TOPOLOGICAL DYNAMICS OF GENERIC PIECEWISE CONTRACTIVE MAPS IN n DIMENSIONS

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, sep- arating different continuity pieces, act as topological saddles with an infinite expanding rate. We prove that C 0 generical 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 struc- turally unstable and bifurcating.