Observed rotation numbers in families of circle maps

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term o f Lebesgue measure on the circle. We study this question numerically for fa milies of circle maps. Both the interval and "observed" rotation numbers ar e computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appea r to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occu r whenever one of the endpoints of the rotation interval changes. We also e xamine the effects of various types of saddle-node bifurcations on the obse rved rotation numbers.