|
satisfying
defines a strictly
convex table T by
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.
