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 . 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  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  p.430), the image of the function is in general a measurable set in the complex plane.