tetration from alternative fixed point

[sheldonison]

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.

Would this refer to the graph of the complex super-function, generated from the secondary fixed point, or to the graph of the real valued super-exponential, after the Kneser solution?

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.

I'd like to graph the 3*pi*i contour line of the super-function generated from the secondary fixed point, 2.0622777296+i*7.5886311785

The super-function "grows" away from the fixed point, and the first n*pi*i contour encountered is the 3*pi*i contour. This is analogous to the primary fixed point, which eventually reaches the pi*i contour. The theory is that the contour line would have real values from -infinity to +infinity, and that the exponent of that contour line, would trace out the real values from -infinity to zero, and the next exponent would trace out the real values from zero to one etc.

It is straightforward to generate the super-function from the secondary fixed point, but the inverse super-function is giving me difficulties, and I need to get the iteration equations for the inverse super-function working before I can graph the 3*pi*i contour, and then perhaps I will understand why this contour line does not allow for Knesser's construction.