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Quasiconformal Surgery in Holomorphic Dynamics by Núria Fagella, Bodil Branner

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4

Introduction to surgery and first occurrences

In this chapter we describe what quasiconformal surgery is about and explain the surgeries which were developed as the first occurrences of this technique: the parametrization of hyperbolic components of the Mandelbrot set and the proof of the No Wandering Domains Theorem.

Generalities What is known as quasiconformal surgery in holomorphic dynamics is a technique commonly used to construct holomorphic maps with prescribed dynamics. The ‘prescribed dynamics’ are given by a map f which in general is not holomorphic, although it may be. We shall refer to f as the model map. The word surgery appears because one may need to ‘cut’ and ‘paste’ different spaces and maps together to construct f. This is ...

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