Multi-dimensional piecewise contractions are asymptotically periodic

TL;DR

This paper extends the understanding of piecewise contractions from one-dimensional to multi-dimensional spaces, showing that most such systems tend to become asymptotically periodic under broad conditions.

Contribution

It provides criteria for asymptotic periodicity in multi-dimensional piecewise contractions and demonstrates that generic polyhedral affine contractions are asymptotically periodic.

Findings

Most multi-dimensional PCs are asymptotically periodic for almost every parameter.

Criteria for asymptotic periodicity are established for families of locally bi-Lipschitz PCs.

Generic polyhedral affine contractions in Euclidean spaces are asymptotically periodic.

Abstract

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.