Dual polygonal billiards and necklace dynamics
type: Item
instance of: scholarly article
title: Dual polygonal billiards and necklace dynamics
zbMATH Open document ID: 0749.58032
DOI: 10.1007/BF02099259
author: Eugene Gutkin
author: Nándor Simányi
published in: Communications in Mathematical Physics
publication date: 1992-09-26T00:00:00Z
review text: For a closed bounded strictly convex domain \(P\) in \(\mathbb{R}^ 2\) with a \(C^ 1\)-boundary, the associated billiard map is \(T: E\to E\), \(T(O)=r_ A(O)\), where \(E=\mathbb{R}^ 2\setminus P\), for any point \(O\in E\), \(A\) and \(A'\) denote the points of tangency of the two rays \(R\) and \(R'\) emanating from \(O\) and tangent to \(P\) where the observer looking at \(P\) from \(O\) sees \(R\) on the left and \(R'\) on the right of \(P\) and, for any \(v\in\mathbb{R}^ 2\), \(r_ v\) denotes the Euclidean reflection about \(v\). The mapping \(T\) is continuous, preserves the Lebesgue measure and is invertible with \(T^{-1}(O)=r_{A'}(O)\). If \(P\) is not strictly convex, \(T\) and \(T^{-1}\) are defined the same way on \(E\setminus \sigma_ 1\) where \(\sigma_ 1\) is the union of straight lines through the sides of \(P\). By induction on \(n\geq 1\) the authors define \(\sigma_ n\), a finite union of straight lines and \(T^ k\), \(-n\leq k\leq n\) are well defined on \(E\setminus \sigma_ n\). The singular set \(\Sigma=\bigcup^ \infty_{n=1}\sigma_ n\) is a countable union of straight lines and, for a regular point \(x\in E\setminus \Sigma\) the infinite orbits \(\{T^ nx: -\infty < n < \infty\}\) are defined. The purpose of this paper is the orbit behavior for dual billiard maps about a polygonal table by using the technique of necklace dynamics. The main result is that for a certain class of tables, called the quasi- rational polygons, the orbits are bounded. This implies that for the subset of rational tables, i.e. polygons with rational vertices, the dual billiard orbits are periodic.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 26B99
zbMATH DE Number: 54721
zbMATH Keywords: dual polygonal billiards
zbMATH Keywords: orbit behaviour
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
cites work: Billiards on almost integrable polyhedral surfaces
cites work: Billiards in polygons
cites work: Q4014108
cites work: Global stability of a class of discontinuous dual billiards
description: scientific article; zbMATH DE number 54721