Hey, Leo.
I'm disagreeing with you--and until I see concrete evidence I will continue to do so. This is not the function in question at all. You are talking about a different function entirely.
You have developed a Taylor series about \(0\). This is a question about the divergent series that Gottfried has developed. You have made a function \(g\) that is holomorphic near zero (which I'm not sure about, as I'm pretty sure this contradicts Baker--Which Gottfried mentions). There exists no half iterate that is holomorphic near zero. So you've made a mistake, or Baker has made a mistake. We cannot expand an iteration in a neighborhood of a parabolic fixed point, unless that function is an LFT. Bo and myself even saw this recently in the Karlin Mcgregor paper, where it is written in the beginning.
By your logic, we've now solved \(f(z) = e^{z}-1\) and found a function \(f^{\circ t}(z)\) which is holomorphic about \(z = 0\). That just doesn't happen. Because you are brushing the paths of the Julia set and the attracting/repelling petal. The abel function cannot be expanded in a neighborhood of zero. And through common transformations, we can turn your iteration into an abel function, and now we have an abel function which is holomorphic for \(z \neq 0\) and \(|z| < \delta\). That doesn't happen. Abel functions are not holomorphic like this at parabolic points.
I can't speak to the accuracy of this graph, because I do not understand how you are constructing it. Additionally, a graph doesn't mean holomorphy. Just because it looks holomorphic, doesn't make it holomorphic. What is far more likely, is that you have expanded an asymptotic series, that converges fairly well, and iterated functional relationships to express how close of an asymptotic it is.
This is equivalent to saying that you've found a holomorphic iteration \(\exp^{\circ t}_{\eta}(z)\) such that this expression is holomorphic in \(z\) about \(e\). It's well known this isn't possible.
I apologize, but until I see some form of hard evidence I'll continue to disagree with you. Baker himself has a paper about no Taylor expansion for a half iterate of \(e^z -1\) about \(z=0\). Straight from the horses mouth. See the top answer here https://mathoverflow.net/questions/4347/...ar-and-exp
So unless you want to contradict baker, I'm sorry.
I'm disagreeing with you--and until I see concrete evidence I will continue to do so. This is not the function in question at all. You are talking about a different function entirely.
You have developed a Taylor series about \(0\). This is a question about the divergent series that Gottfried has developed. You have made a function \(g\) that is holomorphic near zero (which I'm not sure about, as I'm pretty sure this contradicts Baker--Which Gottfried mentions). There exists no half iterate that is holomorphic near zero. So you've made a mistake, or Baker has made a mistake. We cannot expand an iteration in a neighborhood of a parabolic fixed point, unless that function is an LFT. Bo and myself even saw this recently in the Karlin Mcgregor paper, where it is written in the beginning.
By your logic, we've now solved \(f(z) = e^{z}-1\) and found a function \(f^{\circ t}(z)\) which is holomorphic about \(z = 0\). That just doesn't happen. Because you are brushing the paths of the Julia set and the attracting/repelling petal. The abel function cannot be expanded in a neighborhood of zero. And through common transformations, we can turn your iteration into an abel function, and now we have an abel function which is holomorphic for \(z \neq 0\) and \(|z| < \delta\). That doesn't happen. Abel functions are not holomorphic like this at parabolic points.
I can't speak to the accuracy of this graph, because I do not understand how you are constructing it. Additionally, a graph doesn't mean holomorphy. Just because it looks holomorphic, doesn't make it holomorphic. What is far more likely, is that you have expanded an asymptotic series, that converges fairly well, and iterated functional relationships to express how close of an asymptotic it is.
This is equivalent to saying that you've found a holomorphic iteration \(\exp^{\circ t}_{\eta}(z)\) such that this expression is holomorphic in \(z\) about \(e\). It's well known this isn't possible.
I apologize, but until I see some form of hard evidence I'll continue to disagree with you. Baker himself has a paper about no Taylor expansion for a half iterate of \(e^z -1\) about \(z=0\). Straight from the horses mouth. See the top answer here https://mathoverflow.net/questions/4347/...ar-and-exp
So unless you want to contradict baker, I'm sorry.
