"Kneser"-like mapping also for complex base?

[sheldonison]

Both fixed points are fixed points of the associated logarithm. This behavior suggests that it may be possible to form the tetrational at any complex base out of the two regular iterations of the two fixed points of logarithm. Could it be possible that some kind of Kneser-like 1-cyclic transform, or a pair of such transforms, be applied to "bend" the two regular iterations so they flow together in a holomorphic manner on the right half-plane to yield the tetrational function?

I highlighted pair of transforms, and I like the phrase "flow together". I don't know if its possible or not. But your ideas are starting to make sense to me. I like having a pair of kneser mappings, with two different theta functions, one for each repelling fixed point, that have equal sexp(z) for z>-2 at the real axis. The first theta function maps the upper half of the complex plane, and the second theta function maps the lower half of the complex plane. Both theta functions would have singularities at the real axis for integer values of z, corresponding to sexp(0)=1, sexp(1)=B, sexp(2)=B^B. For z>-2, the singularities gets canceled by \( \text{sexp}(z)=\text{superfunction}(z+\theta(z)) \), so that sexp(z) is analytic at the real axis.

For real valued bases, the sexp(z) defining boundary condition is that sexp(z) is real valued at the real axis after the 1-cyclic theta function, and then we use the Schwarz reflection theorem to generate the values for imag(z)<0.

For your complex base, the defining condition would be that the complex contour at the real axis for z>-2 be the same for the upper theta(z) mapping from one fixed point, as the lower theta(z) from the other fixed point. This actually seems like a reasonable and probably unique definition, and probably also works for real valued bases. I think you should pursue it further. Also, you could graph the sexp(z) value at the real axis, for z>-2.

Your example looks like it may have singularities roughly corresponding to z=-2,-3,-4 .... etc, but the singularities look a little off kilter, in that the imag(z) for the singularities should be in a horizontal line, spaced apart by a unit length.

It would be interesting to find some way of computing such a pair of kneser \( \theta(z) \) mappings, and their corresponding analytic complex tetration.
- Sheldon