A complete proof of the Feigenbaum conjectures

Abstract

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map ℳ. This map acts on functions Φ that are jointly analytic in a "position variable" (t) and in the parameter (μ) that controls the period doubling phenomenon. A fixed point Φ* for this map is found. The usual renormalization group doubling operator N acts on this function Φ* simply by multiplication of μ with the universal Feigenbaum ratio δ*= 4.669201..., i.e., (NΦ*(μ,t)=Φ*(δ*μ,t). Therefore, the one-parameter family of functions, Ψμ*, Ψμ*(t)=(Φ*(μ,t), is invariant under N. In particular, the function Ψ0* is the Feigenbaum fixed point of N, while Ψμ* represents the unstable manifold of N. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally. © 1987 Plenum Publishing Corporation.