Let's let \(f(x) = x^9\), then the fractional iteration is precisely \(f^{\circ s}(x) = x^{9^s}\)--whereby \(f^{\circ \frac{1}{2}}(x) = x^3\).
I have not investigated these cases at all. These relate to the cases \(f'(0) = 0\)--and we have to use the Bottcher coordinate. I have never investigated infinite compositions as they relate to Bottcher's coordinates--as they become much more volatile. \(f^{\circ s}(x)\) is holomorphic not holomorphic in a neighborhood of \(x=0\) (generally), unless we invoke a natural iterate.
So, for example, if we write:
\[
\gamma_c(w) = \Omega_{j=1}^\infty \frac{wf(z)}{w+c^j}\bullet z\\
\]
Then:
\[
\gamma_c(cw) = \frac{w f(\gamma_c(w))}{w+1}\\
\]
I highly doubt, choosing \(f^{-1}(z) = \sqrt[9]{z}\), that:
\[
\lim_{k\to\infty} f^{-k} \gamma_c(c^kw)\\
\]
Converges for any \(c\). This is based on the heuristic, when \(|f'(0)| = 1\), this object converges for \(|c| > 1\), and for \(|f'(0)| = a > 1\), that this object only converges for \(|c| > a\). When \(|f'(0)| = a < 1\), then this limit converges for: \(|1/c| < a\). So when \(a =0\), we can expect \(|1/c| < 0\)--which means it should never converge. Though of course, there may be some boundary behaviour. We may be able to get an abel function, but we'd have to finesse a lot.
I haven't done any research on the infinite composition method when dealing with \(f'(0) = 0\) and the Bottcher coordinate. So I can't say forsure. It's never interested me, as this opens us up to a lot of branching talk right off the get go...... And it appears pretty naturally that Mock-Schroder is useless.
We could probably make a mock Bottcher though...
So we maybe able to solve:
\[
H(w^9) = f(H(w))h(w)\\
\]
Which would be written as:
\[
H(w) = \Omega_{j=1}^\infty h(w^{9^{-j}})f(z)\bullet z\\
\]
This may be able to approximate the standard Bottcher coordinate, considering \(h\) is well enough behaved... We would write the limit as:
\[
f^{-j} H(w^{9^j}) \to \Psi\\
\]
Where:
\[
\Psi(w^9) = f(\Psi(w))\\
\]
Which may converge to a Bottcher coordinate, or the standard one.--which would allow us to reconstruct the standard fractional iterate \(f^{\circ s}(x) = x^{9^s}\)...
But since \(f(x) = x^9\) is its own Bottcher coordinate, we'd have that \(\Psi(w) \to w\).... this seems unlikely, or at least difficult to pull off. It could happen though, I don't see why not. I hate Bottcher coordinates, so I've done next to no research on the super attracting case of dynamics. Infinite compositions can definitely aid, but I don't think they shed as much light as in the Schroder/Abel case--geometric/parabolic...
For example, setting:
\[
h(w) = w-1\\
\]
Then \(H(w)\) is holomorphic everywhere:
\[
\sum_{j=1}^\infty |w^{9^{-j}} -1| < \infty\\
\]
Which is for \(w \in \mathbb{C}/(-\infty,0)\). So that:
\[
H(w^9) = H(w)^9 (w-1)\\
\]
Maybe the iterated thing would converge to \(\Psi(w) = w\)... would be an interesting test to run! If it does, it should follow that it works for more complex beasts, where \(f(x) = x^9 + O(x^{10})\). Or for any general \(f(x) = x^n + O(x^{n+1})\).... And reducing to the Bottcher's coordinate in the general sense. Unfortunately, Bottcher's coordinate, doesn't allow for as many natural generalizations. Schroder's generalizations, allow us to create arbitrary multipliers in the Schroder function. Any function \(f(w^9) = f(w)^n\), is well understood and more restrictive. So I can't imagine us having as much freedom.
EDIT:
But, using your formula, we look at:
\[
P_c(x) = \sum_{k=-\infty}^\infty c^k x^{9^k}\\
\]
For \(|c| > 1\) and \(|x| < 1\) and \(x\not \in (-1,0)\), then, absolutely this thing is holomorphic, by which:
\[
P_c(x^9) = cP_c(x)\\
\]
This kind of confirms my suspicion, that this is the complement of the infinite composition/mock schroder/levenstein approach. Where everything fails in my case, your method picks up the slack entirely. And where my method usually works, yours fails, and where mine fails, yours works. And this seems to be universally happening!
Very interesting regardless!!!!
So for instance, using Neutral fixed points \(f'(0) = 1\), I expect your formula NEVER CONVERGES for \(|c|>1\), because my formula converges EVERYWHERE \(|c|>1\). For Geometric fixed points, \(|f'(0)| \neq 0,1\), my formula converges where yours doesnt, and yours converges where mine doesn't. And for super attractive \(f'(0) = 0\), my formula NEVER CONVERGES, but yours converges EVERYWHERE.
So if my formula converges for \(|c| > a\), yours converges for \(1 < | c | <a\). If mine converges for \( |c| >1\), yours never converges (which I can prove). But when mine doesn't converge for all \(|c| > 1\), then yours converges for \(|c|>1\)....
So, considering \(|c| >1\)--where my formula works, yours doesn't, where mine doesn't, yours does... for all \(|c| > 1\). Are they analytic continuations of each other!?!?!?!?!?
