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Using a family of asymptotic tetration functions...
#1
Hey everyone,

I've spent the past nine days hammering out a construction of tetration. This is everything I wanted in (to such an extent that many of the proof layouts I had for could be transferred over). The main focus of this construction is a family of functions which satisfy as .

I worked my ass off on this for nine days, because I'm 99.99999% sure I'm correct. I'm scared there's some tiny flaw I'm not noticing. But, nonetheless, I'm happy to post this here; even more confidently than the method, while using much of the same principles. I've eliminated all the errors and potential errors from the method.

I'm confident I have constructed a holomorphic tetration function which is real-valued. I don't know anything about its behaviour at and no idea of its behaviour at . I do think that as is equivalent to either the julia set or fatou set of ; and this is not a uniform convergence to a fixed point, like with Kneser. I'm pretty certain that as we'll get that should oscillate wildly and behave like orbits of --eventually hitting infinity and back to 0.

I don't know how to summarize this whole construction, but I believe I've done everything I had planned with ; I just mis-stepped in assuming I only need a single function rather than a family. And this family of functions needed to approximate the solution of tetration at infinity; and do so in a uniform manner.

...

A special thanks goes to tommy for the weird infinite compositions he played with...


...

And to Sheldon for running the numbers on ...

...

If you want to read this paper; this requires a lot of infinite compositions, and brushes on Riemann surfaces, and a lot (I mean a lot) of complex function theory.

Regards, James


.pdf   asymptotic_tetration__2.pdf (Size: 375.8 KB / Downloads: 33)

UPDATE!!

So I'm working on an update to the above paper. The things I have changed and the things I am planning to change:

1. I bettered the proof that as . I spend more time explaining this, as I realize the entire paper hinges on this. And it's exactly why is superior to .

2. I realized isn't an exponential series--but it looks enough like one for it not to affect the final result. I went a bit quick here in the first iteration--because I knew it didn't matter if it was either way (but I forgot to double check). The correct asymptotic is . The second iteration of this paper follows similarly; except when talking about convergence we have to be more careful with our compact sets. The functions are not holomorphic for , they're holomorphic for . But! The singularity at is removable in a specific manner-- as . From which, most of the paper isn't really changed, it's just a bit more aggressive (in that there are more arguments). I've finished all this.


3. I'm trying to visualize some of the constructs. And I plan to include some graphics to help me explain some of these things--especially the domain arguments. I've already written out some functor diagrams I'm including. I'm horrible with graphics programming, though--so, it'll take me a while to develop some nice computer produced -plane type graphs.

4. I clarified a lot of the language--at least part way. I haven't fixed everything yet.

5. I'm working on trying to develop a Taylor series at zero for computational purposes, but I lack far too much computational knowledge (especially because pari-gp is too unfamiliar to me, and the only language I ever really worked with was C and python). Everything I do just hits overflow before I'm even at for large arguments. Which says more about my coding than my method...

Again, the main result hasn't changed at all. Just how we get there is a tad more difficult; but I'm explaining it much better. If you felt confused from the first iteration, I apologize. I'll clear everything up. I got ahead of myself--again, I wrote that in 9 days in a light-bulb moment. The proof schema is still the exact same; just some of the brickwork needs to be patched, is all.

Regards, James.


UPDATE! #2

Please see post #3 for all the information I changed in this update. I, about, doubled the length of this paper. But it's much more solid now.


.pdf   asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 11)
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#2
Hey guys,

I've written some supplemental literature to the original paper. This short analysis is meant to explain the general case better. It's fairly short; and only serves to elaborate on everything above. What I'm attaching here; is nothing more than an explanation of variable changes in an infinite composition. And the manner one can understand this around . This brief notice is about solving Schroder equations at infinity; and in a well enough manner. Which explains a lot of this paper; but done in a more blank pattern.

Regards, James


.pdf   Mock_Schroder.pdf (Size: 272.65 KB / Downloads: 23)
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#3
Hey, guys!

So I've fixed the error in the first iteration, which was pretty silly of me actually. I had written that the process,



produces an exponential series. This is not true at all. It just looks like an exponential series (enough for the main theorem to be unaffected); but there are a bunch of singularities. It was really silly of me. The correct statement is that,



This doesn't affect the main theorem at all though. I just have to be more careful when applying Banach's Fixed Point theorem. For the most part, it's much of the same proof.


I've added a total of 12 figures; which took a lot out of me. I am horrible at working out graphs. And since this construction is to do with super-exponential behaviour at infinity; I don't know how to code around the overflow errors. I'm still working on making graphs of the actual tetration (which looks like I might be able to do soonish). I just have to put on my coding hat and find an efficient manner at computing this. I've also added a bunch of commutative diagrams, in an attempt to better explain some of the morphisms I use.

I've expanded a bunch of the arguments and clarified as much of the language as I think I can. Particularly, I made sure the proof that,



Was as solid as I could make it. I had to reference three people to make this argument--all from Milnor's book. This theorem is the crux of the method though. Where, the number one reason the construction with failed was because it oscillated between and very rapidly. I must say, it's very satisfying to see the graphs concur with the divergence of on paper.

I tried to explain the variable changes more clearly in this iteration. I did it a tad off-hand initially. But, I imagine much of you are new to this infinite composition stuff; so the idea of changing variables in an infinite composition may seem odd. I tried to make it simpler to understand by contrasting it with commutative diagrams.

I don't think the paper is quite done yet. I am looking to revamp it one more time once I somehow manage to get some workable code to evaluate these tetrations (of which we want to limit to find the right tetration). At least, as long as I can evaluate it in a rudimentary way.

The more I work with this though, the more I'm convinced this actually constructs tetration. Plus the graphs are behaving exactly as I expected them too (even if they're not the tetration graphs yet).

Regards, James.


.pdf   asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 12)
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