Fixed Point of a complex dynamical spiral system

Question

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!

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.