Hmmmmmmmmmm
I think we can get a little trip up if we think of regular iteration, so I thought I'd do a run through of the non-neutral case to clarify things.
To clarify things, Bo.
The regular iteration for \(b \approx \eta_- + 0.1\) is not real valued, I presume we can agree. It has a fixed point \(p\) and a multiplier \(-1 < c < 0\). Therefore the regular iteration will be
\[
f^{t}(z) = p + c^t(z-p) + O(z-p)^2\\
\]
Now, the beta tetration can construct this. We do this by setting \(2 \pi i/\lambda = -2 \pi i/\log c\). So in a sense, if we match the period of the beta iteration to the period of the regular iteration, then the beta iteration is just the regular iteration.
I liked to phrase this as tetration is unique upto its period. Any two tetrations \(F_1,F_2\) which have the same period, must be the same. So since you can create a beta iteration with this period, it must be the regular iteration. This is a pretty deep theorem, which I had to brush paths with elliptic function theory to properly describe.
BUT!
If I keep \(\lambda > 0\) and real valued, for example, if I set \(\lambda = 1\), then we can make the beta super function which is now real valued. This will make a similar graph as with \(\eta_-\), but now it will be within the interior of Shell Thron.
So, the beta iteration can make a real valued iteration with a purely imaginary period; as opposed to the complex period of the regular iteration--which makes the regular iteration complex valued.
EDIT:
Also, if you want to think of this in terms of \(\theta\) mappings, we can think of it like this.
The regular iteration looks like:
\[
f^{\circ t}(z) = \Psi^{-1}(c^{t}\Psi^{-1}(z))\\
\]
Then what we want is a theta mapping \(\theta(t+1) = \theta(t)\) but additionally:
\[
\theta(t + 2 \pi i/\lambda) = \theta(t)- 2\pi i/\lambda\\
\]
The theta I used is holomorphic almost everywhere on \(\mathbb{C}\), so there are singularities/branch cuts... so remember that.
Then the beta iteration looks like:
\[
f_\beta^{\circ t}(z) = \Psi^{-1}(c^{t + \theta(t)}\Psi^{-1}(z))\\
\]
So that now it is \(2 \pi i/\lambda\) periodic. Obviously this is only a super function, it is not a fractional iteration in any sense because it doesn't satisfy \(f_\beta^{t_0}(f_\beta^{t_1}) = f_\beta^{t_0+t_1}\)--it only satisfies the super function identity.
This plays pretty close to elliptic functions, and you can actually convert \(\theta\) into something that looks nearly like an elliptic function.
In this sense, we can also think of instead of perturbing the base, or perturbing the multiplier--we are perturbing the period.
EDIT:
For the neutral case, the theta mapping has to be looked at like a mapping on the inverse abel function. So take \(b = \eta\), for example, which has a holomorphic super function for \(\mathbb{C}/(-\infty,-2]\). Then we find an almost elliptic mapping \(g\) such that:
\[
\begin{align}
g(z+1) &= g(z)+1\\
g(z+2\pi i/\lambda) &= g(z)\\
\end{align}
\]
Then taking \(\eta \uparrow \uparrow g(z)\), we have the beta iteration at \(\eta\). This is valid so long as \(\Re \lambda >0\).
TO ALSO CLARIFY! THERE IS NO MEROMORPHIC FUNCTION \(g\) WHICH SATISFIES THESE EQUATIONS. BUT, big but, we can have a holomorphic \(g\) which is holomorphic on \(\mathbb{C}/B\) such that:
\[
\int \int_{B}\, dxdy = \int_{B} \,dA = 0\\
\]
So that \(g\) is holomorphic almost everywhere on a Lebesgue area measure; the lebesgue measure in \(\mathbb{R}^2\). This can equivalently be said that the boundary of \(\mathbb{C}/B\) is \(B\).
Fuck! This is hard to get everything out, because I have to summarize so much from my report on the beta method. But we do not get a nice tetration \(F\) for \(\eta_-\) such that \(F : (-2,\infty) \to (-\infty, 1/e)\). We get a far stranger construction. But in the complex plane it still equals tetration almost everywhere.
