Hey, everyone!
So I've been doing a lot of coding lately; and I've coded in regular iteration on the Shell-Thron region. This is done, entirely, using recursive code (I'm a one trick pony). I've coded in the regular iteration to compare the beta iterations to the regular iterations for base values in the Shell-Thron region. The reason I'm doing this, well that's the surprising part. I'm trying to draw out a fascinating relationship between tetration and... Elliptic Functions.
This post is going to be a long one, so it's helpful to talk about what all our symbols will mean. I will try to make an abridged form of the argument, and so I won't use as many symbols as I should, to draw out the full picture. We will always fix \(b \in \mathfrak{S}\), where \(\mathfrak{S}\) is the interior of the Shell Thron region. We will start with the Schroder function, which we always denote with \(\phi\):
\[
\begin{align}
\phi(z) &= \lim_{k\to\infty} \frac{\exp_b^{\circ k}(z) - \omega}{\omega^k\log(b)^k}\\
\phi(\omega) &= 0; \,\, \phi'(\omega) = 1\\
\omega &= \lim_{k\to\infty} \exp_b^{\circ k}(0)\\
\phi(\exp_b(z)) &= \log(b) \omega \phi(z)\\
0 < |\log(b)\omega| &< 1
\end{align}
\]
This is nothing unfamiliar to this forum. From here, we introduce what we'll call our Abel function and our inverse Schroder function, and our tetration function:
\[
\begin{align}
\alpha(z) &= \frac{\log(-\phi(z))}{\log(\log(b)\omega)}\\
\alpha(\exp_b(z)) &= \alpha(z) + 1\\
\Psi &= \phi^{-1}\\
\exp_b(\Psi(z)) &= \Psi(\log(b)\omega z)\\
\text{tet}_b(s) &= \Psi(e^{\log(\log(b)\omega)(s-s_0)}) = \alpha^{-1}(s-s_0)\\
\text{tet}_b(0) &= 1;\,\,\text{tet}_b(s+1) = \exp_b(\text{tet}_b(s))\\
\text{tet}_b(s+2\pi i/\log(\log(b)\omega)) &= \text{tet}_b(s)\\
\end{align}
\]
Now, the \(\beta\) method affords us the following functions: \(\beta,\tau,F\) and the constant \(s^*\), which satisfy the trinity of equations:
\[
\begin{align}
\beta(s+1) &= \exp_b(\beta(s))/(1+e^{-\lambda s}) \\
\tau(s) &= \log_b(1+\frac{\tau(s+1)}{\beta(s+1)}) - \log_b(1+e^{-\lambda s})\\
F(s) &= \beta(s+s^*) + \tau(s+s^*)\\
F(s+1) &= \exp_b(F(s))\\
F(0) &= 1\\
F(s+2\pi i/\lambda) &= F(s)\\
\end{align}
\]
The function \(F\) is holomorphic on \(\mathbb{C}\), upto an area measure. This means there are plenty of discontinuities and singularities, but they make a measure zero effect on \(\mathbb{R}^2\). For example here is \(b = e^{0.3}\) with \(\lambda = 1\). The black areas are over flows, using a super computer you could identify these as a fractal, that is just a series of weird looking branch cuts. It looks like it's not measure zero in this graph, but trust me that it is, doing further iterations than I've done here make this look like a big fractally looking crack:
Nonetheless, the function \(F\) when \(b= e^{0.3}\) and \(\lambda = 1\) is holomorphic on \(\mathbb{C}/\mathcal{E}\) where,
\[
\int_{\mathcal{E}}\,dA = 0\,\,\text{for}\, dA\,\text{the standard Lebesgue area measure in}\,\mathbb{R}^2\\
\]
And, now we can introduce \(\theta\) mappings, and where we can relate the study of \(\theta\) mappings to Elliptic functions. At least, we can do this in the Shell-Thron region. Let us call our \(\theta\) mapping:
\[
\begin{align}
\theta(s) &= \alpha(F(s)) - s - s^*\\
\theta(s+1) &= \alpha(\exp_b(F(s))) - s-1 - s^*\\
\theta(s+1) &= \alpha(F(s)) +1- s-1 - s^*\\
\theta(s+1) &= \theta(s)\\
\end{align}
\]
For example, here is \(\theta\) when \(b = \sqrt{2}\), and \(\lambda = 1\); done over the domain \(0 < \Im(s),\Re(s) < 2\pi\):
This is the value, upon which:
\[
F(s) = \text{tet}_b(s+\theta(s))\\
\]
But, we are squashing, and changing the period. Ensuring that:
\[
\theta(s+2\pi i/\lambda) = \theta(s) - 2\pi i/\lambda\\
\]
Thus enters elliptic functions.
