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

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§9. Böttcher’s Theorem and Polynomial Dynamics

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

image

with n ≥ 2 and an ≠ 0, where the integer n is called the local degree.

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 image wn throughout some neighborhood of zero. Furthermore, ...

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