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Quasiconformal Surgery in Holomorphic Dynamics by Núria Fagella, Bodil Branner

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7

Cut and paste surgeries

In this chapter we treat several different applications of cut and paste surgery. The general principle is to paste together dynamics of different maps, under the condition that they agree on the boundaries of their distinct domains of definition.

Section 7.1 explains one of the earliest uses of quasiconformal mappings in holomorphic dynamics, due to Douady and Hubbard. It is especially relevant as a building block in many other applications. This celebrated result justifies why polynomial Julia sets appear in the dynamical plane of many non-polynomial families. The parameter version of the Straightening Theorem explains why one can also find copies of the Mandelbrot set in parameter spaces other than that of the quadratic ...

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