## Appendix E. Branched Coverings and Orbifolds

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

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