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new results from complex dynamics
#22
(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 .
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Messages In This Thread
new results from complex dynamics - by bo198214 - 01/25/2010, 07:30 PM
RE: new results from complex dynamics - by mike3 - 01/26/2010, 06:21 AM
RE: new results from complex dynamics - by mike3 - 03/10/2010, 03:10 AM
RE: new results from complex dynamics - by mike3 - 03/20/2010, 08:12 AM
RE: new results from complex dynamics - by mike3 - 03/21/2010, 11:03 PM
RE: new results from complex dynamics - by mike3 - 03/22/2010, 10:14 AM
RE: new results from complex dynamics - by mike3 - 03/23/2010, 07:50 AM
RE: new results from complex dynamics - by bo198214 - 02/19/2013, 11:22 PM
RE: new results from complex dynamics - by mike3 - 04/16/2012, 04:21 AM
RE: new results from complex dynamics - by mike3 - 04/17/2012, 07:51 AM
RE: new results from complex dynamics - by mike3 - 04/18/2012, 02:19 AM

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