Lipschitz Shadowing Property for 1-Dimensional Subsystems of $\mathbb{Z}^{k}$-Actions
Received:September 19, 2020  Revised:April 07, 2021
Key Word: $\mathbb{Z}^{k}$-actions   Lipschitz shadowing   1-dimensional subsystem  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11771118; 11801336), the Applied Basic Research Program of Shanxi Province (Grant No.201901D211417) and the Science and Technology Innovation Project of Shanxi Higher Education (Grant No.2019L0475).
Author NameAffiliation
Lin WANG School of Applied Mathematics, Shanxi University of Finance and Economics, Shanxi 030006, P. R. China 
Jinlian ZHANG School of Mathematical Sciences, Hebei Normal University, Hebei 050024, P. R. China 
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      In this paper, the shadowing property for 1-dimensional subsystems of $\mathbb{Z}^{k}$-actions is investigated. The concepts of pseudo orbit and shadowing property for 1-dimensional subsystems of $\mathbb{Z}^{k}$-actions are introduced in two equivalent ways. For a smooth $\mathbb{Z}^k$-action $T$ on a closed Riemannian manifold, we propose a notion of Anosov direction via the induced nonautonomous dynamical system. Adapting Bowen's geometric method to our case, we show that $T$ has the Lipschitz shadowing property along any Anosov direction.
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