## Dynamics in One Complex Variable: Introductory LecturesThese notes will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90 and also in later years. I am grateful to the audiences for a great deal of constructive criticism, and to Branner, Douady, Hubbard, and Shishikura who taught me most of what I know in this field. Also, I want to thank A. Poirier, S. Zakeri, and R. Perez for their extremely helpful criticisms of various drafts. There have been a number of extremely useful surveys of holomorphic dynamics over the years - those of Brolin, Douady, Blanchard, Lyubich, Devaney, Keen, and Eremenko-Lyubich, as well as the textbooks by Bear don, Steinmetz, and Carleson-Gamelin, are particularly recommended to the reader. (Compare the list of references at the end, and see Alexander for historical information. ) This subject is large and rapidly growing. These lectures are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. The necessary material can be found for example in Ahlfors 1966, Hocking and Young, Munkres, and vVillmore. |

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### Contents

Riemann Surfaces | 1 |

Ernst Schroder 18411902 | 7 |

Iterated Holomorphic Maps | 38 |

Copyright | |

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annulus attracting basin attracting fixed point belongs boundary choose closed closure compact set compact subset Compare Problem compute conformal automorphism conformal isomorphism conjugate connected component constant contains coordinate Corollary corresponding covering map Cremer point critical point curve cycles defined definition diameter disjoint equal equation Euclidean example external rays Fatou component Fatou set Figure filled Julia set follows easily half-plane hence holomorphic map homeomorphism identity map inequality infinite integer intersection iterates Lemma linear locally connected map f neighborhood open set open subset orbifold orbit zq parameter periodic point Poincare distance Poincare metric point at infinity point zq polynomial proof prove punctured disk quadratic radius ramified points rational map repelling petal Riemann sphere Riemann surface rotation number satisfies sequence Siegel disk simply connected space superattracting suppose tends to zero Theorem topological torus unique unit circle unit disk universal covering vector