Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shadowing in CR-Dynamical Systems

Andrew Wood

TL;DR

This work extends the shadowing paradigm to CR-dynamical systems by introducing $(i,j)$-shadowing, capturing how pseudo-orbits with a single or multiple successor options relate to actual trajectories. It establishes a unified framework linking metric and uniform formulations, invariance under conjugacy, and connections to Mahavier products and isometric dynamical systems, while yielding precise results for finite nondegenerate sets and diagonal-containing relations. The paper also provides deep links between shadowing on the original system, its Mahavier space, and inverse relations, including a landscape of counterexamples that separate the $(i,j)$-types. These contributions offer a robust foundation for further study of shadowing, inverse limits, and specification in CR-dynamical systems, with several open questions guiding future work.

Abstract

A CR-dynamical system is a pair $(X, G)$, where $X$ is a non-empty compact Hausdorff space with uniformity $\mathscr{U}$ and $G$ is a closed relation on $X$. In this paper we introduce the $(i, j)$-shadowing properties in CR-dynamical systems, which generalises the shadowing property from topological dynamical systems $(X, f)$. This extends previous work on shadowing in set-valued dynamical systems.

Shadowing in CR-Dynamical Systems

TL;DR

This work extends the shadowing paradigm to CR-dynamical systems by introducing -shadowing, capturing how pseudo-orbits with a single or multiple successor options relate to actual trajectories. It establishes a unified framework linking metric and uniform formulations, invariance under conjugacy, and connections to Mahavier products and isometric dynamical systems, while yielding precise results for finite nondegenerate sets and diagonal-containing relations. The paper also provides deep links between shadowing on the original system, its Mahavier space, and inverse relations, including a landscape of counterexamples that separate the -types. These contributions offer a robust foundation for further study of shadowing, inverse limits, and specification in CR-dynamical systems, with several open questions guiding future work.

Abstract

A CR-dynamical system is a pair , where is a non-empty compact Hausdorff space with uniformity and is a closed relation on . In this paper we introduce the -shadowing properties in CR-dynamical systems, which generalises the shadowing property from topological dynamical systems . This extends previous work on shadowing in set-valued dynamical systems.

Paper Structure

This paper contains 14 sections, 43 theorems, 44 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CR-dynamical systems
  4. Uniform spaces
  5. The shadowing property
  6. ${\left( {i, j} \right)}$-shadowing properties
  7. Basic definitions
  8. Distinctness of the ${\left( {i, j} \right)}$-shadowing properties
  9. Basic results
  10. Characterisation when ${\operatorname{nondegenerate}{\left( {G} \right)}}$ is finite
  11. When $G$ contains the diagonal
  12. Shadowing in the infinite Mahavier product
  13. Shadowing in $G^{-1}$
  14. Thoughts for future work

Key Result

Theorem 2.1

Let $X$ be a non-empty compact Hausdorff space, and $f : X \to X$ be a function. Then, $f$ is a continuous self-map on $X$ if, and only if, ${\Gamma{\left( {f} \right)}}$ is closed in $X \times X$.

Figures (6)

  • Figure 1: The Comb Space from Example \ref{['ex:comb-space']}
  • Figure 2: The relation $G$ from Example \ref{['ex:diag-const-11']}
  • Figure 3: The relations $G$ and $G^2$ from Example \ref{['ex:comp-fails']}
  • Figure 4: The relation $G$ in Example \ref{['ex:dom-G-finite-assumption-needed-for-21']}
  • Figure 5: The relation $G$ in Example \ref{['ex:domain-finite']}
  • ...and 1 more figures

Theorems & Definitions (104)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 94 more