View source for Perturbed Fatou coordinate

While a [[Fatou coordinate]] is (the injective part of) a [[principal Abel function]]s of a holomorphic function $f$ at a given fixpoint,
for simplicity this fixpoint is assumed to be at 0:
$$f(z)= z + c_{m+1} z^{m+1} + o(z^{m+1}),$$

the 'perturbed' Fatou coordinate is an [[Abel function]] of a function that is 'disturbed' by some small $c\in\C$:
$$f_c(z)=c + z + c_{m+1} z^{m+1} + o(z^{m+1})$$
which converges to the $2m$ Fatou coordinates of $f$ (on the [[Leau-Fatou flower]]) for $c\to 0$.

The function $f_c$ usually has $2m$ fixpoints in a vicinity of 0.

For the details of the case $m=1$ I refer to [[Shishikura_perturbed_Fatou_coordinates|Shishikuras presentation]].