(07/07/2022, 08:29 PM)MphLee Wrote: I apologize for my fractal "n00bness"... but is there available a diagram of the complex plane showing in addition to the fixed points also the shapes and positions of their respective basin of attraction? I'm interested in visualizing the sets \(A_p=\{z\in\mathbb C\, :\, \lim_{n\to \infty} \exp_b ^n(z)=p\}\).

What is the knowledge about the set \(A=\bigcup_{p\in {\rm fix}(\exp_b)}A_p\) and the set \(\mathbb C\setminus A\). Are they open? Closed? Connected by arcs? Are they punctured/have holes?

Hey, Mphlee

I believe your question isn't the question you intended to ask. So first I'll answer the question you did ask, and then extrapolate what I think you meant to ask.

First of all, the only attracting fixed points are in the Shell-Thron region. So you are asking for what values z are in the basin of the fixed points.

The fixed points come in a set that looks like \(|\log(p)| < 1\), and then \(b = e^{\log(p)/p}\). The basin of these fixed points are very sensitive, and nonsensical. There is not much literature that I've found describing them.

An important fact, is when \(b \in (1,\eta)\), that \(\Re(z) < 0\) is in the basin. As you move in the complex plane, this half plane deforms, I believe there's always a half plane within \(A_p\) for all such \(p\)--not sure how to describe it. But don't quote me on that, I'm not certain.

Then when you ask for your set \(A\), it can be written more clearly as:

$$

A = \{z \in \mathbb{C}\,|\, \exists p\, \text{such that}\,|\log(p)| < 1,\,\lim_{n\to\infty}\exp_{p^{1/p}}^{\circ n}(z) = p\}\\

$$

This set isn't all that interesting, at least I don't think so. It is definitely open, just by looking at this definition. It is probably simply connected, as it's the union of simply connected domains that are deformations of each other. This means there shouldn't be any holes.

The more interesting chaos, and the much crazier idea, would be to invert your definition of \(A_p\). Lets call it \(B_p\):

$$

B_p = \{z \in \mathbb{C}\,|\, \lim_{n\to\infty} \log^{\circ n}_b(z) = p\}\\

$$

This would produce MANY MANY more fixed points. This would be a much more interesting beast, as it'd deal with fixed point pairs of the exponential, and the ridiculous amount of fixedpoints that the exponential has (depending on how you choose the signature of each log in the iteration). The thing is, pretty much all the fixed points are repelling. The only attracting ones are in the shell-thron region. This means, the fixed points are attractive in \(\log\) though. And choosing each branch of \(\log\) as you iterate, produces different fixed points. And these are a much more interesting version of your set. It's kinda like the compliment you brought up. Instead of defining \(A_p\) through attracting fixed points of \(\exp_b\), which are well understood. Look at \(B_p\) defined the exact same manner off of repelling fixed points.

Could I ask what you want to know about this set--what your goals are?