Major references

[Daniel]

I am rewriting my website and need to represent the other approaches to extending tetration. What are the best references for the different tetration methods? I can use https://mizugadro.mydns.jp/t/index.php/Tetration for Kouznetsov and http://go.helms-net.de/math/tetdocs/ for Gottfried. AndyDude's site doesn't appear to have significant material on extending tetration.

[JmsNxn]

https://math.eretrandre.org/publications.html

This is a list of Trappmann's work.

I suggest linking to Ecalle, and Milnor, and Baker's work too; as their work is very foundational to much of the tools. I can give a meagre list if asked.

I suggest Paulsen & Cowgill for Kneser, as I feel it is a slightly stronger paper on the construction of Kneser than Kouznetsov  http://myweb.astate.edu/wpaulsen/tetration2.pdf. Also, it is much more a direct translation of Kneser's work, with some caveats. As there is no english translation of Kneser, this can be frustrating. Also, the Paulsen approach is very much Kneser's approach--where as Kouznetsov uses his own method.

For Sheldon Levenstein (sheldonison), I suggest linking to the megathread on fatou.gp.

For Andy, I do believe there is a similar megathread, where Andy's method is properly described. Which actually dates back to Peter Walker. Walker's construction is pretty brief, and he has priority--it was a bit of an aside in his work, but Andy really crunched the numbers.

Despite my participation on this forum, I don't feel I've really done enough work in tetration to warrant "major references"--though the fractional calculus approach seems like it could fit; as it directly relates these things to analytic number theory. And relates iterations of functions to mellin transforms; which is pretty cool, if I do say so myself

I know mphlee, who's much more organized than me, has a list of major publications in the scope of tetration. He could be more help here, lol.

[Daniel]

https://math.eretrandre.org/publications.html

...
I know mphlee, who's much more organized than me, has a list of major publications in the scope of tetration. He could be more help here, lol.

Thank you very much. I'll quote your whole text and attribute it to you.
Have you seen https://ingalidakis.com/math/IERefs.html? He might let me direct link to the page. I will also be listing Galidakis' work.

[JmsNxn]

https://math.eretrandre.org/publications.html

...
I know mphlee, who's much more organized than me, has a list of major publications in the scope of tetration. He could be more help here, lol.

Thank you very much. I'll quote your whole text and attribute it to you.
Have you seen https://ingalidakis.com/math/IERefs.html? He might let me direct link to the page. I will also be listing Galidakis' work.

Yes! I have read Galidakis--from what I remember his infinite composition theorem was for \(f : \mathbb{D} \to \mathbb{D}\) where \(\mathbb{D}\) is the unit disk. I remember reading some of his work, but he seemed to only have "half-results" while still being fairly right. Him and John Gill are kind of the only two good sources for infinite compositions. And much of my work is just cementing and building a stronger foundation of what they saw... A lot of infinite compositions does not have the \(\epsilon/\delta\) work you'd like to see. Especially in a professional manner. I pride myself with being the first to rigorously prove a lot of the foundations--what Galidakis seems to take for granted in his expansions. Because of this, a lot of their problems and results, are non problems if you understand \(\Omega ... \bullet z\) notation. Additionally everything will converge. I don't talk about these things a lot here, because they aren't directly related to tetration. But I highly suggest:

https://arxiv.org/abs/1910.05111

Which deals with infinite compositions in the general sense

https://arxiv.org/abs/2103.09292

Which deals with infinite compositions in the degenerate case (this is also the beta method case)--while additionally takes infinite compositions while taking infinite compositions.

If you like calculus, then this is the infinite composition representation of Picard and Lindelof's theorem (John Gill did this first, I just did it with more notation, and a stricter dialogue):

https://arxiv.org/abs/2001.04248

---

If you want to learn about the history and players of infinite compositions. I can definitely help there, Daniel. I guess, for me, it's just too indirectly related to tetration--so I don't post these results here.

