Here is the phase portrait of $(\theta, x \; \mathrm{mod} \; 2)$ for the rays that Dimitri suggested
in his response to Q2
(if I have interpreted his suggestion correctly), using the same
data as displayed in the histogram:
Following Dimitri's second suggestion in the comment below, I shot the ray off at a random start angle, e.g.:
Here are the phase portraits of 20 different random rays, with initial angles $\theta_0$ chosen uniformly within $(-\pi/2,\pi/2)$, each reflected through a tube of 200 semicircles. Each ray's phase trajectory is in a different color. The conventions are the same as in the phase portrait above (which corresponds to $\theta_0=0$). (Note: The number of ray reflections is not held constant, but rather the length of the tube is fixed, and the ray bounces until it exits.)
I believe some of the prominently dark bands are caused by $\theta_0 \approx -\pi/2$, when the ray essentially skims around semicircle 1 and then shoots off nearly vertical to hit semicircle 2, and then it again nearly follows semicircle 3, and so on. You can see a hint of this behavior in the $\theta_0=-75^\circ$ example above.
