## Complex DynamicsComplex dynamics is today very much a focus of interest. Though several fine expository articles were available, by P. Blanchard and by M. Yu. Lyubich in particular, until recently there was no single source where students could find the material with proofs. For anyone in our position, gathering and organizing the material required a great deal of work going through preprints and papers and in some cases even finding a proof. We hope that the results of our efforts will be of help to others who plan to learn about complex dynamics and perhaps even lecture. Meanwhile books in the field a. re beginning to appear. The Stony Brook course notes of J. Milnor were particularly welcome and useful. Still we hope that our special emphasis on the analytic side will satisfy a need. This book is a revised and expanded version of notes based on lectures of the first author at UCLA over several \Vinter Quarters, particularly 1986 and 1990. We owe Chris Bishop a great deal of gratitude for supervising the production of course notes, adding new material, and making computer pictures. We have used his computer pictures, and we will also refer to the attractive color graphics in the popular treatment of H. -O. Peitgen and P. Richter. We have benefited from discussions with a number of colleagues, and from suggestions of students in both our courses. |

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### Contents

II | 1 |

III | 5 |

IV | 9 |

V | 11 |

VI | 15 |

VII | 17 |

VIII | 19 |

IX | 27 |

XXV | 91 |

XXVI | 93 |

XXVII | 99 |

XXVIII | 101 |

XXIX | 103 |

XXX | 105 |

XXXI | 106 |

XXXII | 109 |

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### Common terms and phrases

A(oo annulus assume attracting cycle attracting fixed point attracting petal basin of attraction Beltrami coefficient Beltrami equation Blaschke product boundary bounded components branch completely invariant components conformal map connected component containing corresponding critical point cusp cycle of length defined denote depends analytically dynamics ellipse field estimate external angles external rays Fatou set finite fn(z follows functional equation Green's function half-plane Hence Herman ring homeomorphism hyperbolic component hyperbolic metric integral John domain Julia set Lemma locally connected Mandelbrot set Misiurewicz point neutral cycles neutral fixed point obtain parabolic cycle parabolic fixed point parameter periodic point proof of Theorem quasicircle quasiconformal mappings rational function ray terminates repelling arm repelling fixed point repelling periodic point Riemann mapping rotation number satisfies Section sequence shows Siegel disk simple closed Jordan simply connected subhyperbolic superattracting cycle Suppose Theorem 1.1 uniformly bounded univalent zero

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