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Generalized Kneser superfunction trick (the iterated limit definition)
#21
It's back to the drawing board! So I thought I'd post how close I came in the third iteration of this paper.

Call



Note in this space that the transfer operator exists. Then there exists a super function operator on , call it ,



And if



There exists a in which,



BUT I CAN'T GET . I think this might be off base even trying to prove it with this much information. So I nearly have a set which has a conjugate property, but this pesky condition at negative infinity has me stumped. Largely because the behaviour at positive infinity is used to construct and consequently it encodes nothing about the behaviour at negative infinity. Damn it, so close!

As I said, back to the drawing board trying to find a set of functions which satisfies the conjugate property. I was so close, too!

Regards, James
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#22
Honestly, at this point in my evaluation of my ideas (my approach), I don't think constructing a monoid globally on is hopeful. I definitely got ahead of myself in thinking I could get a monoid . I think, doing this in a local setting is easier--and far more probable. Even if we focus on the trivial case. That would be when we can just multiply our Schroder functions. By this I mean, if and and are holomorphic in a neighborhood of and , then,



Is holomorphic in a neighborhood of zero. Here,



And this is of course a group under composition; in which belongs if ... I think this may be a more tractable approach to constructing a general categorical theory.  This set of sheaves do satisfy the conjugate property. But it's a little useless globally (it just means there's a taylor series in a neighborhood of zero). I do think its doable in the global sense, but probably in a local setting, is the correct way to approach the larger theory of conjugation. I definitely can't show it on the real line, but I may be able to do it locally in the complex plane (not necessarily about a fixed point). I'll have to entirely alter my approach though.

Nonetheless the group, under composition, of sheaves ,



Is a very good place to start. Of which the conjugate property is almost satisfied here. And furthermore, this is a GROUP which almost satisfies the conjugate property. We'd just have to allow for and somehow massage this case to allow for the conjugate property--while still staying in the group.

I forgot how useless I am at deep questions in real-analysis, so I'll stick to holomorphy. I'm so angry I can't get it on the real-line ):<

Regards, James
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