The theory of complex dynamics, whose roots lie in 19th-century studies of the iteration of complex function conducted by Koenigs, Schoder, and others, flourished remarkably during the first half of the 20th century, when many of the central ideas and techniques of the subject developed. This book by Alexander, Iavernaro, and Rosa paints a robust picture of the field of complex dynamics between 1906 and 1942 through detailed discussions of the work of Fatou, Julia, Siegel, and several others. A recurrent theme of the authors' treatment is the center problem in complex dynamics. They present its complete history during this period and, in so doing, bring out analogies between complex dynamics and the study of differential equations, in particular, the problem of stability in Hamiltonian systems. Among these analogies are the use of iteration and problems involving small divisors which the authors examine in the work of Poincare and others, linking them to complex dynamics, principally via the work of Samuel Lattes, in the early 1900s, and Jurgen Moser, in the 1960s. Many details will be new to the reader, such as a history of Lattes functions (functions whose Julia set equals the Riemann sphere), complex dynamics in the United States around the time of World War I, a survey of complex dynamics around the world in the 1920s and 1930s, a discussion of the dynamical programs of Fatou and Julia during the 1920s, and biographical material on several key figures. The book contains graphical renderings of many of the mathematical objects the authors discuss, including some of the intriguing fractals Fatou and Julia studied, and concludes with several appendices by current researchers in complex dynamics which collectively attest to the impact of the work of Fatou, Julia, and others upon the present-day study.

Table of Contents:
Preface Acknowledgments Preliminaries A complex dynamics primer Introduction: Dynamics of a complex history Iteration and differential equations I: The Poincaré connection Color plates Iteration and differential equations II: Small divisors The core (1906–1920) Early overseas results: The United States The road to the Grand Prix des Sciences Mathématiques Works written for the Grand Prix Iteration in Italy The giants fall After-maths (1920–1942) Branching out: Fatou and Julia in the 1920s The German wave Siegel, the center problem, and KAM theory Iteratin’ around the globe Tying the future to the past Report on the Grand Prix des Sciences Mathématiques in 1918 A history of normal families Singular lines of analytic functions Kleinian groups Curves of Julia Progress in Julia’s extension of Schwarz’s lemma The Denjoy-Wolff theorem Dynamics of self-maps of the unit disc Koebe and uniformization Permutable maps in the 1920s The last 60 years in permutable maps Understanding Julia sets of entire maps Fatou: A biographical sketch Gaston Julia: A biographical sketch Selected biographies Remarks on computer graphics Glossary Bibliography Index