Universal uniqueness criterion?

[BenStandeven]

.. Well, but \( \exp \) *has* more fixed points. In every strip \( 2\pi i k < \Im(z) < 2\pi i (k+1) \) there is a fixed point of \( \exp \).

Henryk, how about to build up "another" holomorphic tetration that goes to other fixed points at \( \pm i \infty \)?

I dont think that there is an initial region connecting other conjugated fixed point pairs than the one closest to the real axis. (Plot the straight line connecting two such fixed points and plot its image under exp. Both lines intersect.)

On the other hand I asked you som time ago to apply your algorithm to other fixed points, but you somehow did not follow that path.

What about using Kneser's approach to produce the alternate tetration functions?