Perturbed Fatou coordinates

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

'perturbed' Fatou coordinates refer to a function that is 'disturbed' by some small $c_0\in\C$: $$f(z)=c_0 + z + c_{m+1} z^{m+1} + o(z^{m+1})$$

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

For the details of the case $m=1$ I refer to Shishikuras presentation.