Leo, If a function is holomorphic in a neighborhood of \(0\), then it is expandable in a taylor series at \(0\).
If a functions IS NOT holomorphic in a neighborhood of \(0\), then it IS NOT expandable in a taylor series at \(0\).
What Baker is saying, is that there is a divergent series at \(0\)--THEREFORE, it is not holomorphic in a neighborhood of \(0\).
Which is correct, there are two solutions \(g^+\) and \(g^-\). One is analytically continuable to \(\mathbb{C}/(-\infty,0]\) and the other for \(\mathbb{C}/[0,\infty)\). Both have the same asymptotic expansion at \(z=0\). WHICH IS A DIVERGENT SERIES, THEREFORE THE FUNCTION ISN'T HOLOMORPHIC AT ZERO.
As to your question on the construction of these functions, I would point to milnor's book Dynamics in One Complex Variable, which describes Ecalle's construction of Abel functions. Which further, displays the limits of where we can define Abel functions.
I don't have the ability to read off to you every piece of literature I've read, you'd just have to read the literature. We cannot have an iteration \(f^{\circ t}(z)\) which is holomorphic in a \(\epsilon\)-ball about a parabolic fixed point \(f(p) = p,\,f'(p) = e^{2\pi i \theta}\) (*). It just cannot happen. And it is a very deep result to describe, that I would only butcher.
(*) Unless it is an LFT-- by which it wouldn't be a Euclidean mapping
If a functions IS NOT holomorphic in a neighborhood of \(0\), then it IS NOT expandable in a taylor series at \(0\).
What Baker is saying, is that there is a divergent series at \(0\)--THEREFORE, it is not holomorphic in a neighborhood of \(0\).
Which is correct, there are two solutions \(g^+\) and \(g^-\). One is analytically continuable to \(\mathbb{C}/(-\infty,0]\) and the other for \(\mathbb{C}/[0,\infty)\). Both have the same asymptotic expansion at \(z=0\). WHICH IS A DIVERGENT SERIES, THEREFORE THE FUNCTION ISN'T HOLOMORPHIC AT ZERO.
As to your question on the construction of these functions, I would point to milnor's book Dynamics in One Complex Variable, which describes Ecalle's construction of Abel functions. Which further, displays the limits of where we can define Abel functions.
I don't have the ability to read off to you every piece of literature I've read, you'd just have to read the literature. We cannot have an iteration \(f^{\circ t}(z)\) which is holomorphic in a \(\epsilon\)-ball about a parabolic fixed point \(f(p) = p,\,f'(p) = e^{2\pi i \theta}\) (*). It just cannot happen. And it is a very deep result to describe, that I would only butcher.
(*) Unless it is an LFT-- by which it wouldn't be a Euclidean mapping
