new results from complex dynamics

[bo198214]

So does this mean the original hypothesis that it "depends holomorphically on \( b \)" (apparently across \( e^{1/e} \) since you contrast this behavior with that of the "usual" regular iteration) was wrong?

Quite probably. I am still not sure about numeric computation of the bipolar Abel/super function. But my guess that it is not real valued for \( b<e^{1/e} \) stems also from a statement in Shishikura's article (proposition 3.2.3) which quite resembles Dmitrii Kouznetsov's algorithm to compute the superfunction. It says that the inverse of the bipolar Abel function, i.e. the superfunction \( \phi_f \) satisfies:
\( \lim_{\Im(w)\to +\infty}\phi_f(w) = p \) and \( \lim_{\Im(w)\to -\infty} \phi_f(w) = q \)
where \( p \) and \( q \) are the two fixed points of \( f \).
(which is exactly what Dmitrii uses for his construction of the superfunction.)
This implies that \( \phi_f \) is only real-valued if \( \overline{p}=q \) because real-analytic functions \( \phi \) satisfy \( \overline{\phi(z)}=\phi(\overline{z}) \).