Commuting dual billiard maps
type: Item
instance of: scholarly article
title: Commuting dual billiard maps
zbMATH Open document ID: 0813.52003
DOI: 10.1007/BF01264044
published in: Geometriae Dedicata
publication date: 1995-05-29T00:00:00Z
review text: The dual billiard map \(T\) is a map of the exterior \(E(c)\) of a smooth strictly convex closed curve \(c\) in the Euclidean (or, actually: affine) plane into itself, which is defined as follows: given a point \(x\in E(c)\), draw the right (from the view-point of \(x\)) tangent line to \(c\) through it and reflect \(x\) in the point of tangency to obtain its image point \(T(x)\). The author shows that \(T\) is area-preserving and proves the Theorem: Let \(c_ 1\), \(c_ 2\) be smooth strictly convex distinct closed plane curves; then the corresponding dual billiard maps \(T_ 1\), \(T_ 2\) commute: \(T_ 1T_ 2= T_ 2 T_ 1\) (in the domain, where \(T_ 1 T_ 2\) and \(T_ 2 T_ 1\) are defined) if and only if \(c_ 1\), \(c_ 2\) are concentric homothetic ellipses. (To prove that \(c_ 1\), \(c_ 2\) are ellipses, the author shows that these curves have constant affine curvature).
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 51N10
zbMATH DE Number: 701143
zbMATH Keywords: closed convex smooth curve
zbMATH Keywords: affine plane
zbMATH Keywords: dual billiard map
zbMATH Keywords: homothetic ellipses
zbMATH Keywords: concentric
author: S. Tabachnikov
MaRDI profile type: Publication
cites work: Stable and Random Motions in Dynamical Systems
cites work: On the dual billiard problem
cites work: Freund-Witten adelic formulae for Veneziano and Virasoro-Shapiro amplitudes
cites work: Poncelet's theorem and dual billiards
cites work: Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space
cites work: Integrable maps
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description: scientific article; zbMATH DE number 701143