Outer Billiards on Kites

1 Preface Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B.H. Neumann introduced outer billiards in the 1950s, and J. Moser popularized outer billiards in the 1970s as a toy model for celestial mechanics. Outer billiards is an appealing dynamical system because of its simplicity and also because of its connection to such topics as interval exchange maps, piecewise isometric actions, and area-preserving actions. There is a lot left to learn about these kinds of dynamical systems, and a good understanding of outer billiards might shed light on the more general situation. The Moser-Neumann question, one of the central problems in this subject, asks Does there exist an outer billiards system with an unbounded orbit? Until recently, all the work on this subject has been devoted to proving that all the orbits are bounded for various classes of shapes. We will detail these results in the introduction. Recently we answered the Moser-Neumann question in the affirmative by showing that outer billiards has an unbounded orbit when defined relative to the Penrose kite, the convex quadrilateral that arises in the famous Penrose tiling. Our proof involves special properties of the Penrose kite, and naturally raises questions about generalizations. In this monograph we will give a more general and robust answer to the Moser-Neumann question. We will prove that outer billiards has unbounded orbits when defined relative to any irrational kite. A kite is probably best defined as a " kite-shaped " quadrilateral. (See the top of §1.2 for a non-circular definition.) The kite is irrational if it is not affinely equivalent to a quadrilateral with rational vertices. Our proof uncovers some of the deep structure underlying outer billiards on kites, and relates the subject to such topics as self-similar tilings, polytope exchange maps, and the modular group. I discovered every result in this monograph by experimenting with my computer program, Billiard King, a Java-based graphical user interface. For the most part, the material here is logically independent from Billiard King, but I encourage the serious reader of this monograph to download Billiard King from my website 1 and play with it. My website also has an interactive guide to this monograph, in which many of the basic ideas and constructions are illustrated with interactive Java applets. 2 There are a number of people I would like to thank. I especially thank Sergei Tabachnikov, …