07/13/2022, 06:47 PM
Lol, I apologize. I am rather loose with my language, so I'll try to be clearer.
The answer to your first question is no. I did mean in the codomain. So when I said we get a small domain, we usually get a super function \(F : \mathbb{C}_{\Re(z) > 0} \to \mathcal{U}\) where \(\mathcal{U}\) is small. I apologize, that's what I meant. I meant we can get a local solution, so that it's in a tiny domain.
For example take the Schroder iteration about \(|\lambda| < 1\) for a fixed point \(p\), and assume \(\lambda\) is complex valued. There exists a half plane \(\mathcal{H}\) and a neighborhood \(U\), such that we can easily construct \(F: \mathcal{H} \to U\). But as we try to pull back, extend \(\mathcal{H}\), we encounter loads of problems. So this is still what I'd refer to as a local solution, largely because:
\[
F(z) = p + \lambda^z \xi_0 + \mathcal{O}(\lambda^{2z})\\
\]
Is a very accurate approximation.
So when I mean local, I'm referring to the codomain primarily, but remember that \(F(z)\) is also a function of \(\xi_0\), and you can move \(\xi_0\) around, and it too usually lives in the codomain. So, when I refer to local solutions I mean: \(F(z,\xi) : \mathcal{H} \times U \to U\), and \(U\) is the "local" in a local iteration.
Then similarly, when I mean a global iteration, I mean \(F(z,\xi) : \mathcal{H} \times A \to A\) where \(A\) is a "large" set let's say. Now this will usually involve growing the domain \(\mathcal{H}\) to be larger than a half plane. For example with the Schroder iteration, just as above, if \(A\) is the immediate basin of \(p\), then \(F\) can always exist as above. It can actually be extended even further for \(\mathbb{C}/B\) for some branch cuts \(B\) (area measure zero). This is what I would call a global solution.
Another example of global would be kneser's tetration, because it's holomorphic almost everywhere on \(\mathbb{C}\), yet also having a huge codomain.
For transcendental entire functions \(f\), these ideas are fairly synonymous because any super function \(F\) defined for \(\mathbb{C}/B\) for some measure zero set \(B\), will send to a rather large domain. This is intrinsic to how the dynamics of transcendental entire functions behave. If \(f\) has a geometrically attracting fixed point at \(p\), then \(f^{-1}\) has a geometric repelling fixed point at \(p\), and is in the julia set. So as we take \(F(z-n)\), we are iterating \(f^{-1}\) about the Julia set, which is always crazy chaotic, and approaches infinity.
So the two ideas are fairly close "large domain in z" and "non local codomain".
I apologize for the confusion, I can get ahead of myself sometimes. I hope that clears things up?
The answer to your first question is no. I did mean in the codomain. So when I said we get a small domain, we usually get a super function \(F : \mathbb{C}_{\Re(z) > 0} \to \mathcal{U}\) where \(\mathcal{U}\) is small. I apologize, that's what I meant. I meant we can get a local solution, so that it's in a tiny domain.
For example take the Schroder iteration about \(|\lambda| < 1\) for a fixed point \(p\), and assume \(\lambda\) is complex valued. There exists a half plane \(\mathcal{H}\) and a neighborhood \(U\), such that we can easily construct \(F: \mathcal{H} \to U\). But as we try to pull back, extend \(\mathcal{H}\), we encounter loads of problems. So this is still what I'd refer to as a local solution, largely because:
\[
F(z) = p + \lambda^z \xi_0 + \mathcal{O}(\lambda^{2z})\\
\]
Is a very accurate approximation.
So when I mean local, I'm referring to the codomain primarily, but remember that \(F(z)\) is also a function of \(\xi_0\), and you can move \(\xi_0\) around, and it too usually lives in the codomain. So, when I refer to local solutions I mean: \(F(z,\xi) : \mathcal{H} \times U \to U\), and \(U\) is the "local" in a local iteration.
Then similarly, when I mean a global iteration, I mean \(F(z,\xi) : \mathcal{H} \times A \to A\) where \(A\) is a "large" set let's say. Now this will usually involve growing the domain \(\mathcal{H}\) to be larger than a half plane. For example with the Schroder iteration, just as above, if \(A\) is the immediate basin of \(p\), then \(F\) can always exist as above. It can actually be extended even further for \(\mathbb{C}/B\) for some branch cuts \(B\) (area measure zero). This is what I would call a global solution.
Another example of global would be kneser's tetration, because it's holomorphic almost everywhere on \(\mathbb{C}\), yet also having a huge codomain.
For transcendental entire functions \(f\), these ideas are fairly synonymous because any super function \(F\) defined for \(\mathbb{C}/B\) for some measure zero set \(B\), will send to a rather large domain. This is intrinsic to how the dynamics of transcendental entire functions behave. If \(f\) has a geometrically attracting fixed point at \(p\), then \(f^{-1}\) has a geometric repelling fixed point at \(p\), and is in the julia set. So as we take \(F(z-n)\), we are iterating \(f^{-1}\) about the Julia set, which is always crazy chaotic, and approaches infinity.
So the two ideas are fairly close "large domain in z" and "non local codomain".
I apologize for the confusion, I can get ahead of myself sometimes. I hope that clears things up?