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 Infinite tetration of the imaginary unit sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/23/2011, 01:13 PM (This post was last modified: 06/23/2011, 04:53 PM by sheldonison.) (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: $\text{SuperFunction}_{L}(x) = \exp_B^{o m}(L+z)$. And forFor 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, $b\approx 1.7129i$, $\text{period}\approx 2.9883$ $\text{SuperFunction}_{L}(0) = (L+z)$ $\text{SuperFunction}_{L}(1) = b^{(L+z)}$ $\text{SuperFunction}_{L}(2) = \exp_b^{o 2}(L+z)$ $\text{SuperFunction}_{L}(0.0117) = \exp_b^{o 3}(L+z)$ $\text{SuperFunction}_{L}(1.0117) = \exp_b^{o 4}(L+z)$ $\text{SuperFunction}_{L}(2.0117) = \exp_b^{o 5}(L+z)$ $\text{SuperFunction}_{L}(0.0234) = \exp_b^{o 6}(L+z)$ $\text{SuperFunction}_{L}(x) = \exp_b^{o m}(L+z)$ where $x=(m \bmod \text{period})$ 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 « Next Oldest | Next Newest »

 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 Ivars - 02/10/2008, 09:52 AM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 11:35 AM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 01:28 PM RE: Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 08:53 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/10/2008, 09:10 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/14/2008, 08:25 PM RE: Infinite tetration of the imaginary unit - by Ivars - 02/15/2008, 05:35 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/19/2011, 09:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 05:27 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:20 AM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 01:36 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/20/2011, 02:46 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 12:13 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 03:02 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/21/2011, 08:00 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/22/2011, 09:28 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/22/2011, 03:23 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/23/2011, 08:55 AM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/23/2011, 01:13 PM solved -- they're called Siegel discs - by sheldonison - 06/23/2011, 06:10 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/20/2011, 06:11 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/20/2011, 10:22 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/21/2011, 01:58 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/21/2011, 11:19 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 03:15 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/22/2011, 07:05 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 08:07 AM RE: Infinite tetration of the imaginary unit - by tommy1729 - 06/24/2011, 12:25 PM RE: Infinite tetration of the imaginary unit - by sheldonison - 06/24/2011, 04:36 PM RE: Infinite tetration of the imaginary unit - by Gottfried - 06/24/2011, 09:44 PM RE: Infinite tetration of the imaginary unit - by bo198214 - 06/26/2011, 08:06 AM

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