Okay, so I had a very productive day. I sat down and reread Milnor's treatment of Ecalle's construction, to see if there's anything to take from it, but unfortunately not--other than some keywords that helped me on my search. I then went on a search for Fatou coordinates/ and borel summations. I came across a bunch of papers, and much of them ended up being about tangentially related things, but two papers really shone and are definitely relevant to our discussion here.
To begin, let's assume we have a function:
\[
f(z) = z + z^2 + o(z^2)\\
\]
Where the value \(2\) can be replaced with \(n > 1\) but requires more work; and we can add a coefficient here \(az^n\)--but is unnecessary to this explanation.
Then the attracting petal of \(f\) is centered on the direction \(-1\); I.e. \(f^{\circ n}(z) \to 0\) for \( z \in (-\delta, 0)\). If we take the change of variables:
\[
F(w) = \frac{-1}{f(-1/w)}\\
\]
Then this function is holomorphic for \(- \tau < \arg(w) < \tau\) for \(|w| > R\) for large enough \(R\). But aditionally, the Abel function \(\alpha\) of this \(F\) satisfies the expansion:
\[
\begin{align}
\alpha(F(w)) &= \alpha(w) + 1\\
\alpha(w) &= w - A \log(w) + \sum_{k=0}^\infty b_k w^{-k}\\
\end{align}
\]
And, wait for it...
\[
b_k = O(c^kk!)\\
\]
For some \(c > 0\). Which means ITS BOREL SUMMABLE!
The first reference is a beast of a paper--largely laying out all the formal manifestations of borel sums--and goes on to talk about germs and a lot of Ecalle's work. You don't have to read this paper, but I thought I'd put it here, as it does a great job of describing the state of affairs of borel sums.
https://hal.archives-ouvertes.fr/hal-008...nt#cite.LY
The second reference is a holy grail. It explicitly proves the above expansion, and proves that the coefficients \(b_k\) are of "Gevrey class 1"--which is their way of saying the above bound. Which is the equivalent of saying that it is Borel summable.
https://arxiv.org/pdf/1108.2801.pdf
I know this doesn't prove anything about the logit per se. But if we use Bo's above transformation rules--we can relate one borel summable series to the other (it would just be a couple of differentials, no biggie, shouldn't upset the expansion). And it would add the appropriate bound to bo's expansion above.
We'd just be writing:
\[
f^{\circ t}(z) = \alpha^{-1}(t + \alpha(z))\\
\]
And:
\[
\alpha'(z) = 1 + \frac{A}{z} + \sum_{k=1}^\infty c_k z^k
\]
Where \(c_k\) is of Gevrey class 1.
Where \(\alpha(z)\) has a borel summable series at \(z=0\), and this would translate to a borel summable series because \(\alpha^{-1}\) should be regular enough at \(t+\infty\). Actually, a lot of this stuff is covered about 20 pages into the first reference! OH YA!
This won't prove the nice sinusoids that you guys are showing. But It should show we have the Borel sum bound! OH YA! I expect the sinusoids are off shoots of \(n\) and \(a\) where in the general case:
\[
f(z) = e^{2 \pi i k/q} z + az^{n} + o(z^n)\\
\]
And we get a different abel function about each attracting petal (rather than there just being one attracting petal). But they are heavily related to orbits that look like \(f^{\circ m}(z) \sim e^{2 \pi i km/q}m^{-1/(n-1)}\); which would partially explain the sinusoidal behaviour. Especially for half iterates.
EDIT:
Since, Bo and Tommy and Gottfried are in bed, I thought I'd add some more thoughts.
We are definitely going to see a lot of wave like properties, and from there, you can expect the coefficients follow like sin's coefficients--i.e. follow a wave themselves. \(\sin(z)\) is a wave, and \(\frac{d^t}{dz^t} \sin(z) = \sin(z+\frac{t \pi}{2})\) is a wave. This works out pretty similarly for any wave like function; its fractional derivative is a wave like function; therefore so are its coefficients. So I'm less and less surprised by the sinusoidal nature of the coefficients. But I don't know how to explain it properly.
This will also tie in very well with the mellin transform version of this result. \(e^z -1\) was a very nice function that had very nice behaviour in the left half plane. The thing is, more generally we have to reduce this into a sector. And by which, we have to set up the mellin transforms differently. But in the same breath, it will even further argue the wave like nature. Any thing that's mellin transformable, is really a codified way of fourier transformable. And these things are always waves.
I'd also like to add that I mean waves as a quantum physicist means waves, some \(L^2\) integrable constructions. Which quite literally just look like waves, lol.
I really want to understand the wave like structure. Especially now, because of this deep dive.
