Multi-dimensional piecewise contractions are asymptotically periodic

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 ω-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor ω-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family μ\μ∈ U of locally bi-Lipschitz piecewise contractions f_μ:X→ X defined on a compact metric space X is asymptotically periodic for Lebesgue almost every parameter μ running over an open subset U of the M-dimensional Euclidean space RM. As a corollary of our results, we prove that piecewise affine contractions of Rᵈ defined in generic polyhedral partitions are asymptotically periodic.