Discussing fractional iterates of \(f(z) = e^z-1\)

[JmsNxn]

Hey, Everyone!

I am not able to be as present on this site as much as I'd like, but I can still write about all the things we talk about. I am only writing a sketch of a proof right now. But this is a paper which proves everything we need about neutral fixed points, to develop integral transforms which converge. These integral transforms converge, for the same reason the matrix transforms that Helms' talks about.

I will not be able to describe to large extents what I am talking about. I have a rough, sketch, paper of 30 pages which describes everything. There are still some errors, and I need to edit and perfect this paper much more. But this describes a lot about neutral fixed points.

I hope this rough paper explains a lot to everyone. I hope this helps everyone understand neutral iteration,

Nothing but love to everyone here! I'm not going to be available for every comment, or truly here in the forum. I still read a good amount of shit here though. I just want all my questions to be answered with this paper.

So fuck it, read the paper! Hope you guys get it!


  Neutral_Fixed_Points__and_Borel_Sums-4.pdf (Size: 355.38 KB / Downloads: 269)

[MphLee]

Hi James, thx for keeping up the work. I'm happy you were able to synthesize Helms and Trapmann's efforts. The forum and this field will benefit from this. I was able to follow for a bit but then escalated quickly. It seems to me the target of your paper is= "experts of the field".

In this period I have the same problem, I can't be present in the forum as much as I'd like. I travel like 3/4 hours a day reaching my workplace... can't write here but I'm taking lot of notes and developing a grand theory... one day I'll get to your work...

[JmsNxn]

It seems to me the target of your paper is= "experts of the field".

It very much is, in this form. I plan to flesh out everything much more though, and give the "dummy's version", lol. This is still a bit cliff notes version.

I am also writing this because I loathe matrices, but I love Helms' and Trappman's analysis. But where you see infinite matricies, you also see integral transforms. So I'm trying to develop the integral transforms which are the exact same thing that Helms' and Trappman do on the hardy space (though they never mention hardy space). Quite literally it's the difference between Schrodinger's work, or Heisenberg's work. Integrals acting on a Hilbert space--or, Infinite Matrices acting on an infinite vector space \(\{1,z,z^2,z^3,...\}\).

I just want to translate it to integrals. And again, argue for the formula:

\[
f(z) = e^{z}-1\\
\]

\[
f^{\circ s}(z) = \frac{d^{s-1}}{dx^{s-1}}\Big{|}_{x=0} \sum_{n=0}^\infty f^{\circ n+1}(z)\frac{x^n}{n!}\\
\]

And that, the fractional calculus approach, produces the same result as Gottfried and Helms. I'm very excited  

I hope you're doing well MphLee! Good luck on all your work. I don't doubt you're possessed by these problems. I'm super excited to read your work!