Rotation nombers for measure-valued circle maps

The weak and strong topologies on the space of orbits from the unit interval to the set of probability measures are considered. A particular interest is periodic orbits of probability measures on the circle. It is shown that a realvalued rotation number can be defined in a natural way for all smooth enough orbits whose range consists of probability measures supported on the whole circle. Furthermore, this number is a continuous functional with respect to an appropriately defined strong topology. The completion of this space contains as a special case deterministic orbits, whose rotation number is an integer, coinciding with the topological degree.