Multidimensional piecewise contractions

Benito Pires, USP - Ribeirão Preto 

Abstract:

Piecewise contractions (PCs) are piecewise smooth maps that decrease distance between pair of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, a lot of effort has been devoted to proving that in parametrized families of one-dimensional PCs, the attractor of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor attractors. In this seminar, we will discuss a generalisation of some of these results to the multidimensional case. Namely, we will show that piecewise affine contractions defined in generic polyhedral partitions are asymptotically periodic. This is a joint work with José Pedro Gaivão (ISEG - Universidade de Lisboa).