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Infinite tetration of the imaginary unit
#35
(06/23/2011, 08:55 AM)Gottfried Wrote: Hi Sheldon -

likely a typo in the formula:
(06/22/2011, 03:23 PM)sheldonison Wrote: case, Then the period "works". For all integers, we can define the superfunction for x=(m mod period), starting from initial value L+z:
. And for
For the parameter of the superfunction I think you want to write m (or for exp_b the iterator/height x ?)
....
Hey Gottfried,

I don't think its a typo, but I tried to make the original post slightly clearer. x=(m mod period), m would be an integer iteration. The "o m" notation is iteration m times. Lower case b works better; here from your previous example, ,








where

This allows you to use integer iterations to gradually fill in all of the points for the superfunction on the real axis between 0 and the period. Since the function is real periodic, this means we can define the superfunction anywhere on the real axis. Then what I did, was to take the Fourier analysis of the function, using polynomial interpolations from several nearby points on either side to get a fairly exact value for a sampling of evenly spaced points; the results were posted here.

I'm going to read up on what's out there on Fatou sets. Anybody have a suggested reference? Specifically, we're interested in a Periodic Fatou set generated from a complex parabolic fixed point. I'm also trying to formulate a question for math overflow, to verify the existence of real periodic Fatou sets, generated with a superfunction of a function with a periodic parabolic fixed point, similar to what is seen on the Shell-Thron boundary.
- Sheldon
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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 sheldonison - 06/23/2011, 01:13 PM

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