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Dynamics in One Complex Variable

John Milnor
Princeton University Press
Publication Date: 
Number of Pages: 
Annals of Mathematics Studies
[Reviewed by
William J. Satzer
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John Milnor gave an introductory course at SUNY Stony Brook in 1989 on one-dimensional complex dynamics. This book—Dynamics in One Complex Variable—is based on that course. More explicitly, the subject is the dynamics of iterated holomorphic mappings from a Riemann surface to itself. For the most part, the book concentrates on the classical case of rational maps of the Riemann sphere.

Today the subject of complex dynamics is vast and still growing rapidly. Milnor's book provides a solid foundation and the kind of bird's eye view that perhaps only a mathematician of his caliber can offer. The first three chapters of the book provide a quick background in the topology and geometry of Riemann surfaces. However, the reader had better come prepared with experience in complex variable theory and differential geometry in two dimensions as well as some basic topology at the level of Hocking and Young's book. For a reader new to the field, it would probably be desirable to look first, for example, at Devaney's book on chaotic dynamical systems.

Iterated holomorphic mappings are introduced in Chapter 4, where the Fatou and Julia sets are described in the context of dynamics on the Riemann sphere. Succeeding chapters take up dynamics on hyperbolic and Euclidean surfaces. A significant portion of the book is devoted to local fixed point theory and the global theory of periodic points. The final sections concentrate on the structure of the Fatou set and how the Fatou set can be used to study Julia sets.

This third edition is not substantially different from previous editions. However, Chapter 4 now includes a short historical survey that describes the development of the study of iterated holomorphic mappings from its beginnings in the 19th century. There are also other small additions including an expanded definition of the Lattès map, a description of the theory of parabolic points due to Écalle and Voronin and new material on Écalle's résidu itérif (a modified form of the residue index).

There are several appendices that deal with background material such as theorems from classical analysis, branched coverings and orbifolds, and several complex variables. A new appendix in this third edition addresses more recent results on the effective computability of polynomial Julia sets. These results pertain directly to the faithful representation in computer graphics of Julia sets and the Mandelbrot set.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

The table of contents is not available.