Real-analytic tetration uniqueness criterion?

[mike3]

For tetration at the real axis, the nearest singularity is at x=-2, and there are no other singularities to the right of that anywhere in the complex plane.

Well that depends on what type of tetration we use.
Kneser seems to have this property.
But so do many theta variations of Kneser. ( the analytic theta's )

Conjecture; Kneser is the only solution with no singularities in the upper/lower halves of the complex plane, and this is a uniqueness criterion. For all the of the entire theta functions, we know there will be an infinite number of singularities in the upper half of the complex plane, where z+theta(z)=-2,-3,-4 ..... I would also conjecture that these singularities will be in the right half of the complex plane as well. But either way, Kneser has this special property, so Kneser's so at any value of z, there are negative even derivatives for large enough (2n). This doesn't prove that all of the odd derivatives are always positive, but that is also a conjectured uniqueness criterion.

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.

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.