Discrete Holomorphic Local Dynamical Systems

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Editors Prof. Graziano Gentili Università di Firenze Dipartimento di Matematica “U. Dini” viale Morgagni 67/A 50134 Firenze Italy gentili@math.unifi.it Prof. Jacques Guenot Università della Calabria Dipartimento di Strutture Ponte Pietro Bucci 39 87036 Arcavacata di Rende (CS) Edificio 37B - 38B CDE - 39B - 39C Italy guenot@libero.it ISBN: 978-3-642-13170-7 e-ISBN: 978-3-642-13171-4 DOI: 10.1007/978-3-642-13171-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010930280 Prof. Giorgio Patrizio Università di Firenze Dipartimento di Matematica “U. Dini” viale Morgagni 67/A 50134 Firenze Italy patrizio@math.unifi.it Mathematics Subject Classification (2000): 37FXX, 32AXX, 32QXX, 32H50, 30DXX, 31BXX c○ Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper springer.com
Preface The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector field) acting on it. The study of a holomorphic dynamical system consists in describing the asymptotic behavior of the system, associating it with some invariant objects (easy to compute) which describe the dynamics and classify the possible holomorphic dynamical systems supported by a given manifold. The behavior of a holomorphic dynamical system is pretty much related to the geometry of the ambient manifold (for instance, hyperbolic manifolds do no admit chaotic behavior, while projective manifolds have a variety of different chaotic pictures). The techniques used to tackle such problems are of various kinds: complex analysis, methods of real analysis, pluripotential theory, algebraic geometry, differential geometry, topology. To cover all the possible points of view of the subject in a unique occasion has become almost impossible, and the CIME session in Cetraro on Holomorphic Dynamical Systems was not an exception. On the other hand the selection of the topics and of the speakers made it possible to focus on a number of important topics in the discrete and in the continuous setting, both for the local and for the global aspects, providing a fascinating introduction to many key problems of the current research. The CIME Course aimed to give an ample description of the phenomena occurring in central themes of holomorphic dynamics such as automorphisms and meromorphic self-maps of projective spaces, of entire maps on complex spaces and holomorphic foliations in surfaces and higher dimensional manifolds, enlightening the different techniques used and bringing the audience to the borderline of current research topics. This program, with its interdisciplinary characterization, drew the attention and the participation of young researchers and experienced mathematicians coming from different backgrounds: complex analysis and geometry, topology, ordinary differential equations and number theory. We are sure that the present volume will serve the same purpose. We briefly describe here the papers that stemmed from the courses and constitute the Chapters of this volume. v
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Page 7 and 8: Preface vii here are of independent
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44 Marco Abate In [ABT1] we proved
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46 Marco Abate and dimE = 1; but th
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48 Marco Abate It turns out that σ
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50 Marco Abate and the eigenvalues
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52 Marco Abate [CD] Coman, D., Dabi
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54 Marco Abate [Ni] Nishimura, Y.:
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Dynamics of Rational Surface Automo
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166 Tien-Cuong Dinh and Nessim Sibo
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296 Dierk Schleicher made possible
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298 Dierk Schleicher In a brief app
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300 Dierk Schleicher of f are discr
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302 Dierk Schleicher set with at le
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304 Dierk Schleicher logarithmic si
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314 Dierk Schleicher The following
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316 Dierk Schleicher Fig. 1 The wig
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318 Dierk Schleicher Examples of Fa
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320 Dierk Schleicher 8. if f has a
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322 Dierk Schleicher 5 Hausdorff Di
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326 Dierk Schleicher One way to int
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328 Dierk Schleicher indifferent or
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330 Dierk Schleicher Definition 7.1
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332 Dierk Schleicher Conjecture 7.1
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334 Dierk Schleicher Remark 8.13. T
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336 Dierk Schleicher [BH99] Bergwei
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338 Dierk Schleicher [Mi06] Milnor,
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List of Participants 1. Abate Marco
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Lecture Notes in Mathematics For in
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Vol. 1911: A. Bressan, D. Serre, M.
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LECTURE NOTES IN MATHEMATICS Edited
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Discrete Holomorphic Local Dynamical Systems

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