Let us call the set \(\Theta(\lambda)\) the family of functions \(\theta\) such that:
\[
\begin{align}
\theta(s+1) &= \theta(s)\\
\theta(s+2\pi i/\lambda) &= \theta(s) - 2\pi i/\lambda\\
\end{align}
\]
And, if we mod out by a certain operation we can get every tetration function. Taking \(\theta(s),\vartheta(s) \in \Theta\), then:
\[
\begin{align}
\wp(s) &= \theta(s) - \vartheta(s)\\
\wp(s+1) &= \wp(s)\\
\wp(s+2\pi i/\lambda) &= \wp(s)\\
\end{align}
\]
This means, for every \(\theta,\vartheta\) there exists an elliptic function \(\wp\) such that \(\theta = \wp + \vartheta\). Now, these are not exactly elliptic functions. They are not necessarily meromorphic on a parallelogram \(\mathcal{Q}\); they are holomorphic on \(\mathcal{Q}/\mathcal{E}\), where:
\[
\int_{\mathcal{E}} \,dA = 0\\
\]
This, means these functions aren't exactly elliptic; they are elliptic, in the sense that they are doubly periodic in \(\mathbb{C}\). But as these represent what elliptic functions means in \(\mathbb{C}\); we choose this language. They still serve to solve Abel's elliptic equations. Despite we have a fractal in which these things aren't viable. They diverge.
By this exact statement we can say that:
\[
\theta(s) = \alpha(F(s)) - s\\
\]
Is unique upto an elliptic function \(\wp\). This is the last piece of the puzzle covered in chapter 5 of the \(\beta\) method thesis. But I don't think I can elaborate further.
So I've been doing a lot of coding lately; and I've coded in regular iteration on the Shell-Thron region. This is done, entirely, using recursive code (I'm a one trick pony). I've coded in the regular iteration to compare the beta iterations to the regular iterations for base values in the Shell-Thron region. The reason I'm doing this, well that's the surprising part. I'm trying to draw out a fascinating relationship between tetration and... Elliptic Functions.
This post is going to be a long one, so it's helpful to talk about what all our symbols will mean. I will try to make an abridged form of the argument, and so I won't use as many symbols as I should, to draw out the full picture. We will always fix \(b \in \mathfrak{S}\), where \(\mathfrak{S}\) is the interior of the Shell Thron region. We will start with the Schroder function, which we always denote with \(\phi\):
\[
\begin{align}
\phi(z) &= \lim_{k\to\infty} \frac{\exp_b^{\circ k}(z) - \omega}{\omega^k\log(b)^k}\\
\phi(\omega) &= 0; \,\, \phi'(\omega) = 1\\
\omega &= \lim_{k\to\infty} \exp_b^{\circ k}(0)\\
\phi(\exp_b(z)) &= \log(b) \omega \phi(z)\\
0 < |\log(b)\omega| &< 1
\end{align}
\]
This is nothing unfamiliar to this forum. From here, we introduce what we'll call our Abel function and our inverse Schroder function, and our tetration function:
\[
\begin{align}
\alpha(z) &= \frac{\log(-\phi(z))}{\log(\log(b)\omega)}\\
\alpha(\exp_b(z)) &= \alpha(z) + 1\\
\Psi &= \phi^{-1}\\
\exp_b(\Psi(z)) &= \Psi(\log(b)\omega z)\\
\text{tet}_b(s) &= \Psi(e^{\log(\log(b)\omega)(s-s_0)}) = \alpha^{-1}(s-s_0)\\
\text{tet}_b(0) &= 1;\,\,\text{tet}_b(s+1) = \exp_b(\text{tet}_b(s))\\
\text{tet}_b(s+2\pi i/\log(\log(b)\omega)) &= \text{tet}_b(s)\\
\end{align}
\]
Now, the \(\beta\) method affords us the following functions: \(\beta,\tau,F\) and the constant \(s^*\), which satisfy the trinity of equations:
