Early days in complex dynamics : a history of complex dynamics in one variable during 1906-1942
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Early days in complex dynamics : a history of complex dynamics in one variable during 1906-1942
(History of mathematics, v. 38)
American Mathematical Society , London Mathematical Society, c2012
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
ALE||11||2200021324155
Note
Includes bibliographical references (p. 383-432) and index
Description and Table of Contents
Description
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 Poincare 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 Mathematiques
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 Mathematiques 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
by "Nielsen BookData"