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Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)
#17
Ivars Wrote:While studying this, I have become little puzzled about the way converges. When iterated from z_0=I with a any precision available in PARI, it shows a tri-cycle behaviour, as do many other imaginary numbers (e.g 2*I, 3*I etc. ), if I have made the calculations right.
Hi Ivars -
I've discussed this in some initial state earlier here, but I'm currently with my head elsewhere so too lazy to find the thread.
Here are some plots of that tri-furcation, see uncommented list below.
I find the bi-,tri- and multifurcation an interesting subject, as we ask: can we assign an individual value if the iteration oscillates/is furcated - since this is somehow related to the partial evaluation of non-convergent oscillating series, to which we assign a value anyway. But I don't have a special conclusion for this matter, yet, which were worth to write it down here.

Gottfried

http://go.helms-net.de/math/tetdocs/traj..._165_I.png
http://go.helms-net.de/math/tetdocs/traj..._175_I.png
http://go.helms-net.de/math/tetdocs/trajectories_I.png
http://go.helms-net.de/math/tetdocs/traj...rcated.png
http://go.helms-net.de/math/tetdocs/trif...on_1_2.png
http://go.helms-net.de/math/tetdocs/trif...on_1_8.png
http://go.helms-net.de/math/tetdocs/trif...quator.png
http://go.helms-net.de/math/tetdocs/trif..._bases.png
http://go.helms-net.de/math/tetdocs/trif...detail.png
http://go.helms-net.de/math/tetdocs/trif...bases1.png
Gottfried Helms, Kassel
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RE: Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I) - by Gottfried - 05/25/2008, 10:48 AM

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