07/01/2022, 10:35 PM
(07/01/2022, 10:23 PM)tommy1729 Wrote: like 2 black holes tearing up the space between them.
well i was poetic ...
Lmao. It is sort of like two black holes, the orbits become untenable somewhere between the two and cause a short circuit.
Yeah, I've encountered this proof by contradiction so many times trying to do interesting iterations. Fixed points really screw up everything, it's god damned frustrating. The central problem, is you can't move from Julia to Fatou, or Fatou to Julia. It just... breaks down.
You could create a super function though... Not sure what it would look like. So if you were to fix \(z\), and only care about \(s\) this can leave the Julia set/fatou set, but you lose the semi-group property, and only have the super function identity. It kind of quarantines the errors to the abel function (which would allow you to move \(z\)).
I wouldn't give up yet though, iterating \(z + \theta(z)\) with a fixed point at \(\infty\) in the lower and upper half planes is very viable. but then you'd have to restrict to the upper half plane or lower half plane in \(z\). That would probably look pretty weird though. Then you may be able to at least have continuous at the fixed points on \(\mathbb{R}\)...
I didn't mean to discourage you, just that local fixed point iteration is doomed here.
