Oscar Lanford

Oscar Eramus Lanford III (January 6, 1940 – November 16, 2013) was an American mathematician working on mathematical physics and dynamical systems theory.

Born in New York, Lanford was awarded his undergraduate degree from Wesleyan University and the Ph.D. from Princeton University in 1966 under the supervision of Arthur Wightman. He has served as a professor of mathematics at the University of California, Berkeley, and a professor of physics at the Institut des Hautes Études Scientifiques (IHES) in Bures-sur-Yvette, France. As of 1987^[update], he has been with the department of mathematics, Swiss Federal Institute of Technology Zürich (ETH Zürich).

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 ${\displaystyle f(x)=cx(1-x)}$ of maps on the interval [0,1] for example would lead to the same asymptotic law of the ratio of the differences ${\displaystyle b(n)=a(n+1)-a(n)}$ between the bifurcation values a(n) than ${\displaystyle f(x)=c\sin(\pi x)}$. The result is that ${\displaystyle \lim _{n\to \infty }b(n)/b(n+1)}$ converges to the Feigenbaum constants ${\displaystyle d=4.6692016091029...}$ which is a "universal number" independent of the map f. The bifurcation diagram has become an icon of chaos theory.

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