  We consider infinitely renormalizable unimodal mappings with topological type
which is periodic under renormalization. We study the limiting behavior of
fixed points of the renormalization operator as the order of the critical point
increases to infinity. It is shown that a limiting dynamics exists, with a
critical point that is flat, but still having a well-behaved analytic
continuation to a neighborhood of the real interval pinched at the critical
point. We study the dynamics of limiting maps and prove their rigidity. In
particular, the sequence of fixed points of renormalization for finite
criticalities converges, uniformly on the real domain, to a mapping of the
limiting type.
