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 Universal uniqueness criterion? bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 09/29/2008, 01:03 AM As I now see, even if the uniqueness criterion is valid, the precondition: $\lim_{n\to\infty} \exp_b^{\circ n}(z)=\infty$ for each $z\in f(S)$ can not be true for any $f$ for $b=e$ (and probably for each $b>e^{1/e}$)! Michał Misiurewicz [1] showed 1981 that the Julia set of $\exp$ is the whole $\mathbb{C}$ (that is not directly visible from the $e^z$ fractal). The Julia set is the boundary of the set $K$ of all $z$ such that $\lim_{n\to\infty} \exp^{\circ n}(z)=\infty$. That means that the set $K$ and its complement $\mathbb{C}\setminus K$ is dense in $\mathbb{C}$. In every neighborhood of any complex number $z_0$ there is a complex number $z$ such that $\lim_{n\to\infty} \exp^{\circ n}(z)=\infty$ and also a $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$! And that implies that $f(S)$, which contains an open non-empty set, always contains points $w$ such that $\lim_{n\to\infty} \exp^{\circ n}(w)\neq\infty$. [1] Michał Misiurewicz (1981). On iterates of e^z. Ergodic Theory and Dynamical Systems, 1 , pp 103-106, doi:10.1017/S014338570000119X « 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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