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Quasiconformal Surgery in Holomorphic Dynamics

Book Description

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

  1. Cover
  2. Half Title
  3. Series
  4. Title
  5. Copyright
  6. Dedication
  7. Contents
  8. List of contributors
  9. Preface
  10. Acknowledgements
  11. List of symbols
  12. Introduction
  13. 1 Quasiconformal geometry
    1. 1.1 The linear case: Beltrami coefficients and ellipses
    2. 1.2 Almost complex structures and pullbacks
    3. 1.3 Quasiconformal mappings
    4. 1.4 The Integrability Theorem
    5. 1.5 An elementary example
    6. 1.6 Quasiregular mappings
    7. 1.7 Application to holomorphic dynamics
  14. 2 Boundary behaviour of quasiconformal maps: extensions and interpolations
    1. 2.1 Preliminaries: quasisymmetric maps and quasicircles
    2. 2.2 Extensions of mappings from their domains to their boundaries
    3. 2.3 Extensions of boundary maps
  15. 3 Preliminaries on dynamical systems and actions of Kleinian groups
    1. 3.1 Conjugacies and equivalences
    2. 3.2 Circle homeomorphisms and rotation numbers
    3. 3.3 Holomorphic dynamics: the phase space
    4. 3.4 Families of holomorphic dynamics: parameter spaces
    5. 3.5 Actions of Kleinian groups and the Sullivan dictionary
  16. 4 Introduction to surgery and first occurrences
    1. 4.1 Changing the multiplier of an attracting cycle
    2. 4.2 Changing superattracting cycles to attracting ones
    3. 4.3 No wandering domains for rational maps
  17. 5 General principles of surgery
    1. 5.1 Shishikura principles
    2. 5.2 Sullivan’s Straightening Theorem
    3. 5.3 Non-rational maps
  18. 6 Soft surgeries
    1. 6.1 Deformation of rotation rings
    2. 6.2 Branner–Hubbard motion
  19. 7 Cut and paste surgeries
    1. 7.1 Polynomial-like mappings and the Straightening Theorem
    2. 7.2 Gluing Siegel discs along invariant curves
    3. 7.3 Turning Siegel discs into Herman rings
    4. 7.4 Simultaneous uniformization of Blaschke products
    5. 7.5 Gluing along continua in the Julia set
    6. 7.6 Disc-annulus surgery on rational maps
    7. 7.7 Perturbation and counting of non-repelling cycles
    8. 7.8 Mating a group with a polynomial
  20. 8 Cut and paste surgeries with sectors
    1. 8.1 Preliminaries: sectors and opening modulus
    2. 8.2 Creating new critical points
    3. 8.3 Embedding limbs of M into other limbs
    4. 8.4 Intertwining surgery
  21. 9 Trans-quasiconformal surgery
    1. 9.1 David maps and David–Beltrami differentials
    2. 9.2 Siegel discs via trans-quasiconformal surgery
    3. 9.3 Turning hyperbolics into parabolics
  22. References
  23. Index