Forgive me,Bo--I tend to forget exact details.
But you used the word tetrational for a super function \(F(z)\) such that \(b^{F(z)} = F(z+1)\) if \(F(0) = 1\).
It seems that the iteration I have for eta minor, and nearby fixed points is NOT TETRATIONAL. To the extent that there is no normalization constant to make \(g(z) = F(z+z_0)\) s.t. \(g(0) = 1\). I think this speaks leagues as to your fundamental domain problem. We can create a real valued superfunction, BUT IT ISN'T A TETRATION.
I did more and more extensive testing, and this seems pretty clear. Essentially we get an almost periodic sinusoidal wave along \(\mathbb{R}\). This is because this is an iteration about the repelling petals. As opposed to an iteration about the attracting petals (which include \(0\)). So we are effectively mapping the top/bottom petals of the four petals of \(\eta_-\) about \(1/e\). We're making a real valued solution, which has decay at the nearby attracting fixed points of \(\log_{\eta_-}\).
This is very similar to Kneser's mapping theorem, except we've removed the mapping, but additionally, Kneser has no mapping, because no real valued super function exists which is TETRATIONAL. So this means, we're doing something entirely different. We are mapping along the top/bottom petals, which as we limit \(f^{-n}\) we go towards the repelling fixed points on the top/bottom of \(1/e\). And we are doing some kind of mapping, to ensure we paste these two solutions together along the real line.
So this should be a huge disclaimer. The superfunction IS NOT TETRATIONAL. It's just a super-function. And it appears to like the repelling petal, as opposed to the attracting petals.
So, just to clarify. THERE IS NO TETRATION \(\eta_- \uparrow \uparrow z\) that is real valued. But there is a super function!!!! So this is very much, more like an iteration of \(\sqrt{2}^z\) about \(4\), then it is like an iteration about \(2\). Which as you noted, to have a \(\eta_-\) iteration, it would be complex valued. Nonetheless, we can still construct the \(2\) iteration for \(\eta_-\), it's just a different complex period, which approaches \(\infty\) in a specific way. Rather than \(\Re(\lambda) \to 0\) along \(\mathbb{R}^+\), it's along another arc.
God this is making my head spin. I'm thinking of writing a shorter version of beta.gp, which only deals with \(\eta_- + y\) for \(y\) small--so that you can see for yourself how this works. The code I have released does work, but it'll be a bit glitchy near eta minor. So I had to patch work a bunch of the protocols to make them more accurate near eta minor. Let me know if you'd be interested in a pari-gp program that would do that. That way you can see the analytic super function for yourself.
But you used the word tetrational for a super function \(F(z)\) such that \(b^{F(z)} = F(z+1)\) if \(F(0) = 1\).
It seems that the iteration I have for eta minor, and nearby fixed points is NOT TETRATIONAL. To the extent that there is no normalization constant to make \(g(z) = F(z+z_0)\) s.t. \(g(0) = 1\). I think this speaks leagues as to your fundamental domain problem. We can create a real valued superfunction, BUT IT ISN'T A TETRATION.
I did more and more extensive testing, and this seems pretty clear. Essentially we get an almost periodic sinusoidal wave along \(\mathbb{R}\). This is because this is an iteration about the repelling petals. As opposed to an iteration about the attracting petals (which include \(0\)). So we are effectively mapping the top/bottom petals of the four petals of \(\eta_-\) about \(1/e\). We're making a real valued solution, which has decay at the nearby attracting fixed points of \(\log_{\eta_-}\).
This is very similar to Kneser's mapping theorem, except we've removed the mapping, but additionally, Kneser has no mapping, because no real valued super function exists which is TETRATIONAL. So this means, we're doing something entirely different. We are mapping along the top/bottom petals, which as we limit \(f^{-n}\) we go towards the repelling fixed points on the top/bottom of \(1/e\). And we are doing some kind of mapping, to ensure we paste these two solutions together along the real line.
So this should be a huge disclaimer. The superfunction IS NOT TETRATIONAL. It's just a super-function. And it appears to like the repelling petal, as opposed to the attracting petals.
So, just to clarify. THERE IS NO TETRATION \(\eta_- \uparrow \uparrow z\) that is real valued. But there is a super function!!!! So this is very much, more like an iteration of \(\sqrt{2}^z\) about \(4\), then it is like an iteration about \(2\). Which as you noted, to have a \(\eta_-\) iteration, it would be complex valued. Nonetheless, we can still construct the \(2\) iteration for \(\eta_-\), it's just a different complex period, which approaches \(\infty\) in a specific way. Rather than \(\Re(\lambda) \to 0\) along \(\mathbb{R}^+\), it's along another arc.
God this is making my head spin. I'm thinking of writing a shorter version of beta.gp, which only deals with \(\eta_- + y\) for \(y\) small--so that you can see for yourself how this works. The code I have released does work, but it'll be a bit glitchy near eta minor. So I had to patch work a bunch of the protocols to make them more accurate near eta minor. Let me know if you'd be interested in a pari-gp program that would do that. That way you can see the analytic super function for yourself.
