06/03/2022, 02:08 PM
Slightly related. I put this just to not forget.
There are two way to explore: consider \(\Theta_q(z,y)\) is a family of holomorphic solutions for every \(q\in [0,1]_\mathbb Q=[0,1]\cap \mathbb Q\) to
a0) \(\Theta_q(z,w+y)=w+\Theta_q(z,y)\)
a1) \(\Theta_q(z+w,y)=\Theta_q(z,\Theta_q(w,y))\)
b) \(\Theta_0(z,y)=\Theta_1(z,y)=z+y\)
In the above equation the goal is to consider only exotic solutions that are not constant when \(q\in(0,1)_\mathbb Q\). We want to exclude the constant solutions.
We have \(\Theta_q(0,w)=w+\Theta_q(0,0)\)... I hope we can have some periodic behavior or something like that.
The main idea is that for rational ranks \(q\in [0,1]_\mathbb Q\) we define a family of holomorphic \(a_q:\mathbb C\times \mathbb C\to\mathbb C\) s.t. \(\mathbb C\)-equivariance holds.
\[a_q(b,z+y)=\Theta_q(z,a_q(b,y))\]
With boundary conditions \(a_0(b,y)=y+1\) and \(a_1(b,y)=b+y\) and such that \(q\mapsto a_q\) is smooth given a reasonable topological structure (manifold structure?) on \({\mathcal H}(\mathbb C\times \mathbb C)=\mathcal C^\infty (\mathbb C\times \mathbb C)\) in a neighborhood of \(0\) and \(1\).
There are two way to explore: consider \(\Theta_q(z,y)\) is a family of holomorphic solutions for every \(q\in [0,1]_\mathbb Q=[0,1]\cap \mathbb Q\) to
a0) \(\Theta_q(z,w+y)=w+\Theta_q(z,y)\)
a1) \(\Theta_q(z+w,y)=\Theta_q(z,\Theta_q(w,y))\)
b) \(\Theta_0(z,y)=\Theta_1(z,y)=z+y\)
In the above equation the goal is to consider only exotic solutions that are not constant when \(q\in(0,1)_\mathbb Q\). We want to exclude the constant solutions.
We have \(\Theta_q(0,w)=w+\Theta_q(0,0)\)... I hope we can have some periodic behavior or something like that.
The main idea is that for rational ranks \(q\in [0,1]_\mathbb Q\) we define a family of holomorphic \(a_q:\mathbb C\times \mathbb C\to\mathbb C\) s.t. \(\mathbb C\)-equivariance holds.
\[a_q(b,z+y)=\Theta_q(z,a_q(b,y))\]
With boundary conditions \(a_0(b,y)=y+1\) and \(a_1(b,y)=b+y\) and such that \(q\mapsto a_q\) is smooth given a reasonable topological structure (manifold structure?) on \({\mathcal H}(\mathbb C\times \mathbb C)=\mathcal C^\infty (\mathbb C\times \mathbb C)\) in a neighborhood of \(0\) and \(1\).
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)