This is the problem that just keeps on giving!
I'll make another post, from here, which attempts to put all of this together in a more firm manner. Can't tonight, need to think about this more, but I'll separate from bo's thread from here!
To begin, let's assume we have a function:
\[
f(z) = z + z^2 + o(z^2)\\
\]
Where the value \(2\) can be replaced with \(n > 1\) but requires more work; and we can add a coefficient here \(az^n\)--but is unnecessary to this explanation.
Then the attracting petal of \(f\) is centered on the direction \(-1\); I.e. \(f^{\circ n}(z) \to 0\) for \( z \in (-\delta, 0)\). If we take the change of variables:
\[
F(w) = \frac{-1}{f(-1/w)}\\
\]
Then this function is holomorphic for \(- \tau < \arg(w) < \tau\) for \(|w| > R\) for large enough \(R\). But aditionally, the Abel function \(\alpha\) of this \(F\) satisfies the expansion:
\[
\begin{align}
\alpha(F(w)) &= \alpha(w) + 1\\
\alpha(w) &= w - A \log(w) + \sum_{k=0}^\infty b_k w^{-k}\\
\end{align}
\]
And, wait for it...
\[
b_k = O(c^kk!)\\
\]
For some \(c > 0\). Which means ITS BOREL SUMMABLE!
The first reference is a beast of a paper--largely laying out all the formal manifestations of borel sums--and goes on to talk about germs and a lot of Ecalle's work. You don't have to read this paper, but I thought I'd put it here, as it does a great job of describing the state of affairs of borel sums.
https://hal.archives-ouvertes.fr/hal-008...nt#cite.LY
The second reference is a holy grail. It explicitly proves the above expansion, and proves that the coefficients \(b_k\) are of "Gevrey class 1"--which is their way of saying the above bound. Which is the equivalent of saying that it is Borel summable.
https://arxiv.org/pdf/1108.2801.pdf
I know this doesn't prove anything about the logit per se. But if we use Bo's above transformation rules--we can relate one borel summable series to the other (it would just be a couple of differentials, no biggie, shouldn't upset the expansion). And it would add the appropriate bound to bo's expansion above.
We'd just be writing:
\[
f^{\circ t}(z) = \alpha^{-1}(t + \alpha(z))\\
\]
And:
\[
\alpha'(z) = 1 + \frac{A}{z} + \sum_{k=1}^\infty c_k z^k
\]
Where \(c_k\) is of Gevrey class 1.
Where \(\alpha(z)\) has a borel summable series at \(z=0\), and this would translate to a borel summable series because \(\alpha^{-1}\) should be regular enough at \(t+\infty\). Actually, a lot of this stuff is covered about 20 pages into the first reference! OH YA!
This won't prove the nice sinusoids that you guys are showing. But It should show we have the Borel sum bound! OH YA! I expect the sinusoids are off shoots of \(n\) and \(a\) where in the general case:
\[
f(z) = e^{2 \pi i k/q} z + az^{n} + o(z^n)\\
\]
And we get a different abel function about each attracting petal (rather than there just being one attracting petal). But they are heavily related to orbits that look like \(f^{\circ m}(z) \sim e^{2 \pi i km/q}m^{-1/(n-1)}\); which would partially explain the sinusoidal behaviour. Especially for half iterates.
EDIT:
Since, Bo and Tommy and Gottfried are in bed, I thought I'd add some more thoughts.
We are definitely going to see a lot of wave like properties, and from there, you can expect the coefficients follow like sin's coefficients--i.e. follow a wave themselves. \(\sin(z)\) is a wave, and \(\frac{d^t}{dz^t} \sin(z) = \sin(z+\frac{t \pi}{2})\) is a wave. This works out pretty similarly for any wave like function; its fractional derivative is a wave like function; therefore so are its coefficients. So I'm less and less surprised by the sinusoidal nature of the coefficients. But I don't know how to explain it properly.
This will also tie in very well with the mellin transform version of this result. \(e^z -1\) was a very nice function that had very nice behaviour in the left half plane. The thing is, more generally we have to reduce this into a sector. And by which, we have to set up the mellin transforms differently. But in the same breath, it will even further argue the wave like nature. Any thing that's mellin transformable, is really a codified way of fourier transformable. And these things are always waves.
I'd also like to add that I mean waves as a quantum physicist means waves, some \(L^2\) integrable constructions. Which quite literally just look like waves, lol.
I really want to understand the wave like structure. Especially now, because of this deep dive.
This is the problem that just keeps on giving!
I'll make another post, from here, which attempts to put all of this together in a more firm manner. Can't tonight, need to think about this more, but I'll separate from bo's thread from here!
