Major references

[JmsNxn]

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?

I am not aware of Linda Keen, great find. On her paper:

http://comet.lehman.cuny.edu/keenl/forwarditer.pdf

She has used very beautiful, fancy language, to prove the degenerate case of infinite compositions.

So what she has show is if:

\[
F_n = f_1\circ f_2 \circ...\circ f_n\\
\]

Taking A SIMPLY CONNECTED DOMAIN (Again, I never need this) to a smaller simply connected domain, then \(F_n \to F\) a constant.

Contrast this to my result on degenerate case of infinite compositions.

If there is a constant \(A\) such that \(\sum_{n=1}^\infty |f_n - A| < \infty\), then \(F_n \to F\). BUT A HUGE difference I have is that my condition supersedes hers (My condition is actually if and only if--whereby the conditions are either equivalent, or hers is stronger). Especially, my condition is much nicer, because what if \(f_n\) depends on another variable? If I write \(f_n(s,z)\)

\[
F_n(s,z) = f_1(s,f_2(s,...f_n(s,z)))\\
\]

Her condition of convergence will likely work; but it's not a very practical method. Because \(s\) will perturb everything. Where as in my case, just check that:

\[
\sum_{n=1}^\infty |f_n(s,z) - A| < \infty\\
\]

Additionally, she allows for \(A = A(z)\), but I don't like this, because she has required that \(A^{\circ k}(z) \to C\), a constant--so she's kind of hid the constant. I like to be direct and flat out point out there is a constant.

Also, I call this the degenerate case, because it "kills" a variable, per se. But as far as I can tell our conditions are equivalent (at least very comparable). Hers is just dressed up in a suit and tie  

EDIT (AGAIN); I mixed up a detail; she is looking at Outer infinite compositions; not inner (I mistakenly wrote inner as a comparison). But the exact same result holds for outer as for inner; she calls outer compositions "forward iterations"--and inner compositions "backward iterations". To me they follow the exact same rules; and it's simply a matter of orientation. Which is something I'm pretty sure I'm the first to rigorously justify.

Reading through this paper more, it's a real beaut! But it definitely focuses on the DYNAMICS of infinite compositions. I also think she has done a great way of DEFINING degenerate infinite compositions in a topological sense. Where she almost points out the possibility of non-degenerate infinite compositions (The ones where \(F_n(z) \to F(z)\) is non constant). But it's hard to say. I'll have to digest this more; there's a lot of jargon I don't like because she doesn't clarify some terminology. She's also, in my opinion, using overly advanced tools. We don't need the Poincare metric for really any of this--unless you are trying to pull out specifics of the dynamics. A lot of Complex dynamics will use the Poincare metric for the local case, and it's absolutely warranted. If we're simply asking for the convergence, it's a little unnecessary. I think it's necessary for her theorem, but then, I think her theorem is actually pretty weak (in a constructive sense). It is leagues more descriptive though, than anything I've written; about the qualitative difference between degenerate/non-degenerate cases. But, I just wanna make cool functions


I'll have to go through more of her work. She has papers with Devaney, so she is definitely the real deal

EDIT:

To give you a bit of an idea of where Keen's theorem lacks; is with the beta method.

Keen's theorem does not show that:

\[
\beta(s) = \Omega_{j=1}^\infty \frac{e^z}{e^{j-s}+1}\,\bullet z\\
\]

converges.

This is because \(e^z\) takes no simply connected domain to itself (So all of her theorems are instantly disqualified). If we tried to adapt her reasoning, we could probably show that \(\beta(s)\) converges. BUT THEN! You'd be totally shit outta luck trying to prove that \(\beta(s)\) is holomorphic. Totally shit outta luck, lol

It seems most of her work is centered on the dynamics; and for that I commend it. It is definitely more detailed in the nature of dynamics. That's sort of where most infinite composition theory lines up with; the dynamics of these iterated maps. I always shoot straight for the heart and want cool looking functions, lol.

