Hey, Tommy
I haven't fully digested this. But let me see if I get you straight.
Let's write:
\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Now, to make your example concrete, let's put some domains in question. Let's let \(\Im(z) > 0\), where we assume some kind of convergence on \(z \in \mathbb{R}\), and in an analytic manner. Now, let's play your trick, but slightly different. As is always fun, it's better to stick to what we know first. So let's let \(f(z) = \log_{\sqrt{2}}(z+4)-4\) (I promise you, this is going some where). Now, famously, this value tends to \(0\), and for most of the upper half plane does so. By which:
\[
A = \sum_{k=0}^\infty c^k \left(\log_{\sqrt{2}}^{\circ k}(z+4)-4\right)
\]
The values \(\log_{\sqrt{2}}^{\circ k}(z+4)-4 = O(1/2\log(2))^k\)
This value converges for all \(0 < |c| < 2\log(2) > 1\). Now, let's take the negative:
\[
B = \sum_{k=1}^\infty c^{-k} \left(\exp^{\circ k}_{\sqrt{2}}(z+4) - 4\right)\\
\]
Which looks like:
\[
\exp^{\circ k}_{\sqrt{2}}(z+4) - 4 = -2 + O(\log(2)^k)\\
\]
Therefore, this object converges for \(|c| >1\). Therefore in the tiny annulus, of \(1 < |c| < 2\log(2)\) we've constructed:
\[
F(c,z)
\]
Such that:
\[
F(c,\log_{\sqrt{2}}(z+4)-4) = F(c,z)/c\\
\]
And \(z\) is fairly well behaved, it's at least convergent on the interval \((2,4)\), but I suspect much much larger--I can write the exact domains for you if you'd like. It's the \(\beta\) function
...
NOW! I went into this confidently because I knew it'd converge. Because it is the statement of Levenstein's equation in my paper on the beta method. We can actually solve many Weird and Strange Schroder equations. The only unique one, is the one with the multiplier about a fixed point.
For example, I've solved the Schroder equation here, so long as \(1 < |c| < 2\log 2\).
Normally I would write this as follows:
\[
\varphi_\lambda(e^{\lambda}z) = f(\varphi_\lambda(z))\\
\]
Whereby, \(\Re(\lambda) > \log\log(4)\). This is the same thing you've written and discovered. Though we need to do a fair amount of variable changes. I do have uniqueness conditions on the various types of Schroder functions, and what can and can't be a Schroder function. I believe I called it Levenstein's theorem (Or the generalized Levenstein theorem, in my paper). Which is essentially a statement that this thing can only be holomorphic in the annulus \(1 < |c| < 2 \log 2\).
WHAT I WILL SAY IS THIS IS A CRAZY REPRESENTATION! I've tried similar things, but I guess it never clicked. So congratulations Tommy, you've rediscovered a lot of what Levenstein was writing about about the beta method, and the reconstruction of Schroder functions!!!! And you've got a cool expansion to boot!
I had to use infinite compositions to drag out this result; you've done it with sums. I feel embarassed
Also, I opted to calling this Levenstein's equation; as he ran so much code, and really brute forced everything. And determined, for example with:
\[
f(z) = \log_\sqrt{2}(z)\\
\]
That we can only make a Schroder equation with multiplier \(0 < |c| < \frac{1}{2\log(2)}\). You may have heard me talk about how we can "shrink the period of the Schroder iteration, but we can't grow the period". This is the exact same thing.
But this is an awesome formula for the "Inverse Levenstein Equation" (Levenstein and I worked solely on the "inverse Schroder"). It's the exact same thing though.
AWESOME WORK, TOMMY!
I haven't fully digested this. But let me see if I get you straight.
Let's write:
\[
F(c,z) = \sum_{k=-\infty}^\infty c^k f^{\circ k}(z)\\
\]
Now, to make your example concrete, let's put some domains in question. Let's let \(\Im(z) > 0\), where we assume some kind of convergence on \(z \in \mathbb{R}\), and in an analytic manner. Now, let's play your trick, but slightly different. As is always fun, it's better to stick to what we know first. So let's let \(f(z) = \log_{\sqrt{2}}(z+4)-4\) (I promise you, this is going some where). Now, famously, this value tends to \(0\), and for most of the upper half plane does so. By which:
\[
A = \sum_{k=0}^\infty c^k \left(\log_{\sqrt{2}}^{\circ k}(z+4)-4\right)
\]
The values \(\log_{\sqrt{2}}^{\circ k}(z+4)-4 = O(1/2\log(2))^k\)
This value converges for all \(0 < |c| < 2\log(2) > 1\). Now, let's take the negative:
\[
B = \sum_{k=1}^\infty c^{-k} \left(\exp^{\circ k}_{\sqrt{2}}(z+4) - 4\right)\\
\]
Which looks like:
\[
\exp^{\circ k}_{\sqrt{2}}(z+4) - 4 = -2 + O(\log(2)^k)\\
\]
Therefore, this object converges for \(|c| >1\). Therefore in the tiny annulus, of \(1 < |c| < 2\log(2)\) we've constructed:
\[
F(c,z)
\]
Such that:
\[
F(c,\log_{\sqrt{2}}(z+4)-4) = F(c,z)/c\\
\]
And \(z\) is fairly well behaved, it's at least convergent on the interval \((2,4)\), but I suspect much much larger--I can write the exact domains for you if you'd like. It's the \(\beta\) function
...NOW! I went into this confidently because I knew it'd converge. Because it is the statement of Levenstein's equation in my paper on the beta method. We can actually solve many Weird and Strange Schroder equations. The only unique one, is the one with the multiplier about a fixed point.
For example, I've solved the Schroder equation here, so long as \(1 < |c| < 2\log 2\).
Normally I would write this as follows:
\[
\varphi_\lambda(e^{\lambda}z) = f(\varphi_\lambda(z))\\
\]
Whereby, \(\Re(\lambda) > \log\log(4)\). This is the same thing you've written and discovered. Though we need to do a fair amount of variable changes. I do have uniqueness conditions on the various types of Schroder functions, and what can and can't be a Schroder function. I believe I called it Levenstein's theorem (Or the generalized Levenstein theorem, in my paper). Which is essentially a statement that this thing can only be holomorphic in the annulus \(1 < |c| < 2 \log 2\).
WHAT I WILL SAY IS THIS IS A CRAZY REPRESENTATION! I've tried similar things, but I guess it never clicked. So congratulations Tommy, you've rediscovered a lot of what Levenstein was writing about about the beta method, and the reconstruction of Schroder functions!!!! And you've got a cool expansion to boot!
I had to use infinite compositions to drag out this result; you've done it with sums. I feel embarassed
Also, I opted to calling this Levenstein's equation; as he ran so much code, and really brute forced everything. And determined, for example with:
\[
f(z) = \log_\sqrt{2}(z)\\
\]
That we can only make a Schroder equation with multiplier \(0 < |c| < \frac{1}{2\log(2)}\). You may have heard me talk about how we can "shrink the period of the Schroder iteration, but we can't grow the period". This is the exact same thing.
But this is an awesome formula for the "Inverse Levenstein Equation" (Levenstein and I worked solely on the "inverse Schroder"). It's the exact same thing though.
AWESOME WORK, TOMMY!