# Perturbed Fatou coordinate

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While a Fatou coordinate is (the injective part of) a principal Abel functions 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 Shishikuras presentation.