**Contemporary Mathematics**

Volume: 525;
2010;
161 pp;
Softcover

MSC: Primary 30; 37; 47; 76;

**Print ISBN: 978-0-8218-4809-8
Product Code: CONM/525**

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

**Electronic ISBN: 978-0-8218-8204-7
Product Code: CONM/525.E**

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# Five Lectures in Complex Analysis

Share this page *Edited by *
*Manuel D. Contreras; Santiago Díaz-Madrigal*

This volume contains state-of-the-art survey papers in complex
analysis based on lectures given at the Second Winter School on
Complex Analysis and Operator Theory held in February 2008 at the
University of Sevilla, Sevilla, Spain.

Complex analysis is one of the most classical branches of
mathematical analysis and is closely related to many other areas of
mathematics, including operator theory, harmonic analysis, probability
theory, functional analysis and dynamical systems. Undoubtedly, the
interplay among all these branches gives rise to very beautiful and
deep results in complex analysis and its neighboring fields. This
interdisciplinary aspect of complex analysis is the central topic of
this volume.

This book collects the latest advances in five significant areas of
rapid development in complex analysis. The papers are: Local
holomorphic dynamics of diffeomorphisms in dimension one, by
F. Bracci, Nonpositive curvature and complex analysis, by
S. M. Buckley, Virasoro algebra and dynamics in the space of
univalent functions, by I. Markina and A. Vasil'ev,
Composition operators \(\heartsuit\) Toeplitz operators, by
J. H. Shapiro, and Two applications of the Bergman spaces
techniques, by S. Shimorin.

The papers are aimed, in particular, at graduate students with some
experience in basic complex analysis. They might also serve as
introductions for general researchers in mathematical analysis who may be
interested in the specific areas addressed by the authors. Indeed, the
contributions can be considered as up-to-the-minute reports on the current
state of the fields, each of them including many recent results which may
be difficult to find in the literature.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Readership

Graduate students and research mathematicians interested in complex analysis and operator theory.

# Table of Contents

## Five Lectures in Complex Analysis

- Contents vii8 free
- Preface ix10 free
- Local holomorphic dynamics of diffeomorphisms in dimension one 116 free
- Prologue 116
- 1. Formal Normal Forms 217
- 2. Holomorphic Dynamics 1429
- The hyperbolic case 1429
- The elliptic case: Cremer, Siegel diffeomorphisms and small divisors 1631
- The parabolic case: the Leau-Fatou flowers theorem 1833
- Écalle-Voronin holomorphic classification of germs tangent to the identity 2338
- Stability versus Linearizability 2641
- Diffeomorphisms of the circle 2641
- Pérez-Marco's construction 2843
- Naishul’s theorem 3146
- Douady-Hubbard’s Straightening Theorem 3146
- Yoccoz's proof of the qualitative version of the Siegel-Bruno-Yoccoz theorem 3449
- Arithmetic forms of Siegel-Bruno-Yoccoz’s theorem 3752

- 3. Topological normal forms 3853
- References 4156

- Nonpositive curvature and complex analysis 4358
- Virasoro algebra and dynamics in the space of univalent functions 85100
- Composition operators ♡ Toeplitz operators 117132
- Two applications of the Bergman spaces techniques 141156