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.