Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Question

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.

Answer 1

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.

Answer 2

I completely support Rodrigo A. Pérez' answer. I will address the issue of multiple fixed points. In unpublished research from twenty years ago, I reviewed the veracity of my work on extending tetration to the complex numbers. I was looking for evidence that my extension produced the same results regardless of the fixed point used.

Consider two neighboring fixed points $S$ and $T$ for $a^z$, the exponential map. Typically both $S$ and $T$ are hyperbolic fixed points, thus the dynamics in their neighborhood is determined by Schröder's functional equation. Close to $S$ we have some $S+\alpha_S \lambda{_S}^t$, where time zero is $S+\alpha_S$ and $\lambda_S$ is the multiplier. Close to $T$ we have $T+\alpha_T \lambda{_T}^t$.

Usually the region between $S$ and $T$ displays very complex behavior. Therefore the extension to tetration needs to be able to display the simple dynamics close to the fixed points as well. At the same time the complexity of the intervening region indicated that a tetrational extension must be taken to many terms. In order to reduce the entropy of the system I set $a = 1.01$. This provided a system with neighboring fixed points, but minimized the complicated dynamics in between them.

Given $S$ and $\lambda{_S}$ I computed enough terms in the tetration extension that I was able to obtain a close estimate for $T$ and $\lambda{_T}$. In other words I had experimental results indicating extensions of tetration can give the same results regardless of which fixed point was started from.