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Quasiconformal Surgery in Holomorphic Dynamics by Núria Fagella, Bodil Branner

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Exercises Section 7.6

7.6.1  It is shown in Figure 7.21 how one can construct a branched covering of degree two of an annulus onto a disc such that each of the boundaries of the annulus is mapped onto the boundary of the disc by degree one. Generalize this idea to construct a degree 2d branched covering, mapping each of the boundaries of the annulus onto the boundary of the disc by degree d.

Hint: Draw 2d segments in the annulus similar to the two drawn in the figure. Map each connected component of the annulus minus the segments bijectively onto the disc minus the two slits, and map each segment onto one of the two critical slits, alternately. Alternatively, change coordinates so that the annulus separates 0 and ∞ and precompose the degree 2 ...

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