Topological dynamics of generic piecewise continuous contractive maps in n dimensions

Eleonora Catsigeras, Ruben Budelli

Published 2010-03-13Version 1

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.