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Topological Conjugacy Between Induced Non-autonomous Set-Valued Systems and Subshifts of Finite Type

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Abstract

This paper establishes topological (equi-)semiconjugacy and (equi-) conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li–Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li–Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.

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Acknowledgements

This research was supported by the Hong Kong Research Grants Council (GRF Grant CityU11200317) and the NNSF of China (Grant 11571202). The authors would like to thank Dr. Yang Lou for his assistance in simulation.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Hua Shao

  2. Department of Electrical Engineering, City University of Hong Kong, Hong Kong, People’s Republic of China

    Guanrong Chen

  3. Department of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Yuming Shi

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  1. Hua Shao
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  2. Guanrong Chen
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  3. Yuming Shi
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Correspondence to Yuming Shi.

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Shao, H., Chen, G. & Shi, Y. Topological Conjugacy Between Induced Non-autonomous Set-Valued Systems and Subshifts of Finite Type. Qual. Theory Dyn. Syst. 19, 34 (2020). https://doi.org/10.1007/s12346-020-00369-2

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