I plan to read more of these papers though--she seems like a straight shooter



EDIT2:

-2001 Carracedo, Alix - The Theory of Fractional Powers of Operators

I couldn't access this, it appears to be a text book. But from the glimpses I've seen this is about Von Neumann theory on iterating operators on a Hilbert space. Super fun topic. I have a decent understanding of this--mostly because I used some of the common tricks to iterate weird operators back in undergrad. Not sure if this relates too much to iteration theory, as we mean the term. We can only model fractional iterations in L^2 if we stick to \(|\lambda| \neq 0 ,1\) for the multiplier of a function \(f\) (at least as far as I know).

-1999 Belitskii, Lyubich - Abel equation and total solvability of linear functional equations

This is super cool. It is the solution to:

\[
\psi(F(x)) = A(x)\psi(x) + g(x)\\
\]

I have solved these equations (through infinite compositions again). But only in restricted scenarios. Specifically I asked for \(L^1\) conditions on \(A\) and \(g\). I haven't digested this paper yet; but it seems entirely novel to what I do to solve these equations. For example, let \(A(x) = e^{x}+1\) and let \(g(x) = e^x\). Then:

\[
\psi(x) = \Omega_{j=1}^\infty A(F^{-\circ j}(x)) z + g(F^{\circ -j}(x))\,\,\bullet z \Big{|}_{z=0}\\
\]

Assuming that \(F^{\circ -j}(x) \to - \infty\), and we have decently behaved \(F\). Then:

\[
\psi(F(x)) = \Omega_{j=0}^\infty A(F^{-\circ j}(x)) z + g(F^{\circ -j}(x))\,\,\bullet z \Big{|}_{z=0}\\
\]

Which equals: \(\psi(F(x)) = A(x)\psi(x) + g(x)\).

I believe there will be an overlap in our analysis though; there seems to be a kind of common theme.


-1956 - Erdos, Jabotinsky - On analytic iteration

This paper is definitely reiterating what we all already know. But it's not to knock these titans, I'm just pretty sure we've already learnt through osmosis the contents of this paper. And it's largely stating that if:

\[
f^{\circ t}(z) = F(t) : \mathbb{R} \to \mathbb{R}\\
\]

Is expandable as an iterate near \(z \approx 0\)

Then \(F(t)\) is entire. And if it doesn't take the real line to itself...

Well then it takes \(\mathbb{C}/\mathcal{A} \to \mathbb{C}/\mathcal{A}\) where \(\mathcal{A}\) is "measure zero in the two dimensional sense" (It's measure zero in \(\mathbb{R}^2\) under the Lebesgue measure--is how I say it).

This is comparable to our result that when \(|\lambda| > 1\) the iteration is entire; and when \(|\lambda| \le 1\) the iteration has branch cuts.

-1936 Morgan Ward, F. B. Fuller - The continuous iteration of real functions"

Couldn't get a copy of this, but by the looks of it, it looks like all the stuff we already have learned through osmosis. Not to knock the authors; these are again some titans. But still; we've amalgamated this knowledge through the zeitgeist, lol.

[JmsNxn]

Okay, I feel after rereading Linda Keen's paper. I can offer some insight.

She is first of all taking the outer composition of an arbitrary sequence of functions. Let's call this sequence \(\{f_n\}_{n=1}^\infty\) where each function takes \(f_n : \mathbb{D} \to \mathbb{D}\)--and is holomorphic. Now let's write:

\[
F_n = \mho_{j=1}^n f_j(z)\,\bullet z = f_n(f_{n-1}(...f_1(z)))\\
\]

Note first of all, that:

\[
F^{-1}_n = \Omega_{j=1}^n f^{-1}_j(z)\,\bullet z\\
\]

The object \(F^{-1}_n\to \infty\) (which is because it's the degenerate case). If this converged, we would have that \(F_n\) converges to a non-constant, which is the final goal of Linda Keen's result. To continue; how I would check that \(F_n\) converges is a little different than what I wrote. Outer compositions \(\mho\), what she calls a "forward iteration system", has slightly different, easier, but different rules--in comparison to inner compositions \(\Omega\)/"backward iteration systems."

