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 functionsΦ that are jointly analytic in a “position variable” (t) and in the parameter (μ) that controls the period doubling phenomenon. A fixed pointΦ ^* for this map is found. The usual renormalization group doubling operatorN acts on this functionΦ ^* simply by multiplication ofμ with the universal Feigenbaum ratioδ ^*= 4.669201..., i.e., (N Φ ^*(μ,t)=Φ ^*(δ ^* μ,t). Therefore, the one-parameter family of functions,Ψ ^* _μ ,Ψ ^* _μ (t)=(Φ ^*(μ,t), is invariant underN. In particular, the functionΨ ^*₀ is the Feigenbaum fixed point ofN, whileΨ ^* _μ represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.