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 Tetration of complex bases to complex towers andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/29/2009, 05:48 AM (10/26/2009, 03:29 AM)mike3 Wrote: Are there any tetration methods that can tetrate some complex bases whose integer tetrations attract to an n-cycle to real and complex towers? If so, what would the graphs look like for an example base?I tried using regular iteration for base-i tetration once, and it was very slow to converge. So I tried finding a polynomial manually that was differentiable at integers (a piecewise method) basically, making the picture in this article by A.J.Macintyre smooth. It was kinda nice, but it didn't seem close to the regular iteration method... Anyways, I was convinced that I needed a faster computer for a better picture. (10/26/2009, 03:29 AM)mike3 Wrote: For example, I saw the graph here: http://math.eretrandre.org/tetrationforu...926#pid926I made that (10/26/2009, 03:29 AM)mike3 Wrote: What happens if you apply the natural iteration method to a complex base in its wedge, one that converges to an n-cycle on the integers? What do the graphs look like, on the real line and complex plane in the tower (at least on as much plane as can be covered by the series that are obtained)?I will try to make some for you. On another note, I see 2 general approaches to bases with n-cycles (where n > 2). Either (A) find a new method that doesn't choke on n-cycles, or (B), use a method that works on fixed points (regular iteration), and iterate the function $b^{b^{b^x}}$ instead of $b^x$ (for a 3-cycle, the most dominant periodicity in tetration), then use a function like $f(x) = \frac{1 + 2\cos(\frac{2}{3}\pi x)}{3}$ to interpolate between the 3 solutions. This function has the property that $f(x)=1$ if x is integer and divisible by 3, and $f(x)=0$ if x is integer and not divisible by 3. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Tetration of complex bases to complex towers - by mike3 - 10/26/2009, 03:29 AM RE: Tetration of complex bases to complex towers - by andydude - 10/29/2009, 05:48 AM RE: Tetration of complex bases to complex towers - by mike3 - 11/01/2009, 12:37 AM RE: Tetration of complex bases to complex towers - by andydude - 11/03/2009, 08:00 AM RE: Tetration of complex bases to complex towers - by mike3 - 11/03/2009, 08:52 AM RE: Tetration of complex bases to complex towers - by andydude - 11/04/2009, 03:01 AM RE: Tetration of complex bases to complex towers - by mike3 - 11/04/2009, 05:29 AM RE: Tetration of complex bases to complex towers - by andydude - 11/06/2009, 05:13 AM RE: Tetration of complex bases to complex towers - by mike3 - 11/06/2009, 08:46 AM

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