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
\[
g(g(x)) = \eta^x\\
\]
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