riemann surface for tetration and slog?

[bo198214]

Assuming that tetration and slog are holomorphic for all applicable z, what doe the reimann surface for tetration and superlogarithm look like?

The interesting thing with the slog is, that new branchpoints come into existence on different branches (which then create new branches), which you can see in Dmitriis plot.
I am not sure whether anyone has a complete description of this infinitely nested structure.

At least we can say that there must be infinitely many branchpoints of the slog, because each branch point \( z \) occurs again at \( z+2\pi i k \) for all integer \( k \).
\( \operatorname{slog}(z+2\pi i k)=\operatorname{slog}(e^{z + 2\pi i k })-1=\operatorname{slog}(z) \).
However they dont need to occur on the same branch, but they must occur on some branch.

And I never heard about anyone considering the branchpoint structure of sexp at \( z\le -2 \).