tetration from alternative fixed point

[sheldonison]

I can make graphs of the real contour of the superfunction developed from the alternative fixed point, with singularities at 0, 1, e, e^e, e^e^e ....

I think we need to go into the details here.
If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point.
I use something similar to the formula:
\( \text{slog}(z)=\log_c(\log_\ast^{[n]}(z)-e[2])-n \) where c is the derivative at the secondary fixed point e[2] and \( \log_\ast \) is the branch of the logarithm with imaginary part between \( 2\pi \) and \( 4\pi \), which implies that \( \log_\ast^{[n]}(z)\to e[2] \).

Henryk, I'm using the same equation you are. Instead of "-n", I mulitply by c^n before taking the log_c. This is mathematically equivalent, but it seems to help identify a unique branch point independent of the size of "n". Perhaps I can explain what I was trying to say with a graph of the SuperFunction (developed from the secondary fixed point), showing some contour lines. To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from \( +/-\infty \). Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from -\( \infty \) to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points.

The pattern repeats, as shown by the "alt 3pi*i contour line." Each contour line would have alternates repeating endlessly in the diagonal direction. If you follow a path on the Reimann surface of the inverse superfunction from \( -\infty \) to \( +\infty \), the path you take will be along a diagonal. But yes, there will be singularities at 0,1,e,e^e, ... but only two of them will be of interest to the Riemann mapping. I don't think I'm explaining it very well, apologies.

There are many other paths that can be be graphed in the SuperFunction from the alternate fixed point. I included the 2pi*i contour, and the pi*i contour as well, though they aren't my primary concern. They can also be used to generate repeating contour lines. This graph is in stark contrast to the superfunction from the primary fixed point, where the consecutive exponentiations fit each other like a glove, allowing the Riemann mapping to cancel out the singularity.
- Sheldon