Combinations of Complex Dynamical Systems
This work is a research-level monograph whose goal is to develop a general combination, decomposition, and structure theory for branched coverings of the two-sphere to itself, regarded as the combinatorial and topological objects which arise in the classi
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Kevin M. Pilgrim
Combinations of Complex Dynamical Systems
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Author Kevin M. Pilgrim Department of Mathematics Indiana University Bloomington, IN 47401, USA e-mail: [email protected]
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Mathematics Subject Classification (2000): 37F20 ISSN 0075-8434 ISBN 3-540-20173-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10962459
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Preface
The goal of this research monograph is to develop a general combination, decomposition, and structure theory for branched coverings of the two-sphere to itself, regarded as the combinatorial and topological objects which arise in the classification of certain holomorphic dynamical systems on the Riemann sphere. It is intended for researchers interested in the classification of those complex one-dimensional dynamical systems which are in some loose sense tame, though precisely what this constitutes we leave open to interpretation. The program is motivated in general by the dictionary between the theories of iterated rational maps and Kleinian groups as holomorphic dynamical systems, and in particular by the structure theory of compact irreducible three-manifolds. By and large this work involves only topological/combinatorial notions. Apart from motivational discussions, the sole exceptions are (i) the construction of examples which is aided using complex dynamics in §9, and (ii) some familiarity with the Douady-Hubbard proof of Thurston’s characterization of rational functions in §§8.3.1 and §10. The combination and decomposition theory is developed for maps which are not necessarily postcritically finite. However, t
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