Exact and Unique solution for base e^(1/e)

[bo198214]

But I've decided to turn my interests back to tetration and related subjects.

Great.

Can someone point me to anything new that I should read up on before I get too far along? Relevant subjects include not only base \( e^{1/e} \), a.k.a. eta, but also the problem of non-convergence for non-integer iterates of functions, etc.

We have a uniqueness criterion now for the real-analytic iteration at a conjugated fixed point pair, which is however not so easy to verify. Its about injectivity and imaginary unboundedness of the Abel function on an initial region in the complex plane. The construction of Kneser of the arcsuper-exponential - formerly super-logarithm - (base e, or generalized to >e^(1/e)) satisfies this criterion, nothing is known about the other approaches.

Here is the thread of discussion of this paper. Though the proof is from my hand I still doubt if it is correct, perhaps have a look yourself.

Some of us are writing an overview paper of the 5 or more methods we have discovered/developed for real analytic tetration. You might want to see where your cheta function would fit into such a paper.

Yes, I absolutely second that, there is even a section about your method already in the paper, which I originally filled myself extracting your posts, however my computer crashed and I lost most of my data files. So I restarted that article again, but until now the section is empty. So if you want to participate ...

I am also glad to see Sheldon now together with you on this forum as he has perhaps a more intuitive understanding of your ideas than I have.

When you mention that it is a "scaled version" of the regular iteration of exp(x)-1, perhaps you should read an article of Walker, which also based an iteration of exp(x) on the regular iteration of exp(x)-1:

Walker, P. (1991). Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput., 57(196), 723–733.

In this article he also features the by Andrew rediscovered intuitive iteration (we renamed "natural" to "intuitive" as it causes less confusion).

I think he uses also his following articles
Walker, P. L. (1990). The exponential of iteration of \(e\sp x-1\). Proc. Am. Math. Soc., 110(3), 611–620.
Walker, P. L. (1991). On the solutions of an Abelian functional equation. J. Math. Anal. Appl., 155(1), 93–110.
(here he describes the iteration of b^x for b=e^(1/e) )

I have all these articles as pdfs, so if someone does not have access to suitable libraries, I can send it via private e-mail.

For the theoretical properties of the iteration of functions at a parabolic fixed point (f'(p)=1, e.g. f(x)=e^(x/e)) see also the publicly available article "Milnor, dynamics in one complex variable", which I already featured here. You find the discussion in chapter "Local fixed point theory", paragraph 7 "Parabolic fixed points: the Leau-Fatou flower". Though not for beginners its imho very insightful.