Skip to main content
SpringerLink
Log in

The Variational Principle for the Packing Entropy of Nonautonomous Dynamical Systems

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

Let (X, ϕ) be a nonautonomous dynamical system. In this paper, we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems. Moreover, we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler R L, Konheim A G, McAndrew M H. Topological entropy. Trans Amer Math Soc, 1965, 114: 309–319

    Article  MathSciNet  MATH  Google Scholar 

  2. Biś A. Topological and measure-theoretical entropies of nonautonomous dynamical systems. J Dynam Differential Equations, 2018, 30(1): 273–285

    Article  MathSciNet  MATH  Google Scholar 

  3. Bowen R. Topological entropy for noncompact sets. Trans Amer Math Soc, 1973, 184: 125–136

    Article  MathSciNet  MATH  Google Scholar 

  4. Dinaburg E I. A correlation between topological entropy and metric entropy. Dokl Akad Nauk SSSR, 1970, 190: 19–22

    MathSciNet  Google Scholar 

  5. Dou D, Zheng D, Zhou X. Packing topological entropy for amenable group actions. Ergodic Theory Dynam Systems, 2021. DOI:https://doi.org/10.1017/etds.2021.126

  6. Feng D, Huang W. Variational principles for topological entropies of subsets. J Funct Anal, 2012, 263(8): 2228–2254

    Article  MathSciNet  MATH  Google Scholar 

  7. Goodman T N T. Relating topological entropy and measure entropy. Bull London Math Soc, 1971, 3(2): 176–180

    Article  MathSciNet  MATH  Google Scholar 

  8. Goodwyn L W. Topological entropy bounds measure-theoretic entropy. Proc Amer Math Soc, 1969, 23: 679–688

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang X, Wen X, Zeng F. Topological pressure of nonautonomous dynamical systems. Nonlinear Dyn Syst Theory, 2008, 8(1): 43–48

    MathSciNet  MATH  Google Scholar 

  10. Jech T. Set Theory. Berlin: Springer, 2003

    MATH  Google Scholar 

  11. Ju Y, Yang Q. Pesin topological entropy of nonautonomous dynamical systems. J Math Anal Appl, 2021, 500(2): 125125

    Article  MathSciNet  MATH  Google Scholar 

  12. Kawan C. Metric entropy of nonautonomous dynamical systems. Nonauton Dyn Syst, 2014, 1(1): 26–52

    MathSciNet  MATH  Google Scholar 

  13. Kolmogorov A N. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl Akad Nauk SSSR (Russian), 1958, 119: 861–864

    MathSciNet  MATH  Google Scholar 

  14. Kolyada S, Snoha L. Topological entropy of nonautonomous dynamical systems. Random Comput Dynam, 1996, 4(2/3): 205–233

    MathSciNet  MATH  Google Scholar 

  15. Kuang R, Cheng W, Li B. Fractal entropy of nonautonomous systems. Pacific J Math, 2013, 262(2): 421–436

    Article  MathSciNet  MATH  Google Scholar 

  16. Li Z, Zhang W, Wang W. Topological entropy dimension for nonautonomous dynamical systems. J Math Anal Appl, 2019, 475(2): 1978–1991

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu K, Qiao Y, Xu L. Topological entropy of nonautonomous dynamical systems. J Differential Equations, 2020, 268(9): 5353–5365

    Article  MathSciNet  MATH  Google Scholar 

  18. Mattila P. Geometry of Sets and Measures in Euclidean Spaces. Cambridge: Cambridge University Press, 1995

    Book  MATH  Google Scholar 

  19. Xu L, Zhou X. Variational principles for entropies of nonautonomous dynamical systems. J Dynam Differential Equations, 2018, 30(3): 1053–1062

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhou X, Chen E, Cheng W. Packing entropy and divergence points. Dyn Syst, 2012, 27(3): 387–402

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhu Y, Liu Z, Xu X, Zhang W. Entropy of nonautonomous dynamical systems. J Korean Math Soc, 2012, 49(1): 165–185

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Prof. Wen Huang and Prof. Dou Dou for very helpful discussions.

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Ruifeng Zhang  (张瑞丰) & Jianghui Zhu  (朱姜慧)

Authors
  1. Ruifeng Zhang  (张瑞丰)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianghui Zhu  (朱姜慧)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruifeng Zhang  (张瑞丰).

Additional information

Conflict of Interest

The authors declare no conflict of interest.

This work was supported by the National Natural Science Foundation of China (11871188, 12031019).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, R., Zhu, J. The Variational Principle for the Packing Entropy of Nonautonomous Dynamical Systems. Acta Math Sci 43, 1915–1924 (2023). https://doi.org/10.1007/s10473-023-0426-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-023-0426-7

Key words

2010 MR Subject Classification

Search

Navigation