A History of Complex Dynamics: From Schröder to Fatou and Julia

175 Pages·1994·13.023 MB·English

by  Professor Daniel S. Alexander (auth.)

Daniel S. Alexander
AHistory of
Complex Dynamics
Asped~f
Iv\athemati&
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Daniel S. Alexander
A History of
Complex Dynamics
From Schräder to Fatou and Julia
IJ
Vleweg
Professor Daniel S. Alexander
Department of Mathematics
Drake University
203 Howard Hall
Des Moines, Iowa 50311-4505
USA
da 0231 r © acad. drake. edv
Mathematics Subject Classification: 01-02, 01 A 60, 01 A 55, 30 D 05, 30 C XX
All rights reserved
© Springer Fachmedien Wiesbaden 1994
Originally published by Friedr. Vieweg & Sohn Verlags gesellschaft mbH, Braunschweig/Wiesbaden in 1994
Softcover reprint of the hardcover 1s t edition 1994
No part of this publjcation may be reproduced, stored in a retrieval system or
t~ansmitted, mechanical, photocopying or otherwise without prior permission
of the copyright holder.
Cover design: Wolfgang Nieger, Wiesbaden
Printed on acid-free paper
ISSN 0179-2156
ISBN 978-3-663-09199-8 ISBN 978-3-663-09197-4 (eBook)
DOI 10.1007/978-3-663-09197-4
Contents
Preface 1
1 Schröder, Cayley and Newton's Method 3
1.1 Introduction ........... . 3
1.2 Schröder's Study of Iteration . . . . . 4
1.3 Schröder's Fixed Point Theorem ... 6
1.4 A Generalization of Newton's Method 9
1.5 Schröder's Paper [1871] ....... . 11
1.6 Schröder and Functional Equations . . 12
1.7 Schröder and Newton's Method for the Quadratic . 16
1.8 Arthur Cayley and Newton's Method ....... . 19
2 The Next Wave: Korkine and Farkas 23
2.1 Introduction............. 23
2.2 Analytic Iteration. . . . . . . . . . . . 24
2.3 Korkine and the Influence of Abel .. 28
2.4 Abel's Study of Functional Equations 29
2.5 Korkine's Solution to the Abel Equation 31
2.6 Farkas' Solution to the Schröder Equation 34
3 Gabriel Koenigs 37
3.1 Gabriel Koenigs . . . . . . . . . . . . . . . . . . 37
3.2 Koenigs and Darboux .. . . . . . . . . . . . . 38
3.3 The Background to Koenigs' Study of Iteration 40
3.4 Koenigs' Study of Fixed Points . . . . . . . 41
3.5 Koenigs' Solution of the Schröder Equation 46
3.6 Koenigs and Functional Equations . . . . . 47
3.7 Koenigs and the Global Study of Iteration. 50
4 Iteration in the 1890's: Grevy 53
4.1 ABrief Survey of Iteration in the 1890's . . . . . . . . . . . . . . .. 53
4.2 Appell's Application of Koenigs' Work to Hill's Differential Equation 54
4.3 Grevy and the Superattracting Case . . . . . . . . . . . . . . . . .. 55
vi CONTENTS
5 Iteration in the 1890's: Leau 60
5.1 Basic Results in the 14>'(0)1 = 1 Case 60
5.2 Lemeray ............... . 64
5.3 Leau's Work. . . . . . . . . . . . . . 67
5.4 Leau's Anticipation of the Flower Theorem 68
5.5 Leau and Functional Equations . . . . . . 73
6 The Flower Theorem of Fatou and J ulia 75
6.1 The Approaches of Fatou and Julia .. 75
6.2 Julia's Proof of the Flower Theorem . 76
6.3 Fatou's Proof of the Flower Theorem. 78
7 Fatou's 1906 Note 84
7.1 Introduction: Local Versus Global Studies of Iteration 84
7.2 The Lack of Set Theory in Koenigs' Work 85
7.3 Fatou's Application of Set Theory 87
7.4 Pierre Fatou. . . . . . . . . . . . 89
7.5 Fatou's 1906 Note ....... . 90
7.6 Fatou and Functional Equations 94
8 Montel's Theory of Normal Families 96
8.1 Introduction ............ . 96
8.2 Paul Montel and Normal Families . 98
8.3 The Influence of Ascoli and Arzela 99
8.4 Montel's Early Work ....... . 101
8.5 Montel's Study of Convergence Issues 103
8.6 An Important Result from [1912] .... 104
8.7 Applications of Montel's Theory to Picard Theory 105
9 The Contest 108
9.1 Overview ................. . 108
9.2 Biographical Sketches of Lattes and Julia 108
9.3 The 1918 Grand Prix . ... 109
9.4 The Awarding of the Prize . 113
10 Lattes and Ritt 117
10.1 Biographical Sketch of Ritt 117
