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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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Appendix E. Branched Coverings and Orbifolds

This will be an outline of definitions and results due to Thurston. (See Douady and Hubbard [1993].) If

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with n ≥ 1 and c ≠ 0, recall that the integer n = n(z0) is called the local degree of f at the point Z0. Thus n(z0) ≥ 2 if z0 is a critical point, and n(z0) = 1 otherwise. We will use ramified point as a synonym for critical value. Thus if f(z0) = w0 as above with local degree n ≥ 2, then w0 is a ramified point.

A holomorphic map p: S′→S between Riemann surfaces is called a covering map if each point of S has a connected neighborhood U which is evenly covered, in that each connected component of ...

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