I thought I'd add my drawings of tetration for base \(b = \eta_-\).
This is I believe the \( 2 \pi i\) periodic tetration, drawn about the singularity as it moves in the real line. Here's a zoom in on the very sinusoidal curves which spawn as you go further and further to the right (enclosing about \(1/e\)):
And if you increase the imaginary argument you get:
Now when you graph this in the complex plane, it gets a little weird. Essentially in the strip \(- \pi < \Im(z) < \pi\), this the important case, and serves as a "they all kinda work like this". Which you can see in this graph, it's all red, excusing a sea of green or purple. This function is still holomorphic here, it's just a branch cut you can visibly see. Then the goal of "beta can be turned into Kneser" is to stretch this strip into the entire complex plane.
Now the singularities you see in the top and bottom are native to the \(\beta\) method. We always have singularities at precisely \(\Im(z) = \pi i + j\) for all \(j \in \mathbb{Z}\). These are precisely not seen in this graph. But, this function is \(2 \pi i\) periodic. Therefore the top and bottom parts of this function have a direct discontinuity from top to bottom. This results in a branch cut directly along the line \(\Im(z) = \pi i\). These are the singularities inherited by \(\beta\).
Then there are iterated log singularities, which appear at the boundary of the green and red/purple and red sea.
ALSO NOTE THAT GREEEN IS \(a^-\) AND PURPLE IS \(a^+\). Stretch the red domain and we apprach the fixed points.
Thought I'd make a graph explaining what I mean by the singularity is somewhere, and the "false singularity which is really just a julia set" makes it look like the singularity is somewhere else. I'm going to use one of my stock photos of \(b = \sqrt{2}\) and the weak julia set as described in my paper.
The singularity is at the left most point white point, and the white area is how my code fails/describing what I called the weak julia set. This was the same thing as the Julia set but sort of modified in terms for \(\beta\) functions. This is where we hit a fractal boundary. So picture a fractal-like branch cut with a lot of large and small values, but no actual "non-holomorphy" except for on the branching fractal that appears.
This fractal extends from the green sea to the purple sea to the red sea, and that's where we see our "obvious singularities". It's really weird with \(\eta_-\).
Again the white area is as good an approximation as I could get. Sheldon even said that's not the actual fractal right? And he argued for perturbation theory and everything, and we concurred the actual fractal should be a lightning bolt kind of thing.Which as I showed, is actually measure zero, under an Lebesgue area measure.
In the above graph the singularities are precisely on the boundary of the sea of green and sea of purple. And there's precisely a branch parallel to the real line (remember these graphs are \(2 \pi i\) periodic), and this is a graph on a window \(|\Im(z)| < \pi\). So there's a boundary at \(\pi i\) of purple and green. And the singularities are here. But there exists iterated log errors which are branching errors. But they aren't inherent to \(\beta\). They are fractals produced by the \(\beta\) method which result in an explosion half the graph down... Like a lightning fractal.
This is I believe the \( 2 \pi i\) periodic tetration, drawn about the singularity as it moves in the real line. Here's a zoom in on the very sinusoidal curves which spawn as you go further and further to the right (enclosing about \(1/e\)):
And if you increase the imaginary argument you get:
Now when you graph this in the complex plane, it gets a little weird. Essentially in the strip \(- \pi < \Im(z) < \pi\), this the important case, and serves as a "they all kinda work like this". Which you can see in this graph, it's all red, excusing a sea of green or purple. This function is still holomorphic here, it's just a branch cut you can visibly see. Then the goal of "beta can be turned into Kneser" is to stretch this strip into the entire complex plane.
Now the singularities you see in the top and bottom are native to the \(\beta\) method. We always have singularities at precisely \(\Im(z) = \pi i + j\) for all \(j \in \mathbb{Z}\). These are precisely not seen in this graph. But, this function is \(2 \pi i\) periodic. Therefore the top and bottom parts of this function have a direct discontinuity from top to bottom. This results in a branch cut directly along the line \(\Im(z) = \pi i\). These are the singularities inherited by \(\beta\).
Then there are iterated log singularities, which appear at the boundary of the green and red/purple and red sea.
ALSO NOTE THAT GREEEN IS \(a^-\) AND PURPLE IS \(a^+\). Stretch the red domain and we apprach the fixed points.
Thought I'd make a graph explaining what I mean by the singularity is somewhere, and the "false singularity which is really just a julia set" makes it look like the singularity is somewhere else. I'm going to use one of my stock photos of \(b = \sqrt{2}\) and the weak julia set as described in my paper.
The singularity is at the left most point white point, and the white area is how my code fails/describing what I called the weak julia set. This was the same thing as the Julia set but sort of modified in terms for \(\beta\) functions. This is where we hit a fractal boundary. So picture a fractal-like branch cut with a lot of large and small values, but no actual "non-holomorphy" except for on the branching fractal that appears.
This fractal extends from the green sea to the purple sea to the red sea, and that's where we see our "obvious singularities". It's really weird with \(\eta_-\).
Again the white area is as good an approximation as I could get. Sheldon even said that's not the actual fractal right? And he argued for perturbation theory and everything, and we concurred the actual fractal should be a lightning bolt kind of thing.Which as I showed, is actually measure zero, under an Lebesgue area measure.
In the above graph the singularities are precisely on the boundary of the sea of green and sea of purple. And there's precisely a branch parallel to the real line (remember these graphs are \(2 \pi i\) periodic), and this is a graph on a window \(|\Im(z)| < \pi\). So there's a boundary at \(\pi i\) of purple and green. And the singularities are here. But there exists iterated log errors which are branching errors. But they aren't inherent to \(\beta\). They are fractals produced by the \(\beta\) method which result in an explosion half the graph down... Like a lightning fractal.
