Global phase space universality, smooth conjugacies and renormalization. II. The C^{k+ alpha} case using rapid convergence of Markov families

The authors prove that the speed of convergence of two Markov families determines the smoothness of the conjugacy between them. One of the applications of this result that they give is that the attractors of any two quadratic foldings at the Feigenbaum accumulation point of period doubling are C^2+0.11 conjugate. The main result provides the basis for a complete unification of renormalization and smooth conjugacy results which includes both the classical theorems and more recent results about critical systems.