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Rudiments de Dynamique Holomorphe
François Berteloot, Université Paul Sabatier (Toulouse III), France, and Volker Mayer, Université de Lille I, France
A publication of the Société Mathématique de France.
Cours Spécialisés--Collection SMF
2001; 160 pp; softcover
Number: 7
ISBN-10: 2-86883-521-X
ISBN-13: 978-2-86883-521-5
List Price: US$33
Member Price: US$26.40
Order Code: COSP/7
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This book is an introduction to rational iteration theory. In the first four chapters, the authors deal with the classical theory. The basic properties of the Julia set and its complement, the Fatou set, are presented; the highest points of the treatment are the classification of the components of the Fatou set and Sullivan's non-wandering theorem.

The second part of the book studies several topics in more detail. The authors begin by considering at length two classes of rational maps: the chaotic maps and the hyperbolic maps. In the closing chapters, they include respectively a study of holomorphic families of rational maps with a view to discussing Fatou's famous problem concerning the density of hyperbolic maps and an exposition of the methods of potential theory, touching on questions of ergodicity, which may serve as a preparation for generalizations in higher dimensions.

A number of the developments treated here appear for the first time in book form. Several original proofs are presented.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in dynamical systems and ergodic theory.

Table of Contents

  • Introduction
  • La dichotomie dynamique de Fatou et Julia
  • Dynamiques locales et composantes de Fatou
  • Ensemble de Julia
  • Classification des composantes de Fatou
  • Fractions rationnelles chaotiques
  • Fractions rationnelles hyperboliques
  • Familles holomorphes de fractions rationnelles
  • Le point de vu potentialiste
  • Mesure et dimension de Hausdorff
  • Applications quasiconformes et structures conformes
  • Quelques points de théorie du potentiel
  • Bibliographie
  • Index
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