08/18/2022, 04:31 AM
Honestly, no idea Tommy. My knowledge of Riemann surfaces is largely cursory, and more so I learnt pieces of it by way of the link between Elliptic curves and Algebraic curves. In theory, there should be a lift (I believe that's the word), such that we can write:
\[
f^{\circ t} (z) : S \to S\\
\]
Where we are sort of sending the preimages to themselves. Then we can only care about the Riemann surface itself; and in proving things about the Riemann surface we prove things about the iteration on the complex plane--and figure out how and where it is holomorphic if it could be.
For example if \(S = \{ y | y = F(z)\}\) then we define a coordinate \(s_0 \in S\) such that \(f(s_0)\) is the value of \(y\) for \(F(z+1)\). And now we are acting on the preimage. Then we perform iterations as such.
I haven't done much Riemann surface stuff in a while, but this is kind of what you are doing. This isn't very different from how you define the Riemann surface of \(\log\). It's just \(\{y | y = e^z\}\).
Now we are talking about Dynamics on a Riemann surface; and from there, yes, you choose a projection of this riemann surface into the complex plane (you choose your branch of \(\log\)).
\[
f^{\circ t} (z) : S \to S\\
\]
Where we are sort of sending the preimages to themselves. Then we can only care about the Riemann surface itself; and in proving things about the Riemann surface we prove things about the iteration on the complex plane--and figure out how and where it is holomorphic if it could be.
For example if \(S = \{ y | y = F(z)\}\) then we define a coordinate \(s_0 \in S\) such that \(f(s_0)\) is the value of \(y\) for \(F(z+1)\). And now we are acting on the preimage. Then we perform iterations as such.
I haven't done much Riemann surface stuff in a while, but this is kind of what you are doing. This isn't very different from how you define the Riemann surface of \(\log\). It's just \(\{y | y = e^z\}\).
Now we are talking about Dynamics on a Riemann surface; and from there, yes, you choose a projection of this riemann surface into the complex plane (you choose your branch of \(\log\)).