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 $f_n:X_n\to X_{n+1},n\in {\mathbb{Z}}_{+}$. In this paper, we prove that if the compression ratio meets $\prod {\lambda }_i=0$, then there exists ${\delta }_n>0$ such that any ${\delta }_n$ - pseudo-orbit ${\left\{x_n\right\}}_{n\in \mathbb{Z}}$ is $\epsilon$ - shadowed by a unique point $x\in X_0$. For the asymptotic average shadowing property, we prove that if ${\mathcal{F}|}_A$ 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.