eta as branchpoint of tetrational

[sheldonison]

.... Ok, going with the base on the upper halfplane from e to sqrt(2).
As long as we are not landed back on the real axis,
the fixpoints are not conjugate, so the tetration value will not be real.
But this is nothing new.
Now when on the real axis: the Kneser method is not applicable to two real fixpoints.
So we must take the value there as limit from above.

The kneser tetration \( b\mapsto b[4]p \) is real on the real axis \( b>\eta \), which implies that \( \overline{b} [4] p = \overline{b[4]p} \) (conjugation).

So approaching from above or below is just conjugate to each other.
So what one need to show imho is that the imaginary part will not tend to zero when approaching the real axis at \( b<\eta \).

I wonder whether Sheldon could supply some pictures of that, with his fourier algorithm.
....

I made a picture of four different superfunction for B=sqrt(2). The first two are the two in your paper, with Dimitrii, generated from the lower fixed point L=2, and the upper fixed point, L=4. The third is the "new function", I posted about here. The fourth is the hypothetical Perturbed Fatout mapping, which is a merged upper/lower fixed point function. As imag(z) grows, this merged upper/lower Perturbed Fatou mapping gets arbitrarily close to the new function I generated, which is approaching the upper fixed point L=4 function, minus 1.05i, which is half the difference between the two periods. But, as imag(z) gets smaller and smaller, it approaches the lower fixed point function L=2 function. The way I drew it, the merger is occurring at imag(z)~=8.6i, where we have one of the gentle transitions from -4 to -2. Here, the magnitude of the two theta(z) functions would be comparable, but both are very small, with an first harmonic amplitude of ~1E-25. I would like to calculate the two theta(z) functions for Fatou(z), for a couple of harmonics, at least. Both theta(z) would have singularities, but there is a large overlap where both theta(z) functions are analytic.

As you pointed out, this merged upper/lower Perturbed Fatou function wouldn't have Im(F(z))=0, at the real axis or any other horizontal line. I thought this might help the conversation. It would take me a day or two to calculate the upper and lower theta(z) values to generated this new merged upper/lower fixed point Fatou function.

The plots have imag(z)=negative on the left, and imag(z)=positive on the right.
\( \text{Fatou_{\sqrt{2}}(z)=\text{Usexp_{\sqrt{2}}(z+\theta_U(z)+k) \)
\( \text{Fatou_{\sqrt{2}}(z)=\text{Lsexp_{\sqrt{2}}(z+\theta_L(z)) \)
\( \Im(k)=(\text{Period}_U -\text{Period}_L)/2 \)
\( \theta_U(z) \) decays to zero as imag(z) increases, with singularities at the real axis at integer values of z.
\( \theta_L(z) \) decays to zero as imag(z) decreases. It must also have singularities, as imag(z) increases, at approximately imag(z)~=18.1, but I don't understand that as well.
- Sheldon