Reviewing recent developments?

[Ember Edison]

It’s been a long time since I participated in community discussions. Have there been any significant new developments regarding tetration lately? Also, has the community discussed the paper Holomorphic Extension of Tetration to Complex Bases and Heights via Schröder's Equation (https://doi.org/10.13140/RG.2.2.10348.48008)? What are its primary conclusions?
How this method compares to the Kneser?

[MphLee]

First of all, it strikes me that the author seems to build his work purely on the basis of Kneser's original 1950 paper, Paulsen's construction of "Tetration in the complex plane" (2017), and Kouznetsov's "Superfunctions" monograph (2020).
I am highly ignorant of the relevant bibliography, but the literature may be somewhat more extensive. For sure, no results from the Tetration Forum, Trappmann's contribution with Kouznetsov, the other Paulsen papers (e.g., the one coauthored with Cowgill on solving \(F(z+1)=b^{F(z)}\) from 2017 or "Tetration for complex bases" from 2019), nor the vast body of work by Nixon (only available as forum discussions and preprints on the arXiv) are cited.
This is just a superficial observation, of course. I'm sure there are various reasons for this, probably about omitting non–peer-reviewed contributions in the references.

The main contribution of this work seems, as the author suggests, to be 

A) a complete recursive closed-form presentation for the coefficients of the Taylor series of the Schröder function \(\psi_k\) of the exponential \(g_b=\exp_b:\mathbb C\to \mathbb C\) at its fixed point \(L_k=- \frac{W_k(-\ln b}{ \ln b },\,\,k\in\mathbb Z\),  i.e. solutions of the f.e \(\psi_k\circ g_k=s_k \cdot \psi\) for \(s_k=\ln(b)L_k\); the found coefficients are used to construct the inverse power series by Lagrange inversion.

B) Clearly, once we have the Schröder functions (cit. linearizations) and its inverse \(\psi^{-1}\) we can define the superfunction of g(z)=b^z using the Schröder function at the fixed point (see page 6, Theorem 7), and this family of functions \(F_b(z):=\psi^{-1}( s_k^z\psi(1))\), where \(s_k^z:=\exp(z\ln(s_k))\) is proved unique and analytic.

C) the uniqueness proof, modulo translations, of the group \(G_k:=\{g_k^{\circ z}\,|\, z\in\mathbb C\}\) where \( g_k^{\circ z} :=\psi_k^{-1}\circ s_k^z\circ \psi\). Notice that \(G_k\) is proved to be a subgroup of the group of holomorphic automorphism \(\mathcal H(\mathbb C)\) for \(b\neq 0,1\) (theorem 8 ). Here uniqueness in term of group theory is not clear to me... does the author means that for each \(k',k''\in\mathbb Z\), i.e. choice of two branches of the lambert function, the two group of iterates are... isomorphic? i.e. we have that \(G_{k'}\cong G_{k''}\)?


After that, things get way over my head. I hope someone can comment and help me here (maybe the author too). Here, perhaps due to the new closed form for the coefficients or, I don't know, because of classical results (e.g., the holomorphic inverse function theorem + Lagrange inversion?), uniqueness and holomorphy are proved. All of this seems presented as an extension of Kneser's solution to complex numbers and complex bases.
All the discussion of Écalle theory for |s|=1, Borel summability, Liouville numbers, and so on gives me a weird déjà vu feeling, as if this has been discussed on the forum before (by Nixon and other users)… but I won't add more because of my deep ignorance.

Universal condition for the solution.
What puzzles me most is Section 11.2 about future work. There, diagrammatically, the author appears to claim a relationship between his solution and Kneser's.
I hoped to find something intelligible there (I'm deep into category theory), but it is not very helpful. The category \(\bf Tet\) seems to be defined—I say "seems" because no formal definition is given—as consisting of pairs \((b, F)\) as objects, where \(b\) is a complex number and \(F\) is a b‑based superfunction of \(\exp_b\). Morphisms are unclear: they are vaguely described as "change of basis" via \(\beta\) (it is not explained what \(\beta\) is and my be related to multiplication by \({\rm log}_b(b')\)) and some other function \(\phi\) 
I'll try to reverse‑engineer the definition using algebra and informal intuition. Perhaps an object \((b,F)\) should be a bijective solution of \(F(z+1) = \exp_b(F(z))\). Such an object induces an iteration group \(g_{b,F}^z(w) := F(z + F^{-1}(w))\), extending exponentiation to complex times and satisfying \(g_{b,F}^z(F(w)) = F(z+w)\). In other words, every \(F\) is lifted from being a mere superfunction of the discrete exponential dynamics to a \(\mathbb C\)‑equivariant map of complex‑time dynamical systems. I do not think this can hold "on the nose," nor that it matches what the author claims in Theorem 8. Clearly I am missing crucial domain and regularity conditions.

If I can understand how the category \(\bf Tet\) is meant to work, I will be able to assess rigorously the universal/uniqueness property the author claims relative to Kneser's solution.

[Ember Edison]

...

Thank you. Does the Transseries and Resurgence Analysis mentioned in the paper help improve the convergence of our current complex-base and complex-exponent tetration, particularly when the base lies on the boundary of the Shell-Thron region, or is very close to 0 or 1?

[MphLee]

Forgive me, but as I said, I'm not even remotely qualified to answer this. From a bypasser pov I had the impression that that theory was only used for treating the multiplier=1 case.