List of contributors ........................................ ix
Preface ..................................................... xi
Acknowledgements .......................................... xiii
List of symbols ............................................. xv
Introduction ................................................. 1
1 Quasiconformal geometry ...................................... 7
1.1 The linear case: Beltrami coefficients and ellipses ..... 7
1.2 Almost complex structures and pullbacks ................ 14
1.3 Quasiconformal mappings ................................ 20
1.4 The Integrability Theorem .............................. 39
1.5 An elementary example .................................. 49
1.6 Quasiregular mappings .................................. 55
1.7 Application to holomorphic dynamics .................... 60
2 Boundary behaviour of quasiconformal maps: extensions and
interpolations .............................................. 64
2.1 Preliminaries: quasisymmetric maps and quasicircles .... 65
2.2 Extensions of mappings from their domains to their
boundaries ............................................. 69
2.3 Extensions of boundary maps ............................ 77
3 Preliminaries on dynamical systems and actions of Kleinian
groups ...................................................... 92
3.1 Conjugacies and equivalences ........................... 94
3.2 Circle homeomorphisms and rotation numbers ............. 97
3.3 Holomorphic dynamics: the phase space ................. 105
3.4 Families of holomorphic dynamics: parameter spaces .... 126
3.5 Actions of Kleinian groups and the Sullivan
dictionary ............................................ 133
4 Introduction to surgery and first occurrences .............. 147
4.1 Changing the multiplier of an attracting cycle ........ 151
4.2 Changing superattracting cycles to attracting ones .... 162
4.3 No wandering domains for rational maps ................ 169
5 General principles of surgery .............................. 179
5.1 Shishikura principles ................................. 180
5.2 Sullivan's Straightening Theorem ...................... 184
5.3 Non-rational maps ..................................... 186
6 Soft surgeries ............................................. 188
6.1 Deformation of rotation rings
Xavier Buff and Christian Henriksen ................... 189
6.2 Branner-Hubbard motion ................................ 207
7 Cut and paste surgeries .................................... 218
7.1 Polynomial-like mappings and the Straightening
Theorem ............................................... 219
7.2 Gluing Siegel discs along invariant curves ............ 224
7.3 Turning Siegel discs into Herman rings ................ 235
7.4 Simultaneous uniformization of Blaschke products ...... 244
7.5 Gluing along continua in the Julia set ................ 248
7.6 Disc-annulus surgery on rational maps
Kevin M. Pilgrim and Tan Lei .......................... 267
7.7 Perturbation and counting of non-repelling cycles ..... 282
7.8 Mating a group with a polynomial
Shaun Bullett ......................................... 291
8 Cut and paste surgeries with sectors ....................... 307
8.1 Preliminaries: sectors and opening modulus ............ 308
8.2 Creating new critical points .......................... 320
8.3 Embedding limbs of M into other limbs ................. 337
8.4 Intertwining surgery
Adam Epstein and Michael Yampolsky .................... 343
9 Trans-quasiconformal surgery ............................... 364
9.1 David maps and David-Beltrami differentials ........... 365
9.2 Siegel discs via trans-quasiconformal surgery
Carsten Lunde Petersen ................................ 370
9.3 Turning hyperbolics into parabolics
Peter Haïssinsky ...................................... 385
References ................................................. 400
Index ...................................................... 408
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