[Update] Comparision of 5 methods of interpolation to continuous tetration

[sheldonison]

Here ist a picture which shows the mapping of the small positive part of the imaginary axis .... of the powertower with base 4.

The whole segment becomes more and more distorted and shrinks when iterated to the complex fixpoint.

What is especially interesting me are the "eyes" /the whitespace in the inner spiral - and what about of regions of overlap in there if my initial linesegment where longer, say up to 0.8*I. This direction would be related by complex iteration heights - but I've not yet reliable computations of that heights.

http://math.eretrandre.org/tetrationforu...hp?tid=358 is a link to Jay's post on the Chi-Star, which is the distorted part of the limiting behavior of your graph as well. To get the Chi-Star, one has to take the regular Schroeder function, developed from the complex fixed point, of the real axis of the sexp(z) solution. Then the pattern repeats, scaled. And then each of the "eyes" would extend all the way to infinity in a beautiful complicated recursive pattern. Jay's post also shows the relationship with the chi-star and the region which gets Riemann mapped in Kneser's construction, which Tommy asked about.
- Sheldon