1D Mandelbrot set on a circle

[lkmitch]

When I first read the post about the unitary Mandelbrot set, I misunderstood it to be about calculating the Mandelbrot set in a 1D subset of a higher space (like on the real axis of the complex plane).  Clearly, I was wrong about that idea, but that got me to wondering about such things.  Specifically, is there any (visually interesting) way to do the Mandelbrot set (or Mandelbrot-type sets) in 1D that makes sense?  What I came up with is this:

   1. Start with the complex number z = (1, 0).  Take its angle (0 radians) as t.
   2. The parameter c is the angle of another complex number.
   3. Iterate t in the typical Mandelbrot sense (or, more generally, t = f(t; c)).
   4. The new z has a magnitude of 1 and an angle that is the new t.
   5. The new t is the angle of z.
        6. Loop back to step 3.

Aside from steps 4 & 5, this is just 1D dynamics, often expressed in the typical bifurcation plot.  Forcing t to actually be the angle of a complex number in essence reduces t to its residue within a range of 2 pi.  It seems that this restriction introduces the potential for interesting dynamics, and the way in which t is restricted (e.g., from -pi to pi or from 0 to 2 pi) can make a big difference.  Given that this is still a 1D problem, I chose to visualize it using bifurcation diagrams.  In the attached images, the horizontal axis represents c and the vertical axis shows t (t = t^2 + c).  The darkness of the pixel corresponds to how many times that the pixel was visited.  The first image has both c & t restricted to the range [-pi, pi).  In the second, c & t are in [0, 2pi).