  We study the asymptotic dynamics of piecewise contracting maps defined on a
compact interval. For maps that are not necessarily injective, but have a
finite number of local extrema and discontinuity points, we prove the existence
of a decomposition of the support of the asymptotic dynamics into a finite
number of minimal components. Each component is either a periodic orbit or a
minimal Cantor set and such that the $\omega$-limit set of (almost) every point
in the interval is exactly one of these components. Moreover, we show that each
component is the $\omega$-limit set, or the closure of the orbit, of a
one-sided limit of the map at a discontinuity point or at a local extremum.
