Real-analytic tetration uniqueness criterion?

[sheldonison]

However, didn't you disprove this conjecture with the construction of the tetration function from the alternate fixed point here:

http://math.eretrandre.org/tetrationforu...hp?tid=452
http://math.eretrandre.org/tetrationforu...452&page=2

or does this also qualify as a "Kneser"? But it's not a unique function if that's the case.

However, just from looking at the graphs on that second page, it's quite obvious this function fails the criterion given in my OP.

Yeah, we need to add a criteria that not only is the tet(x) function increasing from -2->infinity, but also that the first derivative is positive. The alternative fixed point has a zero derivative at integer values, -1,1,2,3 etc. This is equivalent to requiring that tet(z) have an inverse at the real axis; that the slog be analytic at the real axis.

I wonder what the \( \theta(z) \) mapping carrying the "good" Kneser solution to that thing looks like. I suspect it'll be multivalued, with branch singularities instead of just poles or whatever, which significantly complicates the composition \( \mathrm{tet}(z + \theta(z)) \) in the complex plane -- although on the real line it will, of course, be single-valued.

On the other hand, your "max at the real axis" criterion would seem to rule out this function.

Yeah, \( z+\theta(z)=\text{slog}(\text{tet}_{\text{alt}}(z)) \) would have a cube root branch at integers, so yeah, theta is not analytic at the real axis. Ooops, not correct; edit \( \theta(z)=\text{slog}(\text{tet}_{\text{alt}}(z))-z \) is 1-cyclic analytic function at the real axis.