Complex dynamics studies iterates of complex functions in some domain, in particular the corresponding orbits and stability behavior. The subject has experienced spectacular growth since the late 1970s, thanks in part to the seminal work of Milnor, Douady and others, paired with the use of computer software that allows visualization of the complicated sets that appear. However, as the authors make clear, the theory of complex dynamics has its roots in the nineteenth century, and flourished during the first half of the twentieth century through the work of Fatou, Julia, Siegel and many other mathematicians.
The main purpose of the book under review is to explore the many paths that led to the creation and early development of complex dynamics in one variable in the first three decades of the twentieth century. The book certainly succeeds in all counts.
The authors’ narrative starts in the nineteenth century, with a review of the pioneering work of E. Schröder, apparently inspired by the desire to find algorithms, such as Newton’s, for the solutions of equations. This led him to study iterations of the given function and to discover some convergence properties that are now best formulated in terms of orbits. Details of the work of Schröder and Kœnigs are given in the first chapter of the book under review.
Chapters three and four discuss the role of Poincaré’s applications of iteration in differential equations, celestial mechanics and Kleinian groups, especially his results on the stability of an equilibrium point of an ordinary differential equation and the generalizations obtained by Levi-Civita. These results showed that iterations are a useful tool for studying the stability of solutions of differential equations, paving the way for Samuel Lattès’ thesis on iterations of functions of two and three complex variables, generalizing the work of Kœnigs. Chapter four includes a discussion of the problem of small denominators found by Poincaré in his study of equilibria for systems of differential equations and in the general context of perturbations of a Hamiltonian system, and its relation with complex dynamics.
Chapters five to nine cover the central part of the story, mapping the contributions of several mathematicians around the globe, from 1906 to 1920. The main focus is on the contributions of Pierre Fatou and Gaston Julia. For example, chapter six examines the issues raised by the prize that the Académie in Paris offered for work on iteration of complex functions. As is well known the prize went to Julia and Lattès who had submitted essays for this competition, but also to Fatou who had not entered the competition. Chapter seven examines closely the works written for the competition and the contributions of Fatou, discussing, for example, the role played by Montel’s theory of normal families in the work of Fatou and Julia to conceptualize the iteration of a rational function. (It is important to note that when the authors feel it necessary they use language and concepts that were not available at that time. For example when they describe some work of Julia on the set of (repelling) orbits, they talk about the “fractal” structure of the derived set.)
Chapter nine, the last one of Part 2 of the book, examines some issues that were not solved in Fatou and Julia’s previous work, specifically concerning iteration in neighborhoods of an irrationally neutral point and the existence of centers for functions of degree two or three. Towards the late 1920s there were some advances on the understanding of iteration near irrationally neutral fixed points by H. Cremer and C. L. Siegel who would eventually solve the problem of existence of centers using delicate estimates instead of relying on properties of normal families as in the approach of Julia and Fatou. All of these developments and more are the subject matter of Part three of the book, chapters nine to thirteen, that cover developments from 1920 to 1942. As the authors point out, Siegel’s solution to the center problem in complex dynamics was important for future developments in what is now called KAM theory.
The brief summary given above does little justice to the history of this beautiful subject, but for that we have now this fine book. Lastly I must say that in addition to the thirteen chapters already summarized, there are sixteen appendices, eleven written by mathematicians working on complex dynamics, that put much of the previously discussed history in an actual context, attesting to the impact of the pioneering work of Julia, Fatou, Lattès, Siegel, and others on current research in this area of mathematics. Some of the appendices deal with topics that were just touched on in the main chapters, for example on the history of normal families and the work of Montel, or on the uniformization theorem of Koebe, or on Kleinian groups.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.