tetration from alternative fixed point

[sheldonison]

Has anyone generated the base e tetration from an alternative fixed point? The second fixed point (as far as I can tell) for base e is 2.0622777296+i*7.5886311785 ....

The problem with alternative fixed points is that we dont have a sickel between them. I.e. if we connect two conjugated non-primary fixed points with a straight line, then the image under \( e^z \) of this straight line overlaps itself. This is due to the imaginary part of the fixed point pair is apart more then \( 2\pi \) which makes the image of any connecting line between these fixed points, revolve around 0 at least once.

For Kneser's solution we need a region that is bounded by a line connecting the two fixed points and the image of this line.
Perhaps one (you?) could prove, that regardless how you connect a conjugated fixed point pair, that is not the primary one, the image of this line intersects itself or the connecting line; i.e. both lines never delimit a singly connected region.

Of course, Henryk is correct, although I doubt I understand it well enough to prove anything. 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 .... and with a repeating pattern, but there's a couple of major differences in the behavior of the singularity, as compared to the superfunction developed from the primary fixed point.

(1) there is a gap between consecutive iterations of the contour from the secondary fixed point. For the primary fixed point, the singularity smoothly transitions from one side of the singularity to the other side. If you make a line between the singularity generated from the inverse superfunction on either side of zero+/- delta, the superfunction of that line converges nicely to zero. But this doesn't work for the secondary fixed point!

(2) The inverse superfunction from the secondary fixed doesn't match the repeating real contour with singularities in it. For the inverse superfunction, I can only get the pattern for one iteration, between -inf and 0, or between 0 and 1, or between 1 and e. But I can't get more than one of them at a time. This is really the same problem as (1). I imagine the complete Riemann surface would somehow connect the different regions of the multi-valued inverse superfunction....

As a result of this, any Reimann mapping from the secondary fixed point will have discontinuities in the derivatives, at the integer values of Sexp(-1,0,1,2,3,4....).
- Sheldon