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Multi-dimensional piecewise contractions are asymptotically periodic

Jose Pedro Gaivao, Benito Pires

TL;DR

This work extends the almost-everywhere asymptotic periodicity results from one-dimensional piecewise contractions to multi-dimensional settings by introducing two criteria, Hypotheses (E) and (T), for families {f_μ} of piecewise λ-contractions on a compact space X. The main theorem shows that if (E) and (T) hold at a point μ^*, then f_μ is asymptotically periodic on Z_μ for Lebesgue almost every μ in a neighborhood of μ^*, enabling robust behavior across parameter families. The authors provide a practical sufficiency criterion (via zero multiplicity entropy and stability) and apply the framework to both 1D varying-partition families (NPR2) and multi-dimensional piecewise-affine contractions on polyhedral partitions, with a corollary for homothetic IFS maps. The results yield that, generically, multi-dimensional PCs exhibit a finite union of periodic orbits as attractors, with all periodic points regular, thereby offering a broad, measure-theoretic understanding of the dynamics of complex piecewise systems.

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 $ω$-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.

Multi-dimensional piecewise contractions are asymptotically periodic

TL;DR

This work extends the almost-everywhere asymptotic periodicity results from one-dimensional piecewise contractions to multi-dimensional settings by introducing two criteria, Hypotheses (E) and (T), for families {f_μ} of piecewise λ-contractions on a compact space X. The main theorem shows that if (E) and (T) hold at a point μ^*, then f_μ is asymptotically periodic on Z_μ for Lebesgue almost every μ in a neighborhood of μ^*, enabling robust behavior across parameter families. The authors provide a practical sufficiency criterion (via zero multiplicity entropy and stability) and apply the framework to both 1D varying-partition families (NPR2) and multi-dimensional piecewise-affine contractions on polyhedral partitions, with a corollary for homothetic IFS maps. The results yield that, generically, multi-dimensional PCs exhibit a finite union of periodic orbits as attractors, with all periodic points regular, thereby offering a broad, measure-theoretic understanding of the dynamics of complex piecewise systems.

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 -limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor -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 of locally bi-Lipschitz piecewise contractions defined on a compact metric space is asymptotically periodic for Lebesgue almost every parameter running over an open subset of the -dimensional Euclidean space . As a corollary of our results, we prove that piecewise affine contractions of defined in generic polyhedral partitions are asymptotically periodic.
Paper Structure (11 sections, 28 theorems, 109 equations)

This paper contains 11 sections, 28 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Piecewise contractions on metric spaces
  4. Proof of Theorem \ref{['th:main']}
  5. Sufficient condition for Hypothesis (E)
  6. One-dimensional piecewise contractions
  7. Proof of Theorem \ref{['th:NPR2']}
  8. Completing the proof of Theorem \ref{['th:NPR2']}
  9. Proof of Theorem \ref{['thm:multidim']}
  10. Proof of Corollary \ref{['cor:3']}
  11. Distance to a polyhedron

Key Result

Theorem 2.1

Let $(X,d)$ be a compact metric space whose open balls are connected and let $\lambda \in (0,1)$. If $U\subset\mathbb{R}^M$ is a set of positive Lebesgue measure and $\{f_\mu\}_{\mu\in U}$ is a family of piecewise $\lambda$-contractions on X satisfying the hypotheses (E) and (T) at $\mu^*\in U$, the

Theorems & Definitions (64)

  • Definition 2.1: Piecewise contraction
  • Definition 2.2: Regular point
  • Definition 2.3: Asymptotically periodic
  • Remark 2.1
  • Definition 2.4: Itineraries
  • Definition 2.5: Family of piecewise $\lambda$-contractions
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.1
  • Definition 2.8: Piecewise-affine contraction
  • ...and 54 more