I do believe we have a complimentary relationship going on with this shit!
I have not investigated these cases at all. These relate to the cases \(f'(0) = 0\)--and we have to use the Bottcher coordinate. I have never investigated infinite compositions as they relate to Bottcher's coordinates--as they become much more volatile. \(f^{\circ s}(x)\) is holomorphic not holomorphic in a neighborhood of \(x=0\) (generally), unless we invoke a natural iterate.
So, for example, if we write:
\[
\gamma_c(w) = \Omega_{j=1}^\infty \frac{wf(z)}{w+c^j}\bullet z\\
\]
Then:
\[
\gamma_c(cw) = \frac{w f(\gamma_c(w))}{w+1}\\
\]
I highly doubt, choosing \(f^{-1}(z) = \sqrt[9]{z}\), that:
\[
\lim_{k\to\infty} f^{-k} \gamma_c(c^kw)\\
\]
Converges for any \(c\). This is based on the heuristic, when \(|f'(0)| = 1\), this object converges for \(|c| > 1\), and for \(|f'(0)| = a > 1\), that this object only converges for \(|c| > a\). When \(|f'(0)| = a < 1\), then this limit converges for: \(|1/c| < a\). So when \(a =0\), we can expect \(|1/c| < 0\)--which means it should never converge. Though of course, there may be some boundary behaviour. We may be able to get an abel function, but we'd have to finesse a lot.
I haven't done any research on the infinite composition method when dealing with \(f'(0) = 0\) and the Bottcher coordinate. So I can't say forsure. It's never interested me, as this opens us up to a lot of branching talk right off the get go...... And it appears pretty naturally that Mock-Schroder is useless.
We could probably make a mock Bottcher though...
So we maybe able to solve:
\[
H(w^9) = f(H(w))h(w)\\
\]
Which would be written as:
\[
H(w) = \Omega_{j=1}^\infty h(w^{9^{-j}})f(z)\bullet z\\
\]
This may be able to approximate the standard Bottcher coordinate, considering \(h\) is well enough behaved... We would write the limit as:
\[
f^{-j} H(w^{9^j}) \to \Psi\\
\]
Where:
\[
\Psi(w^9) = f(\Psi(w))\\
\]
Which may converge to a Bottcher coordinate, or the standard one.--which would allow us to reconstruct the standard fractional iterate \(f^{\circ s}(x) = x^{9^s}\)...
But since \(f(x) = x^9\) is its own Bottcher coordinate, we'd have that \(\Psi(w) \to w\).... this seems unlikely, or at least difficult to pull off. It could happen though, I don't see why not. I hate Bottcher coordinates, so I've done next to no research on the super attracting case of dynamics. Infinite compositions can definitely aid, but I don't think they shed as much light as in the Schroder/Abel case--geometric/parabolic...
For example, setting:
\[
h(w) = w-1\\
\]
Then \(H(w)\) is holomorphic everywhere:
\[
\sum_{j=1}^\infty |w^{9^{-j}} -1| < \infty\\
\]
Which is for \(w \in \mathbb{C}/(-\infty,0)\). So that:
\[
H(w^9) = H(w)^9 (w-1)\\
\]
Maybe the iterated thing would converge to \(\Psi(w) = w\)... would be an interesting test to run! If it does, it should follow that it works for more complex beasts, where \(f(x) = x^9 + O(x^{10})\). Or for any general \(f(x) = x^n + O(x^{n+1})\).... And reducing to the Bottcher's coordinate in the general sense. Unfortunately, Bottcher's coordinate, doesn't allow for as many natural generalizations. Schroder's generalizations, allow us to create arbitrary multipliers in the Schroder function. Any function \(f(w^9) = f(w)^n\), is well understood and more restrictive. So I can't imagine us having as much freedom.
EDIT:
But, using your formula, we look at:
\[
P_c(x) = \sum_{k=-\infty}^\infty c^k x^{9^k}\\
\]
For \(|c| > 1\) and \(|x| < 1\) and \(x\not \in (-1,0)\), then, absolutely this thing is holomorphic, by which:
\[
P_c(x^9) = cP_c(x)\\
\]
This kind of confirms my suspicion, that this is the complement of the infinite composition/mock schroder/levenstein approach. Where everything fails in my case, your method picks up the slack entirely. And where my method usually works, yours fails, and where mine fails, yours works. And this seems to be universally happening!
Very interesting regardless!!!!
So for instance, using Neutral fixed points \(f'(0) = 1\), I expect your formula NEVER CONVERGES for \(|c|>1\), because my formula converges EVERYWHERE \(|c|>1\). For Geometric fixed points, \(|f'(0)| \neq 0,1\), my formula converges where yours doesnt, and yours converges where mine doesn't. And for super attractive \(f'(0) = 0\), my formula NEVER CONVERGES, but yours converges EVERYWHERE.
So if my formula converges for \(|c| > a\), yours converges for \(1 < | c | <a\). If mine converges for \( |c| >1\), yours never converges (which I can prove). But when mine doesn't converge for all \(|c| > 1\), then yours converges for \(|c|>1\)....
So, considering \(|c| >1\)--where my formula works, yours doesn't, where mine doesn't, yours does... for all \(|c| > 1\). Are they analytic continuations of each other!?!?!?!?!?
I do believe we have a complimentary relationship going on with this shit!