[MphLee]

I know mphlee, who's much more organized than me, has a list of major publications in the scope of tetration. He could be more help here, lol.

Hahah I hope I was half organized as I'd like to be... My archive is only approximation atm... I had to do a major upgrade of it but had not time. I can share what I have gathered till now, with no claim of completeness. Also I don't claim to understand more then 20% of what follows, nor I claim to have read more than 30% of the following list, but skimmed most of them. Most of the items are in my to read list... some are beyond my math education.


Iteration/Dynamics

1860 Arthur Cayley - On some numerical expansion
1871 Ernst Schroeder - Uber iterirte functionen
1881 Abel
1882 A. Korkine - Sur un probleme d’interpolation. Bulletin des Sciences Math ́ematiques et As-
tronomiques
1884 Koenigs -  Recherches sur le intégrales de certaines équations fonctionnelles

1936 Morgan Ward, F. B. Fuller - The continuous iteration of real functions (Note: classifies all the bounded monoid homomorphism solutions subject to some monotonicity, cites Bennet 1916, points to Korkine 1882 for monoid homorphism iteration, it also proves that each monoid morphism defines a solution to the superfunction problem)
1956 - Erdos, Jabotinsky - On analytic iteration (proves that for complex powerseries \(\sum f_n z^n\) convergent on a disk about zero if there exists a complete complex iteration group then we have an iteration group analytic in the iteration parameter, it claims the existence problem is already settled for \(|f_1|\neq 1\). It claims to solve for \(f_1=1\) and points to Jabotinsky 54, Lewin 60 for "largely open" case of \(f_1\neq 1\) for \(f_1\neq 1\) )
1966 Komatsu - Fractional Power of Operators
1976 Earl Berkson, Horacio Porta - Semigroups of analytic functions and composition operators (note: about infinitesimal generator of a flow/semigroup of hoomorphic mappings)
1978 Lawvere - Categorical dynamics (note: beginning of Lawvere's program to give foundations to continuous dynamics in categorical terms, final goal is to axiomatize continuum mechanics and physiscs)
1980 Lawvere - Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous-body
1982 Nelson - Homomorphisms of monounary algebras (note: from Lawvere's School, she shows sufficient and necessary condition to the existence of a map of dynamical systems, eg. a superfunction, are presented algebraically)
1982 Lawvere - Introduction to categories in continuum physics
1984 Lawvere - Functorial remarks on the general concept of chaos (note: the concept of dynamical chaos is expressed in purely arrow/composition theoretic language, very fascinating)
1986 Lawvere - Taking categories seriously (note: here Lawvere sums up why category theory provides the natural way to treat dynamics and continuous dynamics: actions, dynamics, spectral analysis, periodic points, metrical concepts, cohesivity, philosophical continuity emerges naturally from the pure and fundamental role of composition)
1990 Kuczma, Choczewski, Ger - Iterative functional equations (note: really the most cited)
1990 Milnor - Dynamics in One Complex Varible
1992 Mrozek - Normal functor and retractors in cat of endomorphisms (note: use of normal functors to construct abstractly Conley indices over dynamical systems)
1996 Semeon Bogatyi - On nonexistence of iterative roots
(1997) Lawvere - Toposes of Laws of motion
1997 Woon - Analytic iteration of Operators
1999 Belitskii, Lyubich - Abel equation and total solvability of linear functional equations


