Dual polygonal billiards and necklace dynamics
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.
scientific article; zbMATH DE number 54721