Nonlinear functional equation
rigorous bounds on critical indices
Springer Online Journal Archives 1860-2000
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.
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