Okay, couple things to clear up. I probably should've been clearer. By local iteration, yes I mean the regular iteration, but I sort of mean, only to be considered locally. So when I say "local iteration" I'm trying to emphasize we're only considering \(|z-p| < \delta\) is small, where \(p\) is the fixed point. But yes, the ONLY local iteration is the regular iteration, so we're on the same page. I'll stick to saying regular iteration locally--where I mean we're using the regular iteration, but only care about it locally about the fixed point 
. My apologies, I tend to do poorly at adopting vernacular.
Sorry, yes, THE crescent iteration! I'm well aware it's the only one.
That's the conjecture! The beta iteration is looking like it converges to the Crescent iteration (or for, cases like \(\sqrt{2}\) it converges to the regular iteration (in this case \(\lambda \to -\log\log(2)\)). I can show this but I can't show it for \(b = e\). But I can motivate.
It's largely based on the fact that as you let \(\lambda \to 0\) (which moves the period of the iteration from \(2 \pi i/\lambda \to \infty\)) the iteration looks more and more like \(L^+\) in the upper half plane, and \(L^-\) in the lower half plane, and the branching seems to calm down A LOT. And due to this, the taylor series coefficients start to calm down A LOT--and they start to no longer have a trivial area of convergence (which is what normally happens). But still, very conjectural. This is for the \(b = e\) case, but a similar analysis works for \(b > \eta\). It's also a horrible numerical method though, so I've largely abandoned proving it, or even further justifying through numerical evidence. Trying to limit \(\lambda \to 0\) is a god awful exercise in futility.
There can certainly be a real superfunction of \(\eta_-^z\); but it will not be the regular iteration, as it will not be local (you cannot find a function \(f^{\circ t}(z)\) which is real valued and holomorphic for \(z \approx 1/e\)). So there is no real valued half iterate here--actually wouldn't be surprised if you are correct, there is no real valued half-iterate. But this is largely because the function is not injective for the most part, and as you pointed out no fundamental domain. And to make a half iterate, you would need an abel function. It would be like trying to write \(\sin(1/2 + \arcsin(x))\) which is a very flawed looking half iterate of \(\sin(1+\arcsin(x))\). Doubt you'd be able to even find a domain that this could be called an iterate on.
The beta method is good at one thing, and one thing only, making super functions that are real valued with arbitrary period \(2 \pi i/\lambda\), lol.
But I for the life of me can't think of why, a priori, you'd have that there can't be a super function that is real valued. Naturally, if you wanted it to be injective, you'd have to have it be complex valued, just like how \((-1)^x\) is complex valued.
But also remember, that \(\eta_-\) iterated using the beta method is less than desirable (unless we let \(\lambda \to 0\), which is hard to do numerically); it tends to have a good amount extraneous branches (that look like little fractals), but still it is analytic on the real line. I've run Sheldon's test for analycity on a large sample size, with A bunch of points in Shell Thron, and it all works out the same. Sheldon concurred for the most part--and he helped carve out much of the limits to how we can move \(\lambda\), and for what \(\lambda\) the super functions converge. If your curious, the superfunction converges for \(\Re(\lambda) > -\log |\log(b)p|\), where \(b^p = p\) is the primary fixed point and \(b\) within the shell thron. Outside of Shellthron it becomes more nefarious and difficult--and I'm still not clear exactly how it works.
OH yes! Sorry, the periodic points aren't real valued, they are really close to the real line, they have small imaginary part though. I guess I screwed that explanation up. I meant that since this function is sinusoidal and almost looks like \(\approx 2\) period, there's bound to be a periodic point somewhere in the neighborhood. My apologies. You're also bound to find \(N\)'th order ones.
EDIT:
I guess I should also add that if you choose the standard abel iteration you should have no critical points near \(z \approx p\) in any of the petals. Thereby, the superfunction also has no critical points. You can note instantly, sure we are real valued, but there are critical points everywhere in the above beta iteration of eta minor. So again, trying to invert and grab an abel function is near impossible; we should have singularities like crazy in the abel function. But again, this only says that there's no fractional iteration \(f^{\circ t}\) near the fixed point \(p\) that is real valued (no real valued regular iteration locally). So yes, no half iterate. This doesn't mean that there's no super function though.
Think of it like this.
Regular iteration--done through the Schroder map, is the only holomorphic iteration \(f^{\circ t}(z)\) about the fixed point \(z \approx p\).
Crescent iteration--done through Kneser's riemann mapping, is a holomorphic iteration \(f^{\circ t}(z)\) where \(z\) is holomorphic on a domain with boundary value \(p\)
beta iteration--done through infinite compositions, is only a holomorphic iteration \(f^{\circ t}(z_0)\), and there isn't necessarily an abel function (sometimes there is, sometimes there isn't). Sure you can locally invert this away from critical points, but no one ever said there aren't a hell of a lot of critical points.
