Iteration of Rational Functions: Complex Analytic Dynamical Systems

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Springer Science & Business Media, Sep 27, 2000 - Mathematics - 280 pages
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This book makes available a comprehensive, detailed, and organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The material covered extends from the original memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before. The theory of of dynamical systems and chaos has recently undergone a rapid growth in popularity, in part due to the spectacular computer graphics of Julia sets, fractals, and the Mandelbrot set. This text focuses on the specialized area of complex analytic dynamics, a subject that dates back to 1916 and is currently a very active area in mathematics.
 

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Contents

Examples
1
12 Iteration of Mbius Transformations
3
13 Iteration of z z
6
14 Tchebychev Polynomials
9
15 Iteration of z z 1
13
16 Iteration of z z + c
14
17 Iteration of z z + 1z
19
18 Iteration of z 2z 1z
21
69 The Julia Set and Periodic Points
148
610 Local Conjugacy
150
Appendix III Infinite Products
155
Appendix IV The Universal Covering Surface
157
Forward Invariant Components
160
72 Limit Functions
162
73 Parabolic Domains
165
74 Siegel Discs and Herman Rings
167

19 Newtons Approximation
22
110 General Remarks
25
Rational Maps
27
22 Rational Maps
30
23 The Lipschitz Condition
32
24 Conjugacy
36
25 Valency
37
26 Fixed Points
38
27 Critical Points
43
28 A Topology on the Rational Functions
45
The Fatou and Julia Sets
49
32 Completely Invariant Sets
51
33 Normal Families and Equicontinuity
55
Appendix I The Hyperbolic Metric
60
Properties of the Julia Set
65
42 Properties of the Julia Set
67
43 Rational Maps with Empty Fatou Set
73
Appendix II Elliptic Functions
77
The Structure of the Fatou Set
80
52 Completely Invariant Components of the Fatou Set
82
53 The Euler Characteristic
83
54 The RiemannHurwitz Formula for Covering Maps
85
55 Maps Between Components of the Fatou Set
90
56 The Number of Components of the Fatou Set
93
57 Components of the Julia Set
95
Periodic Points
99
62 The Existence of Periodic Points
101
63 SuperAttracting Cycles
104
64 Repelling Cycles
109
65 Rationally Indifferent Cycles
110
66 Irrationally Indifferent Cycles in F
132
67 Irrationally Indifferent Cycles in J
142
68 The Proof of the Existence of Periodic Points
145
75 Connectivity of Invariant Components
172
The No Wandering Domains Theorem
176
82 A Preliminary Result
177
83 Conformal Structures
179
84 Quasiconformal Conjugates of Rational Maps
183
85 Boundary Values of Conjugate Maps
184
86 The Proof of Theorem 812
186
Critical Points
192
92 The Normality of Inverse Maps
193
93 Critical Points and Periodic Domains
194
94 Applications
199
95 The Fatou Set of a Polynomial
202
96 The Number of NonRepelling Cycles
210
97 Expanding Maps
223
98 Julia Sets as Cantor Sets
227
99 Julia Sets as Jordan Curves
232
910 The Mandelbrot Set
238
Hausdorff Dimension
246
102 Computing Dimensions
248
103 The Dimension of Julia Sets
251
Examples
257
112 Dendrites
258
114 F with Infinitely Connected and Simply Connected Components
260
115 J with Infinitely Many NonDegenerate Components
261
116 F of Infinite Connectivity with Critical Points in J
262
117 A Finitely Connected Component of F
263
118 J Is a Cantor Set of Circles
266
119 The Function z 2z
271
References
273
Index of Examples
278
Index
279
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