Okay, I'm going to approach this a tad more level headed, because I truly believe Tommy is on to something. And I have a sneaking suspicion this analytically continues the Levenstein Schroder function (which is rampant in Sheldon's code), to a larger domain. Or we have some kind of discontinuity when connecting them. If this is an analytic continuation, I think it's absolutely pivotal.
I'm going to change some language here, and start from scratch. So to begin, we are going to fix \(|c| > 1\). We are going to let \(f(z) = \sqrt{2}^{z+2}-2\)--and then we're going to write the first expansion, Tommy has demonstrated.
\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Now, for convenience, assume that \(z \in (0,2)\), or exists in some small neighborhood of this line, without intersecting \((-\infty,0)\) or \((2,\infty)\), which are respective julia sets of \(f^{-1}\) and \(f\). We know that:
\[
f^{\circ k} = O(\log 2 ^k)
\]
For \(z\) in the attracting basin of \(f(0) = 0\). (which the domain I mentioned belonged to). And we know that:
\[
f^{-1}(z) = \log_{\sqrt{2}}(z+2) - 2\\
\]
So that \(f^{-1}\) has an attracting fixed point at \(2\), which looks like \(f^{-1}(z) = 2 + \frac{z-2}{2\log(2)} + O ((z-2)^2)\); and therefore it is attracting at a geometric rate \(\frac{1}{(2\log(2))^k}\) to the fixed point \(2\). Therefore the series:
\[
\sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Is holomorphic for \(1 < |c| < \frac{1}{\log(2)}\) and at least a neighbhorhood of each point \(z \in (0,2)\). This constructs, what I will call the Raes coordinate, or Raes expansion.
Now I want to take a look at something a bit different and show a striking resemblence.
Sheldon was the first to do this change of variables, and utilized it greatly when constructing his program to evaluate the \(\beta\) method. It's an aesthetic change more than anything, but still greatly valuable. Let's write:
\[
g_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\bullet z\\
\]
This is the unique solution to the equation:
\[
g_c(cw) = \frac{wf(g_c(w))}{w+1}\\
\]
Now, if we take:
\[
f^{\circ -k} (g_c(c^k w)) = \Psi_k(c,w)\\
\]
Then this object only converges for \(|c| > \frac{1}{\log(2)}\). And the limit function satisfies:
\[
\Psi(c,cw) = f(\Psi(c,w))\\
\]
This absolutely DOES NOT have an obvious inverse function in the neighborhood of a fixed point. Iconically, it is solving the inverse schroder equation, but with an arbitrary multiplier (though the domains are inverted from Tommy's version).
So, in my humble opinion. We have a GREAT new result. Which I will walk through.
If \(f\) is holomorphic at zero, with a geometric fixed point here; and in the immediate basin \(A_0\), the iterates \(f^{\circ -k} \to L\), another fixed point (\(L\) can be infinite, but that'll require more bounding stuff)--then we can construct a Raes function \(R(z)\) such that:
\[
R(f(z)) = cR(z)\\
\]
If and only if \(1 < |c| < \frac{1}{|f'(0)|}\)
Now, to contrast this with the Levenstein function.... If \(f\) is holomorphic at zero, with a geometric fixed point here--then we can construct a Levenstein function \(L(z)\), such that:
\[
f(L(z)) = L(cz)\\
\]
If and only if \(|c| > \frac{1}{|f'(0)|}\).
The question then is, if we write \(R(c,z)\) and \(L(c,z)\)... IS \(R(c,z)\) an analytic continuation of \(L^{-1}(c,z)\). I'm willing to be this is true. Where a bunch of singularities arrive on the circle \(|c| = \frac{1}{|f'(0)|}\).
To get a good idea what's happening here, let's switch from Levenstein, to Beta. So let's write \(c = e^{\lambda}\), where now Raes' function looks like:
\[
R_\lambda(z) = \sum_{k=-\infty}^\infty e^{\lambda k} f^{\circ k}(z)\\
\]
Where now, \(R_\lambda(f(z)) = e^\lambda R_\lambda(z)\), and:
\[
L_\lambda(e^\lambda z) = f(L_\lambda(z))\\
\]
Where then, if we write:
\[
F_\lambda(s) = L_\lambda(e^{\lambda s})\\
\]
We have:
\[
F_\lambda(s+1) = f(F_\lambda(s))\\
\]
Where this is holomorphic for \(\Re\lambda> - \log\log 2\).
If we write:
\[
F_\lambda(s) = R^{-1}_\lambda(e^{\lambda s})\\
\]
Then this is holomorphic for \(0 < \Re \lambda < -\log\log 2\).
This then allows us to say...
There exists a holomorphic tetration \(F_\lambda(s)\) for \(\Re \lambda \neq -\log\log 2\) and \(\Re\lambda > 0\). Which means, not only can we shrink the period with the beta method... we can grow it!
