Questions about Kneser...

[JmsNxn]

So I've been running through my head a question. But in order to resolve that question, I need to better understand Kneser's construction. The premise of this post is to talk about constructing pentation from Kneser's construction.

Now, I know that,

\(
\text{tet}_{\text{Kneser}}(s) : \mathcal{H} \to \mathbb{C}\\
\)

Where \( \mathcal{H} = \{\Im(s) > 0\} \). And I know it tends to a fixed point \( L \) as \( \Re(s) \to -\infty \). (Or is it multiple fixed points?) Either way, this convergence must be geometric. Giving us, what I'll call,

\(
\Phi_{\text{Kneser}}(s) = \Omega_{j=1}^\infty \text{tet}_{\text{Kneser}}(s-j+z) \bullet z : \mathcal{H} \to \mathbb{C}\\
\)

Satisfying the functional equation,

\(
\Phi(s+1) = \text{tet}(s+\Phi(s))\\
\)

And it tends to \( L \) as \( s \to -\infty \) exactly like \( \text{tet}(s-1 + L) \).

Proving this converges will be a little troublesome if we don't have good control over Kneser's solution; but for the moment assume it's good enough. It'll definitely work for large enough \( T \) with \( \Re(s) < -T \). The idea then being,

\(
\text{slog}^{\circ n} \Phi(s+n) \to \text{pent}(s)\\
\)

Now, I know what you're thinking. Won't the same thing happen that happened with tetration happen with this method? And I'd say probably. Now instead of looking for \( \log(0) \)'s though, we'd be looking for \( \text{slog}(L) \)'s. So where ever \( \Phi(s_0) = L \) we get ourselves into a whole lot of trouble.  The only real benefit I can think in this circumstance is that \( \Phi \) is not periodic. And that \( \Phi(s) \to L \) but it shouldn't equal \( L \), or at least, it should be well controlled where it does. This depends on Kneser though, does it ever attain the value \( L \) other than at \( -\infty \)?

I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_-

Nonetheless, I think this might have a better chance at converging than my tetration function constructed with \( \phi \). It's just a hunch though.

[sheldonison]

So I've been running through my head a question. But in order to resolve that question, I need to better understand Kneser's construction. The premise of this post is to talk about constructing pentation from Kneser's construction.

Now, I know that,

\( \text{tet}_{\text{Kneser}}(s):\mathcal{H}\to\mathbb{C} \)

Where \( \mathcal{H} = \{\Im(s) > 0\} \). And I know it tends to a fixed point \( L \) as \( \Re(s) \to -\infty \). (Or is it multiple fixed points?)
...This depends on Kneser though, does it ever attain the value \( L \) other than at \( -\infty \)?
...
I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_-

Hey James,
The two fixed points are L, L* in the upper/lower halves of the complex plane.  Kneser tends to L as \( \Im(z)\to\infty;\;\;\Re(z)\to-\infty \)
Wherever \( \text{tet}(z)=L+2n\pi i;\;\;\text{tet}(z+1)=L \) and this happens an infinite number of times in the complex plane
See this mathstack post for a good readable overview of Kneser.

[JmsNxn]

Interesting, so we'd have branches of \( \text{slog} \) that are holomorphic in a neighborhood of \( L \) then. Unlike with the logarithm where no branch is holomorphic in a neighborhood of \( 0 \). I think the trouble with this construction will be getting \( \Phi \) to be well behaved as we grow \( \Re(s) \). And not to mention, getting general growth lemmas on the various branches of \( \text{slog} \). The only work around I see would be to take the inverse iteration,

\(
\text{pent} = \text{tet}_{\text{Kneser}}^{\circ n} \Phi(s-n)\\
\)

Where of course, we'd have to modify \( \Phi \) to converge in this circumstance. And here we'd probably lose any chance of it being real-valued. I'm going to keep this on a backburner and come back to it later. I'm going to stay focused on \( \mathcal{C}^\infty \) proofs for the moment.