  We study the dynamics of iteration function systems generated by a pair of
circle diffeomorphisms close to rotations in the $C^{1+\mathrm{bv}}$-topology.
We characterize the obstruction to minimality and describe the limit set. In
particular, there are no invariant minimal Cantor sets, which can be seen as a
Denjoy/Duminy type theorem for iterated systems on the circle.
