Hardcover ISBN:  9780821844649 
Product Code:  HMATH/38 
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eBook ISBN:  9780821883259 
Product Code:  HMATH/38.E 
List Price:  $120.00 
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AMS Member Price:  $96.00 
Hardcover ISBN:  9780821844649 
eBook: ISBN:  9780821883259 
Product Code:  HMATH/38.B 
List Price:  $245.00 $185.00 
MAA Member Price:  $220.50 $166.50 
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Hardcover ISBN:  9780821844649 
Product Code:  HMATH/38 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821883259 
Product Code:  HMATH/38.E 
List Price:  $120.00 
MAA Member Price:  $108.00 
AMS Member Price:  $96.00 
Hardcover ISBN:  9780821844649 
eBook ISBN:  9780821883259 
Product Code:  HMATH/38.B 
List Price:  $245.00 $185.00 
MAA Member Price:  $220.50 $166.50 
AMS Member Price:  $196.00 $148.00 

Book DetailsHistory of MathematicsVolume: 38; 2011; 454 ppMSC: Primary 01; 30
The theory of complex dynamics, whose roots lie in 19thcentury 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 presentday study.
ReadershipGraduate students and research mathematicians interested in complex dynamics, complex analysis, and the history of mathematics.

Table of Contents

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

Aftermaths (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 DenjoyWolff theorem

Dynamics of selfmaps 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


Additional Material

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The theory of complex dynamics, whose roots lie in 19thcentury 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 presentday study.
Graduate students and research mathematicians interested in complex dynamics, complex analysis, and the history of mathematics.

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

Aftermaths (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 DenjoyWolff theorem

Dynamics of selfmaps 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