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Shadowing Property of 1-dimensional Subsystems for Suspension of \(\mathbb {Z}^{k}\)-actions

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Abstract

In this paper, the shadowing property of 1-dimensional subsystems for suspension of \(\mathbb {Z}^{k}\)-actions is investigated. The concepts of pseudo orbit and shadowing property of 1-dimensional subsystems for suspension of \(\mathbb {Z}^{k}\)-actions are introduced. It is shown that the shadowing properties of these subsystems and their induced nonautonomous dynamical systems are equivalent. For the suspension ΦT of a smooth \(\mathbb {Z}^{k}\)-action T, we show that ΦT has the shadowing property along any Anosov subspace. As an application, we show that ΦT is structurally stable along any Anosov subspace.

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Funding

This study was supported by the National Natural Science Foundation of China (No: 11771118,11801336) and the Applied Basic Research Program of Shanxi Province (No: 201901D211417).

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Authors and Affiliations

  1. School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, China

    Lin Wang

  2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Jinlian Zhang

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  1. Lin Wang
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  2. Jinlian Zhang
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Correspondence to Lin Wang.

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Wang, L., Zhang, J. Shadowing Property of 1-dimensional Subsystems for Suspension of \(\mathbb {Z}^{k}\)-actions. J Dyn Control Syst 29, 1203–1217 (2023). https://doi.org/10.1007/s10883-022-09625-x

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  • DOI: https://doi.org/10.1007/s10883-022-09625-x

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