My go to theorem to prove convergence, is that:

\[
F_{n+1} - F_n \to 0\\
\]

Which is writ as:

\[
f_{n+1}(F_n) - F_n \to 0\\
\]


But this just expands as:

\[
\sum_{k=1}^\infty \frac{\partial^k}{\partial z^k}f_{n+1}(F_n) \frac{(z-F_n)^k}{k!}\\
\]

At this point, mine and Keen's work is very similar. It's what we do next that's different. The first thing I do, is prove that \(F_n\) is normal; thereby it's a bounded sequence. Keen does the same thing; but she specifies a very general result which appears to be equivalent to this sequence being normal. At this point, I would introduce a summation condition.

We don't need it with Keen though; as we have included a contraction in her idea. Which is that \(F_{n} : \mathbb{D} \to K\), where \(K\) is precompact in \(\mathbb{D}\). Which is just that \(\overline{K} \subset \mathbb{D}\). This is really the shooting gun that does everything. Essentially if \(F_n\) shrinks the unit disk as a normal family, it must converge to a constant. Which, I've never really thought about before, but makes perfect sense. So what happens is that this Taylor series looks like:

\[
\sum_{k=1}^\infty \rho_n^k (z-\lambda)^k\\
\]

Where the sum \(\sum_n \rho_n < \infty\).

Which gives us convergence. This is sort of a hybridization of how I interpret the first half of her paper, and my own work. The actual better part, and valuable part, is the dynamics portion. This is on the difference between nondegenerate and degenerate infinite compositions.

Let \(K \subset \mathbb{D}\), and let \(f_n : \mathbb{D} \to K\). Then the infinite composition is degenerate. In fact, the only time it can be non degenerate is if \(K = \mathbb{D}\). Which is a well known theorem to me and Gill. It's kind of no duh. But this is a fantastic treatment nonetheless.

Additionally, Keen has definitely made some great novel results I haven't really seen before. But again, it's useless, at say, proving the beta function converges. Or even proving weird outer compositions converge; which there are many.

To be honest, the subject of this result, is a generalization of \(f_n(z) = \lambda z\) (for \(|\lambda| < 1\)) and therefore:

\[
\mho_{j=1}^\infty f_n(z)\,\bullet z = \lim_{n\to\infty} \lambda^n z = 0\\
\]

As she has assumed that \(f_n : \mathbb{D} \to K\), where \(K\) is a contraction of \(\mathbb{D}\)--the two ideas are more than comparable. Again, a very complex dynamical result. Beautiful nonetheless

EDIT: There's another author, I'll try to find them, who proves a similar result to Keen. Which kind of let me coin my own internal meaning "Infinite composition of Blaschke products".

Blaschke products are products of automorphisms of \(\mathbb{D}\); where by when you add infinite compositions; you multiply and compose Blaschke products. I believe, if we add the additional assumption that \(F_n \to 0\), which is always possible because they exist in the unit disk (Just apply an automorphism); then Keen's result simplifies even further--to what is a result dating back a much longer time.

This was my inspiration for section 2 of \(\Delta y = e^{sy}\) Or, How I Learned To Stop Worrying and love the \(\Gamma\) function.

I wanted to do what they had done with Blaschke products/infinite compositions, and generalize it to arbitrary functions \(f : \mathbb{D} \to \mathbb{D}\). This has everything to do with the non-degenerate case.

BUT! in the Blaschke product paper, if memory serves me correctly, they show exactly where we are degenerate or non-degenerate. Keen's paper reminds me of this the more I read it, lol.

[MphLee]

Thanks for this commentary. It will be really useful in the future. Now it did help make my mind a lil bit.

---

A quick search on keywords led me to this recent paper that may bring some useful bibliographic pointer.

