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**Dynamics in one complex variable. Introductory lectures.**
*(English)*
Zbl 0946.30013

Wiesbaden: Vieweg. viii, 257 p. (1999).

The iteration theory of rational functions created by Fatou and Julia in 1918-20 experienced a renaissance in the early 1980s, and has been an area of very active research since then. This book provides an excellent introduction into this subject. It grew out of a course given by the author in 1989-90. The lecture notes of this course appeared in the Stony Brook preprint series and found wide circulation (and high regard) among mathematicians working in this field. This book is an expanded and polished version of these lecture notes. The contents is as follows. First, in §§1-3, some background material is presented, in particular Riemann surfaces, universal coverings, the PoincarĂ© metric, and normal families. Then Fatou and Julia sets of rational functions (and, more generally, analytic selfmaps of compact Riemann surfaces) and their basic properties are introduced in §4. In §5, the iteration of analytic selfmaps of hyperbolic Riemann surfaces is discussed in detail. These results are used later in the classification of periodic Fatou components. There is also a short discussion of the iteration of analytic selfmaps of Euclidean Riemann surfaces in §6 and of smooth Julia sets in §7. Next the local fixed point theory, and the associated functional equations, are described in detail in §§8-11. The global theory of periodic points is then discussed in §§12-14. In particular, §14 is devoted to the result that the repelling periodic points are dense in the Julia set, as well as some consequences of this result. Two different proofs of this fundamental result are given, one following the ideas of Fatou, the other one following Julia. Herman rings are discussed in §15 and the classification of periodic Fatou components is given in §16. Next, §17 contains a discussion of prime ends and local connectivity, concepts needed in the sequel. In §18, external rays of connected polynomial Julia sets are introduced. It is shown that periodic external rays land at repelling or parabolic periodic points, and that such points are in turn landing points of external rays. This section also includes the result that if a Julia set of a polynomial is locally connected, then every periodic point in the Julia set is repelling or parabolic, and every cycle of Siegel disks has a critical point on its boundary. In §19, hyperbolic and subhyperbolic functions are introduced and, among other things, it is shown that if the Julia set of a hyperbolic rational function is connected, then it is locally connected. The extension of this result to subhyperbolic functions is also discussed. The book ends with eight appendices devoted to various topics. In part they deal with further background material from complex analysis and other fields, in part they discuss some additional aspects of the subject. One appendix is devoted to a proof of Sullivan’s no wandering domains theorem (assuming the existence of solutions of the Beltrami equation). When Milnor’s lecture notes first appeared in preprint form, there was no textbook on the subject yet (but there were a number of good surveys). Since then, three very good books on the iteration of rational functions have appeared, by A. F. Beardon [Iteration of rational functions (1991; Zbl 0742.30002)], N. Steinmetz [Rational iteration: Complex analytic dynamical systems. (1993; Zbl 0773.58010)], and L. Carleson and Th. W. Gamelin [Complex dynamics (1993; Zbl 0782.30022)]. Nevertheless the book under review remains a very valuable addition to the literature on the subject, and it is highly recommend to both students and researchers in the field.

Reviewer: W.Bergweiler (Kiel)

### MSC:

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |