https://math.eretrandre.org/tetrationfor...416-1.html
Assuming this is what you are talking about with the Leau-Fatou petal stuff.
Woah cool, I've never seen this before. I assumed when you wrote that, you were referring to Ecalle, but how does Ecalle work for \(b > \eta\)? I didn't realize the idea is to perturb from \(b = \eta\) and watch the petals deform. I'll have to look into that. Just being quick, do you have a link to any articles involved, because I couldn't find anything on that thread.
If what you are saying is that there exists a holomorphic iteration \(1 < b < \infty\), without the usual branching/spike at \(b = \eta\), that would be really cool. I've always reserved myself into believing that Tetration naturally has a branching problem at \(b = \eta\), to avoid that would be god like!
As not to derail this thread further, I'll refrain from pursuing this further here. Thanks though, never heard of this method before!
EDIT:
Aww damnnit, I read through further in that post and I see the statement is still correct that no such analytic continuation exists which is holomorphic at \(b=\eta\), to think otherwise would be detrimental to the foundations of how I've done complex dynamics, so there's at least that. That would've saved me a lot of headaches though. Oh well, back to continuing my current drawing board.
Assuming this is what you are talking about with the Leau-Fatou petal stuff.
Woah cool, I've never seen this before. I assumed when you wrote that, you were referring to Ecalle, but how does Ecalle work for \(b > \eta\)? I didn't realize the idea is to perturb from \(b = \eta\) and watch the petals deform. I'll have to look into that. Just being quick, do you have a link to any articles involved, because I couldn't find anything on that thread.
If what you are saying is that there exists a holomorphic iteration \(1 < b < \infty\), without the usual branching/spike at \(b = \eta\), that would be really cool. I've always reserved myself into believing that Tetration naturally has a branching problem at \(b = \eta\), to avoid that would be god like!
As not to derail this thread further, I'll refrain from pursuing this further here. Thanks though, never heard of this method before!
EDIT:
Aww damnnit, I read through further in that post and I see the statement is still correct that no such analytic continuation exists which is holomorphic at \(b=\eta\), to think otherwise would be detrimental to the foundations of how I've done complex dynamics, so there's at least that. That would've saved me a lot of headaches though. Oh well, back to continuing my current drawing board.