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The Shadowing Properties Of Nonautonomous Dynamical System

Min An

Abstract

Let $\left(X_n, d_n\right)$ be a sequence of metric spaces and let $\mathcal{F}=\left\{f_n\right\}_{n \in \mathbb{Z}}$ be a sequence of continuous and onto maps $f_n: X_n \rightarrow X_{n+1}, n \in \mathbb{Z}_{+}$. In this paper, we prove that if the compression ratio meets $\prod λ_i=0$, then there exists $δ_n>0$ such that any $δ_n$ - pseudo-orbit $\left\{x_n\right\}_{n \in \mathbb{Z}}$ is $\varepsilon$ - shadowed by a unique point $x \in X_0$. For the asymptotic average shadowing property, we prove that if $\left.\mathcal{F}\right|_A$ has asymptotically average shadowing property, then $\mathcal{F}$ also has a asymptotically average shadowing propertywhen a density-related condition is satisfied. Additionally, the conclusion that the shadowing performance of strong equicontinuity and pseudo-shadowing property implies limit shadowing is also obtained. Furthermore, the shadowing property of non-autonomous product space is also discussed.

The Shadowing Properties Of Nonautonomous Dynamical System

Abstract

Let be a sequence of metric spaces and let be a sequence of continuous and onto maps . In this paper, we prove that if the compression ratio meets , then there exists such that any - pseudo-orbit is - shadowed by a unique point . For the asymptotic average shadowing property, we prove that if has asymptotically average shadowing property, then also has a asymptotically average shadowing propertywhen a density-related condition is satisfied. Additionally, the conclusion that the shadowing performance of strong equicontinuity and pseudo-shadowing property implies limit shadowing is also obtained. Furthermore, the shadowing property of non-autonomous product space is also discussed.
Paper Structure (6 sections, 11 theorems, 78 equations)

This paper contains 6 sections, 11 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Shadowing Properties of Nonautonomous Dynamical System
  4. Description of Theorem
  5. Asymptotic Average Shadowing
  6. Shadowing For Product Systems

Key Result

Theorem 2.1

(1) Let $\left(X_n, d_n\right)$ be a sequence of metric spaces and let $\mathcal{F}=\left\{f_n\right\}_{n \in \mathbb{Z}_{+}}$ be a sequence of expanding maps and onto maps $f_n: X_n \rightarrow X_{n+1}, n \in \mathbb{Z}_{+}$. If the contraction rates $\left\{\lambda_n\right\}_{n \in \mathbb{Z}_{+}}

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.1
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.2
  • ...and 12 more