x↑↑x = -1

[sheldonison]

Somewhat of a followup :

We can say D(z + theta_A(z)) = z + theta_B(z).
....

Here's my interpretation of that equation.
\( \theta_2(z) = \alpha_2(\alpha_1^{-1}(z+\theta_1(z)))-z\;\;\; \alpha_1, \theta_1 \) are the \( \Im(z)>0 \) Abel and theta functions, \( \;\;\alpha_2, \theta_2 \) are the \( \Im(z)<0 \) functions

For Tetration, involving a pair of repelling fixed points (real bases are in this category), my conjecture is that this equation is only 1-cyclic at the real axis at the analytic boundary of both theta functions, and isn't useful anywhere else. Once you cross the real axis, I don't think the theta(z) branches follow the same path as the tet(z) branches.

Basically, if you make a circle around 0, and follow \( \alpha(z) \), you don't get back to where you started because there is a logarithmic branch at zero, since the Abel function is an iterated logarithm, followed by renormalization. So the path you take, for \( \Im(z)<0 \) and \( \Im(\exp(z))<0 \) for the \( \Im(z)>0 \; \alpha(z) \) function has to go back into the upper half of the complex plane between z and exp(z), or else the logarithmic branches become inconsistent with a 1-cyclic theta mapping, because theta(z) has a really complicated Riemann surface branch singularity for integers at the real axis.

I could try to come up with a numeric example, tet(-0.5-0.1i) and tet(0.5-0.1i), and help explain this, if you think that would help you.