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+--- Thread: Are there any noteworthy developments about tetration in the last six months? (/showthread.php?tid=1701)
Are there any noteworthy developments about tetration in the last six months? - Ember Edison - 02/09/2023
I didn't resist the effects of COVID-19 as easily as I thought I would. I hope I didn't miss anything interesting.
RE: Are there any noteworthy developments about tetration in the last six months? - JmsNxn - 02/10/2023
I think the only real "tetration-y" stuff that has happened here--at least on mine and Gottfried's part has been discussing Neutral fixed points. Where we have stumbled across a bunch of literature which confirms that there is no holomorphic function:
\[
g(g(x)) = \eta^x\\
\]
For \(x\) in a neighborhood of \(e\). And we were able to find proofs of very tight bounds on this series; where:
\[
g(x) = e + \sum_{k=1}^\infty a_k (x-e)^k\\
\]
Where:
\[
a_k = O(c^k k!)
\]
This led me to write a half finished paper, I should get back to, about iterating around neutral fixed points in general. It didn't really go anywhere though really. But we did cover a lot of ground on asymptotic series expansions near neutral fixed points; and their relation to Abel functions, so that's cool !
The rest of the stuff that's happened here has been the usual whacky "tetration adjacent" stuff, lol
RE: Are there any noteworthy developments about tetration in the last six months? - Ember Edison - 02/11/2023
(02/10/2023, 03:27 AM)JmsNxn Wrote: I think the only real "tetration-y" stuff that has happened here--at least on mine and Gottfried's part has been discussing Neutral fixed points. Where we have stumbled across a bunch of literature which confirms that there is no holomorphic function:
\[
g(g(x)) = \eta^x\\
\]
For \(x\) in a neighborhood of \(e\). And we were able to find proofs of very tight bounds on this series; where:
\[
g(x) = e + \sum_{k=1}^\infty a_k (x-e)^k\\
\]
Where:
\[
a_k = O(c^k k!)
\]
This led me to write a half finished paper, I should get back to, about iterating around neutral fixed points in general. It didn't really go anywhere though really. But we did cover a lot of ground on asymptotic series expansions near neutral fixed points; and their relation to Abel functions, so that's cool !
The rest of the stuff that's happened here has been the usual whacky "tetration adjacent" stuff, lol
Fortunately, I really didn't miss much. And congratulations on your discovery.
The neutral fixed points is quite mysterious, and it's a good thing to have more understanding.