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Complex Dynamics and Geometry
Dominique Cerveau, Université de Rennes, France, Étienne Ghys, Ecole Normale Supérieure de Lyon, France, and Nessim Sibony and Jean-Christophe Yoccoz, Université de Paris-Sud, Orsay, France
A co-publication of the AMS and Société Mathématique de France.
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SMF/AMS Texts and Monographs
2003; 197 pp; softcover
Volume: 10
ISBN-10: 0-8218-3228-X
ISBN-13: 978-0-8218-3228-8
List Price: US$65
Member Price: US$52
Order Code: SMFAMS/10
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In the last twenty years, the theory of holomorphic dynamical systems has had a resurgence of activity, particularly concerning the fine analysis of Julia sets associated with polynomials and rational maps in one complex variable. At the same time, closely related theories have had a similar rapid development, for example the qualitative theory of differential equations in the complex domain.

The meeting, "Etat de la recherche", held at Ecole Normale Supérieure de Lyon, presented the current state of the art in this area, emphasizing the unity linking the various sub-domains. This volume contains four survey articles corresponding to the talks presented at this meeting.

D. Cerveau describes the structure of polynomial differential equations in the complex plane, focusing on the local analysis in neighborhoods of singular points. E. Ghys surveys the theory of laminations by Riemann surfaces which occur in many dynamical or geometrical situations. N. Sibony describes the present state of the generalization of the Fatou-Julia theory for polynomial or rational maps in two or more complex dimensions. Lastly, the talk by J.-C. Yoccoz, written by M. Flexor, considers polynomials of degree \(2\) in one complex variable, and in particular, with the hyperbolic properties of these polynomials centered around the Jakobson theorem.

This is a general introduction that gives a basic history of holomorphic dynamical systems, demonstrating the numerous and fruitful interactions among the topics. In the spirit of the "Etat de la recherche de la SMF" meetings, the articles are written for a broad mathematical audience, especially students or mathematicians working in different fields. This book is translated from the French edition by Leslie Kay.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students and research mathematicians interested in complex geometry and holomorphic dynamical systems.

Table of Contents

  • Holomorphic dynamical systems
  • Codimension-one holomorphic folations, reduction of singularities in low dimensions, and applications
  • Riemann surface laminations
  • Dynamics of rational maps on \(\mathbb{P}^k\)
  • Dynamics of quadratic polynomials
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