Billiards in polygons: Survey of recent results

Abstract

In the ten or so years since the publication of ref. 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature. As a result, our understanding of the subject, although still far from being complete, is much better than it was ten years ago. This survey attempts to give a broad overview of the dynamics of billiards in polygons, with an emphasis on the material that was not in the literature in 1985. This is not a survey of the publications on polygonal billiards, and we apologize to authors whose work has not been included. The selection of topics has been strongly influenced by the personal taste of the author, and by space limitations. Thus, we do not discuss quantum polygonal billiards (in particular, quantum chaos; see, e.g., refs. 41, 44, and 45). We hope that somebody will write a survey on this important subject in the near future. A fascinating aspect of the subject is the interplay between the geometric shape of the billiard table (i.e., a planar curve) and the qualitative features of the billiard dynamics. Hence it is instructive to compare polygonal billiard tables with other classes of billiard tables. In particular, the smooth, strictly convex tables and the dispersing (or Sinai) tables produce strikingly different types of billiard dynamics, We recommend refs. 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables.