Universal uniqueness criterion?

[Kouznetsov]

@Kouznetsov: how are the other tetrations relevant to our discussion of uniqueness?

I expect, all "other" tetrations have additional singularities and cutlines.

@Kouznetsov: Also I just have a quick question to ask about tetration: What is the mathematical reason that the tetrational approaches the fixed points \( L \) or \( L* \) as you go to \( \pm i \infty \)? I think instead, that \( \lim_{x \rightarrow -\infty} x\pm ci = L\mathrm \) or \( L* \), where c is a nonzero real number, should be the limit that corresponds to the fixed point.

You have no need to type "instead", both should be correct. The reason in not so mathematical, but computational: it is easier to work with a function, holomorphic in wide range, and create all other tetrations, just modifying the argument, if necessary.
In principle, "all animals are equal"; but as soon as you begin to plot graphics, it happens, that "some animals are more equal than others".
For me, "more equal" are animals with wide range of holomorphizm.

.. to get from 1 to a non-real number by iteration of exp, you need complex iteration (real iterations always give real numbers)

Yes.

But then since \( L,L* \) are repelling with respect to the exponential, you need infinite negative iterations of exp (positive iterations of log).

Yes.

If you infinitely iterate log on a nonreal number, you get closer to L, why isn't this reflected in the tetrational graph?

? In the left hand side of the plot of tetration, values approach \( L \) in the upper halfplane and \( L^* \) in the lower halfplane; this corresponds to the graphic you supply.

If you mean our discussion with Bo about a super-exponential that approaches \( L_1 \) instead of \( L=L_0 \), then it is not correct:
In some region, at the iteration of logarithm, we have to add \( 2 \pi i \) in the upper halfplane and \( -2 \pi i \) in the lower halfplane.
The question is, wether we can match them for large enough real part of the argument.
I am not yet successful to construct or plot such a function.
I expect such function to have two additional horizontal cutlines, going to -infinity.

Bo said something like log is not in the initial region anymore, could you clarify that for me?

As I understand, the initial region is \(
C=\{z \in \mathbf{C} : \Re(z)\ge 1 , \|z|\le |L| \} \)
if some point \( z \) is inside the initial region (not at the margin), then
\( \log(z) \) in outside the initial region.
The same about \( \log(z)+2 \pi i \) .

Is this that you were asking for?

I have attached a schematic diagram to represent my reasoning:
how does the tetrational for base b>exp(1/e) actually behave at large values of real part of non-real arguments?

There are pictures at http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf
The tetrational shows complicated quasi-periodic behavior.