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.
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.