10.2 The Approach of Lattes and Ritt 118
11 Fatou and J ulia 124
11.1 Introduction to the Studies of Fatou and Julia 124
11.2 Iteration and the Theory of Normal Families 125
'e •
11.3 The Julia Set ....... . 126
11.4 Some Interesting Julia Sets '" ...... . 128
CONTENTS vii
11.5 Further Properties of the J ulia Set . . . . . . 131
11.6 Iteration on the Fatou Set .......... . 134
11.7 A Limit on the Number of Attracting Orbits 136
11.8 The Number of Components of the Fatou Set 139
11.9 Newton's Method Again ........... . 140
Bibliography 143
Index 163
List of Figures
5.1 A representation of attracting and repelling fixed points. . . 61
5.2 A representation of the Flower Theorem in the m = 1 case. 62
5.3 The Flower Theorem in the m = 2 case. . . . . . . . . .. . 63
5.4 An example of graphical analysis from Lemeray's paper. . . 65
5.5 Lemeray's decomposition of a neighborhood of a fixed point. . 67
5.6 Leau's depiction of attracting and repelling cardioids. 69
5.7 Leau's approximation of iteration in the m = 2 case. 72
5.8 The formation of an attracting petal for A(Z). 73
6.1 Julia's construction of a cardioid. . . . . . . . 77
6.2 Julia's formation of an attracting petal in the m = 2 case. 78
6.3 Iteration of tP*(z) in Fatou's proof of the Flower Theorem. 80
6.4 Fatou's depiction of repelling and attracting cardioids. 81
6.5 Fatou's depiction of D*. . . . . . . . . . . . . . . 82
7.1 The hyperbola y2 - Z2 = 1 viewed from infinity. . 93
7.2 The preimages of runder tP2(Z), 94
11.1 The Koch Snowflake. . . . . . . 129
11.2 Julia's schematic of a Julia set. 130
Preface
In late 1917 Pierre Fatou and Gaston Julia each announced several results regarding
the iteration ofrational functions of a single complex variable in the Comptes rendus
of the French Academy of Sciences. These brief notes were the tip of an iceberg.
In 1918 Julia published a long and fascinating treatise on the subject, which was
followed in 1919 by an equally remarkable study, the first instalIment of a three
part memoir by Fatou. Together these works form the bedrock of the contemporary
study of complex dynamics.
This book had its genesis in a question put to me by Paul Blanchard. Why did
Fatou and Julia decide to study iteration? As it turns out there is a very simple
answer. In 1915 the French Academy of Sciences announced that it would award its
1918 Grand Prix des Sciences mathematiques for the study of iteration. However,
like many simple answers, this one doesn't get at the whole truth, and, in fact,
leaves us with another equally interesting question. Why did the Academy offer
such a prize?
This study attempts to answer that last question, and the answer I found was
not the obvious one that came to mind, namely, that the Academy's interest in
iteration was prompted by Henri Poincare's use of iteration in his studies of celestial
mechanics. While this may have played a part in the Academy's decision, it also
turns out that there was a longstanding French interest in the iteration of complex
maps, beginning with the studies of Gabriel Koenigs in the mid-1880's. However,
he was not the first to become intrigued by the dynamics of complex maps. That
honor seems to belong to a German mathematician, not unknown by any means,
but one who deserves more renown than he seems to have at the present moment,
Ernst Schröder, who in 1870 articulated the following theorem.
Let 4>ß(z) denote the n-fold composition of 4>(z) with itself. If 4>(z) is a
complex analytic map satisfying 4>(z) and 14>'(z)1 < 1 for some point z,
then there exists a neighborhood D of z on which 4>ß(z) converges to z
for all z in D.
This book traces the history of the iteration of complex maps from Schröder's
first paper to the studies of Fatou and Julia. I have tried to keep myself focused
on that devel~pment, and as a consequence decided not to include a number o[