-2022, Ferreira - A note on forward iteration of inner
functions

To be honest I got a ton of interesting pointers... since it came to my mind that there is a term for dealing with iterated function systems and is non-autonomous dynamical systems, opposed to autonomous ones. Autonomous systems are just representation of time semigroups, i.e. semigroup homomorphisms. While non-autonomous systems are representations of categories... i.e. functors... just like your omega notation.

-2018, Bracci et al. - BACKWARD ORBITS AND PETALS OF SEMIGROUPS OF HOLOMORPHIC
SELF-MAPS OF THE UNIT DISC
-2018, Bracci et al. - ASYMPTOTIC BEHAVIOR OF ORBITS OF HOLOMORPHIC
SEMIGROUPS
-2020, Bracci, Roth - SEMIGROUP-FICATION OF UNIVALENT SELF-MAPS OF THE UNIT DISC
-2022, Benini et al. - The Denjoy–Wolff set for holomorphic sequences,
non-autonomous dynamical systems and wandering domains

I'm always positively surprised by how much of different keywords there are on almost the same topic. Also is descriptive of how much I'm still ignorant on the subject after all these years wandering in the literature.

I remember asking myself few months ago, when I discovered the theorem, if the Denjoy–Wolff point was important or useful. Idk if I ever heard of it on this forum before but I'm sure its a pretty standard result.

[Daniel]

The Ultimate Story
Back in the early days of the Internet a gentleman referred to as Dr. Chaos had the ultimate fascinating posting. He alleged that every few years Stephen Smale, Ralph Abrams and a few other uber elites of dynamics met at Berkeley for a few days to discuss progress in fractional iteration.

[JmsNxn]

Thanks for this commentary. It will be really useful in the future. Now it did help make my mind a lil bit.

---

A quick search on keywords led me to this recent paper that may bring some useful bibliographic pointer.

-2022, Ferreira - A note on forward iteration of inner
functions

To be honest I got a ton of interesting pointers... since it came to my mind that there is a term for dealing with iterated function systems and is non-autonomous dynamical systems, opposed to autonomous ones. Autonomous systems are just representation of time semigroups, i.e. semigroup homomorphisms. While non-autonomous systems are representations of categories... i.e. functors... just like your omega notation.

-2018, Bracci et al. - BACKWARD ORBITS AND PETALS OF SEMIGROUPS OF HOLOMORPHIC
SELF-MAPS OF THE UNIT DISC
-2018, Bracci et al. - ASYMPTOTIC BEHAVIOR OF ORBITS OF HOLOMORPHIC
SEMIGROUPS
-2020, Bracci, Roth - SEMIGROUP-FICATION OF UNIVALENT SELF-MAPS OF THE UNIT DISC
-2022, Benini et al. - The Denjoy–Wolff set for holomorphic sequences,
non-autonomous dynamical systems and wandering domains

I'm always positively surprised by how much of different keywords there are on almost the same topic. Also is descriptive of how much I'm still ignorant on the subject after all these years wandering in the literature.

I remember asking myself few months ago, when I discovered the theorem, if the Denjoy–Wolff point was important or useful. Idk if I ever heard of it on this forum before but I'm sure its a pretty standard result.

HAHAHAHAHA!

I HAVE PRIORITY ON FEREIRA

They hide this by saying:

\[
\sum_{n=0}^\infty |1 - f_n'(0)| < \infty
\]

But this is my condition:

\[
\sum_{n=0}^\infty |f_n(z) -z| < \infty\\
\]

When we restrict \(f_n : \mathbb{D} \to \mathbb{D}\)--these are equivalent statements! THANK GOD I PUBLISHED A LOT IN 2020!!!


