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Quasiconformal Surgery in Holomorphic Dynamics by Núria Fagella, Bodil Branner

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9

Trans-quasiconformal surgery

In this chapter we shall refer to the classical Integrability Theorem of Morrey, Bojarski, Ahlfors and Bers (see Section 1.4) as the qc Integrability Theorem in order to distinguish it from its generalization formulated in Theorem 9.7. We shall refer to the latter as the trans-qc Integrability Theorem or the David Integrability Theorem. The qc Integrability Theorem is the key element that makes quasiconformal surgery such a successful technique in holomorphic dynamics. Recall that this remarkable result establishes conditions under which a Beltrami differential μ can be integrated by a quasiconformal homeomorphism ϕ. This means that ϕ must satisfy the Beltrami equation

or equivalently, the pullback of μ0 ≡ 0 under ...

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