. My apologies, I tend to do poorly at adopting vernacular.Sorry, yes, THE crescent iteration! I'm well aware it's the only one.
That's the conjecture! The beta iteration is looking like it converges to the Crescent iteration (or for, cases like \(\sqrt{2}\) it converges to the regular iteration (in this case \(\lambda \to -\log\log(2)\)). I can show this but I can't show it for \(b = e\). But I can motivate.
It's largely based on the fact that as you let \(\lambda \to 0\) (which moves the period of the iteration from \(2 \pi i/\lambda \to \infty\)) the iteration looks more and more like \(L^+\) in the upper half plane, and \(L^-\) in the lower half plane, and the branching seems to calm down A LOT. And due to this, the taylor series coefficients start to calm down A LOT--and they start to no longer have a trivial area of convergence (which is what normally happens). But still, very conjectural. This is for the \(b = e\) case, but a similar analysis works for \(b > \eta\). It's also a horrible numerical method though, so I've largely abandoned proving it, or even further justifying through numerical evidence. Trying to limit \(\lambda \to 0\) is a god awful exercise in futility.
There can certainly be a real superfunction of \(\eta_-^z\); but it will not be the regular iteration, as it will not be local (you cannot find a function \(f^{\circ t}(z)\) which is real valued and holomorphic for \(z \approx 1/e\)). So there is no real valued half iterate here--actually wouldn't be surprised if you are correct, there is no real valued half-iterate. But this is largely because the function is not injective for the most part, and as you pointed out no fundamental domain. And to make a half iterate, you would need an abel function. It would be like trying to write \(\sin(1/2 + \arcsin(x))\) which is a very flawed looking half iterate of \(\sin(1+\arcsin(x))\). Doubt you'd be able to even find a domain that this could be called an iterate on.
The beta method is good at one thing, and one thing only, making super functions that are real valued with arbitrary period \(2 \pi i/\lambda\), lol.
But I for the life of me can't think of why, a priori, you'd have that there can't be a super function that is real valued. Naturally, if you wanted it to be injective, you'd have to have it be complex valued, just like how \((-1)^x\) is complex valued.
But also remember, that \(\eta_-\) iterated using the beta method is less than desirable (unless we let \(\lambda \to 0\), which is hard to do numerically); it tends to have a good amount extraneous branches (that look like little fractals), but still it is analytic on the real line. I've run Sheldon's test for analycity on a large sample size, with A bunch of points in Shell Thron, and it all works out the same. Sheldon concurred for the most part--and he helped carve out much of the limits to how we can move \(\lambda\), and for what \(\lambda\) the super functions converge. If your curious, the superfunction converges for \(\Re(\lambda) > -\log |\log(b)p|\), where \(b^p = p\) is the primary fixed point and \(b\) within the shell thron. Outside of Shellthron it becomes more nefarious and difficult--and I'm still not clear exactly how it works.
OH yes! Sorry, the periodic points aren't real valued, they are really close to the real line, they have small imaginary part though. I guess I screwed that explanation up. I meant that since this function is sinusoidal and almost looks like \(\approx 2\) period, there's bound to be a periodic point somewhere in the neighborhood. My apologies. You're also bound to find \(N\)'th order ones.
EDIT:
I guess I should also add that if you choose the standard abel iteration you should have no critical points near \(z \approx p\) in any of the petals. Thereby, the superfunction also has no critical points. You can note instantly, sure we are real valued, but there are critical points everywhere in the above beta iteration of eta minor. So again, trying to invert and grab an abel function is near impossible; we should have singularities like crazy in the abel function. But again, this only says that there's no fractional iteration \(f^{\circ t}\) near the fixed point \(p\) that is real valued (no real valued regular iteration locally). So yes, no half iterate. This doesn't mean that there's no super function though.
Think of it like this.
Regular iteration--done through the Schroder map, is the only holomorphic iteration \(f^{\circ t}(z)\) about the fixed point \(z \approx p\).
Crescent iteration--done through Kneser's riemann mapping, is a holomorphic iteration \(f^{\circ t}(z)\) where \(z\) is holomorphic on a domain with boundary value \(p\)
beta iteration--done through infinite compositions, is only a holomorphic iteration \(f^{\circ t}(z_0)\), and there isn't necessarily an abel function (sometimes there is, sometimes there isn't). Sure you can locally invert this away from critical points, but no one ever said there aren't a hell of a lot of critical points.
