(03/20/2010, 08:12 AM)mike3 Wrote: However, for it to be not real valued in and real valued in would imply there is a singularity/branchpoint at , hence not holomorphic there, eh?

Ya I figured out that the holomorphy mentioned in Shishikuras article is only for functions that have the fixed points rather vertically aligned.

I added the corresponding text from Shishikuras chapters here: http://math.eretrandre.org/hyperops_wiki...sition_A.1

Proposition A.1 is about this holomorphy. The whole proposition A.1 is applicable only for functions which are in the class , which corresponds to having the fixpoints rather vertically aligned.

In our case this probably means: as long as the base is outside the Shell-Thron-Region it depends holomorphically on the base.

But as soon as we pass the Shell-Thron-Boundary the both fixpoints collapse and hence the function is no more . And inside the STR I guess the fixpoints are rather horizontally aligned which also means its not in .

(Of course I always mean here the corresponding meaning of if the first fixpoint is not set to 0.)

On the other hand if I remember, Sheldon posted somewhere that he also found a solution for the non-primary fixpoints. The secondary fixpoints however do not collapse into a horizontal fixpoint pair, but they remain vertical when passing on the real axis. Though I really wonder whether there can exist a sickle between these fixpoints, i.e. an area bounded by an (injective) curve between both fixpoints and its image under .