## Iteration of Rational Functions: Complex Analytic Dynamical SystemsThis 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 Möbius Transformations | 3 |

13 Iteration of z z² | 6 |

14 Tchebychev Polynomials | 9 |

15 Iteration of z z² 1 | 13 |

16 Iteration of z z² + c | 14 |

17 Iteration of z z² + 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 2²z² | 271 |

273 | |

278 | |

279 | |

### Other editions - View all

Iteration of Rational Functions: Complex Analytic Dynamical Systems Alan F. Beardon Limited preview - 2000 |

### Common terms and phrases

analytic map argument assume attracting fixed point Beltrami coefficient bounded Cantor set Chapter closed curve closure compact subset complement completely invariant completes the proof complex sphere component F0 component of F conjugacy conjugate construct contains covering map deduce define deg(K deg(R denote disjoint equicontinuous Euclidean Euler characteristic example Exercise Fatou set finite critical point finite number follows forward orbit given hence Herman ring homeomorphism hyperbolic metric indifferent fixed point infinitely connected iterates Jordan curve Julia set lies map of degree Mobius map neighbourhood non-constant normal obtain open set origin periodic points petals polynomial positive integer positive number proof is complete proof of Theorem prove Theorem rational function rational map rationally indifferent cycle reader regular subdomain repelling fixed point result Riemann satisfies Siegel disc simply connected simply connected domain super)attracting cycle Suppose Theory topological unit circle unit disc wandering domain zeros