Rank-Wise Approximations of Hyper-Operations

[JmsNxn]

The trouble then becomes, which value of \(s_0\) and which fixed point, but there are countable solutions to tetration. None will be real valued though.

They should all produce the same Shröder iteration.

Hmm, I'm not sure what you mean here. They all spawn from the same Schroder iteration (I'm using this word a little differently than I guess how it should be used).

So,

\[
\exp^{\circ s}(z) = \Psi^{-1}(e^{Ls}\Psi(z))\\
\]

Is the one and only Schroder iteration about \(z \approx L\). What I was saying, is that, there exists countable \(s_0\) such that:

\[
\Psi^{-1}(e^{-Ls_0}) = 1\\
\]

So, there are countable functions:

\[
F(s) = \Psi^{-1}(e^{L(s-s_0)})\\
\]

Such that:

\[
\begin{align}
F(0) = 1\\
F(s+1) = e^{F(s)}\\
F(s+2\pi i/L) = F(s)\\
\end{align}
\]

But there is only one Schroder iteration of the exponential. There are countable TETRATIONS which satisfy this. I apologize if I talked too loosely.

Just a quick note, on this forum a Schroder iteration will always refer to standard/regular iteration about a fixed point. BUT Tetration is to me something that satisfies the functional equation and an initial condition of \(F(0)=1\). So the standard Schroder iteration of \(e^z\) about \(L\) creates countably many tetrations....