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Iteration of rational functions. Complex analytic dynamical systems. (English) Zbl 0742.30002
Graduate Texts in Mathematics. 132. New York etc.: Springer-Verlag. xvi, 280 p. (1991).
In the past decade, there has been an explosion of interest in the iteration theory of rational functions, the study of which was initiated by Fatou and Julia between 1918 and 1920. This renewed interest is partially due to the beautiful and complicated computer pictures related to the theory, but partially also to new mathematical methods introduced into the subject. Beardon’s book, although illustrated with a few pictures, concentrates on the mathematical aspects of the subject and presents the classical theory of Fatou and Julia as well as more recent results. It appears to be the first textbook wholly devoted to this topic.
The book begins with a chapter of examples, which (because no iteration theory is developed yet) are discussed on an elementary level. Chapter 2 provides the mathematical background about rational maps that is needed in the following chapters. The Fatou and Julia sets, the basic objects of the theory, are defined in Chapter 3 and their basic properties are discussed in Chapters 4 and 5. Here Beardon follows Fatou’s approach and defines the Fatou set as the set where the iterates are normal and the Julia set as its complement. That the Julia set is equal to the closure of the set of repelling periodic points (Julia’s definition of the set that bears his name) is derived much later (§6.9 and §9.6), but it is not essential for the development of the theory. Chapter 6 is devoted to the local behavior of iterates near fixed points. Chapter 7 gives the classification of (forward) invariant components. Sullivan’s theorem that rational functions do not have wandering domains is proved in chapter 8. The behaviour of the critical points under iteration is of fundamental importance, and their role is discussed in chapter 9. This chapter also contains Shishikura’s sharp bound for the number of non-repelling periodic cycles. After a chapter on Hausdorff dimension the book ends as it started, with a chapter of examples. But now a lot of theory is available and the examples can be discussed on a more advanced level.
The book is very well written, and accessible to anyone familiar with the basic results of complex analysis, say a first course in this subject. (As already mentioned, some basic facts are recalled in Chapter 2.) More advanced results that are needed are carefully introduced and explained at the appropriate places. Beardon’s book is suitable as a textbook for lectures or for seminars as well as for private study and can be recommended to anybody who wants to learn about this interesting subject.

30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
37B99 Topological dynamics
26C15 Real rational functions