Proof of The Rigidity Conjectures
Lanford gave the first proof that the Feigenbaum-Cvitanovic functional equation has an even analytic solution g and that this fixed point g of the Feigenbaum renormalisation operator T is hyperbolic with a one dimensional unstable manifold.This provided the first mathematical proof of the rigidity conjectures of Feigenbaum. The proof was computer assisted. The hyperbolicity of the fixed point is essential to explain the Feigenbaum universality observed experimentally by Mitchell Feigenbaum and Coullet-Tresser. Feigenbaum has studied the logistic family and looked at the sequence of Period doubling bifurcations. Amazingly the asymptotic behavior near the accumulation point appeared universal in the sense that the same numerical values would appear. The logistic family of maps on the interval for example would lead to the same asymptotic law of the ratio of the differences between the bifurcation values a(n) than . The result is that converges to the Feigenbaum constants which is a "universal number" independent of the map f. The bifurcation diagram has become an icon of chaos theory. Campanino and Epstein also gave a proof of the fixed point without computer assistance but did not establish its hyperbolicity. They cite in their paper Lanfords computer assisted proof. There are also lecture notes of Lanford from 1979 in Zurich and announcements in 1980. The hyperbolicity is essential to verify the picture discovered numerically by Feigenbaum and independently by Coullet and Tresser. Lanford later gave a shorter proof using the Leray-Schauder fixed point theorem but establishing only the fixed point without the hyperbolicity. Lyubich published in 1999 the first not computer assisted proof which also establishes hyperbolicity. Work of Sullivan later showed that the fixed point is unique in the class of real valued quadratic like germs.
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