Lets say I have $f(z)=z^2+c$, with $c=0.35676274578 + 0.32858194507i$. Then $f(z)$ has a fixed point $\kappa_0=0.15450849719 + 0.47552825815i$, which is rationally indifferent with a period $m=5$. The "Complex Dynamics" book by Lennart Carleson says you can normalize $f^{[5]}(z)$, which is parabolic with $5$ attracting petals, and $5$ repelling petals, but what about generating a Fatou Coordinate for $f(z)$ itself, for the rational case?

I am interested in what is known about $g(z)=f^{[\circ z]}(0)$, where $g(z)$ is generated from both fixed points of $f(z)$. *(The other fixed point of $f(z)$ is repelling, $\kappa_1=0.84549150281 - 0.47552825815i$).* 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.

Developed around the neutral $\kappa_0$ fixed point, we have $\lambda_0= \exp(\frac{2\pi i}{5})$, which would is a neutral fixed point with pseudo periodicity of 5. There is an Abel function, but only for $f^{\circ 5}(z)$, not for $f(z)$. Since 5 is a rational number, there is no Schroder function from the neutral fixed point. $$f(x+\kappa_0)= \lambda_0(x+\kappa_0) + (x+\kappa_0)^2 + \kappa_0$$

Developed around the other repelling $\kappa_1$ fixed point, we have $\lambda_1 \approx
1.691 - 0.9511i$, with a periodicity of $\approx 4.588 - 5.934i$. For this repelling fixed point, there is a well defined Schroder equation, which can be used to develop the the iterated function in the neighborhood of the fixed point.

$$f(x+\kappa_1)=\lambda_1(x+\kappa_1) + (x+\kappa_1)^2 + \kappa_1$$

The superfunction is the inverse of the Abel function; $$\;f^{[\circ z]}=\alpha^{-1}(z)\;\;\text{where} \;\;\alpha(f(z)=\alpha(z)+1$$

The desired $g(x)=f^{[\circ z]}$ function would combine both of these fixed points into a single function. $g(x)$ would approach the $\kappa_1$ fixed point and the $\lambda_1$ periodicity as $\Im(z) \to -i\infty$, and as $\Im(z) \to +\infty$, $g(z)$ would approach the $\kappa_0$ neutral fixed point with the periodicity approaching 5.

Another way this question might be viewed, is as perturbation of the $f(x)=x+x^2$ parabolic fixed point, where $f(x)=x+x^2+\delta;\;\;\;\delta\approx 0.1068 + 0.3286i$, for the equivalent $f(x)$ for this question. I think the relevant mathematical term would be a perturbed Fatou coordinate (for the Abel function); but at that point I am over my mathematics skill level. The reason why I asked the question, is because unproven methods used for Tetration can also be used to calculate such a $g(x)=f^{\circ z}$ function without being able to prove a solution should exist, or converge.