This section studies the case of a *superattracting* fixed point, with multiplier λ equal to zero. As usual, we can choose a local uniformizing parameter *z* with fixed point *z* = 0. Thus our map takes the form

with *n* ≥ 2 and *a _{n}* ≠ 0, where the integer

**Theorem 9.1 (Böttcher [1904]).*** *With f as above, there exists a local holomorphic change of coordinate w* = *φ*(*z*), *with φ;*(0) = 0, *which conjugates f to the nth power map w* *w ^{n} throughout some neighborhood of zero. Furthermore, ...*

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