A complete proof of the Feigenbaum conjectures
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations,J. Stat. Phys. 19:25–52 (1978);21:669–706 (1978).
- P. Collet, J.-P. Eckmann, and O. E. Lanford III, Universal properties of maps on the interval,Commun. Math. Phys. 76:211–254 (1980).
- E. B. Vul and K. M. Khanin, The unstable separatrix of Feigenbaum's fixed-point,Russ. Math. Surveys 37(5):200–201 (1982).
- E. B. Vul, Ya. G. Sinai, and K. M. Khanin, Feigenbaum universality and the thermodynamic formalism,Russ. Math. Surveys 39(3):1–40 (1984).
- J.-P. Eckmann, A. Malaspinas, and S. Oliffson Kamphorst, to be published.
- O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures,Bull. AMS N. S.6:127 (1984).
- H. Koch and P. Wittwer, A non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories,Commun. Math. Phys., to appear.
- J.-P. Eckmann, H. Koch, and P. Wittwer, A computer-assisted proof of universality for area-preserving maps,Mem. AMS 47:289 (1984).
- J.-P. Eckmann and P. Wittwer, Computer methods and Borel summability applied to Feigenbaum's equation,Lecture Notes in Physics (Springer-Verlag, Berlin, 1985).
- R. E. Moore,Interval Analysis (Prentice-Hall, 1966).
- R. E. Moore,Methods and Applications of Interval Analysis (SIAM, Philadelphia, 1979).
- R. de la Llave and O. E. Lanford III, to be published.
- 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).
- A complete proof of the Feigenbaum conjectures
Journal of Statistical Physics
Volume 46, Issue 3-4 , pp 455-475
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Nonlinear functional equation
- renormalization group
- Feigenbaum phenomenon
- computer-assisted proof
- rigorous bounds on critical indices
- Industry Sectors