It is interesting to compare the probability density at band merging with the analytically known probability at the "most chaotic" value of a=4
At band merging:
It is again useful to look at the 4th iterate of the map (enlarge the visited portion of the x-axis using the mouse):
and compare this with the behavior over the whole unit interval at a=4:
The histograms have essentially the same form (and then the first plot is made up of the 4 bands with the same distribution, but with different scales).