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 \(\omega\)-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.