s derivation of the Liouville property from the Ivory property and his proof of Weihnacht’s theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations.

This suggests the following generalization of Birkhoff’s conjecture: If an interior neighborhood of a closed strictly geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.

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Acknowledgement

We are grateful to D. Burago for a consultation.

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Correspondence to Ivan Izmestiev.

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The author is partially supported by Laboratory of Dynamical Systems and Applications, HSE University, of the Ministry of science and higher education of the RF grant ag. No 075-15-2019-1931 and by RFBR and JSPS (research project 19-51-50005).

Supported by SNCF grants 200021_169391 and 200021_179133.

Supported by NSF grant DMS-1510055.

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Glutsyuk, A., Izmestiev, I. & Tabachnikov, S. Four equivalent properties of integrable billiards. Isr. J. Math. 241, 693–719 (2021). https://doi.org/10.1007/s11856-021-2110-8

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  • DOI: https://doi.org/10.1007/s11856-021-2110-8

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