Journal of Statistical Physics

, Volume 46, Issue 3–4, pp 455–475

A complete proof of the Feigenbaum conjectures

  • Jean-Pierre Eckmann
  • Peter Wittwer
Articles

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., (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 equation renormalization group Feigenbaum phenomenon computer-assisted proof rigorous bounds on critical indices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations,J. Stat. Phys. 19:25–52 (1978);21:669–706 (1978).Google Scholar
  2. 2.
    P. Collet, J.-P. Eckmann, and O. E. Lanford III, Universal properties of maps on the interval,Commun. Math. Phys. 76:211–254 (1980).Google Scholar
  3. 3.
    E. B. Vul and K. M. Khanin, The unstable separatrix of Feigenbaum's fixed-point,Russ. Math. Surveys 37(5):200–201 (1982).Google Scholar
  4. 4.
    E. B. Vul, Ya. G. Sinai, and K. M. Khanin, Feigenbaum universality and the thermodynamic formalism,Russ. Math. Surveys 39(3):1–40 (1984).Google Scholar
  5. 5.
    J.-P. Eckmann, A. Malaspinas, and S. Oliffson Kamphorst, to be published.Google Scholar
  6. 6.
    O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures,Bull. AMS N. S.6:127 (1984).Google Scholar
  7. 7.
    H. Koch and P. Wittwer, A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories,Commun. Math. Phys., to appear.Google Scholar
  8. 8.
    J.-P. Eckmann, H. Koch, and P. Wittwer, A computer-assisted proof of universality for area-preserving maps,Mem. AMS 47:289 (1984).Google Scholar
  9. 9.
    J.-P. Eckmann and P. Wittwer, Computer methods and Borel summability applied to Feigenbaum's equation,Lecture Notes in Physics (Springer-Verlag, Berlin, 1985).Google Scholar
  10. 10.
    R. E. Moore,Interval Analysis (Prentice-Hall, 1966).Google Scholar
  11. 11.
    R. E. Moore,Methods and Applications of Interval Analysis (SIAM, Philadelphia, 1979).Google Scholar
  12. 12.
    R. de la Llave and O. E. Lanford III, to be published.Google Scholar
  13. 13.
    D. Stevenson, IEEE Computer Society. A proposed standard for binary floating-point arithmetic, Draft 8.0 of IEEE Task P754, Computer, 51–62 (March 1981).Google Scholar

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

Personalised recommendations