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Bounds, quadratic differentials, and renormalization conjectures. (English) Zbl 0936.37016
Browder, Felix E. (ed.), Mathematics into the twenty-first century. Proceedings of the AMS centennial symposium, Providence, RI, USA, August 8-12, 1988. Providence, RI: American Mathematical Society. Am. Math. Soc. Centen. Publ. 2, 417-466 (1992).
The description renormalization comes from statistical physics when a cascade of period doubling bifurcations in any complete family was analyzed. The physicists Feigenbaum and Coullet with Tresser, working numerically and independently, found universal numerical characteristics about this cascade and the limit geometry of the \((\tau, \tau,\dots)\) orbital Cantor set. These characteristics were universal in the sense that they were computed numerically to be independent of the choice of complete family of mappings with quadratic turnings. The paper gives a survey of modern (at that time) results on the dynamic of bounded time renormalization on smooth quadratic-like mappings. In particular, the author proves that if \[ \dots\to f_{n} \to f_{n-1}\to \dots \to f_{2}\to f_{1}\to f_{0} \] be any inverse chain of bounded smooth quadratic-like mappings related by renormalizations, \(R(\sigma_{n})[f_{n}]=f_{n-1}\), where degree of shuffle permutation \(\sigma_{n}\) is bounded then the mapping \(f_{n}\) are analytic functions and determined uniquely by the combinatorics \(\sigma_{i}\) and a real number \(c\in [-2,1/4]\). The real number \(c\) is determined by the complex analytic extension of \(f_{0}\) and is the unique element of \([-2,1/4]\) such that \(f_{0}\) is conjugate to \(z\to z^{2}+c\) on a neighborhood of some invariant set. Besides information about complex analytic structure of renormalization limits is obtained.
For the entire collection see [Zbl 0921.00016].

37E20 Universality and renormalization of dynamical systems
37F25 Renormalization of holomorphic dynamical systems
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30F60 Teichmüller theory for Riemann surfaces