2001 Carracedo, Alix - The Theory of Fractional Powers of Operators (note: vast theory)
2003 Marco Abate - Discrete local holomorphic dynamics (note: survey about complex dynamics about a fixed point)
2005 Marco Abate - Index theorems for holomorphic self-maps (note: generalizing index theorems, eg. Lefschetz theorem, from self max with isolated fixpoints to maps with fixpoints forminf positive dimension)
2005 Muller, Schleicher - How to add a non-integer number of terms, and how to produce unusual infinite summations
2006 Hilberdink - Orders of Growth of real functions (note: using the concept of growth rate a natural criterion for the uniqueness of fractional iterates of continuous functions is studied)
2007 Muller, Schleicher - Fractional Sums and Euler-like Identities
2008 Marco Abate - Discrete holomorphic local dynamical systems (note: survey with big bibliography)
2009 Cheritat - Parabolic implosion a mini-course (Amazing illustrations!!!)
2009 Curtright, Zachos - Evolution profiles and functional equations
2009 Robbins - On analytic iteration and iterated exponentials
2009 Trappmann, Kouznetsov - Uniqueness of Analytic Abel Functions without fixed points
2011 Muller, Schleicher - How to add a non-integer number of terms - From Axioms to New Identities
2011 G A Edgar - Fractional Iteration of series and transseries (note: points back to Korkine 1882 for iteration of powerseries and to Cayley 1860 about discussion on real iteration groups, i.e. fractional iteration)
2011 Trappmann, Kouznetsov - Uniqueness of holomorphic abel functions
2012 Giunti,  Mazzola - Dynamical systems on monoids - toward a general theory of deterministic systems and motion (note: basic framework of dynamics over arbitrary monoids of time, it should point to Lawevere but it doesn't!)
2013 Behrisch, Kerkhoff, Poschel, Shneider, Siegmund- Dynamical systems in Categories (note: dynamical systems can be translated as coalgebras/algebras over monads/comonads, really serious and deep article)
2014/03 Daniel Geisler - Bell polynomials of iterated functions
2014 Domoradzki, Stawiska - LUCJAN EMIL BOTTCHER AND HIS MATHEMATICAL LEGACY (note: seems worth to check, some unpubblished material is presented along with historical survey of Bottcher life and mathematical achievements)
2015 Marco Abate - Fatou flowers and parabolic curves (note: survey of multivariable generalizations of the Leau-Fatou flower theorem)
2015 Aschenbrenner, Bergweiler - Julia equation and differential trascendence (note: Jabotinsky ilog is shown to be differentially trascendental over the ring of entire functions)
2016/08 Geisler - The Existence and Uniqueness of the TaylorSeries of Iterated Functions (note: proof attempt, retracted claim?)
2017 Kouznetsov - Superfunctions (note: monography)
2019 Rogers - Toposes of discrete monoid Actions (note: characterization of topoi of actions over arbitrary monoids of time, up to equivalence)
2020 Rogers, Hamelaer - Monoid properties as invariants of toposes of monoid actions (note: super deep, how the semigroup property of the time monoid translates into properties of all the dynamical systems over that monoid, topos theoretically)
2021 Arnauld Maret - RTG Seminar
2021/04 Nixon - Infinite composition and complex dynamics - Generalizing Schroeder and abel functions
2021/05 Rogers - Toposes of topological monoid actions (note: same enterprise but jump from discrete time to topological, ie continuous, monoids of time)
2021/12 Rogers - Toposes of monoid actions, phd Thesis (note: from Olivia Caramello's School, big thesis summing up all the Topos theoretic recasting of dynamical systems theory over arbitrary monoids of time... this was the phd thesis I had to do if I wasn't so fucking stupid to lose my chance at the university...)
2022 Jacopo Garofali - Dynamical Sheaves, phd thesis (note: from Olivia Caramello's School, recasting of holomorphic dynamics in the language of topos theory)


Iteration of exponential/tetration

1986 Clenshaw, Lozier, Olver, Turner - Generalized exponential and logarithmic functions (note: study of solution to inverse abel equation of exp, application to computer arithmetic)
1990 Clement Frappier - Iterations of a kind of exponentials (note: a kind of nested exp)
1991 Daniel Geisler -  Algebraic Exponential Dynamics
1991 Peter Walker - Infinitely differentiable generalized logarithmic and exponential functions

