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

A spectral decomposition of the attractor of piecewise contracting maps of the interval