We study the 4 to 2 band merging process at a=59257 in more detail. First we straddle the merging:
At a=3.58 there are 4 separate bands, and the power spectrun shows sharp peaks at 0 (the fundamental), 1 (the period 2 peak) and 0.5 (the period 4 peak), corresponding to the regular jumping between the bands, as well as a broad band background corresponding to the chaotic motion within the bands. At a=3.60 the 4 bands have merged to 2 and the peak in the power spectrum at 0.5 has broadened.
It is informative to look at this behavior in the 4th iterate f 4:
For a=3.59 below the band merging the 4th iterate is confined to a band near the centre around x=0.4 to x=0.6 (with a suitable choice of the phase of the "strobing" determined by the initial choice of x). For a=3.595 the iterations "escape" into a wider band.