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Indexed by:Journal Papers

Date of Publication:2019-11-01

Journal:INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS

Included Journals:EI、SCIE

Volume:29

Issue:12

ISSN No.:0218-1274

Key Words:Average shadowing; asymptotic average shadowing; hyperbolicity

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 T 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 T is hyperbolic, then T 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.