Some "Theorem" on the generalized superfunction

[Leo.W]

Wow, a very long thread has formed in my absence, lol. I figure I should add a long thread in return, so buckle up, lol.

I'd like to add a couple details. Mphlee did a much better job at explaining the concerns I have with this notation. The more I'm rereading through this thread, I've come to the conclusion, in order for Leo's idea to make sense; there's no two ways around it; we need to introduce Riemann Surfaces. As Leo seems dead set on multi-valued functions; it's imperative we introduce a more rigorous framework. Leo makes mention of choosing the "worthy" iteration (lol, which I understand entirely; but he hasn't really given a way to find said worthy iteration; besides sticking to Schroder or Abel about a fixed point).

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But this is tricky to use this how we want... manipulating how you're describing. I do it in a limited manner in that paper. And this gives us a way of relating superfunctions to a differential relationship. But it doesn't do what you're asking your theory to do. That sounds super hard. Especially if you keep with this multi-valued language; you're going to have to talk about riemann surfaces somewhere in your discussion.

Regards.

Thank you, James!

The theorem you state is related to iteration velocity(or Koenig's function) in wikipedia, this idea is the same to use the differential operator to manipulate the functional iteration, with respect to index t. It's for sure more general.
I've read it 3 years ago. It's a wonderful theory and I collected a relative article：https://projecteuclid.org/journals/michi...02009.full
Also you may find similar comments in the wikipedia article Iterated functions, it states the iteration velocity is the solution to Julia's equation... And amongst all these theories, the notations are not unified yet, it's kinda messy
But honestly I don't see there's a relation between these stuffs and my statements... My statements are more related to the existence of multivalued functional iterations, their conjugacies and transformations between them, and without any proofs on the existence yet. Also my laws include ways to generate and calculate them, so they're more practical in computation.

About Riemann Surface, it is more a way to descirbe multivalued functions, and branch cuts is another way, maybe you can consider riemann surface as a set of branch cuts, where each are joint to another. You can say the logarithm function has a riemann surface, and it's okay if you say that logarithm function has infinitely many branch cuts. A branch cut can make us more easily to cope with multivalued functions by considering only a branch of values, which is same as singlevalued.
Why I'm so obsessed around the branch cuts is that, it's everywhere! You can see that, if we use the second pairs of fixed points~{2.062+7.589i,2.062-7.589i} instead of the original one~{0.318+1.337i,0.318-1.337i}, the original branch cut, or the original defined logarithm function contains only the first pair. When we use the double-dagger track, and calculate the line segment integral, this original logarithm fails to converge, and we'll fail to arrive at the merged tetration of the second pairs. But we should notice the two branches:{log(z)+2*pi*i,log(z)-2*pi*i} contains the second pairs, then we can use these two branch cuts to calculate the integral, and only then can we manage to merge the second tetration.

Choosing a branch cut is difficult to describe, so I only claimed the existence. The choice will be discussed in another thread, I guess. But you can see how it works. The first example is included in #2 Part 2, Section III 1.Miscellaneous Results. It shows if you choose a proper branch, the schroder function can be entire. The same thing happens to Kneser's iterated exponentiation. Most still believe the schroder function is only defined in the upper half plane, but you should notice, while we were generating the schroder function at L~0.318+1.337i, we used the limit \( \sigma(z)=\lim_{n\to\infty}s^n(\sigma_0(\log^n(z))) \) where log function has infinitely many branch cuts.
We use a slight modification: \( \sigma(z)=\lim_{n\to\infty}s^n(\sigma_0(f^n(z))) \), in which f is defined as:
if \( \Im(z)\ge0 \), \( f(z)=\log(z) \); if \( \Im(z)<0 \), \( f(z)=\log(z)+2\pi*i \). And then the schroder function is defined only except the points {0,1,e,e^e,...}, so the Kneser's iteration is extended now to the whole complex plane.
Another example is when you attempt to generate supersine from the pair ~{7+2i,7-2i} the same thing happens, since arcsin(z) doesn't contain these 2 points, but its another branch cut does.
Another example is when you try to generate the merged superfunction of f(z)=z^2+i, the original inverse function g(z)=sqrt(z-i) fails to make the schroder function convergent, so we should modify the branch cut, and finally it turned out as a successful method.

And without multivalued function, most iterations are unavailable, whenever you have a pair of f(a)=b and f(b)=a, it's proved that there's no singlevalued second iterative root of f, so do forth ,eighth and so on. Also it's proved similarly, if one have f^n(a)=a but f(a),f^2(a),f^3(a),...,f^(n-1)(a)=/=a, then f have no singlevalued nth iterative root, and hence f^s(f^t)=f^(s+t) is impossible. The multivalued functions play a crucial role in iteration theory, and its multivalued property may mend these holes. And you can check that L~0.883998+6.92228i who satisfies exp(exp(L))=L but exp(L)=/=L, thus tetration has to be multivalued.

Another example is also worth thinking, you can see if f(z)=-z+z^2, and g(z)=f(f(z))=z-2*z^3+z^4.
You probably say f(z) is a second iterative root of g(z), but if we generate the superfunction of g(z) in the traditional method, it'll result in that g^(1/2)(z)=/=f(z). So it forces you to conclude that there lies many branch cuts of g^(1/2)(z).

I considered a way to more rigorously define my declaration in section I, and the description is below.

And lastly the three pics show the 3 different branches of half-iteration of tan(z), each satisfies f(f(z))=tan(z) in only a subset of C, wish you enjoy it.

Regards, Leo