# Complex Dynamics and Geometry

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*Dominique Cerveau; Étienne Ghys; Nessim Sibony; Jean-Christophe Yoccoz*

A co-publication of the AMS and Société Mathématique de France

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.