07/24/2022, 12:43 PM
Is I am not as good with reading complex function graphs I reduced the problem to the real line.
The function \(\eta_-^x\) has a parabolic fixed point at 1/e with multiplier -1 which is a/the second root of unity.
In general it is a standard consideration if you have a m-th root of unity as multiplier to just consider the m-th iteration of the function.
Because if we have an iteration \((f\circ f)^t\) then we also have an iteration \(f^t = (f\circ f)^{\frac{t}{2}}\).
So I did that and was graphing how the parabolic fixed point behaved under perturbation:
And we see this is quite different from \(b=\eta\).
But particularly I was wondering: where do those extra attracting fixed points - left and right for \(b=e^{-e}-0.005\) - come from?!
These are not fixed points of \(b^z\), because it does not have any attracting fixed points (b is outside the Shell-Thron region).
And then it became clear to me, these are periodic points!
So one can iterate at periodic points! Gottfried, sorry that I never attentively read your posts about the periodic points!
Seems they are quite important as another source of iterating the exponentials.
(beside regular iteration, and - how about we call Kneser/perturbed Fatou coordinates "double regular iteration"?! Because the iteration is so to say done at 2 fixed points - double regular iteration.)
Indeed these two extra fixed points of \(b^{b^x}\) are a 2-cycle of \(b^x\):
\(b^{0.527385987834363}= 0.228746810954529\)
The function \(\eta_-^x\) has a parabolic fixed point at 1/e with multiplier -1 which is a/the second root of unity.
In general it is a standard consideration if you have a m-th root of unity as multiplier to just consider the m-th iteration of the function.
Because if we have an iteration \((f\circ f)^t\) then we also have an iteration \(f^t = (f\circ f)^{\frac{t}{2}}\).
So I did that and was graphing how the parabolic fixed point behaved under perturbation:
And we see this is quite different from \(b=\eta\).
But particularly I was wondering: where do those extra attracting fixed points - left and right for \(b=e^{-e}-0.005\) - come from?!
These are not fixed points of \(b^z\), because it does not have any attracting fixed points (b is outside the Shell-Thron region).
And then it became clear to me, these are periodic points!
So one can iterate at periodic points! Gottfried, sorry that I never attentively read your posts about the periodic points!
Seems they are quite important as another source of iterating the exponentials.
(beside regular iteration, and - how about we call Kneser/perturbed Fatou coordinates "double regular iteration"?! Because the iteration is so to say done at 2 fixed points - double regular iteration.)
Indeed these two extra fixed points of \(b^{b^x}\) are a 2-cycle of \(b^x\):
\(b^{0.527385987834363}= 0.228746810954529\)
