Complex dynamics. (English) Zbl 0782.30022

Universitext: Tracts in Mathematics. New York: Springer-Verlag. ix, 175 p. (1993).
The present text is much more than an introduction to complex dynamics – rational iteration. It gives a comprehensive account of the field, starting with the basic theory of Julia and Fatou, and ending up with results found not earlier than 1992.
The books starts with the discussion of the most important tools: Montel’s concept of normality, hyperbolic metric, the invariant Lemma of Schwarz, and quasiconformal mappings. The latter play (up to now) an undispensible role in the proof of Sullivan’s Theorem.
The second chapter contains the local fix-point theory, dealing with analytic functions in a neighbourhood of (super)attracting, repelling and neutral fix-points. Most of this part of the theory dates back to the nineteenth century, and is related to certain functional equations (due to Abel, Schröder and Böttcher). The importance of these functional equations for rational iteration becomes clear only in chapter IV, where, after Sullivan’s No-Wandering-Domain-Theorem, the Fatour-Cremer classification of periodic stable domains is proved. The preceding chapter includes an account of the more or less elementary parts of the Julia-Fatou theory.
In two chapters, V and VII, the key role of the critical points and their orbits becomes apparent. Several theorems are proved concerning local connectedness of boundaries of stable domains and accessibility of periodic points on the Julia set. These include work of Fatou, Douady, Hubbard, Eremenko-Levin, and the authors.
In chapter IV it is shown how quasiconformal mappings apply in various problems, which obviously have nothing to do with this kind of mappings. This method – due to Douady and Hubbard and called quasiconformal surgery – was refined by Shishikura to obtain the precise bound \(2(d-1)\) for the number of periodic cycles of stable domains. The last chapter is devoted to a study of the Mandelbrot set and the related quadratic family \(z^ 2+c\). The authors give a proof of Douady’s result (the Mandelbrot set is connected) and discuss to a certain extend several topics, in particular external rays in the dynamical (\(z\)-) plane and the parameter (\(c\)-) plane.
Although the text sometimes is hard to read (the reader has to fill several gaps, which is always possible) the book of Carleson and Gamelin is a valuable guide for everybody who wants to become familiar with the old and new, anyway exciting and beautiful part of complex analysis.
[Reviewer’s remark: I sympathized with the authors when reading in the preface that it “[…] required a great deal of work going through preprints and papers and in some cases even finding a proof”].


30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics