Tetration below 1

[Gottfried]

Therefore, as I promised to Enryk, I submit to your attention the attached pdf notes, hoping that you will not be annoyed and that find some ideas to be developed. Please see also what should, in my opinion, happen in the range 0 < b < e^(-e). It is a simulation but, just tell me what you think. In fact, I agree that that interval corresponds to the negative bases, for exponentiation. The appearence of oscillations, for b<1 must be admitted, if we accept to go out of the "reality". I think to understand a little bit better what Euler meant by saying that the "infinite towers" don't converge
for b < e ^(-e). Perhaps, they just oscillate -> oo.

Please also investigate, in the 1 < b < e^(1/e) interval, what could be the role of the second (upper) "unreachable" asymptote.

The notation of the limits of b in my eigen-decomposition concept is very intriguing.
Assume b=h^(1/h), or h=h(b) , where h() is the function described by Ioannis Galidakis, then the log of admissible h, hl=log(h) is -1< hl <1.
The eigenvalues of the operator for tetration are the consecutive powers of hl, so the diagonal contains a convergent sequence 1,hl,hl^2,hl^3,... if hl is in the admissible limit, and a divergent sequence if hl is outside. There is a degeneration if hl is exactly 1 or -1.

So the tetration behaves properly if we begin by selecting -1<hl<1, use then h and s when computed from this. Thus we are bound for e^-e < s < e^e^-1.
If we assume an s outside of this bounds, we need solutions of h(s) for this assumption. The eigenvectors can be determined formally anyway (seemingly with some interesting exceptions, I'll be looking at this next days), but the needed complex solutions for h(s) determine the "extravagant" behave because of
a) the divergence of the sequence of eigenvalues,
b) the occurence of divergent series, when the operator [*1] is actually computed by matrix-multiplication of its components.

Since the evaluation of the matrix-multiplication of the eigensystem-components implies a powerseries in hl, I think, the radius of convergence is best described by the complex unit disk (except of its bounding circle) for hl. I'll check this with my formulae and the messages here in the forum so far.

(Consequences of the analytical solution for the eigenvectors, as far as this is not in error)

Gottfried
[*1] edit: "eigensystem" exchanged by the better choice "operator"