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Chain Mixing, Shadowing Properties and Multi-transitivity

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Abstract

We prove that a map f is chain mixing if and only if \(f^r\times f^s\) is chain transitive for some positive integers r, s. We prove that a map which has the average shadowing property with dense \(\underline{0}\)-recurrent points is transitive, and by this result we point out that a map is multi-transitive if it has the average shadowing property and an invariant Borel probability measure with full support. Moreover, we show that \(\Delta \)-mixing, the completely uniform positive entropy and the average shadowing property are equivalent mutually for a surjective map which has the shadowing property.

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Acknowledgements

This research was supported by National Nature Science Funds of China (Grant Nos. 11471125, 11771149). The authors would like to thank the referees for their helpful suggestions and remarks.

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Authors and Affiliations

  1. Department of Mathematics, Guangzhou University, Guangzhou, 510006, People’s Republic of China

    Huoyun Wang, Sulan Diao & Peng Zeng

  2. School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, 526061, People’s Republic of China

    Heman Fu

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  1. Huoyun Wang
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  2. Heman Fu
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  3. Sulan Diao
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  4. Peng Zeng
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Correspondence to Heman Fu.

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Communicated by Mohammad Reza Koushesh.

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Wang, H., Fu, H., Diao, S. et al. Chain Mixing, Shadowing Properties and Multi-transitivity. Bull. Iran. Math. Soc. 45, 1605–1618 (2019). https://doi.org/10.1007/s41980-019-00218-2

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  • DOI: https://doi.org/10.1007/s41980-019-00218-2

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