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Rudiments of holomorphic dynamics. (Rudiments de dynamique holomorphe.) (French) Zbl 1051.37019
Cours Spécialisés (Paris) 7. Paris: Société Mathématique de France (ISBN 2-86883-521-X/pbk). 160 p. (2001).
The monograph under review is an introduction to holomorphic dynamics in the sense of iterations of rational maps of the Riemann sphere. According to the book itself, the authors have aimed at offering both a panoramic view of the most classical aspects of the subject, and a sketch of some of the more recent developments. In my opinion, they have accomplished these two goals elegantly, and the text should be excellent for an advanced undergraduate or beginning graduate course. The first four chapters (roughly half the book) contain the most fundamental material: basic facts about the Fatou and Juila sets. While the results here are classical, they are nicely presented. A novelty is the systematic use of the renormalization lemma attributed to Zalcman: this is used to produce quick proofs of Montel’s theorem and Fatou’s theorem on the density of repelling periodic orbits in the Julia set. Sullivan’s celebrated no-wandering theorem is also proved here. The last four chapters are largely independent from one another. In Chapter 5, the focus is on “chaotic” rational maps, i.e.; maps with empty Fatou set. The discussion progresses from Lattés examples to more general (strictly) critically finite maps. The chapter ends with a nice discussion of Lyubich’s construction of chaotic rational maps with dense postcritical orbit. Chapter 6 deals with hyperbolic rational maps. It contains both classical material, such as the characterization of hyperbolicity in terms of the postcritical set, but also more recent results, e.g., the porosity of Julia sets of hyperbolic maps. In Chapter 7, the authors discuss holomorphic families of rational maps, in particular the Ma\(\{n\}\)é-Sad-Sullivan theory. Chapter 8 is dedicated to potential theoretic methods. This part is of particular interest to anyone interested in the higher-dimensional complex dynamics. The reader is assumed to have a basic knowledge in complex analysis but the book is otherwise self-contained. Some more advanced material — Hausdorff dimension and measure, quasiconformal mappings, and potential theory — is collected in three appendices. While the book by Berteloot and Mayer does not replace existing references such as the ones by A. F. Beardon [Iteration of rational functions. Complex analytic dynamical systems, Graduate Texts in Mathematics, 132,(Springer, New York (1991; Zbl 0742.30002)], L. Carleson and T. W. Gamelin [Complex dynamics, Universitext: Tracts in Mathematics, New York etc.: Springer-Verlag (1993; Zbl 0782.30022)], M. Yu. Lyubich [Russ. Math. Surv. 41, 43–117 (1987; Zbl 0619.30033)], J. Milnor [Dynamics in one complex variable. Introductory lectures, Wiesbaden: Vieweg (1999; Zbl 0946.30013)], and N. Steinmetz [Rational iteration: complex analytic dynamical systems, Berlin: de Gruyter Studies in Mathematics, 16 (1993; Zbl 0773.58010)] it is a welcome addition to the literature.

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B12 Iteration theory, iterative and composite equations
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics