Dual billiard

The first volume of the Mathematical Intelligencer contains an article by Jurgen Moser “Is the solar system stable?” [23]. As a toy model for planetary motion, Moser proposed the system illustrated in figure 1 and called the dual (or outer) billiard. The dual billiard table P is a planar oval. Choose a point x outside P . There are two tangent lines from x to P ; choose one of them, say, the right one from x’s view-point, and reflect x in the tangency point z. One obtains a new point, y, and the transformation T : x 7→ y is the dual billiard map. Like the planetary motions, the dual billiard dynamics is easy to define but hard to analyze, in particular, it is difficult to reach conclusions about its global properties, such as boundedness or unboundedness of orbits. In this article we survey results on the dual billiard problem obtained since the publication of Moser’s article. We hope that the reader will share our fascination with this beautiful subject. We do not assume familiarity with a much better studied subject of the conventional, inner billiards; an interested reader is referred to [12, 16, 27]. The definition of the dual billiard map has a shortcoming: T is not defined if the tangency point z is not unique. This is the case if the dual billiard curve γ, the boundary of P , contains a straight segment, for example, if γ is a polygon. The