HOLY FUCK NICE JOB TOMMY!!!!!!!!
These things should definitely be continuations of each other, but I'm not seeing at the moment how. Things can get tricky, because we need to identify where the singularities of the Raes function are. It's no surprise that \(F_\lambda(s)\) should have singularities at \(2\pi i/\lambda + j\) for \(\Re(\lambda) > -\log\log 2\). But this would require a change of variables in some way shape or form to map where the singularities are for the Raes function. This will have something to do with theta mappings, but for the life of myself I can't see it obviously.
Quite clearly we need to show there's a theta function \(\theta_\lambda\), which is holomorphic for \(\Re\lambda > 0\) with singularities somewhere on \(\Re \lambda = -\log \log 2\). And show that, using the normal Schroder function:
\[
F_\lambda(z) = \Psi^{-1}\left( e^{\log\log 2( z + \theta_\lambda(z))}\right)\\
\]
Where \(\theta_\lambda(z+1) = \theta_\lambda(z)\) and \(\theta_\lambda(z+2\pi i/\lambda) = \theta_\lambda(z) - 2\pi i/\lambda\).
Jesus, this looks like it's gonna be a headache
....
I think Elliptic functions is going to save us here! But I need to dust off some old books. Essentially, I think I can prove Tommy's theta function for his Raes function is the same as the beta function UPTO an elliptic function. Finding the elliptic function is gonna be a bitch and a half. But I think I see how we might get it. Fuck!!!
This is one for the books Tommy!!!! You've successfully constructed a function \(F_\lambda(z)\) such that \(\sqrt{2}^{F_\lambda(z)} = F_\lambda(z+1)\) and \( 0 < \Re \lambda < -\log \log 2\). It is real valued for \(0 < \lambda < -\log \log 2\) and it has period \(2 \pi i / \lambda\). And additionally, it is highly non trivial! Have a glass of champagne on me!
!!!!!
OH FUCK! I SEE IT NOW!!!
So I forgot a negative sign! We can prove absolutely that Raes coordinate is an analytic continuation of the Levenstein coordinate. This is really nuanced though, and you gotta let me work through some shit! YES! We can have \(F_\lambda(s)\) for \(\Re(\lambda) > 0\) with singularities only on the line \(\Re\lambda = -\log\log 2\). I forgot that \(R_\lambda(f(z)) = e^{-\lambda}R_\lambda(z)\) where we use \(1/c = e^{-\lambda}\) rather than \(c = e^{\lambda}\)--this changes everything. Much of the discussion is still pretty good, but I forgot a whole layer of complexity
I'm going to change some language here, and start from scratch. So to begin, we are going to fix \(|c| > 1\). We are going to let \(f(z) = \sqrt{2}^{z+2}-2\)--and then we're going to write the first expansion, Tommy has demonstrated.
\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Now, for convenience, assume that \(z \in (0,2)\), or exists in some small neighborhood of this line, without intersecting \((-\infty,0)\) or \((2,\infty)\), which are respective julia sets of \(f^{-1}\) and \(f\). We know that:
\[
f^{\circ k} = O(\log 2 ^k)
\]
For \(z\) in the attracting basin of \(f(0) = 0\). (which the domain I mentioned belonged to). And we know that:
\[
f^{-1}(z) = \log_{\sqrt{2}}(z+2) - 2\\
\]
So that \(f^{-1}\) has an attracting fixed point at \(2\), which looks like \(f^{-1}(z) = 2 + \frac{z-2}{2\log(2)} + O ((z-2)^2)\); and therefore it is attracting at a geometric rate \(\frac{1}{(2\log(2))^k}\) to the fixed point \(2\). Therefore the series:
\[
\sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Is holomorphic for \(1 < |c| < \frac{1}{\log(2)}\) and at least a neighbhorhood of each point \(z \in (0,2)\). This constructs, what I will call the Raes coordinate, or Raes expansion.
Now I want to take a look at something a bit different and show a striking resemblence.
Sheldon was the first to do this change of variables, and utilized it greatly when constructing his program to evaluate the \(\beta\) method. It's an aesthetic change more than anything, but still greatly valuable. Let's write:
\[
g_c(w) = \Omega_{j=1}^\infty \frac{w f(z)}{w+c^j}\bullet z\\
\]
This is the unique solution to the equation:
\[
g_c(cw) = \frac{wf(g_c(w))}{w+1}\\
\]
Now, if we take:
\[
f^{\circ -k} (g_c(c^k w)) = \Psi_k(c,w)\\
\]
Then this object only converges for \(|c| > \frac{1}{\log(2)}\). And the limit function satisfies:
\[
\Psi(c,cw) = f(\Psi(c,w))\\
\]
This absolutely DOES NOT have an obvious inverse function in the neighborhood of a fixed point. Iconically, it is solving the inverse schroder equation, but with an arbitrary multiplier (though the domains are inverted from Tommy's version).
