Appendix: Is there a nonintegrable billiard with an algebraic return map?

For the illustrations in Figure 3 and Figure 4, we constructed the map by finding so that is real. This problem is equivalent to solving

Numerically, the second root of this equation can be found efficiently with a Newton method. (For three-dimensional billiards, the situation is similar but slightly more complicated). In the simplest case , one has to solve

for . We do not know, whether the second root is an algebraic expression in and . One has the problem to find roots of a specific class of polynomials of degree eight. More generally, one can ask, whether there exists a table of constant width different from the circle, for which the return map is algebraic in some coordinates. We do not know of any smooth convex billiard table different from an ellipse, where the return map is an algebraic expression in some coordinates. This question is related to the open Birkhoff-Poritsky conjecture [20] which claims that the ellipse is the only integrable smooth convex billiard. In [7] it was suggested that curves of equal thickness might provide counter examples to the Birkhoff-Poritsky conjecture.

Acknowledgments. I want to thank E. Gutkin for discussions on billiards and caustics, P. Gruber for literature references, M. Wojtkowski for useful remarks on an earlier version of the manuscript and E. Amiran for helpful suggestions on the final draft. A first version of this paper was written at Caltech