\[
\begin{align}
\beta(s+1) &= \exp_b(\beta(s))/(1+e^{-\lambda s}) \\
\tau(s) &= \log_b(1+\frac{\tau(s+1)}{\beta(s+1)}) - \log_b(1+e^{-\lambda s})\\
F(s) &= \beta(s+s^*) + \tau(s+s^*)\\
F(s+1) &= \exp_b(F(s))\\
F(0) &= 1\\
F(s+2\pi i/\lambda) &= F(s)\\
\end{align}
\]
The function \(F\) is holomorphic on \(\mathbb{C}\), upto an area measure. This means there are plenty of discontinuities and singularities, but they make a measure zero effect on \(\mathbb{R}^2\). For example here is \(b = e^{0.3}\) with \(\lambda = 1\). The black areas are over flows, using a super computer you could identify these as a fractal, that is just a series of weird looking branch cuts. It looks like it's not measure zero in this graph, but trust me that it is, doing further iterations than I've done here make this look like a big fractally looking crack:
Nonetheless, the function \(F\) when \(b= e^{0.3}\) and \(\lambda = 1\) is holomorphic on \(\mathbb{C}/\mathcal{E}\) where,
\[
\int_{\mathcal{E}}\,dA = 0\,\,\text{for}\, dA\,\text{the standard Lebesgue area measure in}\,\mathbb{R}^2\\
\]
And, now we can introduce \(\theta\) mappings, and where we can relate the study of \(\theta\) mappings to Elliptic functions. At least, we can do this in the Shell-Thron region. Let us call our \(\theta\) mapping:
\[
\begin{align}
\theta(s) &= \alpha(F(s)) - s - s^*\\
\theta(s+1) &= \alpha(\exp_b(F(s))) - s-1 - s^*\\
\theta(s+1) &= \alpha(F(s)) +1- s-1 - s^*\\
\theta(s+1) &= \theta(s)\\
\end{align}
\]
For example, here is \(\theta\) when \(b = \sqrt{2}\), and \(\lambda = 1\); done over the domain \(0 < \Im(s),\Re(s) < 2\pi\):
This is the value, upon which:
\[
F(s) = \text{tet}_b(s+\theta(s))\\
\]
But, we are squashing, and changing the period. Ensuring that:
\[
\theta(s+2\pi i/\lambda) = \theta(s) - 2\pi i/\lambda\\
\]
Thus enters elliptic functions.
Let us call the set \(\Theta(\lambda)\) the family of functions \(\theta\) such that:
\[
\begin{align}
\theta(s+1) &= \theta(s)\\
\theta(s+2\pi i/\lambda) &= \theta(s) - 2\pi i/\lambda\\
\end{align}
\]
And, if we mod out by a certain operation we can get every tetration function. Taking \(\theta(s),\vartheta(s) \in \Theta\), then:
\[
\begin{align}
\wp(s) &= \theta(s) - \vartheta(s)\\
\wp(s+1) &= \wp(s)\\
\wp(s+2\pi i/\lambda) &= \wp(s)\\
\end{align}
\]
This means, for every \(\theta,\vartheta\) there exists an elliptic function \(\wp\) such that \(\theta = \wp + \vartheta\). Now, these are not exactly elliptic functions. They are not necessarily meromorphic on a parallelogram \(\mathcal{Q}\); they are holomorphic on \(\mathcal{Q}/\mathcal{E}\), where:
\[
\int_{\mathcal{E}} \,dA = 0\\
\]
This, means these functions aren't exactly elliptic; they are elliptic, in the sense that they are doubly periodic in \(\mathbb{C}\). But as these represent what elliptic functions means in \(\mathbb{C}\); we choose this language. They still serve to solve Abel's elliptic equations. Despite we have a fractal in which these things aren't viable. They diverge.
By this exact statement we can say that:
\[
\theta(s) = \alpha(F(s)) - s\\
\]
Is unique upto an elliptic function \(\wp\). This is the last piece of the puzzle covered in chapter 5 of the \(\beta\) method thesis. But I don't think I can elaborate further.