So, in my humble opinion. We have a GREAT new result. Which I will walk through.
If \(f\) is holomorphic at zero, with a geometric fixed point here; and in the immediate basin \(A_0\), the iterates \(f^{\circ -k} \to L\), another fixed point (\(L\) can be infinite, but that'll require more bounding stuff)--then we can construct a Raes function \(R(z)\) such that:
\[
R(f(z)) = cR(z)\\
\]
If and only if \(1 < |c| < \frac{1}{|f'(0)|}\)
Now, to contrast this with the Levenstein function.... If \(f\) is holomorphic at zero, with a geometric fixed point here--then we can construct a Levenstein function \(L(z)\), such that:
\[
f(L(z)) = L(cz)\\
\]
If and only if \(|c| > \frac{1}{|f'(0)|}\).
The question then is, if we write \(R(c,z)\) and \(L(c,z)\)... IS \(R(c,z)\) an analytic continuation of \(L^{-1}(c,z)\). I'm willing to be this is true. Where a bunch of singularities arrive on the circle \(|c| = \frac{1}{|f'(0)|}\).
To get a good idea what's happening here, let's switch from Levenstein, to Beta. So let's write \(c = e^{\lambda}\), where now Raes' function looks like:
\[
R_\lambda(z) = \sum_{k=-\infty}^\infty e^{\lambda k} f^{\circ k}(z)\\
\]
Where now, \(R_\lambda(f(z)) = e^\lambda R_\lambda(z)\), and:
\[
L_\lambda(e^\lambda z) = f(L_\lambda(z))\\
\]
Where then, if we write:
\[
F_\lambda(s) = L_\lambda(e^{\lambda s})\\
\]
We have:
\[
F_\lambda(s+1) = f(F_\lambda(s))\\
\]
Where this is holomorphic for \(\Re\lambda> - \log\log 2\).
If we write:
\[
F_\lambda(s) = R^{-1}_\lambda(e^{\lambda s})\\
\]
Then this is holomorphic for \(0 < \Re \lambda < -\log\log 2\).
This then allows us to say...
There exists a holomorphic tetration \(F_\lambda(s)\) for \(\Re \lambda \neq -\log\log 2\) and \(\Re\lambda > 0\). Which means, not only can we shrink the period with the beta method... we can grow it!
HOLY FUCK NICE JOB TOMMY!!!!!!!!
These things should definitely be continuations of each other, but I'm not seeing at the moment how. Things can get tricky, because we need to identify where the singularities of the Raes function are. It's no surprise that \(F_\lambda(s)\) should have singularities at \(2\pi i/\lambda + j\) for \(\Re(\lambda) > -\log\log 2\). But this would require a change of variables in some way shape or form to map where the singularities are for the Raes function. This will have something to do with theta mappings, but for the life of myself I can't see it obviously.
Quite clearly we need to show there's a theta function \(\theta_\lambda\), which is holomorphic for \(\Re\lambda > 0\) with singularities somewhere on \(\Re \lambda = -\log \log 2\). And show that, using the normal Schroder function:
\[
F_\lambda(z) = \Psi^{-1}\left( e^{\log\log 2( z + \theta_\lambda(z))}\right)\\
\]
Where \(\theta_\lambda(z+1) = \theta_\lambda(z)\) and \(\theta_\lambda(z+2\pi i/\lambda) = \theta_\lambda(z) - 2\pi i/\lambda\).
Jesus, this looks like it's gonna be a headache
....I think Elliptic functions is going to save us here! But I need to dust off some old books. Essentially, I think I can prove Tommy's theta function for his Raes function is the same as the beta function UPTO an elliptic function. Finding the elliptic function is gonna be a bitch and a half. But I think I see how we might get it. Fuck!!!
This is one for the books Tommy!!!! You've successfully constructed a function \(F_\lambda(z)\) such that \(\sqrt{2}^{F_\lambda(z)} = F_\lambda(z+1)\) and \( 0 < \Re \lambda < -\log \log 2\). It is real valued for \(0 < \lambda < -\log \log 2\) and it has period \(2 \pi i / \lambda\). And additionally, it is highly non trivial! Have a glass of champagne on me!
!!!!!OH FUCK! I SEE IT NOW!!!
So I forgot a negative sign! We can prove absolutely that Raes coordinate is an analytic continuation of the Levenstein coordinate. This is really nuanced though, and you gotta let me work through some shit! YES! We can have \(F_\lambda(s)\) for \(\Re(\lambda) > 0\) with singularities only on the line \(\Re\lambda = -\log\log 2\). I forgot that \(R_\lambda(f(z)) = e^{-\lambda}R_\lambda(z)\) where we use \(1/c = e^{-\lambda}\) rather than \(c = e^{\lambda}\)--this changes everything. Much of the discussion is still pretty good, but I forgot a whole layer of complexity