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Lets us define $f(z)=z^2+z+c$ with real valued c>0 and iterate the function f(z). Then the Abel function for $f(z)$ is $$\alpha(z)\;\; \text{where}\;\; \alpha(f(z))=\alpha(z)+1$$ $$ f^{[\circ z]} = \alpha^{-1}(z)$$

Then $f(z)$ has two fixed points $\pm \sqrt{-c}\;$ If c<0, then we could generate the $\alpha(z)$ Abel function from either fixed point. Both Abel functions would be real valued at the real axis.

But I am more interested in the case c>0, where the two fixed points are complex conjugates of each other and both fixed points are repelling. Might there be some literature on a real valued $\alpha(z)$ function extending on a sickle between the two fixed complex points? For example, consider c=0.01, with repelling fixed points of $\pm \frac{i}{10}$.

And then the iterated function $f^{[\circ z]}=\alpha^{-1}(z)$ would also be real valued even though the two fixed points are complex. I believe this $\alpha(z)$ is called the perturbed fatou coordinate. It can be proven to be unique. See Trapman Another similar problem exists in uniquely extending Tetration to real values for bases>exp(1/e), where the two primary fixed points are complex conjugates, which is what Trapmann's paper is about. That is usually referred to as Kneser's solution.

Calculating these "Perturbed" fatou coordinates is a little bit tricky, so there might be some interesting literature in that area. Also, one can also ask what happens when we consider complex values of c, and then generate the Abel function between both fixed points. I attempted to ask a question about that some years ago without much success. It occurs to me that I asked too complicated a question. https://mathoverflow.net/questions/93411/fatou-coordinate-for-function-with-rationally-indifferent-fixed-point-and-repel

For c=0.01, here is a graph of the function of interest, extended to real values of z, and ranging from $f^{[\circ z]}=-0.5 \to f^{[\circ z]}=+1\;\;$ To the left of this graph, the $f^{[\circ z]}$ is no longer real valued. I can supply the Taylor series of the Abel function as well; which has been normalized so that $\alpha(-0.5)=0$.

{abel_function=
        11.70110299224329
+x^ 1*  99.83328272224278
+x^ 2*  50.16661903263718
+x^ 3* -3344.664073143450
+x^ 4* -2491.349225554010
+x^ 5*  200657.7567965004
+x^ 6*  166121.7650600110
+x^ 7* -14332744.02701906
+x^ 8* -12459064.34166609
+x^ 9*  1114768889.027114
+x^10*  996725250.9915228
+x^11* -91208363731.48674
+x^12* -83060437574.16705
+x^13*  7717630777381.277
+x^14*  7119466077684.030
+x^15* -668861334039999.9
+x^16* -622953281796252.0
+x^17*  5.901717653294016 E16
+x^18*  5.537362504855256 E16
+x^19* -5.280484216105162 E18
+x^20* -4.983626254369730 E18 }

iterating x^2+x+0.01

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    $\begingroup$ I just found a rather recent Ph.D. thesis on the subject of Fatou coordinates (math.toronto.edu/graduate/Dudko-thesis.pdf). $\endgroup$ – Jean Marie Jan 15 '17 at 6:59
  • $\begingroup$ Thanks, I will read that. It looks like it might be on topic. A few years ago, I wrote a pari-gp program to compute these Abel functions for arbitrary complex values of c, or for tetration.... $\endgroup$ – Sheldon L Jan 15 '17 at 7:04
  • $\begingroup$ @JeanMarie I don't think Dudko's paper is applicable. However in a book publiched in 2002, "The Mandelbrot Set, Theme and Variations", there is an article by "M. Shishikura", "Bifurcation of parabolic fixed points", which would seem to be the correct reference. Perhaps someone is familiar with Shishikura's work. I have the book, but I can't follow the details. $\endgroup$ – Sheldon L Jun 25 '17 at 23:47

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