Journal of Statistical Physics

, Volume 46, Issue 3, pp 455-475

First online:

A complete proof of the Feigenbaum conjectures

  • Jean-Pierre EckmannAffiliated withPhysique Théorique, Université de Genéve
  • , Peter WittwerAffiliated withDepartment of Mathematics, Rutgers University

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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.

Key words

Nonlinear functional equation renormalization group Feigenbaum phenomenon computer-assisted proof rigorous bounds on critical indices