I think we can get a little trip up if we think of regular iteration, so I thought I'd do a run through of the non-neutral case to clarify things.
To clarify things, Bo.
The regular iteration for \(b \approx \eta_- + 0.1\) is not real valued, I presume we can agree. It has a fixed point \(p\) and a multiplier \(-1 < c < 0\). Therefore the regular iteration will be
\[
f^{t}(z) = p + c^t(z-p) + O(z-p)^2\\
\]
Now, the beta tetration can construct this. We do this by setting \(2 \pi i/\lambda = -2 \pi i/\log c\). So in a sense, if we match the period of the beta iteration to the period of the regular iteration, then the beta iteration is just the regular iteration.
I liked to phrase this as tetration is unique upto its period. Any two tetrations \(F_1,F_2\) which have the same period, must be the same. So since you can create a beta iteration with this period, it must be the regular iteration. This is a pretty deep theorem, which I had to brush paths with elliptic function theory to properly describe.
BUT!
If I keep \(\lambda > 0\) and real valued, for example, if I set \(\lambda = 1\), then we can make the beta super function which is now real valued. This will make a similar graph as with \(\eta_-\), but now it will be within the interior of Shell Thron.
So, the beta iteration can make a real valued iteration with a purely imaginary period; as opposed to the complex period of the regular iteration--which makes the regular iteration complex valued.
EDIT:
Also, if you want to think of this in terms of \(\theta\) mappings, we can think of it like this.
The regular iteration looks like:
\[
f^{\circ t}(z) = \Psi^{-1}(c^{t}\Psi^{-1}(z))\\
\]
Then what we want is a theta mapping \(\theta(t+1) = \theta(t)\) but additionally:
\[
\theta(t + 2 \pi i/\lambda) = \theta(t)- 2\pi i/\lambda\\
\]
The theta I used is holomorphic almost everywhere on \(\mathbb{C}\), so there are singularities/branch cuts... so remember that.
Then the beta iteration looks like:
\[
f_\beta^{\circ t}(z) = \Psi^{-1}(c^{t + \theta(t)}\Psi^{-1}(z))\\
\]
So that now it is \(2 \pi i/\lambda\) periodic. Obviously this is only a super function, it is not a fractional iteration in any sense because it doesn't satisfy \(f_\beta^{t_0}(f_\beta^{t_1}) = f_\beta^{t_0+t_1}\)--it only satisfies the super function identity.
This plays pretty close to elliptic functions, and you can actually convert \(\theta\) into something that looks nearly like an elliptic function.
In this sense, we can also think of instead of perturbing the base, or perturbing the multiplier--we are perturbing the period.
EDIT:
For the neutral case, the theta mapping has to be looked at like a mapping on the inverse abel function. So take \(b = \eta\), for example, which has a holomorphic super function for \(\mathbb{C}/(-\infty,-2]\). Then we find an almost elliptic mapping \(g\) such that:
\[
\begin{align}
g(z+1) &= g(z)+1\\
g(z+2\pi i/\lambda) &= g(z)\\
\end{align}
\]
Then taking \(\eta \uparrow \uparrow g(z)\), we have the beta iteration at \(\eta\). This is valid so long as \(\Re \lambda >0\).
TO ALSO CLARIFY! THERE IS NO MEROMORPHIC FUNCTION \(g\) WHICH SATISFIES THESE EQUATIONS. BUT, big but, we can have a holomorphic \(g\) which is holomorphic on \(\mathbb{C}/B\) such that:
\[
\int \int_{B}\, dxdy = \int_{B} \,dA = 0\\
\]
So that \(g\) is holomorphic almost everywhere on a Lebesgue area measure; the lebesgue measure in \(\mathbb{R}^2\). This can equivalently be said that the boundary of \(\mathbb{C}/B\) is \(B\).
Fuck! This is hard to get everything out, because I have to summarize so much from my report on the beta method. But we do not get a nice tetration \(F\) for \(\eta_-\) such that \(F : (-2,\infty) \to (-\infty, 1/e)\). We get a far stranger construction. But in the complex plane it still equals tetration almost everywhere.
