[question] Local to global and superfunctions

[JmsNxn]

So the two ideas are fairly close "large domain in z" and "non local codomain".

Just note that I mean this statement fairly imprecisely. That is to mean it's a loose description of what things look like. I like to think using half planes, so let's use half planes.

Take an arbitrary local iteration: \(f^{\circ s}(z)\), and let's let \(\Re(s) > K\) for \(K\) large. Let's call the fixed point \(p\), and the multiplier \(\lambda\)--which for expositional purposes we'll assume is between \(0 < \lambda < 1\). Then:

\[
|f^{\circ s}(z) - p| \le r^K\\
\]

For some \(0 < |\lambda| < r < 1\). So the codomain of this function looks a lot like a disk about \(p\). We haven't mentioned \(z\) yet, but since we've defined our codomain, let \(z\) exist there too; inside this tiny disk. Call it \(E\).

Now this is still a very large domain that \(s\) lives in, it is a half plane. But! It's a rather restrictive half plane. What happens when we limit \(K \to -\infty\)? We'll we are effectively taking \(f^{-1}\) repeatedly of this little disk. This little disk behaves oddly underneath \(f^{-1}\) though. First of all, since \(f\) has an attracting fixed point at \(p\), we must have \(f^{-1}\) has a repelling fixed point. Any repelling fixed point is in the Julia set of its function, so call \(J\) the julia set of \(f^{-1}\), then \(J \cap E \neq \emptyset\).  Here is where things get interesting.

Pick a neighborhood \(\mathcal{N}\) of a point \(p\) in the julia set \(J\); then the orbits \(\bigcup_{n \ge 0} f^{-n}(\mathcal{N})\) is dense in \(J\).

Now what is very common, is that \(J\) includes the point at infinity, this is almost always, though there are exceptions. So this means, as we take repeated \(f^{-1}\)'s we are getting closer and closer to eventually hitting infinity (on the Riemann sphere I mean)--or we hit wild essential singularity chaos. All this is happening as we are letting \(K \to -\infty\), as we are growing the half plane (and by consequence growing the codomain from a tiny disk to a wild chaotic fractal that will eventually include infinity, or at least a singularity).

I guess what I'm trying to say, is there is a correlation to just how local we are, and the size of the domain of \(s\). The more it looks like a disk, the "smaller" the half plane of the iteration, the more local it is. The more it looks like chaos and fractally, the "larger" the half plane of the iteration, the less local it is.

You can see this in almost all of our iterations too, they happen even with Kneser which is pretty far from a fixed point iteration. But for \(\Re(s) < -K\) and \(\Im(s) > 0\), then more and more tetration just looks like \(L + e^{Ls}\). So if you were to do a local tetration in this quarter plane with very large \(K\), the codomain would just look like a disk around \(L\).  But as you grow the domain of \(s\), we let \(K \to -\infty\), the more chaotic we get, and more and more we get closer to the codomain being exactly \(\mathbb{C}\).