• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Does tetration take the right half plane to itself? JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 05/16/2017, 04:09 AM (This post was last modified: 05/16/2017, 04:11 AM by JmsNxn.) Your second response confuses me. That's VERY close to what I am asking, but it is slightly off. It is DEFINITELY NOT 1-1, first of all. Tetration is periodic, so there's no hope in hell of that, I wouldn't dare say that. Tetration IS injective modulo a period (which is easy to prove). I do know that tetration does live in the right half plane, using experimental arguments. And I can prove it if the Schroder function of $\alpha^z$ is fully monotone about zero, then tetration lives in a small disk about the fixed point of the exponential (that small disk residing in the right half plane).  Yes, I reread a bunch of your posts regarding it. I read them very slow, and I now know why the inverse Schroder function is fully monotone. It is much simpler than I thought! MUCH simpler than what I was thinking. We just have to use that the composition of two fully monotone functions is fully monotone, and that the uniform limit of fully monotone functions is fully monotone. Where "fully monotone functions" are defined as having "completely monotone first derivative." It easily follows from this that $\log^{\circ n}(\lambda^n z + L)$ is fully monotone, it uniformly converges to the inverse Schroder function, and therefore the inverse Schroder function is a fully monotone function. Therefore, by the above arguments $^z\alpha : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}$, it actually sends the right half plane to a small disk in the right half plane (but that's extraneous). Therefore $\alpha \uparrow^n z : \mathbb{C}_{\Re(z) > 0} \to \mathbb{C}_{\Re(z) > 0}$.  If perhaps you are confused. I'm betting when you read the paper it'll be a lot clearer. I'll be sure to make a head nod and an accreditation to you. I'll PM you to ask how to reference you. I'm rebuilding a paper I wrote two years ago, and adding grout to everything, so there are no leaks and no holes. « Next Oldest | Next Newest »

 Messages In This Thread Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 07:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 08:22 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 08:16 PM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/15/2017, 09:00 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:09 AM RE: Does tetration take the right half plane to itself? - by sheldonison - 05/16/2017, 03:34 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 08:46 PM RE: Does tetration take the right half plane to itself? - by JmsNxn - 05/16/2017, 04:46 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Why the beta-method is non-zero in the upper half plane JmsNxn 0 462 09/01/2021, 01:57 AM Last Post: JmsNxn Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 2,222 09/09/2019, 10:55 PM Last Post: tommy1729 Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 3,216 09/09/2019, 10:50 PM Last Post: tommy1729 Half-iteration of x^(n^2) + 1 tommy1729 3 8,076 03/09/2017, 10:02 PM Last Post: Xorter Uniqueness of half-iterate of exp(x) ? tommy1729 14 28,800 01/09/2017, 02:41 AM Last Post: Gottfried [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 67,048 03/27/2015, 11:28 PM Last Post: tommy1729 [entire exp^0.5] The half logaritm. tommy1729 1 4,513 05/11/2014, 06:10 PM Last Post: tommy1729 Does the Mellin transform have a half-iterate ? tommy1729 4 7,553 05/07/2014, 11:52 PM Last Post: tommy1729 Simple method for half iterate NOT based on a fixpoint. tommy1729 2 6,446 04/30/2013, 09:33 PM Last Post: tommy1729 half-iterates of x^2-x+1 Balarka Sen 2 6,940 04/30/2013, 01:14 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)