2009/01 Kouznetsov, Trappmann - Protrait of the four regular super-exponentials
2009/04 Trappmann, Kouznetsov - 5+ methods for real analytic tetration (note: older version)
2009 Trappmann, Kouznetsov - Uniqueness of Holomorphic superlogarithms
2010/06 Trappmann, Kouznetsov - 5+ methods for real analytic tetration
2010 Kouznetsov - Tetrational as special function
2010 Kouznetsov, Trappmann - portrait of the four regular super-exponentials to base sqrt2
2011 Trappmann - The intuitive logarithm
2014/10 Aldrovandi - Tetration an iterative approach
2016 Paulsen - Finding the natural solution to \(f(f(x))=\exp(x)\)
2017 Cowgill - Exploring tetration in the complex plane
2017 Paulsen, Cowgill - Solving \(F(z+1)=b^{F(z)}\) in the complex plane
2020 Helms - Determining of and finding patterns in n-perios of exp-function
2020 Ripa - The congruence speed formula
2021/02 Nixon - A tetration function by unconventional means [v2]
2021/04 Nixon - The limits of a family of asymptotic solutions to the tetration equation [v1]
2021/05 Ueda - Extension of tetration to real and complex heights


Hyperoperations

(1915) Bennet - Note on an Operation of the Thrid Grade
(1947) Goodstein - Transfinite ordinals in recursive number theory
(1947) Robinson - primitive recursive functions
(1953) Arcidiacono - Sulla Estensione delle operazioni aritmetiche
(1969) Doner, Tarski - An extended arithmetic of ordinal numbers
(1975) Raspletin - Hyperoperations
(1989) Rubtsov - A component R0 in R3
(1989) Rubtsov - Algorithms ingredients in a set of algebraic operations
(1990) Rubtsov - A complement of a set of real numbers and his application in cybernetics
(1990) Rubtsov - A hypothetical reflexive complement of a set of real numbers
(1990) Rubtsov - An image by derivative obtained by replacement of operations
(1994) Rubtsov - Integro-differential objects of a new nature
(1995) Muller - Reihenalgebra - What comes beyond exponentiation
(1998) Rubtsov - New Mathematical Objects

(2001 09) Geisler - Recurring digits in the ackermann function
(2001 12) Loday - Arithmetree
(2001) Micheal L. Carroll - The natural chain of binary arithmetic operations and generalized derivatives (note: basically part of Rubstov but formally)
(2004 09) Rubtsov, Romerio - Ackermanns function and new arthmetical operations
(2006 07 25) Rubtsov, Romerio - Notes on Hyperoperations - Progress Report - NKS Forum III
(2006 08) Rubtsov, Romerio - Hyperopertions as a tool for science and engineering, for ICM-06
(2007 06) Romerio - NKS Forum - Hyper-operations. Progress Report. Zeration.
(2007 08 09) Tetration FAQ
(2007 11 04) Romerio - Incomplete Towers
(2007) Trappmann - Arborescent Numbers, Hyperoperations,Division trees
(2008 07 10) Tetration REF

(2010) Williams - What Lies Between + and x (and beyond)
(2014 02 17) Reale - Operazione di rango zero e numeri non transitivi
(2014 06 11) Barrette - Hyperoperator manuscript 02
(2014 11) Kouznetsov - Evaluation of Holomorphic ackermanns
(2015 07) MphLee - On fractional ranks Mathematics Stack Exchange
(2016) Crespo, Montàs - Fractional Mathematical Operators and Their Computational Approximation
(2017) Altman - Intermediate arithmetic operations on ordinal numbers
(2018 06) Tezlaf - On ordinal dynamics and the multiplicity of transfinite cardinality
(2019 02) Barrett - The fundamental theorems of Hyperoperations
(2019 03) Leonardis, d'Atri, Caldarola - Byond Knuth notation for Unimaginable numbers
(2019) Caldaro, d'Atri, Maiolo - What are the unimaginable numbers
(2019) Dalthorp - Hyperoperations - introduction to the theory and potential solutions
(2020 07) Rubtsov - Application of hyperoperations for engineering practice
(2020 12) Aguilera, Freund, Rathjen, Weiermann - Ackermann and goodstein go functorial
(2020 12) Salazar - Hyperoperations in exponential fields
(2020) Judijasa - The logarithmic chain complex
(2021 03 13) Geisler - Extension of hyperoperators
(2021 03 03) Nixon - Hyper-operations By Unconventional Means
(2021 05) Jaramillo - Hyperoperations in exponential fields
(2021 05 24) Nixon - A family of bounded and analytic hyper-operators
(2021) Andonov - Constructing a hyperoperation sequence-pisa hyperoperations


