Commuting dual billiard maps
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).
scientific article; zbMATH DE number 701143
scientific article