To generalize, the beginning of my paper: "The Compositional Integral: The Narrow And The Complex Looking Glass"; gave the result:

\[
f_n: G \to G\\
\]

Then if:

\[
\sum_{n=0}^\infty \sup_{z \in K }|f_n(z) -z| < \infty\\
\]

For all compact \(K \subset G\); then:

\[
\begin{align}
g(z) &= \Omega_{j=1}^\infty f_j(z)\bullet z\\
h(z) &=\mho_{j=1}^\infty f_j(z) \bullet z\\
\end{align}
\]

Were both analytic functions taking \(G \to G\). This was originally proved in \(\Delta y = e^{sy}\). I gloss over the "forward iteration part", but The narrow and complex looking glass completely justifies this once you can rigorously invert; which you can so long as \(f'(z) \neq 0\), which always happens because \(f_n \to z\)...\(f'_n(z) \to 1\), for large enough \(n\).

I apologize Mphlee, but Everything in this paper is work I did 2-5 years ago. God I love being right!


Sorry for being snarky, but some of this stuff just appears as old news to me, lol. I never did too much with forward iteration systems; just for the non-degenerate case, I mapped it back to backwards iteration systems. Especially because forward iteration systems have training wheels. They are very simple. Backwards iteration systems are the real OG; but much harder to work with.

If you solve forward iteration systems; it tells you nothing about backwards iteration systems.
If you solve backward iteration systems; it tells you everything about forward iteration systems.


Either way I'm excited for this stuff to hit the mainstream more, and more people work on it. Just saying, "hey mphlee, this is like 2 pages from my 90 page thesis," lol. Not to knock Fereira, who clearly came to this independently. But in 2015 I had his condition, and communicated to a few people at U of T; which culminated to my paper in 2019 which proved a much much more general result. Then in 2020, I added the differential calculus stuff; and I stated the theorem

Theorem 1.2.1--The Compactly Normal Convergence Theorem

Which appears in Through the looking glass... (2020). The actual details of this theorem are handled by \(\Delta y = e^{sy}\) ; Or How I Learned To Stop Worrying and Love the \(\Gamma\)-function (2019).

And it states a very broad generalization of Fereira's work.

EDIT:

If you'll hear me out. In 2015 I had the condition, if \(f_n : \mathbb{D} \to \mathbb{D}\) where \(\mathbb{D}\) is the unit disk. Then:

\[
\begin{align}
g(z) &= \Omega_{j=1}^\infty f_j(z)\\
h(z) &= \mho_{j=1}^\infty f_j(z)
\end{align}
\]

Converged to functions \(g,h:\mathbb{D} \to \mathbb{D}\) so long as:

\[
\sum_{j=0}^\infty |f'_j(0) - 1| < \infty\\
\]

This is essentially Ferreira's result, but he's only shown it for \(\mho\). My breakthrough in around 2017-2018, was that, on the unit disk:

\[
|z||f'_j(0) - 1| < |f_j(z) -z| \le |f_j(0)|+ |z||f_j'(0) - 1 + \sum_{k=1}^\infty b_kz^k|\\
\]

Where then, if you work with the right hand side; and only worry about:

\[
\sum_{j=0}^\infty |f_j(z) - z| < \infty\\
\]

We do kickflips on what Ferreira does  

Now this reduces into a specific equation:

\[
\sum_{j=0}^\infty |f_j(z) - z| < \sum_j^\infty |f_j(0)| + |z||f_j'(0) - 1|\\
\]

This is Ferreira's approach; which is me in 2015. Obviously this is garbage. You cannot expand from the unit disk to arbitrary domains. For fuck's sake; Mphlee, please understand I am better than everyone at fucking infinite compositions. My work just hasn't been fully published yet.

Where we assume that \(\sum_j f_j(0) \) converges. If \(\Omega_j f_j(0)\) doesn't converge--then we are degenerate. If it does converge, then that means \(\sum_j f_j(0)\) converges (proving this is really tricky; again, mphlee, I'll have my moment. I'll have my moment of confirmation, U of T professors have worked with me a lot. And I will and will not say that I met donald Knuth ). Additionally it implies \(f'_j(0)-1\) must converge in some form.

