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Caustics of convex billiards

A bounded measurable positive function tex2html_wrap_inline1012 satisfying tex2html_wrap_inline1014 defines a strictly convex table T by


tex2html_wrap_inline1018 is the radius of curvature at the point T(s).

Douady remarked that a k-times differentiable invariant curve tex2html_wrap_inline878 of a (k+1)-times differentiable table defines a (k-1)-times differentiable caustic [8]. In order to reformulate this result also for intermediate degrees of smoothness, it is convenient to pick tex2html_wrap_inline1030 in one of the Hilbert spaces


which are all contained in the Hilbert space tex2html_wrap_inline1034 of square integrable periodic functions. For tex2html_wrap_inline1036 , a function in tex2html_wrap_inline1038 has a uniformly convergent Fourier series tex2html_wrap_inline1040 . Moreover, if tex2html_wrap_inline1042 with tex2html_wrap_inline1044 , then tex2html_wrap_inline1046 is r times continuously differentiable. We use here tex2html_wrap_inline1050 mainly for notational reason in order to control with a single parameter tex2html_wrap_inline1052 both nowhere-differentiable and smooth situations. Not much is lost for a reader who decides to ignore the spaces tex2html_wrap_inline1050 .



In [14] the authors ask how to define, in general, a caustic of a convex two times differentiable table whose invariant curve is only Lipschitz. The equation in Lemma 3.1 does this for a table with invariant circle tex2html_wrap_inline878 and bounded measurable curvature function tex2html_wrap_inline1138 . Because the derivative tex2html_wrap_inline1140 is also a bounded measurable function by Birkhoff's theorem (for a proof see [16] p.430), the image of the function tex2html_wrap_inline1142 is in general a measurable set in the complex plane.

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