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 new results from complex dynamics bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/19/2013, 11:22 PM (This post was last modified: 02/19/2013, 11:27 PM by bo198214.) (03/20/2010, 08:12 AM)mike3 Wrote: However, for it to be not real valued in $(1, e^{1/e}]$ and real valued in $(e^{1/e}, \infty)$ would imply there is a singularity/branchpoint at $b = e^{1/e}$, 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 $\mathcal{F}_1$, 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 $\mathcal{F}_1$. And inside the STR I guess the fixpoints are rather horizontally aligned which also means its not in $\mathcal{F}_1$. (Of course I always mean here the corresponding meaning of $\mathcal{F_1}$ 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 $b=e^{1/e}$ 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 $b^z$. « Next Oldest | Next Newest »

 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 bo198214 - 03/08/2010, 11:59 AM RE: new results from complex dynamics - by mike3 - 03/10/2010, 03:10 AM RE: new results from complex dynamics - by bo198214 - 03/10/2010, 11:13 AM RE: new results from complex dynamics - by mike3 - 03/20/2010, 08:12 AM RE: new results from complex dynamics - by bo198214 - 03/21/2010, 01:04 PM RE: new results from complex dynamics - by mike3 - 03/21/2010, 11:03 PM RE: new results from complex dynamics - by bo198214 - 03/22/2010, 12:07 AM RE: new results from complex dynamics - by mike3 - 03/22/2010, 10:14 AM RE: new results from complex dynamics - by Gottfried - 03/22/2010, 01:32 PM RE: new results from complex dynamics - by bo198214 - 03/22/2010, 01:37 PM RE: new results from complex dynamics - by mike3 - 03/23/2010, 07:50 AM RE: new results from complex dynamics - by bo198214 - 03/23/2010, 10:32 AM RE: new results from complex dynamics - by bo198214 - 02/19/2013, 11:22 PM RE: new results from complex dynamics - by sheldonison - 02/20/2013, 05:25 AM RE: new results from complex dynamics - by sheldonison - 02/21/2013, 09:01 AM RE: new results from complex dynamics - by bo198214 - 02/24/2013, 08:57 PM RE: new results from complex dynamics - by sheldonison - 02/25/2013, 01:42 AM RE: new results from complex dynamics - by sheldonison - 03/21/2012, 08:58 PM RE: new results from complex dynamics - by sheldonison - 04/10/2012, 10:37 PM RE: new results from complex dynamics - by sheldonison - 04/15/2012, 04:30 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 sheldonison - 04/17/2012, 12:48 PM RE: new results from complex dynamics - by mike3 - 04/18/2012, 02:19 AM

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