Dynamics in One Complex Variable

Beschreibung

This text studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. It is based on introductory lectures given by the author at Stony Brook, NY, in the past ten years.
The subject is large and rapidly growing. These notes 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 exposition is clear and enriched by many beautiful illustrations.


These 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.
Dies ist ein Buch eines bekannten amerikanischen Mathematikers, das aus seiner langjährigen Vorlesungspraxis und seinen Forschungen am Institute for Mathematical Sciences in Stony Brook, NY, als Lehrbuch für Studenten konzipiert wurde. Elementare Kenntnisse aus der Funktionentheorie, Differentialgeometrie, Topologie sind nur erforderlich, um dieses sehr gut geschriebene Buch verstehen zu können. Der Stil von John Milnors Büchern wird allgemein sehr geschätzt. Das Gebiet ist aktuell, es steht in Verbindung mit der mathematischen Theorie der Fraktale. Stichpunkte aus dem Inhalt: Riemannsche Flächen, Iterierte holomorphe Abbildungen, Lokale Fixpunkttheorie, Periodische Punkte: Globale Theorie, Struktur der Fatou Menge, Julia Menge.
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Milnor's Textbook on Dynamics

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John Milnor

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