Lets say I have f(z)=z^2+c, with c=0.35676274578 + 0.32858194507i. Then f(z) has a fixed point=0.15450849719 + 0.47552825815i, which is rationally indifferent with a period=5. The "Complex Dynamics" book by Lennart Carleson says you can normalize $f^5(z)$, which is parabolic with five attracting petals, and 5 repelling petals, but what about generating a Fatour Coordinate for f(z) itself, for the rational case?

I am interested in what is known about $g(z)=f^{z}(0)$, where g(z) is generated from both fixed points of f(z). The other fixed point of f(z) is repelling, =0.84549150281 - 0.47552825815i. I am interested in learning what the current theory says about such a Superfunction/Fatou coordinate, developed from both fixed points. I have a hunch that such a super function or Abel function of f(z) exists, and has interesting properties, but I'm trying to find references for such problems.

Sheldon