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Quasiconformal Surgery in Holomorphic Dynamics by Núria Fagella, Bodil Branner

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5

General principles of surgery

The previous chapters contain examples of quasiregular maps for which an invariant almost complex structure with bounded dilatation exists. It is a natural question to ask under which conditions this can be accomplished, so that a holomorphic dynamical copy is obtained by means of the Integrability Theorem.

We first present two statements, both due to Shishikura, describing typical scenarios in surgery constructions. The first one was called the Fundamental Lemma of Quasiconformal Surgery in [Sh1]. It applies to most procedures of cut and paste surgery, where we paste together holomorphic and quasiregular mappings. This is the case, for example, in Section 4.2, where we glue a Blaschke product into a basin of attraction. ...

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