Multi-dimensional piecewise contractions are asymptotically periodic

Piecewise contractions (PCs) are piecewise smooth maps that decrease distance between pairs 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 $\omega$-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega$-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 $\{f_{\mu }{\}}_{\mu \in U}$ of locally bi-Lipschitz piecewise contractions $f_{\mu }:X\to X$ defined on a compact metric space $X$ is asymptotically periodic for Lebesgue almost every parameter $\mu$ running over an open subset $U$ of the $M$-dimensional Euclidean space ${\mathbb{R}}^M$. As a corollary of our results, we prove that piecewise affine contractions of ${\mathbb{R}}^d$ defined in generic polyhedral partitions are asymptotically periodic.