Half-iterate exp(z)-1: hypothese on growth of coefficients

[bo198214]

Also, you can't define julia's function only by \(j(f)=jf'\), to distinct a julia function, you'd need to define an initial value, and claim it's not multiplied by any of the form \(\theta(\alpha(z))\) where theta is 1-periodic and alpha is the abel funct.

The Julia function is considered as a formal powerseries here. The coefficients recursively obtained from the above equation are unique if we  set \(j_k=0\)  for \(k< m\) and \(j_m=f_m\) where  \(f(z) = z + f_m z^m + f_{m+1} z^{m+1}+ ...,\quad f_m\neq 0\). There is no theta-ambiguity here - like with the regular iteration at a fixed point. For parabolic iteration the condition is that the iterates are asymptotically analytic at the fixed point.

But the normal thing is that these Abel-Functions/Fatou-Coordinates are different from each other (and hence the iterates). Only in exceptional cases (e.g. LFTs) they agree.

Please lemme know what is this and why~ 

Its written in Milnor "Dynamics in one complex variable"
And btw I can not read your quotes when you use [size=1]!