Quasiconformal Surgery in Holomorphic Dynamics

Bodil Branner, Núria Fagella

• Publisher: Cambridge University Press
• Year: 2014
• Print ISBN: 9781107042919
• eBook ISBN: 9781107597204
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Since its introduction in the early 1980s quasiconformal surgery has become a major tool in the development of the theory of holomorphic dynamics, and it is essential background knowledge for any researcher in the field. In this comprehensive introduction the authors begin with the foundations and a general description of surgery techniques before turning their attention to a wide variety of applications. They demonstrate the different types of surgeries that lie behind many important results in holomorphic dynamics, dealing in particular with Julia sets and the Mandelbrot set. Two of these surgeries go beyond the classical realm of quasiconformal surgery and use trans-quasiconformal surgery. Another deals with holomorphic correspondences, a natural generalization of holomorphic maps. The book is ideal for graduate students and researchers requiring a self-contained text including a variety of applications. It particularly emphasises the geometrical ideas behind the proofs, with many helpful illustrations seldom found in the literature.

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Imprints
• Dedication
• Contents
• List of contributors
• Preface
• Acknowledgements
• List of symbols
• Introduction
• 1 Quasiconformal geometry
• 2 Boundary behaviour of quasiconformal maps: extensions and interpolations
• 3 Preliminaries on dynamical systems and actions of Kleinian groups
• 4 Introduction to surgery and first occurrences
• 5 General principles of surgery
• 6 Soft surgeries
• 7 Cut and paste surgeries
• 8 Cut and paste surgeries with sectors
• 9 Trans-quasiconformal surgery
• References
• Index