## Rational Iteration: Complex Analytic Dynamical SystemsThe series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. Please submit any book proposals to Niels Jacob. |

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### Contents

7 The Centre Problem | 78 |

8 Rotation Domains | 82 |

Chapter 4 The Existence of Rotation Domains | 85 |

1 Siegels Theorem | 89 |

2 The BryunoRüssmann Theorem | 90 |

3 Arnolds Theorem | 96 |

Chapter 5 The Geometry of the Julia Set | 102 |

1 Critical Points | 114 |

2 Notation | 20 |

3 The Julia Set | 24 |

4 Montels Criterion | 28 |

5 Repelling Cycles | 30 |

6 Stable Domains | 35 |

7 The DenjoyWolff Theorem | 39 |

Chapter 3 The Fatou Set | 42 |

1 Sullivans Theorem | 46 |

2 The FatouCremer Classification | 47 |

3 Böttcher Domains | 54 |

4 Schröder Domains | 60 |

5 Leau Domains and Leau Flowers | 66 |

6 Indifferent Cycles | 72 |

2 Symbolic Dynamics | 115 |

3 Smooth Julia Sets | 120 |

4 Completely Invariant Stable Domains | 125 |

5 Boundaries of Stable Domains | 132 |

Chapter 6 Miscellanea | 137 |

1 Polynomials | 146 |

2 The Mandelbrot Set | 147 |

3 Lyubichs Invariant Measure | 157 |

4 Stable Julia Sets | 163 |

5 Permutable Rational Functions | 173 |

Bibliography | 177 |

Index | 183 |