Iterated composition

2003 Keen, Lakic - Forward iterated Function Systems (note: points to Gill 1988)
2005/09 Keen, Lakic - Accumulation constants of iterated function systems with Bloch target
2006/01 Keen, Lakic - Limit points of iterated function systems domains
2014/02 Kyriakos Kefalas - On smooth solutions of non-linear dyynamical systems \(f_{n+1}=u(f_n0)\), PART I (note: cites Hooshmand 06, Walker 91, Szekeres 58)
2020/01 Nixon - The compositional integral - A brief introduction
2020/11 Nixon - The compositional integral - The narrow and the complex looking glass

I've many more papers collected but I don't have time to organize them.


EDIT: +added essential bibliography about the algebraic/categorial perspective on dynamics.

[Daniel]

OMG MphLee, your list is beyond amazing! While I would be happy to have the list on my web site I feel this should probably be managed as a collective resource. Also the list is damn valuable. The best previous list I had seen was Galidakis' list which he inherited from David Renfro. Galidakis had me link to the root of his math site so that folks would be encouraged to look at his research. Having an accessible list like this can improve a site's standing in Google searches.
I need to give this topic significant thought and return to this posting. I'd love to hear other's thoughts.

[JmsNxn]

OMG MphLee, your list is beyond amazing! While I would be happy to have the list on my web site I feel this should probably be managed as a collective resource. Also the list is damn valuable. The best previous list I had seen was Galidakis' list which he inherited from David Renfro. Galidakis had me link to the root of his math site so that folks would be encouraged to look at his research. Having an accessible list like this can improve a site's standing in Google searches.
I need to give this topic significant thought and return to this posting. I'd love to hear other's thoughts.

I will say, of Mphlee's list--it is very foundational. And focuses I'd say 60% on the foundations of hyper-operational "structures". So it's not a list on analytic approaches. I'm not discrediting this--it's just plain to see Mphlee is focused on the categorical nature of hyper-operations.

I agree with you about Galidakis, Daniel. The math isn't all there also. I suggest looking at John Gill's work, which is rather poorly documented (he's on Research Gate, and his typesetting leaves nothing but things to be desired). But John has done much more than Galidakis, and much of Galidakis is rephrasing of Gill's work. My work was actually in response to a multitude of these sources, which, seemed so damn incomplete--that I tried to prove everything from scratch in purely rigorous terms. Especially when it comes to infinite compositions. And not to toot my own horn, I proved things both of these parties claimed because their numbers worked out. And I did so with much greater generality--again, Galidakis worked on \(f:\mathbb{D} \to \mathbb{D}\), and I can generalize that to \(f:G \to G\) for arbitrary domain \(G\)--and that's just the first step in my first paper, lol.

[MphLee]

OMG MphLee, your list is beyond amazing! While I would be happy to have the list on my web site I feel this should probably be managed as a collective resource. Also the list is damn valuable. The best previous list I had seen was Galidakis' list which he inherited from David Renfro. Galidakis had me link to the root of his math site so that folks would be encouraged to look at his research. Having an accessible list like this can improve a site's standing in Google searches.
I need to give this topic significant thought and return to this posting. I'd love to hear other's thoughts.

