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Compartmentalizing tetration proofs
#1
Howdy,
My limited understanding of people's progress at extending tetration on this forum is that proof of convergence is the main problem. Would it be correct to say that once there is a solid convergence proof, that there would then be a number of useful proofs on extending tetration. 

What progress has been made at identifying which approaches extend tetration in the same manner? While I don't have a formal proof, I note I have the same experimental results as the following:
 
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps,
J. Math. Phys. 39, 5324 ( 1998 )

Thanks,
Daniel
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#2
Hey, Daniel.

I'd say yes to your question, but a defining difference, is that each extension faces its own convergence problems. Alors, there is no one convergence problem we can solve that would fix everything. For instance, I came up with using a function which almost looks like tetration and tried to generate a limit which would give tetration; this only worked on the real-line unfortunately, and it seems resolved this solution was non-analytic. Contrast that to Kouznetsov's method, which needs an entirely different method of convergence; but the general consensus is that it does converge to an analytic function (I don't believe this has been proven yet though). But the resolution of the method is unnecessary for the resolution of Kouznetsov's method, and vice versa.

A better way would be to say that we can formally express tetration in many different manners; where it looks like this thing should converge to tetration; but tetration just has a knack at either being too hard to prove convergence, or throws a wrench in the gears at the last second forcing non-analycity. So in this sense, there are many ways to construct candidate tetration functions; but making sure they converge is a totally other problem; and then further, making sure they're real valued and satisfy analycity, or certain behaviour at infinity, produces even more problems.

That's an interesting paper. I'd have comments on it but I despise working with matrices, and I despise proofs with matrices; but I gather it's just the matrix form of the flow operation; and the construction of flows by iterating matrices. Not to diminish matrix proofs though; I just personally hate matrices, lol. But nonetheless in the case of tetration, this paper will necessarily produce complex values for real values. Kneser's method has all of the above encoded in it, but Kneser's solution requires a unique Riemann mapping to produce a tetration we want. As such, any solution with matrices and flow of matrices is less developed than Kneser's solution. Not less right, just... it doesn't look as nice.

Regards, James
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