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 new results from complex dynamics bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 03/10/2010, 11:13 AM (This post was last modified: 03/10/2010, 11:19 AM by bo198214.) (03/10/2010, 03:10 AM)mike3 Wrote: Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in $(1, e^{1/e}]$ (after all, if it's analytic at $e^{1/e}$ and the regular isn't, the two can't be equal if they're both analytic in that interval.). Oh, my explanation was misunderstandable, the bipolar Abel function is always holomorphic and injective on a sickel between the two fixed points. But it may not be real on the real axis, or perhaps not even defined on the real axis, in the case of two real fixed points. You know the sickel would be above or below the real axis or possibly even wind around the fixed points, so being defined on pieces of the real axis; and it is not clear whether it can be extended to the real axis between the fixed points. On the other hand to obtain alternative solutions you can always build up linear combinations of the two regular Abel functions $\alpha$, $\beta$: $\gamma(z)=c\alpha(z)+(1-c)\beta(z)$ which is again an Abel function: $\gamma(f(z))=c(\alpha(z)+1)+(1-c)(\beta(z)+1)=\gamma(z)+1$ « 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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