Iteration with two analytic fixed points

[Daniel]

Sorry, but what are the definitions for local and regular iteration?

As I've defined Local iteration, is a little sketchy; but it is essentially the idea that the iterate \(f^{\circ t}(z)\) can be split into:

\[
f^{\circ t}(z) : D \times A \to A\\
\]

For some fixed domains \(t \in D\) and \(z\in A\). This only ever happens "locally" or what we have a Schroder function. It can't happen anywhere else; and then, it only happens in a half plane and near the fixed point.

Regular iteration is much more general, and means to solve the Taylor expansion similar to what you've done, Daniel. We are just solving a problem with the coefficients of the \(N\)'th order polynomial. Additionally Regular iteration handles pretty much everything about a fixed point. To remind you, much of your Matrix equations are actually just producing regular iteration.

Local iteration is a term I am arguing for; in that it means that we can split the domains like I wrote above. And they are still "regular iterations" but they are specific kinds of regular iterations.

Hope that helps

Awesome, thanks for the info.