Quasiconformal Surgery in Holomorphic Dynamics

A comprehensive introduction to quasiconformal surgery in holomorphic dynamics. Contains a wide variety of applications and illustrations.

Bodil Branner (Author), Núria Fagella (Author)

9781107042919, Cambridge University Press

Hardback, published 23 January 2014

428 pages, 30 b/w illus. 105 colour illus. 85 exercises
23.3 x 15.6 x 2 cm, 0.85 kg

'This worthwhile book, written by two of the main experts in this field with some contributions from some well-known researchers, gives a comprehensive introduction to the subject, from the foundations of the theory up to several important and representative applications … All in all, this book is a very welcome addition to the literature, and an excellent entrance point to the theory for any researcher interested in this subject.' Marco Abate, Zentralblatt MATH

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.

Preface
Introduction
1. Quasiconformal geometry
2. 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 with a contribution by X. Buff and C. Henriksen
7. Cut and paste surgeries with contributions by K. M. Pilgrim, Tan Lei and S. Bullett
8. Cut and paste surgeries with sectors with a contribution by A. L. Epstein and M. Yampolsky
9. Trans-quasiconformal surgery with contributions by C. L. Petersen and P. Haïssinsky
Bibliography
Symbol index
Index.

Subject Areas: Geometry [PBM], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB], Mathematics [PB]