We consider a minimal action of a finitely generated semigroup by
homeomorphisms of a circle, and show that the collection of translation numbers
of individual elements completely determines the set of generators (up to a
common continuous change of coordinates). One of the main tools used in the
proof is the synchronization properties of random dynamics of circle
homeomorphisms: Antonov's theorem and its corollaries.Comment: 15 pages, 1 figure, submitte