Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture

Abstract

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type.This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws.We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and "hairiness" of the Mandelbrot set near the corresponding parameter values.We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1.In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.