Hmmm, that's very interesting. So since this fundamental domain isn't injective, the idea of the riemann mapping would fail, and the Kneser construction would be nulled right? At least, that's how I interpret this.
Very fascinating. So you'd want to take the attracting petal of \(b = e^{-e}\) which includes \(0\) and just construct the Abel function about an indifferent fixed point and call it that then? So that there wouldn't even be a Kneser construction.
I imagine you'd be able to derive the decay to the fixed points in the upper-lower planes, so it'd almost be like a pseudo Kneser? You never actually use a Kneser Riemann Mapping, you just do the Abel iteration and:
\[
\begin{align*}
\alpha^{-1}(0) &= 1\\
\alpha^{-1}(z) \,\,&\text{is holomorphic on}\, \mathbb{C}/(-\infty,-2]\\
\alpha^{-1} &: (-2,\infty) \to (-\infty, 1/e)\\
\alpha^{-1}(z) &\to 1/e \,\text{as } \Re(z) \to \infty\\
\alpha^{-1}(z) &\to a^{\pm}\,\text{as}\, \Im(z) \to \pm \infty\\
e^{-e\alpha^{-1}(z)} &= \alpha^{-1}(z+1)\\
\end{align*}
\]
Where \(a^\pm\) are the fixed points of \(e^{-ez}\) with nonzero imaginary part, with the least imaginary part.
I mean, this construction should be entirely possible, and it looks an awful like Kneser, but if the riemann mapping approach is nonsensical cause there's no fundamental domain, it sounds more like a "pseudo"-Kneser.
I guess the main question would be how to ensure holomorphy at precisely this point in \(b\). Would we get another branching problem like at \(\eta\)? That doesn't really make sense though, unless the branch cut formed by \(\eta\) were more disastrous than we think, which maybe spawns at \(\eta\) but continues to \(\eta_-\). Perhaps it goes all the way to \(0\).
Paulsen never explains how to take the branch cut in his construction, but I wouldn't be surprised if it resulted in something like:
\[
b \in [0,\eta]\\
\]
This would be really odd though, as holomorphy, about say, \(\sqrt{2}\) shouldn't be a problem for Kneser... right? But maybe there would be a branching problem in \(b\) about these points that we haven't encountered yet.
I also haven't familiarized myself enough with Kneser, so I may be saying stupid things. But no fundamental domain means no Riemann mapping... right?
What if, Kneser is perfectly continuable at the point \(b = \eta_-\), but there exists no fundamental domain? In the sense that, the standard Abel iteration is Kneser at \(\eta_-\), this would also follow for \(\eta\) as a limit, but there's trouble doing that, because it's a branching point in not only the iterated exponential, but also in \(b = y^{1/y}\). \(\eta_-\) has all the benefits of being a real valued, neutral fixed point, without the trouble of being a critical point.
I mean, what if for complex tetration \(\eta_-\) gave us a view of how standard petal/Ecalle iteration about an attracting basin, relates to the Kneser method about two fixed points. In a similar manner that \(\eta\) does. But in this case \(\eta \uparrow \uparrow z \to \infty\) as we let \(\Re(z) \to - \infty\) (the iterated log largely diverges). With \(\eta_-\) we have nearby attracting fixed points for \(\log\) and if we iterate we aproach them. And not only that.
\[
\log_{\eta_-}^{\circ n}(z) = a^{\pm}\,\,\text{depending on if}\,\,\pm\Im(z) > 0\\
\]
But not only that, there exists a real valued abel iteration about an indifferent fixed point. THERE'S NO NEED TO DO THE RIEMANN MAPPING, HENCE NO FUNDAMENTAL DOMAIN!
But nonetheless \(b\) is holomorphic at \(\eta_-\)! Because, the standard Abel iteration about the attracting petal that includes \(0\) (I mean this as there are 4 petals about \(1/e\) for the \(\eta_-^z\), and therefore 4 Abel functions on each petal, which can and cannot be continued together--there's one petal that \(0\) is in, use that Abel iteration (Think of this like if we choose the eta iteration or the cheta iteration, but now we have 4 choices--choose the one with 0)).
Think of it like this. Take the standard Abel iteration of \(b = \eta_-\) and let's force it to be real as \(b\) moves on the real line. What if a way of talking about Kneser is a perturbation of the abel solution at \(b = \eta_-\). Very much how you spoke of perturbed parabolic points (forget the exact words). But we do it at \(b = \eta_-\) instead. And then \(b = \eta\) is a branching problem.
The problem I forsee with this approach is that the branch cuts will look like \(b \in (-\infty,0) \cup \(\eta,\infty)\). This might be weird but this was something I was very interested in, made a thread recently.
