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Asymptotic Measure Expansive Flows

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Abstract

In this paper, we extend the notion of asymptotic measure expansivity for diffeomorphisms to flows on a compact metric space, and prove that there exists an asymptotic measure expansive flow in that space. Then we consider the hyperbolicity of the asymptotic measure expansive vector fields on a closed smooth Riemannian manifold M. More precisely, we prove that any C1 stably asymptotic measure expansive vector field on M satisfy Axiom A without cycles, and it is quasi-Anosov. On the other hand, we show that C1 generically, every asymptotic measure expansive vector field on M satisfies Axiom A without cycles. Moreover, we study the asymptotic measure expansivity of the homoclinic classes which are the delegate of the invariant subsets for given systems, prove that C1 stably and C1 generically, the asymptotic measure expansive homoclinic class is hyperbolic. Furthermore, we consider the hyperbolicity of asymptotic measure expansive divergence-free vector fields for C1 stably and C1 generic point of view. We also apply the our results to the divergence-free vector fields.

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Acknowledgements

The authors wish to express their appreciation to Professor Keonhee Lee for his valuable comments.

Funding

Manseob Lee is supported by NRF-2020R1F1A1A01051370. And Jumi Oh is supported by the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159 and NRF 2019R1A2C1002150.

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Authors and Affiliations

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

    Manseob Lee

  2. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, South Korea

    Jumi Oh

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  1. Manseob Lee
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  2. Jumi Oh
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Correspondence to Jumi Oh.

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Lee, M., Oh, J. Asymptotic Measure Expansive Flows. J Dyn Control Syst 29, 293–318 (2023). https://doi.org/10.1007/s10883-022-09598-x

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  • DOI: https://doi.org/10.1007/s10883-022-09598-x

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