Rational Iteration: Complex Analytic Dynamical Systems

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Editorial Reviews

A self-contained exposition of the theory of Fatou and Julia sets and of more recent developments, which may serve as a textbook for a course following a one-year introduction to analytic function theory, and should prove useful for advanced students as well as mathematicians who want to become acquainted with this field. Research workers will also likely find some aspects new, many of the results and some of the proofs of known results appearing for the first time in book form (if at all). Each section includes a list of exercises. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Table of Contents

List of Special Symbols
Ch. 1 Preliminaries 1
1 Basic Notation 2
2 Proper Mappings 4
3 The Riemann-Hurwitz Formula 7
4 The Poincare Metric 10
5 Capacity and Green's Function 14
Ch. 2 The Dynamical Dichotomy 19
1 Two Examples 20
2 Notation 24
3 The Julia Set 28
4 Montel's Criterion 30
5 Repelling Cycles 35
6 Stable Domains 39
7 The Denjoy-Wolff Theorem 42
Ch. 3 The Fatou Set 46
1 Sullivan's Theorem 47
2 The Fatou-Cremer Classification 54
3 Bottcher Domains 60
4 Schroder Domains 66
5 Leau Domains and Leau Flowers 72
6 Indifferent Cycles 78
7 The Centre Problem 82
8 Rotation Domains 85
Ch. 4 The Existence of Rotation Domains 89
1 Siegel's Theorem 90
2 The Bryuno-Russmann Theorem 96
3 Arnold's Theorem 102
Ch. 5 The Geometry of the Julia Set 114
1 Critical Points 115
2 Symbolic Dynamics 120
3 Smooth Julia Sets 125
4 Completely Invariant Stable Domains 132
5 Boundaries of Stable Domains 137
Ch. 6 Miscellanea 146
1 Polynomials 147
2 The Mandelbrot Set 157
3 Lyubich's Invariant Measure 163
4 Stable Julia Sets 173
5 Permutable Rational Functions 177
Bibliography 183
Index 188
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