Carlson's theorem and tetration

[mike3]

I guess it can be generalized to arbitrary regular superfunctions as they are always of the form \( \eta(\pm e^{\kappa z}) \) for some function \( \eta \) analytic at 0.

Yes, provided the fixed point is attracting and positive real.

i believe we need oo to be repelling and f^^n(z) converging for lim n-> oo and any z too.

You mean \( f^n(z) \), right? For "any" z seems too restrictive: \( f(z) = \eta^z \), for example, does have many \( z \)-values for which its iteration diverges, but these do not show up in the range of the tetrational \( ^z \eta \).