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 Tetration of complex bases to complex towers mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 10/26/2009, 03:29 AM 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? For example, I saw the graph here: http://math.eretrandre.org/tetrationforu...926#pid926 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)? 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 mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/01/2009, 12:37 AM So did you try applying the natural iteration method to some of those bases where it was applicable? What were the results? andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/03/2009, 08:00 AM (11/01/2009, 12:37 AM)mike3 Wrote: So did you try applying the natural iteration method to some of those bases where it was applicable? What were the results? Yes. The coefficients from intuitive/natural iteration do not converge for base-i. Also, the coefficients from regular iteration do not converge for base (-3+4i). mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/03/2009, 08:52 AM (This post was last modified: 11/03/2009, 10:26 PM by mike3.) (11/03/2009, 08:00 AM)andydude Wrote: Yes. The coefficients from intuitive/natural iteration do not converge for base-i. Also, the coefficients from regular iteration do not converge for base (-3+4i). Neither of those look to be within the triangular or kidneybean region shown on the graph, however I was thinking of what happens if you tried one in the triangular region, but not in the kidneybean. Correction: base "i" is in the kidneybean, sorry. But it's not in the triangle, and base -3+4i is out in the left half-plane well away from either. andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/04/2009, 03:01 AM (This post was last modified: 11/04/2009, 03:02 AM by andydude.) (11/03/2009, 08:52 AM)mike3 Wrote: ... the kidneybean ... lol, I call it the Shell-Thron region, or "the shell" for short, which oddly enough describes both the author and the shape. So how about the base $b = 3 + i$. The intuitive/natural Abel function of exponentials associated with this base is: $\text{slog}_b(x) = -1 + (0.984001 + 0.124042 i)z + (0.246854 - 0.0364779 i)z^2 + \cdots$ when expanded at 0. It seems to be fairly close to being real-valued for arguments around 0 to 1. I have attached 2 plots, the critical function above, and the piecewise extension to all R. Blue is the real part and red is imaginary part. Andrew Robbins Attached Files Image(s)         mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/04/2009, 05:29 AM (This post was last modified: 11/04/2009, 05:40 AM by mike3.) Hmm. What about $\mathrm{tet}_b(x)$ for this base? And where is this on the "tetration fractal" (this thing: http://www.tetration.org/Fractals/Atlas/index.html)? Edit: Just saw the answer to the last question, there was a picture with the grid over it. Looks like it'd be in one of the large, long period 4 lobes coming off to the right. andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/06/2009, 05:13 AM (11/04/2009, 05:29 AM)mike3 Wrote: Hmm. What about $\mathrm{tet}_b(x)$ for this base? And where is this on the "tetration fractal" (this thing: http://www.tetration.org/Fractals/Atlas/index.html)? Edit: Just saw the answer to the last question, there was a picture with the grid over it. Looks like it'd be in one of the large, long period 4 lobes coming off to the right. I've had the hardest time making those pictures. The closest I could come was using xaos and under "Fractal", "User formula" I entered "c^z". I made some pictures for you. In the exact center of each is 3+i. Sorry I can't find the grid button. Attached Files Image(s)             mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/06/2009, 08:46 AM But what does the graph of $\mathrm{tet}_{3 + i}(x)$ look like for real x using the natural/intuitive tetration? You provided a graph for $\mathrm{slog}_{3 + i}(x)$, so how about $\mathrm{tet}_{3 + i}(x)$? « Next Oldest | Next Newest »

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