Cours SpécialisésCollection SMF 2001; 160 pp; softcover Number: 7 ISBN10: 286883521X ISBN13: 9782868835215 List Price: US$33 Member Price: US$26.40 Order Code: COSP/7
 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 nonwandering 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. Readership 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
