Rank-Wise Approximations of Hyper-Operations

[JmsNxn]

I'm doubtful it'd be possible though, it'd be really cool if it was. The trouble is every tetration induces an iteration, and as I remarked that iterations \(f^{\circ s}(z)\) can't be holomorphic in the neighborhood of two fixed points.

What if the different Schröder iterations flowed together nicely?

That would be quite the function!

I'm doubtful it'd be possible though, it'd be really cool if it was. The trouble is every tetration induces an iteration, and as I remarked that iterations \(f^{\circ s}(z)\) can't be holomorphic in the neighborhood of two fixed points.

The only way I can interpret "flowed together nicely" is Kneser. Kneser makes sure \(L\) and it's complex conjugate \(L^*\) flow together properly. Though, they are never allowed to produce an iteration holomorphic in the neighborhood of both points.

And I mean this absolutely, you can't have holomorphy in \(s\) on some domain, and holomorphy in \(z\) on some domain which includes \(L,L^*\)--the closest you get is Kneser.

1.) \( \tau \)(x,y) = x. If y = 1.
2.) \( \tau \)(x,y+1) = x^\( \tau \)(x,y). For -y-1 ∉ \( \mathbb{N} \).
3.) For a given k, using \( \tau \) to do sexp(slog(x)+k) produces the same iteration of exponentials. (Or principled logarithms if k is negitave.) For any branch of slog. (If slog happens to branch.)
4.) If \( \tau \)(x,y) approaches any of the fixed points or n-cycles of the function f(k) = x^k, it will approach continuously iterating y*m+b.
5.) \( \tau \)(x,y) is mostly holomorphic.
Does \( \tau \)(x,y) exist?
If so is it unique?

The key you've added here is is \(\tau\) mostly holomorphic. Well, it isn't holomorphic at the fixed points/ periodic points. Then Kneser satisfies your solution.

Think of it this way:

\[
\exp^{\circ s}(z) = F(s,z)\\
\]

Then \(F(s,z)\) can't be holomorphic on intersecting domains for different fixed points. We cannot construct something near multiple fixed points.

Think \(\sqrt{2}\). Let's try:

\[
\exp^{\circ s}_{\sqrt{2}}(4+h)\\
\]

And:

\[
\exp^{\circ s}_{\sqrt{2}}(4-h)\\
\]

Despite looking like similar functions, there is no way to make the transition between the two functions to be holomorphic at \(4\) (\(h=0\)). This CANNOT BE DONE. These are two different analytic functions, and at \(4\) there is a singularity or the value \(4\). You cannot paste the two together.