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., (NΦ*(μ,t)=Φ*(δ*μ,t). Therefore, the one-parameter family of functions,Ψμ*,Ψμ*(t)=(Φ*(μ,t), is invariant underN. In particular, the functionΨ0* is the Feigenbaum fixed point ofN, whileΨμ* represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.
Nonlinear functional equation renormalization group Feigenbaum phenomenon computer-assisted proof rigorous bounds on critical indices