  We study topological properties of semi-group actions on the circle by
orientation-preserving homeomorhisms. We prove that a generic action either
possesses a forward-invariant interval-domain (i.e. a finite union of disjoint
circle arcs), or is two-sided minimal (and, moreover, by a small perturbation
one can create a map with arbitrary rotation number in the semigroup). For the
minimal case, we also study conditions required for global synchronization.
