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Last semester I finished my first class on complex variables and of course we had to show that $i^i$ was real. That got me wondering about quantities like $i^{i^i}$ and similar power towers.

For my investigation, I let $f:\mathbb{C} \to \mathbb{C}$ where $f(z)=(ui)^z$ with $u \in \mathbb{R}$ and let $f_n(z)$ denote the quantity $f(f(\cdots f(z)$ where $f$ occurs $n$ times. I then plotted the points generated by $\{f(ui),f_2(ui),f_3(ui),\ldots,f_k(ui)\}$ for various values of $u$. The plots that I obtained are quite interesting!

complex spiral points

complex spiral lines

The top one is a plot of the points $\{f(ui),f_2(ui),f_3(ui),\ldots,f_{100}(ui)\}$ with the real axis on the horizontal and imaginary on the vertical, and $u$ going from .05 to 2.05 in increments of .1. Before .05 it blows up and after 2, the points seem to settle into 3 groups near $(0,u),(0,0)$, and $(1,0)$. The second picture is the same as the first, but with lines connecting $f_k(ui)$ to $f_{k+1}(ui)$.

Just a note, the dots do spiral inward with successive nestings, so my inkling of convergence is well-founded, and $|f_k(ui)|$ seems to converge only for $0 < u <2$. Does anyone have any insights into the values of $u$ for which this system converges or if this has been written about before? Even reference to a method for determining the existence of a fixed point would be appreciated.

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How did you generate the plots? – copper.hat Oct 22 '12 at 23:34
@copper.hat With Mathematica. Here is the code:ListPlot[Map[{Re[#], Im[#]} &, NestList[(u*I)^# &, u*I, 100]], PlotRange -> {{0, 1}, {-.3, 10}}, Axes -> True] – Jeremy Oct 22 '12 at 23:39
Cool, thanks very much! – copper.hat Oct 23 '12 at 0:42
I know not your background, so I will say things you may already know. The first thing one looks for in a dynamical system is equilibria. Here they are of the form $f(z)=z$. Unfortunately it seems they can only be found numerically. So the first step is to find for what range of $u$ equilibria exist. Then to show that one is stable you have to expand $f$ around it by its series and check its eigenvalues, which in this case is the value of $f'$ at the equilibrium. If the norm of the value is less than 1, the equilibrium is stable, if it is more than 1 it is unstable. Have you done any of these? – tst Nov 11 '12 at 3:44
If im not mistaken, Gottried or Sheldon have considered this before. You can find them on the tetration forum or here at Mse by searching for members/users. Btw added to favorites :) – mick Feb 11 '13 at 22:19

You are going ( on parameter u plane ) u=0.05 to u=2.05 in increments of .1.

IMHO you are ( on parameter plane) inside component where period 1 point is attracting ( regular spiral) and you move toward parabolic point where period 3 coincide with period 1 ( 3 arm distorted star) The star is distorted because u point is not on the internal ray 1/3 of this component

Compare with this animation ( 1 to 6 ) and this image ( 1 to 2)) :


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