Hi.

I was wondering about this. I noticed that the tetrational constructed with the continuum sum for the base , shown here:

http://math.eretrandre.org/tetrationforu...hp?tid=514

looks in the upper half-plane like the regular iteration at the attracting fixed point, while in the lower half-plane it looks different. The lower half-plane actually looks like the regular iteration at a repelling fixed point:

Compare to:

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 was wondering about this. I noticed that the tetrational constructed with the continuum sum for the base , shown here:

http://math.eretrandre.org/tetrationforu...hp?tid=514

looks in the upper half-plane like the regular iteration at the attracting fixed point, while in the lower half-plane it looks different. The lower half-plane actually looks like the regular iteration at a repelling fixed point:

Compare to:

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?