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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.08444 |
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Table of Contents:
- 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 $ω$-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 $\{f_μ\}_{μ\in U}$ of locally bi-Lipschitz piecewise contractions $f_μ:X\to X$ 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 $\mathbb{R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb{R}^d$ defined in generic polyhedral partitions are asymptotically periodic.