## |

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

Douady remarked that a *k*-times differentiable invariant curve
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 in one of the Hilbert spaces

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

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 and bounded measurable curvature function . Because the derivative is also a bounded measurable function by Birkhoff's theorem (for a proof see [16] p.430), the image of the function is in general a measurable set in the complex plane.

Fri Jun 12 13:34:37 CDT 1998