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Quasiconformal surgery in holomorphic dynamics. With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. (English) Zbl 1319.37003

Cambridge Studies in Advanced Mathematics 141. Cambridge: Cambridge University Press (ISBN 978-1-107-04291-9/hbk; 978-1-107-33760-2/ebook). xvii, 413 p. (2014).
Quasiconformal surgery is an essential tool in one-dimensional holomorphic dynamics, with many various applications. Roughly speaking, quasiconformal surgery is a technique allowing to build holomorphic maps with prescribed dynamics, by putting together certain model maps which are locally quasiconformal. 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.
To be as self-contained as possible, the book starts with three background chapters, two on quasiconformal maps and one on dynamical systems and Kleinian groups, recalling the necessary material and providing proofs when not readily available in the literature. The next two chapters introduce the main ideas in quasiconformal surgery, first discussing in detail the (historically) first main results proved by means of quasiconformal surgery, and then presenting general statements describing when quasiconformal surgery can be performed. The last four chapters form the core of the book. Including contributions from other experts, the authors present several important applications of quasiconformal surgery, starting from relatively easy surgeries (the so-called soft surgeries) and arriving up to applications of techniques (trans-quasiconformal surgeries) going even beyond the classical realm of quasiconformal maps.
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.
Reviewer: Marco Abate (Pisa)

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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