**History of Mathematics**

Volume: 38;
2011;
454 pp;
Hardcover

MSC: Primary 01; 30;

**Print ISBN: 978-0-8218-4464-9
Product Code: HMATH/38**

List Price: $111.00

AMS Member Price: $88.80

MAA Member Price: $99.90

**Electronic ISBN: 978-0-8218-8325-9
Product Code: HMATH/38.E**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

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#### Supplemental Materials

# Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906–1942

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*Daniel S. Alexander; Felice Iavernaro; Alessandro Rosa*

A co-publication of the AMS and the London Mathematical Society

The theory of complex dynamics, whose roots lie in 19th-century studies of
the iteration of complex function conducted by Kœnigs, Schröder, 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 Poincaré and others, linking them to
complex dynamics, principally via the work of Samuel Lattès, in the
early 1900s, and Jürgen Moser, in the 1960s.

Many details will be new to the reader, such as a history of Lattès
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.

#### Readership

Graduate students and research mathematicians interested in complex dynamics, complex analysis, and the history of mathematics.

#### Table of Contents

# Table of Contents

## Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906-1942

- Cover Cover11
- Title page iii4
- Contents v6
- List of figures xi12
- Preface xiii14
- Acknowledgments xv16
- Preliminaries 120
- A complex dynamics primer 322
- Introduction: Dynamics of a complex history 2342
- Iteration and differential equations I: The Poincaré connection 3352
- Color plates 6382
- Iteration and differential equations II: Small divisors 6786
- The core (1906–1920) 93112
- Early overseas results: The United States 95114
- The road to the Grand Prix des Sciences Mathématiques 113132
- Works written for the Grand Prix 131150
- Iteration in Italy 181200
- The giants fall 197216
- After-maths (1920–1942) 211230
- Branching out: Fatou and Julia in the 1920s 213232
- The German wave 229248
- Siegel, the center problem, and KAM theory 253272
- Iteratin’ around the globe 267286
- Tying the future to the past 279298
- Report on the Grand Prix des Sciences Mathématiques in 1918 281300
- A history of normal families 285304
- Singular lines of analytic functions 291310
- Kleinian groups 295314
- Curves of Julia 297316
- Progress in Julia’s extension of Schwarz’s lemma 301320
- The Denjoy-Wolff theorem 303322
- Dynamics of self-maps of the unit disc 307326
- Koebe and uniformization 313332
- Permutable maps in the 1920s 315334
- The last 60 years in permutable maps 325344
- Understanding Julia sets of entire maps 331350
- Fatou: A biographical sketch 341360
- Gaston Julia: A biographical sketch 351370
- Selected biographies 359378
- Remarks on computer graphics 371390
- Glossary 379398
- Bibliography 383402
- Index 433452
- Back Cover Back Cover1474