Universal uniqueness criterion?

[bo198214]

Cool, now we have to prove the biholomorphicity of the different tetration methods between G and the strip, and according to the theorem, since they are all tetration/tetralogarithm (superfunction/Abel function of exponential), they are all the same.

But it could also occur that one slog is biholomorphic on one initial region A but not on B, while another slog is biholomorphic on the initial region B, but not on A.

I think if both initial regions belong to the same fixed point pair, then both slogs are analytic continuations of each other, i.e. basically the same holomorphic function. But this is not yet proven.

There should also be an example of a quartic, that has two pairs of conjugated fixed points, and that has two different Abel functions, each biholomorphic on an initial region belonging to one fixed point pair.

Does the applicabilty of theorem 1 only require that the function F (in the case of tetration, exp) is holomorphic, and F (exp_b, b>e^1/e) has no real fixed points?

Well you just need a fixed point pair and an initial region belonging to it.
If both fixed points are on the real axis, then there is no initial region between them.
Better chances have (complex) conjugated fixed points.

Then what are you guys griping about in threads like Bummer!?

About different regular Abel/super functions developed at different fixed points on the real axis.