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Infinite tetration of the imaginary unit
#21
(06/20/2011, 05:27 AM)sheldonison Wrote: My conjecture is for bases on the Shell Thron boundary, there is an analytic superfunction with a real period, whose structure depends on what the continued fraction representation of the real period is. As long as the period is a real number (with an infinite continued fraction representation), then I suspect the superfunction is analytic. If the period is a rational number, then I don't think there is an analytic superfunction. For example, this base, with a real period=3, probably doesn't have an analytic superfunction, developed from the neutral fixed point, because starting with a point near L, and iterating the function x=B^x three times, doesn't get you back to the initial starting point.
Base= 0.030953557167612060 + 1.7392241043091316i
L= 0.39294655583435517 + 0.46203078407110528i
Hi Sheldon -

I've inserted your base-parameter and got the following plot for the orbit/for the three partial trajectories in the same style of my previous plots. I seem to have problems to understand your comment correctly. For instance, isn't that fixpoint attracting instead of neutral?

[Image: br10.png]

Having seen this I assume, that also with a starting-point near the fixpoint we get something converging to the fixpoint, however slow. But, well, that would be now another job to prove.

[Image: br_01.png]

In my initial plot it seemed, that there is only one base b0, whose orbits are between converging to the fixpoint and diverging, and because the base at 1.71290*I is such a base I assume, that we get either convergence here or divergence to a triplett of cumulation points.

What do you think?

Gottfried

[Update]
A startingpoint x0=0.41*(1+I)=b^^0, even nearer at the fixpoint L, exhibits now repelling properties of the fixpoint. So I think, that in fact there are three "oscillating" fixpoints in the near of the orbit of the last experiment and the trajectories of the first picture do not approach the fixpoint L but that triplett of accumulation(?) points.

[Image: br_041.png]


Gottfried Helms, Kassel
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Messages In This Thread
Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM
RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM

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