zbMATH — the first resource for mathematics

The dynamics of rational transforms: the topological picture. (English. Russian original) Zbl 0619.30033
Russ. Math. Surv. 41, No. 4, 43-117 (1986); translation from Usp. Mat. Nauk 41, No. 4(250), 35-95 (1986).
The study of the dynamics of rational transformations of the sphere starts essentially with the work of Fatou and Julia, summarized, for example, in the great paper by P. Fatou in Bull. Soc. Math. France 47, 161-271 (1919) and ibid. 48, 33-94 and 208-314 (1920). In recent years there has been a vigorous revival of the field owing to the data supplied by computer experiments and to the introduction of new techniques such as quasiconformal maps and Teichmüller spaces. The article under review summarizes the results of the Fatou-Julia theory, including the important recent supplements on the existence of Siegel discs and Herman rings and the absence of wandering domains. It also includes a chapter on the dynamics of holomorphic families of rational maps, the problem of structural stability, the role of Teichmüller theory and the special quadratic families \(1+cz^{-2}\), \(z^ 2+c\). Most results are proved. There is a list of 94 references. This paper is a remarkably complete introduction to the present state of the theory.
Reviewer: I.N.Baker

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37A99 Ergodic theory
Full Text: DOI