Do what you like with my list but take it with a grain of salt: I'm not an expert on dynamics. That's just a list I collected during the years of things that seems related and interesting to me.

While for the hyperoperations list, well I'm pretty proud of it... I believe it is almost a complete bibliography on the argument... even if some players are missing... like Nambiar's paper, Galidakis stuff and few other items... But Its all there in my database... but not time to organize the files...

I will say, of Mphlee's list--it is very foundational. And focuses I'd say 60% on the foundations of hyper-operational "structures". So it's not a list on analytic approaches. I'm not discrediting this--it's just plain to see Mphlee is focused on the categorical nature of hyper-operations.

It is but not categorical. It is actually three lists... the one about hyper-operations, I claim, is pretty much complete. It is all there is around about it imho. The one about Tetration/iterated exp is pretty complete as well, even if some of the oldest papers may be missing it contains almost everything that came out of this forum. No categorical bullshit!

About the list on iteration/dynamics, It is pretty analytic imho. Originally I meant to post it like that, I just added the blue items later, the blue items are about foundational/categorical approach... just for completeness. It is actually two different lists merged.

And the last list is not even an attempt. I know you have the right literature on that topic.

[JmsNxn]

OMG MphLee, your list is beyond amazing! While I would be happy to have the list on my web site I feel this should probably be managed as a collective resource. Also the list is damn valuable. The best previous list I had seen was Galidakis' list which he inherited from David Renfro. Galidakis had me link to the root of his math site so that folks would be encouraged to look at his research. Having an accessible list like this can improve a site's standing in Google searches.
I need to give this topic significant thought and return to this posting. I'd love to hear other's thoughts.

Do what you like with my list but take it with a grain of salt: I'm not an expert on dynamics. That's just a list I collected during the years of things that seems related and interesting to me.

While for the hyperoperations list, well I'm pretty proud of it... I believe it is almost a complete bibliography on the argument... even if some players are missing... like Nambiar's paper, Galidakis stuff and few other items... But Its all there in my database... but not time to organize the files... 

I will say, of Mphlee's list--it is very foundational. And focuses I'd say 60% on the foundations of hyper-operational "structures". So it's not a list on analytic approaches. I'm not discrediting this--it's just plain to see Mphlee is focused on the categorical nature of hyper-operations.

It is but not categorical. It is actually three lists... the one about hyper-operations, I claim, is pretty much complete. It is all there is around about it imho. The one about Tetration/iterated exp is pretty complete as well, even if some of the oldest papers may be missing it contains almost everything that came out of this forum. No categorical bullshit!

About the list on iteration/dynamics, It is pretty analytic imho. Originally I meant to post it like that, I just added the blue items later, the blue items are about foundational/categorical approach... just for completeness. It is actually two different lists merged.

And the last list is not even an attempt. I know you have the right literature on that topic.

Oh, I apologize Mphlee. The reason I consider your list "not very analytic"--is because there are no analytic papers on these subjects. There really are ZERO hard well developed papers on the hyperoperators, as analytic objects. The closest you'll find is Kouznetsov, who definitely leaves things to be desired (Just because his calculator works, and his intuition/ad hoc reasoning is very correct; doesn't mean it's rigorous).

So, I think I misspoke a tad. What I mean is that there are no good analytic papers on these subjects, and the few you will find, are severely lacking in rigor. Even myself, I only consider 80-90% of my results to be "proven rigorously"--but then, they've never truly been vetted, other than my calculations--which brings us back to Kouznetsov's level of "truth".

I hope you don't think I'm calling these papers categorical, I should've been clearer--I meant these papers look like a good foundation for a categorical approach. Whereby, some of these papers, do not have the quality of rigor, to actually call an analytic solution. But rather, a numerical solution. Where as the good papers, that I identified, are very... not sure the word, "foundational," "about the structure of hyper-operators". Which definitely lead to a "categorical" understanding. I'm aware you're probably the most eminent person on this planet on "Hyper-operators & category theory", because no ones ever touched this before. (No one cares about these subjects, and I love it, because it allows me to work without fear of rediscovering some 100 year old formula no one cares about --doing this, at least it's a new formula... no one cares about).

