(03/21/2010, 11:03 PM)mike3 Wrote: So does this mean the original hypothesis that it "depends holomorphically on " (apparently across 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 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 satisfies:

and

where and are the two fixed points of .

(which is exactly what Dmitrii uses for his construction of the superfunction.)

This implies that is only real-valued if because real-analytic functions satisfy .