On minimal invariant sets and stationary measures for random dynamical systems on the circle

In this paper, we study random dynamical systems (RDSs) of homeomorphisms of the circle without a finite orbit. We characterize the topological dynamics of the associated semigroup by identifying the existence of invariant sets that are finite unions of intervals. We describe the accumulation points of the average orbit of the transfer operator. For each ergodic stationary measure, we establish structural properties of its weight function on the circle. We also establish relationships between the minimal sets of an RDS and its inverse RDS.