05/08/2022, 09:30 PM
The fact that it seems a plane is amazing... even if I don't know exactly what this means.
I skimmed your paper for the fourth time, it's starting to make more and more sense. In my poor understanding, switching from controlling the period \(\lambda\) to controlling \(\varphi\) you switched from beta method to something more mundane, just perturbating the exponent/fixedpoint, but that has same effect but better behaviour. Is this a good sketch of it?
Also the part where you "de-synchronize" the three \(\varphi\)s turning it into a surface and making the coordinates implicitly functions of different arguments and subject to some relations... that's the most tricky part imho. I need to go back to your forum post and read them more. But now I understand why I was slow at getting a full picture... your methods are really rich of details and layers.
What I don't get at this point is... for a triple \((b,y,s)\) you get a surface \({\bf \Phi}_{(b,y,s)}\subseteq \mathbb C^3\)... this amounts to have a family of surfaces. You are interested only in a single point for each surface? How to select them? All of this reminds of me of the fiber bundles/sections business.
If all of them are homeomorphic to a complex plane(but curved), i.e. \(\mathbb C^2\) via a parametrization then, do you get a fiber bundle \[\bigsqcup_{(b,y,s)\in \mathbb C^2\times (0;2)} {\bf \Phi}_{(b,y,s)} \simeq \mathbb C^4\times (0,2) \overset{\bar{\bf \Phi}}{\longrightarrow} \mathbb C^2\times (0;2)?\]
Each fiber at a point \(P=(b,y,s)\) is the desired surface \( \bar{\bf \Phi}^{-1}\{P\}={\bf \Phi}_{P} \). If it is a fiber bundle then probably you get some lifting properties... yea I'm just inventing things here but... sections must have something to do with vector fields...
I skimmed your paper for the fourth time, it's starting to make more and more sense. In my poor understanding, switching from controlling the period \(\lambda\) to controlling \(\varphi\) you switched from beta method to something more mundane, just perturbating the exponent/fixedpoint, but that has same effect but better behaviour. Is this a good sketch of it?
Also the part where you "de-synchronize" the three \(\varphi\)s turning it into a surface and making the coordinates implicitly functions of different arguments and subject to some relations... that's the most tricky part imho. I need to go back to your forum post and read them more. But now I understand why I was slow at getting a full picture... your methods are really rich of details and layers.
What I don't get at this point is... for a triple \((b,y,s)\) you get a surface \({\bf \Phi}_{(b,y,s)}\subseteq \mathbb C^3\)... this amounts to have a family of surfaces. You are interested only in a single point for each surface? How to select them? All of this reminds of me of the fiber bundles/sections business.
If all of them are homeomorphic to a complex plane(but curved), i.e. \(\mathbb C^2\) via a parametrization then, do you get a fiber bundle \[\bigsqcup_{(b,y,s)\in \mathbb C^2\times (0;2)} {\bf \Phi}_{(b,y,s)} \simeq \mathbb C^4\times (0,2) \overset{\bar{\bf \Phi}}{\longrightarrow} \mathbb C^2\times (0;2)?\]
Each fiber at a point \(P=(b,y,s)\) is the desired surface \( \bar{\bf \Phi}^{-1}\{P\}={\bf \Phi}_{P} \). If it is a fiber bundle then probably you get some lifting properties... yea I'm just inventing things here but... sections must have something to do with vector fields...
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)