  We consider the outer billiards map with contraction outside polygons. We
construct a 1-parameter family of systems such that each system has an open set
in which the dynamics is reduced to that of a piecewise contraction on the
interval. Using the theory of rotation numbers, we deduce that every point
inside the open set is asymptotic to either a single periodic orbit (rational
case) or a Cantor set (irrational case). In particular, we deduce existence of
an attracting Cantor set for certain parameter values. Moreover, for a
different choice of a 1-parameter family, we prove that the system is uniquely
ergodic; in particular, the entire domain is asymptotic to a single attractor.