"Since the linearization converges, the actual composition converges"

Since \(f_j(z) = a_j + b_j (z-z_0) + O(z-z_0)^2\) and:

\[
\begin{align}
\sum_j |a_j| &< \infty\\
\sum_j | b_j - 1| &< \infty\\
\end{align}
\]

Where, with a normality condition:

\[
|f_j(z) - z| < a_j + |z-z_0||b_j -1| + O(z-z_0)^2\\
\]

Which is the mathematics Fereira is using. This never works as a proof system. The math is too hard for that. But this is a good heuristic "it looks something like this".

Consider this sum "compactly normally"; then this object converges for any set \(G\); not just the unit disk \(\mathbb{D}\). This is about the half way marker of \(\Delta y = e^{sy}\). From there, I was able to derive holomorphy in something like \(\Omega f_j(s,z)\); so long as the above sum converged "compactly normally". Not to toot my own horn, but it's nice to see social empirical justification of your work. It's like "reproducibility of the experiment" but for mathematics

Not gonna lie or brag. I didn't talk about it a lot; and I don't; but I will now. I met Donald Knuth in 2019. And we talked for an hour--highlight of my life. I got money shit, Vittorio. We met on a professional sense.

Ima go full torch on everything now. I've proved a lot. I'm going to explain as much as I can.

[JmsNxn]

I just realized you may not be familiar with Schwarz's lemma.

If \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f(0) =0 \); then:

\[
|f(z)| \le |f'(0)||z|
\]

In the infinite composition case; since \(f(z) : \mathbb{D} \to \mathbb{D}\); there is a unique fixed point \(z_0\) (the benefit of using a simply connected domain).

Thereby:

\[
g(z) = h(f(h^{-1}(z))\\
\]

Where \(h: \mathbb{D} \to \mathbb{D}\) biholomorphically; and sends \(z_0 \to 0\). Then:

\[
g'(0) = f'(z_0)\\
\]

The value \(h'(z_0)\) is the value of a Blashcke product (too lazy to do all the fine details); where it acts as the derivative of an automorphism. The following is an equivalent statement. If:

\[
\sum_{j=0}^\infty |f_j'(0) -1| < \infty\\
\]

Then the sum \(\sum_{j=0}^\infty |f(z) -z| < \infty\). BUT!!!! This is only true for the unit disk. If you change into a different simply connected domain; things get much more complicated; and these statements are not equivalent. You'll have to modify some steps...

This is because \(|h(f(h^{-1}(z))| \le |z| |f'(z_0)|\). Now the infinite compositions cancel out... at least to a point. The fixed points \(z_0\) can move around; but since \(f_j'(0) \to 1\), we are guaranteed \(f_j(z) \to z\), because we are guaranteed \(f\) fixes the unit disk. The unique function to satisfy \(f(z) : \mathbb{D} \to \mathbb{D}\) and \(f'(0) = 1\) is \(f(z) = z\). Therefore \(z_0 \to 0\).

And we are just checking that:

\[
0 \neq \prod_{j=1}^\infty f_j'(0) \neq \infty
\]

The quick and easy way is to just say:

\[
\sum_{j=1}^\infty |f_j'(0) -1| < \infty\\
\]

Which dates to Weierstrass...

Which is my old condition (Fereira's condition).  But this is equivalent to just asking:

\[
\sum_{j=1}^\infty |f_j(z) - z| < \infty\\
\]

On the Unit Disk, Fereira's condition looks easier and nicer. But it's a very special case. And not open to generalization.

