Rotation Numbers for Measure-valued Circle Maps

We consider strong and weak topologies on the space of orbits from the unit interval to the set of probability measures on a given domain. In particular, we are interested in periodic orbits of probability measures on the circle. We show that a real-valued rotation number can be defined in a natural way for all smooth enough orbits whose range is composed of probability measures supported on the whole circle. Next, we show that this number is a continuous functional with respect to an appropriately defined strong topology. The completion of this space contains deterministic orbits as a special case, whose rotation number is an integer, coinciding with the topological degree.