Major references

[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.