The iteration theory of rational (and entire) functions goes back to Fatou and Julia, and it has experienced a remarkable renaissance in the last two decades, partially due to the beautiful computer graphics related to it, but also due to new and powerful mathematical methods developed by various people. Most of the research has concentrated on the dynamics of rational functions, and there are now a number of very good introductions to this part of the theory, in particular the books by A. F. Beardon [Iteration of rational functions. Complex analytic dynamical systems (1991; Zbl 0742.30002)], L. Carleson and Th. W. Gamelin [Complex dynamics (1993; Zbl 0782.30022)] and N. Steinmetz [Rational iteration (1993; Zbl 0805.30030)], and the lecture notes by J. Milnor [Dynamics in one complex variable: introductory lectures, StonyBrook preprint 1990]. There has also been considerable research concerning the dynamics of transcendental functions, but so far there was no book covering this aspect of the theory.
The book under review is intended to fill this gap. It starts with a brief review of the facts from complex analysis that are needed. This covers some topics like conformal and quasiconformal mappings which are also essential for rational iteration, but also includes topics like Nevanlinna theory or Wiman-Valiron theory which are very useful in the iteration of transcendental functions, but rarely occur in rational dynamics.
The second chapter, after a discussion of fixed points of iterated and composite functions, contains the classical results on the local behavior near fixed points. The remaining chapters then deal with the main concepts and results of the global iteration theory of meromorphic functions.
Chapter 3 introduces Fatou and Julia sets and their basic properties. Chapter 4 discusses components of the Fatou set. It contains the classification of periodic components, their relation to singularities of the inverse function, a discussion of wandering domains and Baker domains as well as some classes of functions which do not have such domains, and some further topics.
Chapter 5 is devoted to the geometry of the Julia sets. For example, it is shown that certain Julia sets are Cantor sets or Cantor bouquets. Chapter 6 is on Hausdorff dimension of Julia sets. After the basic definitions it presents the fundamental results of McMullen on the exponential and sine families. Most of this chapter is devoted to a deep theorem of Stallard which says that the Hausdorff dimension of the Julia set of a transcendental entire function is strictly greater than one, if the set of singularities of the inverse function is bounded. A final chapter discusses miscellaneous topics such as measurable dynamics, permutable functions, etc.
The book collects a number of interesting results on various aspects of the theory. Unfortunately, however, the authors have not made much effort to give a good exposition of these results and to explain the underlying ideas. Instead, they have often just copied (sometimes word by word) the results and proofs from the original papers (giving references to the literature whenever this is done in the original paper). So not much is gained from reading the book instead of the original papers.
The book also contains a number of errors, both in the presentation of classical results and in the description of current research. For example, Theorem 2.9 on conjugacy near rationally indifferent points is not correct in the form it is stated. This topic (and in fact the whole local theory) is treated better in the books cited above. This also applies to the Denjoy-Wolff theorem, which does not follow from the classification of periodic components as claimed on p. 64. Concerning new results, I mention that there is an error in the proof of Theorem 7.14.
This incomplete sample of errors should suffice to caution the reader that this book has serious defects. Nevertheless it may serve as a useful guide to some of the progress that has been made on the dynamics of transcendental functions in recent years. In conclusion I mention that some of the problems raised in the book have been solved. The answer to Question 4.2 is “no” [M. E. Herring, Ann. Acad. Sci. Fenn., Math. 23, 263-274 (1998; Zbl 0910.30025); A. Bolsch, Iteration of meromorphic functions with countably many essential singularities. Thesis Berlin (1997; Zbl 0901.30022) and Bull. Lond. Math. Soc. 31, No. 5, 543-555 (1999; Zbl 0932.30022)].
The answer to Question 4.4 is “yes” [I. N. Baker, Ann. Acad. Sci. Fenn., Ser. A I 12, No. 2, 191-198 (1987; Zbl 0606.30029)]. Question 4.5 has been answered by Bolsch [loc.cit.].