On the Hyperbolicity of the Period-doubling Fixed Point

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: such proof uses the non existence of invariant line fields in the perioddoubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.