| Lipschitz Shadowing Property for 1-Dimensional Subsystems of $\mathbb{Z}^{k}$-Actions |
| Received:September 19, 2020 Revised:April 07, 2021 |
| Key Words:
$\mathbb{Z}^{k}$-actions Lipschitz shadowing 1-dimensional subsystem
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| Fund Project: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). |
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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2021.06.006 |
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