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Vector Fields with the Asymptotic Orbital Pseudo-orbit Tracing Property

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Abstract

In this paper, we introduce the notion of the asymptotic orbital pseudo-orbit tracing property for a vector field X of a compact smooth manifold M. We show that if a vector field X has the \(C^1\) robust asymptotic orbital pseudo-orbit tracing property, then it is Anosov, and if a \(C^1\) generic vector field X has the asymptotic orbital pseudo-orbit tracing property, then it is Anosov. Moreover, we also show our results for divergence-free vector fields and Hamiltonian systems.

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Acknowledgements

The authors thank the referee for the careful reading and helpful comments that improved the quality of the paper.

Funding

Manseob Lee is supported by National Research Foundation of Korea (NRF) 2017R1A2B4001892 and Junmi Park is supported by Korea (NRF) 2017R1D1A1B03032148.

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Authors and Affiliations

  1. Department of Mathematics, Mokwon University, Daejeon, 35349, Korea

    Manseob Lee

  2. Department of Mathematics, Chungnam National University, Daejeon, 34134, Korea

    Junmi Park

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  1. Manseob Lee
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  2. Junmi Park
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Correspondence to Manseob Lee.

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Lee, M., Park, J. Vector Fields with the Asymptotic Orbital Pseudo-orbit Tracing Property. Qual. Theory Dyn. Syst. 19, 52 (2020). https://doi.org/10.1007/s12346-020-00388-z

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  • DOI: https://doi.org/10.1007/s12346-020-00388-z

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