Can't stress enough that this only happens because of super nice Unit Disk behaviour, small well behaved area. This will not follow on general simply connected domains (though it'll be something similar), and will not follow on domains (Open and connected sets). But it's a great start

EDIT:

Also, since I mentioned Donald Knuth in the last post. He was the one to suggest to me, instead of writing:

\[
\Omega_{j=1}^n f_j(z) \bullet z \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z = \Omega_{j=1}^\infty f_j(z) \bullet z\\
\]

I should just write:

\[
\Omega_{j=1}^n f_j(z) \bullet \Omega_{j=n+1}^\infty f_j(z) \bullet z= \Omega_{j=1}^\infty f_j(z) \bullet z\\\\
\]

So he helped with the bullet notation !!!!!! The whole \(f \bullet g \bullet z\) was kinda his idea in some respects. I had a rough sketch; but it didn't fit right, and was clunky. He helped me stream line some shit. He really added a "functional programming" element to it, lmao.

[marcokrt]

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
(199 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 0 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 198
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 5
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.

Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).
After that, I finally managed to provide an inverse map of this new function (assuming that radix-10 is given by hypothesis), the congruence speed of any integer tetration with the base which is not a multiple of 10, by publishing the mentioned paper entitled "The congruence speed formula" (available also on the arXiv at https://arxiv.org/abs/2208.02622).

Then, me and Luca have finally provided also the direct map of the "congruence speed of tetration" by considering the valuation function (applied to a few very simple manipulations of the given base) of the divisors of the squarefree value of the considered numerical system (i.e., 2 and 5, since 10 = 2 · 5) thanks to the paper titled "Number of stable digits of any integer tetration", which closes the bounds that I previously gave in "The congruence speed formula", by providing extended proofs based on the theorems published in "The congruence speed formula", so I think that it would be the best to mention both "The congruence speed formula" and "Number of stable digits of any integer tetration" (arXiv version: https://arxiv.org/abs/2210.07956) in order to provide the full map of this peculiar property of the integer tetration, named "constant congruence speed".

Just my two cents.

[JmsNxn]

.........

Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).
After that, I finally managed to provide an inverse map of this new function (assuming that radix-10 is given by hypothesis), the congruence speed of any integer tetration with the base which is not a multiple of 10, by publishing the mentioned paper entitled "The congruence speed formula" (available also on the arXiv at https://arxiv.org/abs/2208.02622).

Then, me and Luca have finally provided also the direct map of the "congruence speed of tetration" by considering the valuation function (applied to a few very simple manipulations of the given base) of the divisors of the squarefree value of the considered numerical system (i.e., 2 and 5, since 10 = 2 · 5) thanks to the paper titled "Number of stable digits of any integer tetration", which closes the bounds that I previously gave in "The congruence speed formula", by providing extended proofs based on the theorems published in "The congruence speed formula", so I think that it would be the best to mention both "The congruence speed formula" and "Number of stable digits of any integer tetration" (arXiv version: https://arxiv.org/abs/2210.07956) in order to provide the full map of this peculiar property of the integer tetration, named "constant congruence speed".

Just my two cents.

I am very interested in your work; as it deals with digit analysis. You've uncovered a general structure that the digts play. Have you ever tried \(\sqrt{2}\), and dealing with similar modular results? As an analyst myself; I tend to not be so worried about the digit patterns that appear in \(^52\). But if such a digit pattern were to appear in \(\sqrt{2}\), this would define an algebraic result.

Not to spoil what you are doing. I have followed your posts closely. I suggest reading about \(p\)-adic analysis. I cannot reduce your results to \(p\)-adic results. But for fucks sakes; it smells like it. There is a \(p\)-adic interpretation of your result. I do not know it; but I could probably work a guess. In this language you should find a clearer version of your formula. Not to degrade your result; you have done great work. Just to suggest--I believe we can transplant this idea.

Either way; I apologize if I'm being presumptuous. I'm just trying to help  

[marcokrt]

.........

Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).
After that, I finally managed to provide an inverse map of this new function (assuming that radix-10 is given by hypothesis), the congruence speed of any integer tetration with the base which is not a multiple of 10, by publishing the mentioned paper entitled "The congruence speed formula" (available also on the arXiv at https://arxiv.org/abs/2208.02622).

