09/17/2021, 06:00 AM
I know three posts in a row is poor etiquette. But each post is on a vastly different problem. I'd like to look at Sheldon's idea of a theta mapping much more closely. Using the analytic implicit function theorem and the monodromy theorem,
\(
\text{tet}_K(z+\theta) = \text{tet}_\beta(z)\\
\)
We have a locally holomorphic function \( \theta \) everywhere,
\(
\frac{d}{d\theta}\text{tet}_K(z+\theta) \neq 0\\
\text{we have existence of points}\\
\)
Each tetration sends \( \mathbb{C}/(-\infty,-2] \to \mathbb{C} \) surjectively. They both have a nonzero derivative; so let's try to paste these local solutions using the monodromy theorem.
Now, what Sheldon is thinking (as far as I can tell), is that this should be able to be made into a single function (with singularities or without (without this program would be dead in the water; as he correctly noted)). The way I'd look at this, is that we have a riemann surface defined by \( \theta \) and there is no projection to \( \mathbb{C}/(-\infty,-2] \) that fits perfectly; without branching out in some manner.
This does not mean that the beta method diverges or has singularities. It means the theta mapping does. This is to say, there is no entire/meromorphic function,
\(
\text{tet}_K(z+\theta) = \text{tet}_\beta(z)\\
\)
This is because as \( \Im \theta \to \pm \infty \) the left hand side limits to \( L,L^* \); but on the right hand side, this doesn't force \( z \to \infty \); it forces \( z \to z_0 \) to \( \text{tet}_\beta(z_0)=L,L^* \) a fixed point.
Now, very importantly, Sheldon is calling on:
\(
\theta(z) = \text{slog}_K(\text{tet}_\beta(z)) - z\\
\)
And, if you're wondering what happens at singularities; it's rather plain.
\(
\theta(z_0) = \infty\\
\lim_{\delta \to 0}\text{tet}_K(z_0+ \delta + \theta(z_0 +\delta)) = q\\
\)
Where \( q \) is a point in which \( \exp^{\circ n}(q) \) is recurrent. This means it's either part of a cycle or in the preimage of a cycle. To think about this, the moment that \( \text{tet}_\beta(z_0) = q \) and \( \text{tet}_K(z_0) \neq q \) we're going to run into enormous trouble by trying to just compare them functionally. It doesn't really resonate that,
\(
\theta(z) = \text{slog}_K(\text{tet}_\beta(z)) - z\\
\)
BUT!!!! locally this makes perfect sense. So without assuming this is one function; locally this expression will work. So long as we're away from cyclic points and the such. As to such, Sheldon is very much correct in his observation. But he's assumed that when we have a singularity in \( \theta \) we aren't just limiting towards infinity in some manner, where we approach (completely valid values) recurrent values.
I just thought I'd post this because I agree with Sheldon's points, I think he's just assumed this must create an EVERYWHERE perfect version of \( \text{tet}_\beta \). The point of the beta method is to be an exception to these theta mappings.
\(
\text{tet}_K(z+\theta) = \text{tet}_\beta(z)\\
\)
We have a locally holomorphic function \( \theta \) everywhere,
\(
\frac{d}{d\theta}\text{tet}_K(z+\theta) \neq 0\\
\text{we have existence of points}\\
\)
Each tetration sends \( \mathbb{C}/(-\infty,-2] \to \mathbb{C} \) surjectively. They both have a nonzero derivative; so let's try to paste these local solutions using the monodromy theorem.
Now, what Sheldon is thinking (as far as I can tell), is that this should be able to be made into a single function (with singularities or without (without this program would be dead in the water; as he correctly noted)). The way I'd look at this, is that we have a riemann surface defined by \( \theta \) and there is no projection to \( \mathbb{C}/(-\infty,-2] \) that fits perfectly; without branching out in some manner.
This does not mean that the beta method diverges or has singularities. It means the theta mapping does. This is to say, there is no entire/meromorphic function,
\(
\text{tet}_K(z+\theta) = \text{tet}_\beta(z)\\
\)
This is because as \( \Im \theta \to \pm \infty \) the left hand side limits to \( L,L^* \); but on the right hand side, this doesn't force \( z \to \infty \); it forces \( z \to z_0 \) to \( \text{tet}_\beta(z_0)=L,L^* \) a fixed point.
Now, very importantly, Sheldon is calling on:
\(
\theta(z) = \text{slog}_K(\text{tet}_\beta(z)) - z\\
\)
And, if you're wondering what happens at singularities; it's rather plain.
\(
\theta(z_0) = \infty\\
\lim_{\delta \to 0}\text{tet}_K(z_0+ \delta + \theta(z_0 +\delta)) = q\\
\)
Where \( q \) is a point in which \( \exp^{\circ n}(q) \) is recurrent. This means it's either part of a cycle or in the preimage of a cycle. To think about this, the moment that \( \text{tet}_\beta(z_0) = q \) and \( \text{tet}_K(z_0) \neq q \) we're going to run into enormous trouble by trying to just compare them functionally. It doesn't really resonate that,
\(
\theta(z) = \text{slog}_K(\text{tet}_\beta(z)) - z\\
\)
BUT!!!! locally this makes perfect sense. So without assuming this is one function; locally this expression will work. So long as we're away from cyclic points and the such. As to such, Sheldon is very much correct in his observation. But he's assumed that when we have a singularity in \( \theta \) we aren't just limiting towards infinity in some manner, where we approach (completely valid values) recurrent values.
I just thought I'd post this because I agree with Sheldon's points, I think he's just assumed this must create an EVERYWHERE perfect version of \( \text{tet}_\beta \). The point of the beta method is to be an exception to these theta mappings.