The Shadowing Properties Of Nonautonomous Dynamical System

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.