Journal of Statistical Physics, Vol. 46, Nos. 3/4, 1987, p. 455-475.
The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map M. This map acts on functions PHI that are jointly analytic in a "position variable" (t) and in the parameter (mu) that controls the period doubling phenomenon. A fixed point PHI* for this map is found. The usual renormalization group doubling operator N acts on this function PHI* simply by multiplication of mu with the universal Feigenbaum ratio delta* =4.669201..., i.e., (NPHI*)(mu, t)= PHI*(delta*mu, t). Therefore, the one-parameter family of functions PSI*_mu, PSI*_mu(t) = PHI*(mu, t), is invariant under N. In particular, the function PSI*_0 is the Feigenbaum fixed point of N, while PSI*_mu represents the unstable manifold of N. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.