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 Universal uniqueness criterion? Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 06/20/2009, 07:27 PM (This post was last modified: 07/06/2009, 12:08 AM by Base-Acid Tetration.) Theorem. Let $f$ be a function that is biholomorphic on both of the initial regions $D_1$ and $D_2$ that share as boundaries the same conjugate pair of fixed points of f. Also let f have no other fixed points. Let there be two Abel functions of f, $A_1$ and $A_2$, that are biholomorphic on these initial regions and satisfy A(d) = c. For each f, there exists exactly one biholomorphism $A_{cont}(z)$ on a single simply connected open set $C \supseteq D_1,\,D_2$ such that $\forall z \in D_i\, A_{cont}(z) = A_i (z).$ (i is an index that can be 1 or 2) i.e. there is an analytic continuation, and it's unique. Proof. 1. Let $D_1, D_2$ be disjoint, simply connected domains that have as boundaries: (1) $\partial_1 D,\, \partial_2 D \not \subset D$, disjoint curves which are homeomorphic to (0,1); (2) $L$ and $\bar{L}$, which are boundaries of, but not contained in, $\partial _1 D_1, \, \partial_2 D_1,\, \partial_1 D_2, \,\partial_2 D_2$. 2. Let $f$ be a biholomorphism on $S := \lbrace z:|\Im(z)| < \Im(L) + \epsilon i \rbrace$, where $\epsilon>0$, (tried to make the domain of biholomorphism into an open set) that: (1) bijects $\partial_1 D$ to $\partial_2 D$; (2) has a conjugate pair of fixed points $L$ and $\bar{L}$; (3) has no other fixed points in the domain of biholomorphy. 3. (1) Let $A_1$, a biholomorphism on $D_1$, and $A_2$, a biholomorphism on $D_2$, both satisfy $A(f(z))=A(z)+1$ for all applicable z. (for all z such that A(f(z)) is defined) (2) Let $A(d)=c$ for some $d \in S$. to be continued « Next Oldest | Next Newest »

 Messages In This Thread Universal uniqueness criterion? - by bo198214 - 05/21/2008, 06:24 PM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 05:19 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 06:42 AM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 11:25 AM RE: Universal uniqueness criterion? - by andydude - 05/22/2008, 03:11 PM RE: Universal uniqueness criterion? - by bo198214 - 05/22/2008, 05:55 PM RE: Universal uniqueness criterion? - by bo198214 - 05/23/2008, 12:07 PM RE: Universal uniqueness criterion? - by Gottfried - 06/25/2008, 06:15 AM Uniqueness of analytic tetration - by Kouznetsov - 09/30/2008, 07:58 AM RE: Uniqueness of analytic tetration - by bo198214 - 09/30/2008, 08:17 AM RE: Universal uniqueness criterion? - by bo198214 - 10/04/2008, 11:19 PM RE: Universal uniqueness criterion? - by Kouznetsov - 10/05/2008, 12:22 AM RE: Universal uniqueness criterion? - by Kouznetsov - 06/19/2009, 08:45 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 02:04 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 02:51 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 04:19 PM RE: miner error found in paper - by bo198214 - 06/19/2009, 04:53 PM i don't think it will work - by Base-Acid Tetration - 06/19/2009, 05:17 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 06:25 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/19/2009, 06:27 PM RE: Universal uniqueness criterion? - by bo198214 - 06/19/2009, 07:59 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/20/2009, 02:01 PM RE: Universal uniqueness criterion? - by bo198214 - 06/20/2009, 02:10 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 02:39 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/23/2009, 05:46 PM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 06/23/2009, 09:28 PM RE: Universal uniqueness criterion? - by Kouznetsov - 06/24/2009, 05:02 AM RE: Universal uniqueness criterion? - by Base-Acid Tetration - 07/04/2009, 11:17 PM RE: Universal uniqueness criterion? - by Kouznetsov - 07/05/2009, 08:28 AM RE: Universal uniqueness criterion? - by bo198214 - 07/05/2009, 06:54 PM

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