Then, me and Luca have finally provided also the direct map of the "congruence speed of tetration" by considering the valuation function (applied to a few very simple manipulations of the given base) of the divisors of the squarefree value of the considered numerical system (i.e., 2 and 5, since 10 = 2 · 5) thanks to the paper titled "Number of stable digits of any integer tetration", which closes the bounds that I previously gave in "The congruence speed formula", by providing extended proofs based on the theorems published in "The congruence speed formula", so I think that it would be the best to mention both "The congruence speed formula" and "Number of stable digits of any integer tetration" (arXiv version: https://arxiv.org/abs/2210.07956) in order to provide the full map of this peculiar property of the integer tetration, named "constant congruence speed".

Just my two cents.

I am very interested in your work; as it deals with digit analysis. You've uncovered a general structure that the digts play. Have you ever tried \(\sqrt{2}\), and dealing with similar modular results? As an analyst myself; I tend to not be so worried about the digit patterns that appear in \(^52\). But if such a digit pattern were to appear in \(\sqrt{2}\), this would define an algebraic result.

Not to spoil what you are doing. I have followed your posts closely. I suggest reading about \(p\)-adic analysis. I cannot reduce your results to \(p\)-adic results. But for fucks sakes; it smells like it. There is a \(p\)-adic interpretation of your result. I do not know it; but I could probably work a guess. In this language you should find a clearer version of your formula. Not to degrade your result; you have done great work. Just to suggest--I believe we can transplant this idea.

Either way; I apologize if I'm being presumptuous. I'm just trying to help  

Thank you for your interest, I would be very glad if you (or anyone else on this forum) will publish further results about the congruence speed concept, generalizing what I have written and going beyond.
Basically, my starting idea is just to "count" how many digits frozes each unitary increment of the hyperexponent (and we could apply this also to hyper-\(3\) or hyper-\(5\), in general we have a function of the base and the (hyper)-exponent, which turns to not depend on the (hyper)-exponent under certain circumstances).
Then, by assuming radix-\(10\) and that the base is a positive integer which is not congruent to \(0\) modulo \(10\) by hypotesis, everything comes from the \(15\) solutions of the fifth degree equation \(y^t=y\) in the ring of decadic integers, since \(10\) is not a prime.
Now, since I am just a self-taught amateur in number theory, I am aware that the behaviour of \(p\)-adic numbers is "good" and smooth if compared to the behaviour of \(g\)-adic ones (where \(g\) is not prime). In general, I strongly believe that we can reply what we already did in "Number of stable digits of any integer tetration" (see equation 16) for any \(g\) that is a squarefree semiprime and also if \(g\) is a generic squarefree composite number, maybe... but it would be a great step forward if you can write something that shows this by using only \(p\)-adic analysis, best wishes!

[MphLee]

Awesome list! I would like to just point out that my real surname is "Ripà" (instead of "Ripa", a very common mistake indeed), mentioning also the last paper that I have written together with the other TetrationForum user "Luca Onnis", entitled "Number of stable digits of any integer tetration", which completes the trilogy on the congruence speed that I started by submitting to Notes on Number Theory and Discrete Mathematics, in 2019, the manuscript entitled "On the constant congruence speed of tetration" (see NNTDM, Vol. 26(3), pp. 245-260, DOI: 10.7546/nntdm.2020.26.3.245-260).

Hi Marco, forgive me for the typo. I'm actually well aware of your surname, I'm also italian so I don't have the excuse for being not used to accents. The reason for the error is, I was copy-pasting the name of the files in my folder on the forum, and windows do not accept special symbols naming the files then I had no enough time to double check the post.

Thanks for the contribution btw. I'll update my folder with your work. Some work on number theory related to hyperoperations is always welcome imho... since it is an underdeveloped topic. We need to face the problem from multiple viewpoints.

Cordialmente

ps: mi sono preso la liberta di usare i miei poteri da mod per editare il tuo ultimo messaggio e sistemare la sintassi LaTex, adesso aprendo il post puoi controllare come va fatta. You have to write "\ (" before a formula and "\ )" as closure.