The Different Fixed Points of Exponentials

[JmsNxn]

The paper relies on the initial paper with him and Cowgill. In that paper they reconstruct Kneser on \(b > \eta\). From that point, the paper I linked he analytically continues in \(b\). The paper with Cowgill is more indepth, and the one I linked is a little loose.

Yeah, I think I read the one with Cowgill. But I think we really must be specific here:
They have a numerical method, that is not proven to converge (though it practically seems to).
Then they have in the paper with Cowgill a uniqueness proof.
But with a numerical method, that is not proven to converge, you can not claim that it satisfies any (uniqueness) criterion.
And what does it mean to analytically continue? It just means the calculated plots look good.
(Correct me if I am wrong!)
Sheldons method is insofar as good as theirs (though maybe he had a problem with some bases if I remember), and other methods that have a good looking plot for complex bases. But Sheldon, was already close about a proof of convergence of his method, so in the race he is my favourite . And you were also close to a proof of your method, if I remember! So what happened with that?!

That is very true. So if I'm being very specific.

Cowgill and Paulsen describe how the fourier series is constructed, and designed--and Paulsen analytically continued that. I believe nestled in that paper is a proof of convergence for the Kneser case, but I could be mistaken... But I'm pretty sure they describe the solution, but you are correct, they could be divergent fourier series in Paulsen's case. I guess we'd have to rigorously prove that the Fourier series converges. I'll have to read the paper again, I believe there was some work put into it.

My method (the beta method) turns out to only be an asymptotic solution in the general case, but converges perfectly fine within the Shell-Thron region. I wrote a long report which details how everything works:

The essential statement was to construct the analytic function:

\[
\beta_{\mu,\lambda}(s)\\
\]

Such that:

\[
\beta_{\mu,\lambda}(s+1) = \frac{e^{\mu \beta_{\mu,\lambda}(s)}}{1+e^{-\lambda s}}\\
\]

This function is holomorphic for \(\mu \in \mathbb{C}\) and \(\Re\lambda > 0\) so long as \(e^{\lambda(j-s)} \neq -1\) for all \(j \ge 1\) with \(j \in \mathbb{N}\). Then the idea was to correct the beta function to be tetration. This was done with the tau error, so that:

\[
\beta_{\mu,\lambda}(s) + \tau_{\mu,\lambda}(s) = F_{\mu,\lambda}(s)\\
\]

Where then:

\[
\begin{align}
F_{\mu,\lambda}(s+1) &= e^{\mu F_{\mu,\lambda}(s)}\\
F_{\mu,\lambda}(s + 2 \pi i/\lambda) &= F_{\mu,\lambda}(s)\\
\end{align}
\]

Deriving where this converges is very difficult, and is the sum of 100 pages. But the basic answer is that when \(b=e^{\mu}\) is in the shell thron region, so long as \(\Re(\lambda) > \Re\kappa\), where \(2 \pi i/\kappa\) is the period of the regular iteration about the principal (attracting/neutral) fixed point of \(b\). Then the object converges outside of a measure zero set in \(\mathbb{C}\) (so a bunch of branches or fractals). These tended to be pretty small, so it was safe to say (even mathematically) that \(F_{\mu,\lambda}\) converges almost everywhere on \(\mathbb{C}\).

This essentially is mostly valuable because you can make a tetration base \(b=\sqrt{2}\) with arbitrary period \(2 \pi i / \lambda\), so long as \(\Re(\lambda) > -\log\log(2)\). And on the boundary of shell thron, at \(\eta\) you can make a tetration of arbitrary period. All of this, upto a measure zero set in \(\mathbb{C}\) of course (here I mean an area measure in \(\mathbb{R}^2\), so upto 1 dimensional lines and fractals).


Then when you let \(b = e^{\mu}\) outside of the Shell-thron region, you can make an arbitrarily accurate asymptotic expansion (too lengthy to draw out that statement without reading the paper), but it will never be analytic. So the function:

\[
F_{1,\lambda}(s)\\
\]

Is analytic nowhere, this was supported by Sheldon too, who followed my proof, and it was his own conjecture. I believe I proved it, it essentially just relied on the julia set of \(e^z\) being the entire plane. So essentially for \(b > \eta\) you can make arbitrarily accurate asymptotic approximations, but they'll never be analytic. In the complex plane (out side of shell thron) it essentially devolved into small little patches of analycity, but most of the domain only admitted an asymptotic expansion.

Additionally the program beta.gp is a great asymptotic tool for \(b \approx 0\) and \(b \approx 1E1E10\) so it works very well for very large numbers. Essentially I've reserved the solution to being an asymptotic solution for studying the orbits of the various exponential.

Here's the finished pdf, it's a long read, still a little rough around the edges, but I tried to keep myself to as high a standard as I could. Plus, lots of pretty graphs!


  Asymptotic_Solutions_Of_The_Tetration_Equation_In_The_Style_Of_Sterling.pdf (Size: 7.68 MB / Downloads: 558)

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Also, I should add, that my conjecture, and Tommy's conjecture, is that the \(\beta\) method can reproduce Kneser's method for \( b =e\).

To me, I have maybe 60-70% of the proof, a good amount of numerical evidence that:

\[
\text{tet}_K(s+c) = \lim_{\lambda\to 0, n\to\infty} \log^{\circ n}\beta_\lambda(s+n)\,\,\text{while}\,\,\lambda = \mathcal{O}(n^{-1/2})\\
\]

Tommy's Gaussian method is equivalent to this construction, though Tommy says other wise. These are the same. I conjecture this is Kneser. The problem is that the code caps at about \(\lambda = 0.05\) and we can't test the asymptotics. But mathematically this is what should happen. I could elaborate more, but I leave it at that.

EDIT: Not to discredit tommy's work. He boiled it down into an absolute and a much simpler construction with the Gaussian method. But both methods are equivalent, is all I'm saying. tommy's gaussian is far better/more efficient.

Regards, James