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A Note on Specification Property of Dynamical Systems

Received: 20 April 2018     Accepted: 22 May 2018     Published: 28 June 2018
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Abstract

The paper is discussed the sensitive and transitive property of a dynamical system with strong specification property. It is proved that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Further, it is given another condition such that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Meanwhile, it is stated that if a dynamical system has shadowing property, then it is totally syndetically transitive.

Published in Mathematics Letters (Volume 4, Issue 2)
DOI 10.11648/j.ml.20180402.12
Page(s) 34-38
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Sensitive, Specification Property, Syndetically Transitive

References
[1] Bowen R. (1971). Periodic points and measures for axiom a diffeomorphisms, trans, Trans. Amer. Math. Soc. 154, 377–397.
[2] Lampart M., Oprocha P. (2009). Shift spaces, ω-chaos and specification property, Topology & Its Applications 156 (18), 2979–2985.
[3] Kulczycki M., Kwietniak D., Oprocha P. (2013). On almost specification and average shadowing properties, Fundamenta Mathematicae 224 (3), 241–278.
[4] Lidong. W., Hui. W., Guifeng. H. (2014). Minimal sets and ω-chaos in expansive systems with weak specification property, Discrete & Continuous Dynamical Systems 35 (3), 1231–1238.
[5] Dong Y. (2015). Systems with almost specification property may have zero entropy, Fuzzy Sets & Systems 364 (10), 5395–5414.
[6] Kwietniak D., Lacka M., Oprocha P. (2015). A panorama of specification-like properties and their consequences, Contemporary Mathematics arXiv: 1503.07355v2 [math. DS] 26 May. doi:10.1016/S0031-8. 914 (53) 80099-6.
[7] Kwietniak D., Oprocha P., Rams M. (2016). On entropy of dynamical systems with almost specification, Israel Journal of Mathematics 213 (1), 475–503.
[8] Shah S., Das R., Das T. (2016). Specification property for topological spaces, Journal of Dynamical & Control Systems 29 (7), 1–8.
[9] Pfister C., Sullivan W. (2007). On the topological entropy of saturated sets, Ergodic Theory and Dynamical Systems 27 (3), 919–956.
[10] Akin E., Glasner E. (2001). Residual properties and almost equicontinuity, Journald’ Analyse Mathmatique 84 (1), 243–286.
[11] Wen. H., H. Li, Xiangdong. Y. (2012). Family independence for topological and measurable dynamics, Transactions of the American Mathematical Society 364 (10), 5209–5242.
[12] Furstenberg H. (1981). Recurrence in ergodic theory and combinatorial number theory, Princeton University Press.
[13] Xinxing W., Oprocha P., Guanrong C. (2016). On various definitions of shadowing with average error in tracin, Eprint Arxiv 29 (7), 942–1972.
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  • APA Style

    Nan Li, Lidong Wang, Fengchun Lei. (2018). A Note on Specification Property of Dynamical Systems. Mathematics Letters, 4(2), 34-38. https://doi.org/10.11648/j.ml.20180402.12

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    ACS Style

    Nan Li; Lidong Wang; Fengchun Lei. A Note on Specification Property of Dynamical Systems. Math. Lett. 2018, 4(2), 34-38. doi: 10.11648/j.ml.20180402.12

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    AMA Style

    Nan Li, Lidong Wang, Fengchun Lei. A Note on Specification Property of Dynamical Systems. Math Lett. 2018;4(2):34-38. doi: 10.11648/j.ml.20180402.12

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  • @article{10.11648/j.ml.20180402.12,
      author = {Nan Li and Lidong Wang and Fengchun Lei},
      title = {A Note on Specification Property of Dynamical Systems},
      journal = {Mathematics Letters},
      volume = {4},
      number = {2},
      pages = {34-38},
      doi = {10.11648/j.ml.20180402.12},
      url = {https://doi.org/10.11648/j.ml.20180402.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20180402.12},
      abstract = {The paper is discussed the sensitive and transitive property of a dynamical system with strong specification property. It is proved that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Further, it is given another condition such that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Meanwhile, it is stated that if a dynamical system has shadowing property, then it is totally syndetically transitive.},
     year = {2018}
    }
    

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    T1  - A Note on Specification Property of Dynamical Systems
    AU  - Nan Li
    AU  - Lidong Wang
    AU  - Fengchun Lei
    Y1  - 2018/06/28
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    N1  - https://doi.org/10.11648/j.ml.20180402.12
    DO  - 10.11648/j.ml.20180402.12
    T2  - Mathematics Letters
    JF  - Mathematics Letters
    JO  - Mathematics Letters
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    UR  - https://doi.org/10.11648/j.ml.20180402.12
    AB  - The paper is discussed the sensitive and transitive property of a dynamical system with strong specification property. It is proved that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Further, it is given another condition such that if a dynamical system is sensitive, then it is syndetically sensitive with the same constant of sensitivity. Meanwhile, it is stated that if a dynamical system has shadowing property, then it is totally syndetically transitive.
    VL  - 4
    IS  - 2
    ER  - 

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Author Information
  • School of Mathematical Science, Dalian University of Technology, Dalian, China

  • School of Mathematical Science, Dalian University of Technology, Dalian, China

  • School of Mathematical Science, Dalian University of Technology, Dalian, China

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