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Shadowing Property and Invariant Measures Having Full Supports

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Abstract

We consider the shadowing property from the measure theoretical view point. Let f be a homeomorphism of a compact metric space. Then f has the shadowing property on the chain recurrent set if and only if there exists a shadowable f-invariant measure having full support. Suppose that f does not have the shadowing property. Then, if there exists a f-invariant measure having full support, then, generically, every f-invariant measure is not shadowable.

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Acknowledgements

K.S. is supported by JSPS KAKENHI Grant Numbers 16K05167 and 19K03578, and N.S. is supported by JSPS KAKENHI Grant Numbers 15K04902 and 19K03585.

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Authors and Affiliations

  1. Department of Mathematics, Tokushima University, Tokushima, 770-8502, Japan

    Kazumine Moriyasu

  2. Department of Mathematics, Utsunomiya University, Utsunomiya, 321-8505, Japan

    Kazuhiro Sakai

  3. Department of Mathematics, Kumamoto University, Kumamoto, 860-8555, Japan

    Naoya Sumi

Authors
  1. Kazumine Moriyasu
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  2. Kazuhiro Sakai
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  3. Naoya Sumi
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Correspondence to Kazuhiro Sakai.

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Moriyasu, K., Sakai, K. & Sumi, N. Shadowing Property and Invariant Measures Having Full Supports. Qual. Theory Dyn. Syst. 19, 3 (2020). https://doi.org/10.1007/s12346-020-00338-9

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  • DOI: https://doi.org/10.1007/s12346-020-00338-9

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