Rank-Wise Approximations of Hyper-Operations

[Daniel]

Tetration for a global function doesn't necessarily behave like that. That's a uniqueness criterion you haven't defined though. All you've said with this is that:

\[
\text{Tet}(s-k) \approx L + e^{-Lk}\text{Tet}(s)\,\,\text{for}\,\,\Re(s) < -R\,\,\text{for}\,\,R\,\,\text{Large}\\
\]

Yes ABSOLUTELY THAT's TRUE!!!!! There are countably infinite solutions to that though...

A tetration function must do that about all of the fixed points. You could iterate from any of the fixed points, and continue back to where you want it. Like to calculate 2^^.5 you could use any of the fixed points. Tetration needs to be unique.

Daniel
Using a base a bit larger than 1 produces simple dynamics and fractals with chaos only in small areas. I was able to compute the dynamics from one fixed point that was able to give the location of a neighboring fixed point and it's Lyapunov multiplier. This indicates that the Taylors series of an iterated function can give the position of all the other fixed points and consistent Taylor series.