  We study the topological dynamics by iterations of a piecewise continuous,
non linear and locally contractive map in a real finite dimensional compact
ball. We consider those maps satisfying the "separation property": different
continuity pieces have disjoint images. The continuity pieces act as stable
topological manifolds while the points in the discontinuity lines, separating
different continuity pieces, act as topological saddles with an infinite
expanding rate. We prove that C0 generically such systems exhibit one and at
most a finite number of persistent periodic sinks attracting all the orbits. In
other words, the chaotic behaviors that this class of mappings may exhibit, are
structurally unstable and bifurcating.
