07/25/2022, 04:01 PM
Because I was always wondering why is there no parabolic implosion/explosion at all the other parabolic fixed points (except b=eta)?!
The reason seems to be that we did not consider fixed cycles.
This becomes visible when we take \(f^{\circ N}\) as the function in question, suddenly it has much more fixed points, because a fixed N-cycle turns into a fixed point under \(f^{\circ N}\).
So I was looking at parabolic fixed points for bases on the Shell-Thron boundary and perturbed them perpendicular to the Shell-Thron boundary.
But not considering \(f(z)=b^z\), but considering \(f^{\circ N}\) for \(e^{2\pi i /N}\) being the multiplier of the fixed point.
And voilá we see the ex-/im-plosion there!
This is plosion at \(b_2=\eta_-\), i.e. \(b_2=\exp\left(e^{\phi i - e^{\phi i}}\right)\) with \(\phi=2\pi/2 = \pi\).
The fixed point \(z_2 = e^{e^{\pi i}}=1/e\) has multiplier \(-1=e^{\pi i}\)
The animation shows \(f^{\circ 2}(z) -z\) while \(b\) breaks through the STB at \(b_2\) from inside to outside (however with such a small amount that you can not see it in the animation, the middle fixed point breaks through the red curve from inside to outside).
I think you need to reload the page if you want to see the animation again.
The read curve are the fixed points with absolute value of the multiplier being 1.
Additionally to the two real fixed points (we saw in the previous image) we can see in this image also the two complex fixed points:
This is the plosion for \(\phi=2\pi / 3\) showing \(f^{\circ 3}(z)-z\):
And this is the plosion for \(\phi = 2\pi /4 = \pi/2 \) showing \(f^{\circ 4}(z)-z\):
The fixed point turns into N+1 fixed points.
So theoretically one could do an iteration at any of these fixed points/cycles maybe even at two of them (if they have a fundamental region).
This again opens the question whether one can continue a regular iteration through the STB.
But actually I am already again deviating from the thread's topic XD.
The reason seems to be that we did not consider fixed cycles.
This becomes visible when we take \(f^{\circ N}\) as the function in question, suddenly it has much more fixed points, because a fixed N-cycle turns into a fixed point under \(f^{\circ N}\).
So I was looking at parabolic fixed points for bases on the Shell-Thron boundary and perturbed them perpendicular to the Shell-Thron boundary.
But not considering \(f(z)=b^z\), but considering \(f^{\circ N}\) for \(e^{2\pi i /N}\) being the multiplier of the fixed point.
And voilá we see the ex-/im-plosion there!
This is plosion at \(b_2=\eta_-\), i.e. \(b_2=\exp\left(e^{\phi i - e^{\phi i}}\right)\) with \(\phi=2\pi/2 = \pi\).
The fixed point \(z_2 = e^{e^{\pi i}}=1/e\) has multiplier \(-1=e^{\pi i}\)
The animation shows \(f^{\circ 2}(z) -z\) while \(b\) breaks through the STB at \(b_2\) from inside to outside (however with such a small amount that you can not see it in the animation, the middle fixed point breaks through the red curve from inside to outside).
I think you need to reload the page if you want to see the animation again.
The read curve are the fixed points with absolute value of the multiplier being 1.
Additionally to the two real fixed points (we saw in the previous image) we can see in this image also the two complex fixed points:
This is the plosion for \(\phi=2\pi / 3\) showing \(f^{\circ 3}(z)-z\):
And this is the plosion for \(\phi = 2\pi /4 = \pi/2 \) showing \(f^{\circ 4}(z)-z\):
The fixed point turns into N+1 fixed points.
So theoretically one could do an iteration at any of these fixed points/cycles maybe even at two of them (if they have a fundamental region).
This again opens the question whether one can continue a regular iteration through the STB.
But actually I am already again deviating from the thread's topic XD.
