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

For the illustrations in Figure 3 and Figure 4, we constructed the map tex2html_wrap_inline1648 by finding tex2html_wrap_inline1650 so that tex2html_wrap_inline1652 is real. This problem is equivalent to solving

displaymath1654

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 tex2html_wrap_inline1656 , one has to solve

displaymath1658

for tex2html_wrap_inline1660 . We do not know, whether the second root tex2html_wrap_inline1662 is an algebraic expression in tex2html_wrap_inline1158 and tex2html_wrap_inline864 . 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 tex2html_wrap_inline894 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



Oliver Knill
Fri Jun 12 13:34:37 CDT 1998