Universal uniqueness criterion? BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 07/05/2009, 08:57 PM (06/22/2009, 09:46 PM)bo198214 Wrote: (06/22/2009, 07:19 PM)Kouznetsov Wrote: (06/22/2009, 02:24 PM)bo198214 Wrote: .. Well, but $\exp$ *has* more fixed points. In every strip $2\pi i k < \Im(z) < 2\pi i (k+1)$ there is a fixed point of $\exp$.Henryk, how about to build up "another" holomorphic tetration that goes to other fixed points at $\pm i \infty$?I dont think that there is an initial region connecting other conjugated fixed point pairs than the one closest to the real axis. (Plot the straight line connecting two such fixed points and plot its image under exp. Both lines intersect.) On the other hand I asked you som time ago to apply your algorithm to other fixed points, but you somehow did not follow that path. What about using Kneser's approach to produce the alternate tetration functions? bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 07/05/2009, 09:13 PM (This post was last modified: 07/05/2009, 09:15 PM by bo198214.) (07/05/2009, 08:57 PM)BenStandeven Wrote: What about using Kneser's approach to produce the alternate tetration functions? There is no initial region connecting an alternative fixed point pair. The image of the straight line connecting a secondary fixed point pair overlaps with itself. Secondary fixed points lie in a range with imaginary part greater or less than pi. The vertical line connecting a pair is longer than 2*pi. This means the image revolves more than once around 0 with constant radius, hence overlapping itself. I tried to construct different connecting lines of a secondary fixed point pair and failed. I believe there is no initial region connecting two secondary fixed points. BenStandeven Junior Fellow Posts: 27 Threads: 3 Joined: Apr 2009 07/06/2009, 12:53 AM (07/05/2009, 09:13 PM)bo198214 Wrote: (07/05/2009, 08:57 PM)BenStandeven Wrote: What about using Kneser's approach to produce the alternate tetration functions? There is no initial region connecting an alternative fixed point pair. The image of the straight line connecting a secondary fixed point pair overlaps with itself. Secondary fixed points lie in a range with imaginary part greater or less than pi. The vertical line connecting a pair is longer than 2*pi. This means the image revolves more than once around 0 with constant radius, hence overlapping itself. I tried to construct different connecting lines of a secondary fixed point pair and failed. I believe there is no initial region connecting two secondary fixed points. Yeah, that's right; the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point. bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 07/06/2009, 08:56 AM (This post was last modified: 07/06/2009, 09:01 AM by bo198214.) (07/06/2009, 12:53 AM)BenStandeven Wrote: the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point. Say the curve $\gamma: [0,1]\to \mathbb{C}$ is injective and connects two points $\gamma(0)=a$ and $\gamma(1)=b$ with equal real part and with $\Im(b)-\Im(a)>2\pi$. One needs to show that then there is always a pair of points $c_1=\gamma(t_1)$ and $c_2=\gamma(t_2)$ with equal real part and with $\Im(c_2)-\Im(c_1)=2\pi$. This sounds very plausible but I couldnt prove it except for certain simple shapes of $\gamma$, e.g. convex. bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 07/06/2009, 09:20 AM (This post was last modified: 07/06/2009, 09:30 AM by bo198214.) (07/06/2009, 08:56 AM)bo198214 Wrote: One needs to show that then there is always a pair of points $c_1=\gamma(t_1)$ and $c_2=\gamma(t_2)$ with equal real part and with $\Im(c_2)-\Im(c_1)=2\pi$. This is equivalent to that $\gamma$ and $\gamma+2\pi i$ intersect. If $\gamma$ only extend to the right, i.e. $\Re(\gamma(t))>\Re(a)\forall t\in (0,1)$, then this is a consequence of the Jordan curve theorem. We have $\Im(a) < \Im(a+2\pi i) < \Im(b) < \Im(b+2\pi i)$. The closed Jordan curve $[a,b]\cup \gamma$ has the point $\gamma(0+\epsilon)+2\pi i\approx a+2\pi i$ in its interior and the point $\gamma(1-\epsilon)+2\pi i\approx b+2\pi i$ in its exterior. Hence there must be an intersection of $\gamma+2\pi i$ and $\gamma$ (as $\gamma+2\pi i$ does not pass [a,b] for $t\in (0,1)$.) Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022   06/26/2022, 08:49 AM (This post was last modified: 06/26/2022, 08:49 AM by Catullus.) I do not understand why there must be that kind of boundedness with the universal uniqueness criterion. Why? Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 06/27/2022, 05:15 PM (06/26/2022, 08:49 AM)Catullus Wrote: I do not understand why there must be that kind of boundedness with the universal uniqueness criterion. Why? I don't understand the question ... why? Because this is how this proof works ... JmsNxn Ultimate Fellow Posts: 1,176 Threads: 123 Joined: Dec 2010 06/28/2022, 12:00 AM (06/27/2022, 05:15 PM)bo198214 Wrote: (06/26/2022, 08:49 AM)Catullus Wrote: I do not understand why there must be that kind of boundedness with the universal uniqueness criterion. Why? I don't understand the question ... why? Because this is how this proof works ...  Lmao. Bo is out here with the patience of a saint. Catullus, it's okay to accept when something is outside of your purview, and beyond your scope. But it's also a tad rude to bring up old threads and ask a question like this. This thread more than already answers the question you are asking. « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Uniqueness of fractionally iterated functions Daniel 7 1,225 07/05/2022, 01:21 AM Last Post: JmsNxn A question concerning uniqueness JmsNxn 4 10,413 06/10/2022, 08:45 AM Last Post: Catullus [Exercise] A deal of Uniqueness-critrion:Gamma-functionas iteration Gottfried 6 7,912 03/19/2021, 01:25 PM Last Post: tommy1729 Semi-exp and the geometric derivative. A criterion. tommy1729 0 3,562 09/19/2017, 09:45 PM Last Post: tommy1729 A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 26,713 02/17/2017, 05:21 AM Last Post: JmsNxn Uniqueness of half-iterate of exp(x) ? tommy1729 14 32,898 01/09/2017, 02:41 AM Last Post: Gottfried Removing the branch points in the base: a uniqueness condition? fivexthethird 0 3,647 03/19/2016, 10:44 AM Last Post: fivexthethird [2014] Uniqueness of periodic superfunction tommy1729 0 4,125 11/09/2014, 10:20 PM Last Post: tommy1729 Real-analytic tetration uniqueness criterion? mike3 25 46,147 06/15/2014, 10:17 PM Last Post: tommy1729 exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 5,076 06/03/2014, 09:58 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)