Beautiful thread, LEO!
These are the same problems I am facing now. And I've only managed to solve it in the parabolic case. So as you, use your example \(-z + z^2\), I use \(e^{-z}+1\) (I like dealing with non-polynomial constructions off the bat; in case you're just proving something special for polynomials). And you have \(q=2\), which is exactly what I write...
The only difference I am trying to observe in this construction you've beautifully exposited: I'm trying to watch this reaction beneath the integral. This is truly fascinating. And I'm all ears to your experiments. My experiments are all with integrals; any different angle is a plus for me, so keep hitting it.
So, for example, let's take:
\[
f_\pi(z) = e^{-z}+1\\
\]
Then \(f_{\pi}^2(z) = e^{-e^{-z}-1}+1 = z + O(z^2) = h(z)\).
There are two petals for \(f_\pi\), (and for all \(f_\theta\)), but there is a transposition in each application of \(f_\pi\) (an order \(q=2\) permutation). So when we take \(h^{\circ k}(z)\), then if \(k \equiv 0\) \(h(z) \approx 0\) when \(\Re(z) < 0\). And if \(k \equiv 1\) \(h(z) \approx 0\) when \(\Re(z) > 0\). Where the closeness to \(0\) is asymptotically \(\frac{1}{\sqrt{k z}}\) (or something close to this, it can vary a bit).
Now when I write:
\[
\vartheta[h](x,z) = \sum_{n=0}^\infty h^{\circ n+1}(z) \frac{x^n}{n!}\\
\]
We can prove that the differintegral at zero converges, and that we get:
\[
h^{\circ s}(z) \Gamma(1-s) = \int_0^\infty \vartheta[h](x,z) x^{-s}\,dx\\
\]
But how do we transform \(h^{\circ s} = (f_\pi^{\circ 2})^{\circ s}\) into \(f_\pi^{\circ s}\)? Something you've so clearly articulated. And I'm super excited to everything you can bring to the table!!!
My goal, and how I think. Is that this problem can be solved by operations on \(\vartheta\) and "how we take the integral". For the parabolic case I am fairly confident. But I am not confident for \(f_{2\pi \xi}(z)\) where \( \xi\) is irrational. The general idea seems like it could work, but things don't seem to be working--in as straight forward a manner, at least.
So I have a question for you.
Let \(\xi = \sqrt{2} -1\), What are your comments on:
\[
p(z) = e^{2\pi i\xi} z + z^2\\
\]
Siegel disks tend to be needed here, and they develop fairly good Schroder "similar" constructions. But Abel functions still seem super mysterious in the general sense. I'm very curious if you can run your ideas on this
Nice to have you back, Leo!
These are the same problems I am facing now. And I've only managed to solve it in the parabolic case. So as you, use your example \(-z + z^2\), I use \(e^{-z}+1\) (I like dealing with non-polynomial constructions off the bat; in case you're just proving something special for polynomials). And you have \(q=2\), which is exactly what I write...
The only difference I am trying to observe in this construction you've beautifully exposited: I'm trying to watch this reaction beneath the integral. This is truly fascinating. And I'm all ears to your experiments. My experiments are all with integrals; any different angle is a plus for me, so keep hitting it.
So, for example, let's take:
\[
f_\pi(z) = e^{-z}+1\\
\]
Then \(f_{\pi}^2(z) = e^{-e^{-z}-1}+1 = z + O(z^2) = h(z)\).
There are two petals for \(f_\pi\), (and for all \(f_\theta\)), but there is a transposition in each application of \(f_\pi\) (an order \(q=2\) permutation). So when we take \(h^{\circ k}(z)\), then if \(k \equiv 0\) \(h(z) \approx 0\) when \(\Re(z) < 0\). And if \(k \equiv 1\) \(h(z) \approx 0\) when \(\Re(z) > 0\). Where the closeness to \(0\) is asymptotically \(\frac{1}{\sqrt{k z}}\) (or something close to this, it can vary a bit).
Now when I write:
\[
\vartheta[h](x,z) = \sum_{n=0}^\infty h^{\circ n+1}(z) \frac{x^n}{n!}\\
\]
We can prove that the differintegral at zero converges, and that we get:
\[
h^{\circ s}(z) \Gamma(1-s) = \int_0^\infty \vartheta[h](x,z) x^{-s}\,dx\\
\]
But how do we transform \(h^{\circ s} = (f_\pi^{\circ 2})^{\circ s}\) into \(f_\pi^{\circ s}\)? Something you've so clearly articulated. And I'm super excited to everything you can bring to the table!!!
My goal, and how I think. Is that this problem can be solved by operations on \(\vartheta\) and "how we take the integral". For the parabolic case I am fairly confident. But I am not confident for \(f_{2\pi \xi}(z)\) where \( \xi\) is irrational. The general idea seems like it could work, but things don't seem to be working--in as straight forward a manner, at least.
So I have a question for you.
Let \(\xi = \sqrt{2} -1\), What are your comments on:
\[
p(z) = e^{2\pi i\xi} z + z^2\\
\]
Siegel disks tend to be needed here, and they develop fairly good Schroder "similar" constructions. But Abel functions still seem super mysterious in the general sense. I'm very curious if you can run your ideas on this
Nice to have you back, Leo!