My main grievance, with saying "it's not a very analytic list"--is, I guess, it's not really a very "RIGOROUS ANALYTIC" list. If anything, a lot of work in tetration/hyper-operation/nested function theory tends to be numerical based. So I would call it more so, numerical analytic. And much of the results end up being effective calculators; but not rigorously proven constructs.

This reminds me of an old joke I heard once. "Of course \(\pi\) is rational, I plugged it in my calculator and out came: \(3.14159265359\)"

So I struggle to call much of hyper-operation/tetration/iteration theory actual analysis. When usually, at best, it is Numerical Analysis. This would definitely fall under Kouznetsov's work. Where the exception is Kouznetsov & Trappman, which is absolutely Analysis in the hard rigorous sense.

Also, don't take too much weight to what I say. I've only had time to post in my free time at night; and I've already started drinking (had too many drinks when I made the comment before), lmao

[MphLee]

Oh, I apologize Mphlee. The reason I consider your list "not very analytic"--is because there are no analytic papers on these subjects.

idk man, u don't leave me fully convinced. OFC there are not analytic rigorous papers on hyperops and tetration at the moment. ofc that makes you claim trivially true. But in my iteration list, that is not complete for sure, there are items that I can't imagine how you could call non-hard-analytic. There is also Milnor there.
Try to check that first list better when you have time. Maybe there are some gems that you can use.

There really are ZERO hard well developed papers on the hyperoperators, as analytic objects. The closest you'll find is Kouznetsov, who definitely leaves things to be desired (Just because his calculator works, and his intuition/ad hoc reasoning is very correct; doesn't mean it's rigorous).

So, I think I misspoke a tad. What I mean is that there are no good analytic papers on these subjects, and the few you will find, are severely lacking in rigor. Even myself, I only consider 80-90% of my results to be "proven rigorously"--but then, they've never truly been vetted, other than my calculations--which brings us back to Kouznetsov's level of "truth".

Ok Agree.

Where as the good papers, that I identified, are very... not sure the word, "foundational," "about the structure of hyper-operators". Which definitely lead to a "categorical" understanding. I'm aware you're probably the most eminent person on this planet on "Hyper-operators & category theory", because no ones ever touched this before. (No one cares about these subjects, and I love it, because it allows me to work without fear of rediscovering some 100 year old formula no one cares about --doing this, at least it's a new formula... no one cares about).

My main grievance, with saying "it's not a very analytic list"--is, I guess, it's not really a very "RIGOROUS ANALYTIC" list. If anything, a lot of work in tetration/hyper-operation/nested function theory tends to be numerical based. So I would call it more so, numerical analytic. And much of the results end up being effective calculators; but not rigorously proven constructs.

...

So I struggle to call much of hyper-operation/tetration/iteration theory actual analysis. When usually, at best, it is Numerical Analysis. This would definitely fall under Kouznetsov's work. Where the exception is Kouznetsov & Trappman, which is absolutely Analysis in the hard rigorous sense.

For example, I'd like to have a quick opinion from you, maybe in a separate thread, on
-1936 Morgan Ward, F. B. Fuller - The continuous iteration of real functions"
-1956 - Erdos, Jabotinsky - On analytic iteration
-1999 Belitskii, Lyubich - Abel equation and total solvability of linear functional equations
-2001 Carracedo, Alix - The Theory of Fractional Powers of Operators
-2003 Keen, Lakic - Forward iterated Function Systems

In particular Linda Keen (http://comet.lehman.cuny.edu/keenl/publications.html ) seems to have continued to build lot on the theory of Gill. Do your work fully subsume it, were you aware of them?