Average Shadowing Property and Asymptotic Average Shadowing Property of Linear Dynamical Systems

Abstract

This paper investigates the average shadowing property and the asymptotic average shadowing property of linear dynamical systems in Banach spaces. Firstly, necessary and sufficient conditions for an invertible operator [Formula: see text] on a Banach space to have the average shadowing property and the asymptotic average shadowing property are given, respectively. Then, it is concluded that both the average shadowing property and the asymptotic average shadowing property are preserved under iterations. Furthermore, if [Formula: see text] is hyperbolic, then [Formula: see text] has the (asymptotic) average shadowing property. However, the inverse implication fails in infinite-dimensional Banach spaces. Finally, it is proved that the (asymptotic) average shadowing property is equivalent to the hyperbolicity for dynamical systems in a finite-dimensional Banach space.