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Piecewise Contractions

Sakshi Jain, Carlangelo Liverani

TL;DR

This work analyzes piecewise contractions on compact subsets of $\mathbb{R}^d$ with piecewise injective structure. By linking the dynamics to associated iterated function systems (IFS) and employing Sard-type perturbations and transversality arguments, the authors prove that generically the attractor $\Lambda(f)$ is disjoint from the discontinuity set $\Delta(f)$ and hence is a finite union of periodic orbits; they also establish topological stability for these systems. They show openness of the Markov property in a coarse topology and, via a two-step perturbation scheme, density of piecewise smooth Markov contractions in the corresponding space, establishing that Cantor-type attractors are non-generic. The results extend the understanding of high-dimensional discontinuous dynamics beyond the separation property, providing a robust framework for the existence and stability of periodic attractors in piecewise contracting maps.

Abstract

We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of \(\bR^d\). We prove that generically the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.

Piecewise Contractions

TL;DR

This work analyzes piecewise contractions on compact subsets of with piecewise injective structure. By linking the dynamics to associated iterated function systems (IFS) and employing Sard-type perturbations and transversality arguments, the authors prove that generically the attractor is disjoint from the discontinuity set and hence is a finite union of periodic orbits; they also establish topological stability for these systems. They show openness of the Markov property in a coarse topology and, via a two-step perturbation scheme, density of piecewise smooth Markov contractions in the corresponding space, establishing that Cantor-type attractors are non-generic. The results extend the understanding of high-dimensional discontinuous dynamics beyond the separation property, providing a robust framework for the existence and stability of periodic attractors in piecewise contracting maps.

Abstract

We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of . We prove that generically the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.
Paper Structure (12 sections, 24 theorems, 131 equations)

This paper contains 12 sections, 24 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Settings and results
  3. Openness and Topological stability
  4. Density
  5. IFS's associated to the map and their properties
  6. A simple perturbation and the Proof of Proposition \ref{['prop:markov-dense']}
  7. Fixed points in a generic position
  8. Pre-images of the boundary manifolds and how to avoid them
  9. Perturbations with low complexity and the proof of Proposition \ref{['prop:psc-dense']}
  10. Extension Theorem
  11. Transversality
  12. Technical Lemmata

Key Result

Theorem 2.8

A piecewise contraction $f$ with $\Lambda(f)$ as the attractor and $\Delta(f)$ as the union of the set of discontinuities and $\partial X$, satisfies that $\Lambda(f)\cap\Delta(f)=\emptyset$ if and only if it is Markov. Moreover, the attractor of a Markov map consists of periodic orbits.

Theorems & Definitions (71)

  • Definition 2.1: Piecewise contraction
  • Remark 2.2
  • Definition 2.3: Maximal Partition
  • Remark 2.4
  • Definition 2.5: Attractor
  • Definition 2.6: Stabilisation of Partition
  • Definition 2.7: Markov Map
  • Theorem 2.8
  • Remark 2.9
  • Definition 2.10: Piecewise injective contraction
  • ...and 61 more