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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.


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Please explain the exact meaning of the expression "generated from both fixed points", and what is a "superfunction". –  Alexandre Eremenko Aug 11 '12 at 14:03

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I just taught Leau's Flower Theorem today...

Your best bet to understand this map is to read Milnor's book. Click here for a copy of the original notes (which are less polished than the book). The chapter on parabolic points is perhaps the most pedestrian, but even that is clearer than other books!

To give you an idea of the flow of ideas, Milnor considers an analytic function $f$ with a parabolic fixed point at 0. He first assumes the multiplier is $\lambda = 1$, and proves the attraction/repulsion picture directly from the power series (call this result T1). Some corollaries follow, and then he describes the case $\lambda = {\rm e}^{2\pi{\rm i}p/q}$, which is the same as before, except that the petals are permuted by $f$ instead of determining independent basins of attraction. THIS IS YOUR MAP. The multiplier is $\lambda = {\rm e}^{2\pi{\rm i}/5}$, so the petals map counterclockwise onto each other.

The fifth iterate of your $f$ has a fixed point with multiplier 1 so each petal maps into itself. This is the situation where the Abel function makes sense. Milnor shows how to construct explicitly this coordinate change by refining the computations from the proof of T1. THIS IS THE ANSWER TO YOUR QUESTION.

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