Either way, I guess, I'm asking if we could think of \(\eta_-\) as a complex valued \(\eta\). I think this might help a lot. It's like Kneser's \(\eta\).
Very fascinating. So you'd want to take the attracting petal of \(b = e^{-e}\) which includes \(0\) and just construct the Abel function about an indifferent fixed point and call it that then? So that there wouldn't even be a Kneser construction.
I imagine you'd be able to derive the decay to the fixed points in the upper-lower planes, so it'd almost be like a pseudo Kneser? You never actually use a Kneser Riemann Mapping, you just do the Abel iteration and:
\[
\begin{align*}
\alpha^{-1}(0) &= 1\\
\alpha^{-1}(z) \,\,&\text{is holomorphic on}\, \mathbb{C}/(-\infty,-2]\\
\alpha^{-1} &: (-2,\infty) \to (-\infty, 1/e)\\
\alpha^{-1}(z) &\to 1/e \,\text{as } \Re(z) \to \infty\\
\alpha^{-1}(z) &\to a^{\pm}\,\text{as}\, \Im(z) \to \pm \infty\\
e^{-e\alpha^{-1}(z)} &= \alpha^{-1}(z+1)\\
\end{align*}
\]
Where \(a^\pm\) are the fixed points of \(e^{-ez}\) with nonzero imaginary part, with the least imaginary part.
I mean, this construction should be entirely possible, and it looks an awful like Kneser, but if the riemann mapping approach is nonsensical cause there's no fundamental domain, it sounds more like a "pseudo"-Kneser.
I guess the main question would be how to ensure holomorphy at precisely this point in \(b\). Would we get another branching problem like at \(\eta\)? That doesn't really make sense though, unless the branch cut formed by \(\eta\) were more disastrous than we think, which maybe spawns at \(\eta\) but continues to \(\eta_-\). Perhaps it goes all the way to \(0\).
Paulsen never explains how to take the branch cut in his construction, but I wouldn't be surprised if it resulted in something like:
\[
b \in [0,\eta]\\
\]
This would be really odd though, as holomorphy, about say, \(\sqrt{2}\) shouldn't be a problem for Kneser... right? But maybe there would be a branching problem in \(b\) about these points that we haven't encountered yet.
I also haven't familiarized myself enough with Kneser, so I may be saying stupid things. But no fundamental domain means no Riemann mapping... right?
What if, Kneser is perfectly continuable at the point \(b = \eta_-\), but there exists no fundamental domain? In the sense that, the standard Abel iteration is Kneser at \(\eta_-\), this would also follow for \(\eta\) as a limit, but there's trouble doing that, because it's a branching point in not only the iterated exponential, but also in \(b = y^{1/y}\). \(\eta_-\) has all the benefits of being a real valued, neutral fixed point, without the trouble of being a critical point.
I mean, what if for complex tetration \(\eta_-\) gave us a view of how standard petal/Ecalle iteration about an attracting basin, relates to the Kneser method about two fixed points. In a similar manner that \(\eta\) does. But in this case \(\eta \uparrow \uparrow z \to \infty\) as we let \(\Re(z) \to - \infty\) (the iterated log largely diverges). With \(\eta_-\) we have nearby attracting fixed points for \(\log\) and if we iterate we aproach them. And not only that.
\[
\log_{\eta_-}^{\circ n}(z) = a^{\pm}\,\,\text{depending on if}\,\,\pm\Im(z) > 0\\
\]
But not only that, there exists a real valued abel iteration about an indifferent fixed point. THERE'S NO NEED TO DO THE RIEMANN MAPPING, HENCE NO FUNDAMENTAL DOMAIN!
But nonetheless \(b\) is holomorphic at \(\eta_-\)! Because, the standard Abel iteration about the attracting petal that includes \(0\) (I mean this as there are 4 petals about \(1/e\) for the \(\eta_-^z\), and therefore 4 Abel functions on each petal, which can and cannot be continued together--there's one petal that \(0\) is in, use that Abel iteration (Think of this like if we choose the eta iteration or the cheta iteration, but now we have 4 choices--choose the one with 0)).
Think of it like this. Take the standard Abel iteration of \(b = \eta_-\) and let's force it to be real as \(b\) moves on the real line. What if a way of talking about Kneser is a perturbation of the abel solution at \(b = \eta_-\). Very much how you spoke of perturbed parabolic points (forget the exact words). But we do it at \(b = \eta_-\) instead. And then \(b = \eta\) is a branching problem.
The problem I forsee with this approach is that the branch cuts will look like \(b \in (-\infty,0) \cup \(\eta,\infty)\). This might be weird but this was something I was very interested in, made a thread recently.
Either way, I guess, I'm asking if we could think of \(\eta_-\) as a complex valued \(\eta\). I think this might help a lot. It's like Kneser's \(\eta\).
