Journal of Statistical Physics

, Volume 46, Issue 3, pp 455–475

A complete proof of the Feigenbaum conjectures

  • Jean-Pierre Eckmann
  • Peter Wittwer

DOI: 10.1007/BF01013368

Cite this article as:
Eckmann, JP. & Wittwer, P. J Stat Phys (1987) 46: 455. doi:10.1007/BF01013368


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.

Key words

Nonlinear functional equationrenormalization groupFeigenbaum phenomenoncomputer-assisted proofrigorous bounds on critical indices

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Jean-Pierre Eckmann
    • 1
  • Peter Wittwer
    • 2
  1. 1.Physique ThéoriqueUniversité de GenéveGeneva 4Switzerland
  2. 2.Department of MathematicsRutgers UniversityNew Brunswick