Hey, Tommy
I'll point you to this report:
https://arxiv.org/pdf/2208.05328.pdf
And specifically Section 2 On Mock Abel And Mock Schroder coordinates.
I only analytically prove the Mock Abel coordinate becoming an Abel coordinate here, but they can be transformed into a mock Schroder equation by a change of variables. You'll probably have to read the whole report to get a good grasp on what I mean by the Weak Julia set, and the Weak Fatou set.
But, just a quick refresher.
A mock Abel equation looks like:
\[
F(s+1) = f(F(s)) h(s)\\
\]
(Here, I always use a logistic function \(h(s) = \frac{1}{1+e^{-\lambda s}}\) for some \(\Re \lambda > 0\))
Where \(h(s) \to 1\) as \(\Re(s) \to \infty\). And a Mock Schroder equation looks like:
\[
P(c w) = f(P(w)) h(\log_c(w))\\
\]
(I've set \(c = e^\lambda\))....
Then, in this paper, I show that on the Weak Fatou set; we have that:
\[
G(s) = \lim_{n\to\infty} f^{-n} F(s+n)\\
\]
Converges--and similarly:
\[
H(w) = \lim_{n\to\infty} f^{-n} P(c^n w)\\
\]
Converges, on the weak fatou set.
What happens that Sheldon was the first to notice, is that if \(|c| \ge |f'(0)|\), this object has a nontrivial domain of convergence, and the process I had written fails for \( 0 \le |c| \le |f'(0)|\). And it appeared that no such function should exist--least ways, my method of construction fails...
The weak Fatou set is characterized as follows: If \(s_0\) is in the weak Fatou set, then there exists an open neighborhood \(\mathcal{N}\) about \(s_0\) such that
\[
\limsup_{n\to \infty} \sup_{s\in\mathcal{N}}\frac{1}{|F(s+n)|} < \infty\\
\]
And if \(s_0\) is not in the weak Fatou set, it's in the weak Julia set. The weak Fatou set is open, and the weak Julia set is closed, and they partition the complex plane. This definition is equivalent to:
\[
\limsup_{n\to\infty} \sup_{s\in\mathcal{N}} \frac{1}{|f^{\circ n}(F(s))|} < \infty\\
\]
And if we make the change of variables \(s = \log_c(w)\), you have to account for this in the definition, but it works the same manner.
For example, letting \(\beta(s+1) = \sqrt{2}^{\beta(s)}/(1+e^{-s})\) produces this fractal for the weak Julia set:
Which appears at every point \(s = (2k+1)\pi i + j\) for \(k,j \in \mathbb{Z}\). The white fractal is the weak Julia set, and the black area is the weak Fatou set.
This means that:
\[
\lim_{n\to\infty} \log_{\sqrt 2} \beta(s+n) = G(s)\\
\]
converges uniformly in the black area, and nowhere in the white area (which is sort of the central thesis of this report). And this creates a tetration that satisfies \(G(s+1) = \sqrt{2}^{G(s)}\) and \(G(s + 2\pi i) = G(s)\). Towards the end of this report I describe how this induces a theta mapping \(\theta(s+1) = \theta(s)\); where the regular tetration \(\text{tet}_{\sqrt{2}}(s)\) can be turned into \(G\) by the equation \(G(s) = \text{tet}_{\sqrt{2}}(s + \theta(s))\). Aditionally \(\theta\) is very close to an elliptic function. It's 1 periodic, and linear in a second period. Which is given in this instance as \(\theta(s+2 \pi i) = \theta(s) - 2\pi i\).
Mind you, this paper only deals with exponential functions \(e^{\mu z}\) for \( \mu \neq 0\), but it extends for any function. There's nothing special about the exponential function being used. So if we were to switch this to arbitrary functions, nothing really changes; especially near attracting/repelling fixed points. I just wrote the work for tetration
I'll point you to this report:
https://arxiv.org/pdf/2208.05328.pdf
And specifically Section 2 On Mock Abel And Mock Schroder coordinates.
I only analytically prove the Mock Abel coordinate becoming an Abel coordinate here, but they can be transformed into a mock Schroder equation by a change of variables. You'll probably have to read the whole report to get a good grasp on what I mean by the Weak Julia set, and the Weak Fatou set.
But, just a quick refresher.
A mock Abel equation looks like:
\[
F(s+1) = f(F(s)) h(s)\\
\]
(Here, I always use a logistic function \(h(s) = \frac{1}{1+e^{-\lambda s}}\) for some \(\Re \lambda > 0\))
Where \(h(s) \to 1\) as \(\Re(s) \to \infty\). And a Mock Schroder equation looks like:
\[
P(c w) = f(P(w)) h(\log_c(w))\\
\]
(I've set \(c = e^\lambda\))....
Then, in this paper, I show that on the Weak Fatou set; we have that:
\[
G(s) = \lim_{n\to\infty} f^{-n} F(s+n)\\
\]
Converges--and similarly:
\[
H(w) = \lim_{n\to\infty} f^{-n} P(c^n w)\\
\]
Converges, on the weak fatou set.
What happens that Sheldon was the first to notice, is that if \(|c| \ge |f'(0)|\), this object has a nontrivial domain of convergence, and the process I had written fails for \( 0 \le |c| \le |f'(0)|\). And it appeared that no such function should exist--least ways, my method of construction fails...
The weak Fatou set is characterized as follows: If \(s_0\) is in the weak Fatou set, then there exists an open neighborhood \(\mathcal{N}\) about \(s_0\) such that
\[
\limsup_{n\to \infty} \sup_{s\in\mathcal{N}}\frac{1}{|F(s+n)|} < \infty\\
\]
And if \(s_0\) is not in the weak Fatou set, it's in the weak Julia set. The weak Fatou set is open, and the weak Julia set is closed, and they partition the complex plane. This definition is equivalent to:
\[
\limsup_{n\to\infty} \sup_{s\in\mathcal{N}} \frac{1}{|f^{\circ n}(F(s))|} < \infty\\
\]
And if we make the change of variables \(s = \log_c(w)\), you have to account for this in the definition, but it works the same manner.
For example, letting \(\beta(s+1) = \sqrt{2}^{\beta(s)}/(1+e^{-s})\) produces this fractal for the weak Julia set:
Which appears at every point \(s = (2k+1)\pi i + j\) for \(k,j \in \mathbb{Z}\). The white fractal is the weak Julia set, and the black area is the weak Fatou set.
This means that:
\[
\lim_{n\to\infty} \log_{\sqrt 2} \beta(s+n) = G(s)\\
\]
converges uniformly in the black area, and nowhere in the white area (which is sort of the central thesis of this report). And this creates a tetration that satisfies \(G(s+1) = \sqrt{2}^{G(s)}\) and \(G(s + 2\pi i) = G(s)\). Towards the end of this report I describe how this induces a theta mapping \(\theta(s+1) = \theta(s)\); where the regular tetration \(\text{tet}_{\sqrt{2}}(s)\) can be turned into \(G\) by the equation \(G(s) = \text{tet}_{\sqrt{2}}(s + \theta(s))\). Aditionally \(\theta\) is very close to an elliptic function. It's 1 periodic, and linear in a second period. Which is given in this instance as \(\theta(s+2 \pi i) = \theta(s) - 2\pi i\).
Mind you, this paper only deals with exponential functions \(e^{\mu z}\) for \( \mu \neq 0\), but it extends for any function. There's nothing special about the exponential function being used. So if we were to switch this to arbitrary functions, nothing really changes; especially near attracting/repelling fixed points. I just wrote the work for tetration