Skip to main content
SpringerLink
Log in

Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We investigate the relations between Pesin-Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions. Let (X, \({\cal G}\)) be a system, where X is a compact metric space and \({\cal G}\) is a finite family of continuous maps on X. Given a continuous function f on X, we define Pesin-Pitskel topological pressure \({P_{\cal G}}(Z,f)\) for any subset Z ⊂ X and measure-theoretical pressure \({P_{\mu ,{\cal G}}}(X,f)\) for any \(\mu \in {\cal M}(X)\), where \({\cal M}(X)\) denotes the set of all Borel probability measures on X. For any non-empty compact subset Z of X, we show that

$${P_{\cal G}}(Z,f) = \sup \{ {P_{\mu ,{\cal G}}}(X,f):\mu \in {\cal M}(X),\mu (Z) = 1\} .$$

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. Transactions of the American Mathematical Society, 114(2), 309–319 (1965)

    Article  MathSciNet  Google Scholar 

  2. Biś, A.: Entropies of a semigroup of maps. Discrete & Continuous Dynamical Systems A, 11(2–3), 639–648 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Transactions of the American Mathematical Society, 153, 401–414 (1971)

    Article  MathSciNet  Google Scholar 

  4. Bowen, R.: Topological entropy for noncompact sets. Transactions of the American Mathematical Society, 184, 125–136 (1973)

    Article  MathSciNet  Google Scholar 

  5. Bufetov, A.: Topological entropy of free semigroup actions and skew-product transformations. Journal of Dynamical and Control Systems, 5(1), 137–143 (1999)

    Article  MathSciNet  Google Scholar 

  6. Carvalho, M., Rodrigues, F. B., Varandas, P.: Quantitative recurrence for free semigroup actions. Nonlinearity, 31(3), 864–886 (2018)

    Article  MathSciNet  Google Scholar 

  7. Carvalho, M., Rodrigues, F. B., Varandas, P.: A variational principle for free semigroup actions. Advances in Mathematics, 334, 450–487 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dinaburg, E. I.: A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk Sssr, 190, 19–22 (1970)

    MathSciNet  Google Scholar 

  9. Federer, H.: Geometric Measure Theory, Springer-Verlag, New York, 1969

    MATH  Google Scholar 

  10. Feng, D. J., Huang, W.: Variational principles for topological entropies of subsets. Journal of Functional Analysis, 263(8), 2228–2254 (2012)

    Article  MathSciNet  Google Scholar 

  11. Huang, Y., Zhong, X.: Topological entropy of switched systems. Journal of the Korean Mathematial Society, 55(5), 1157–1175 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Hui, H., Ma, D.: Some remarks on measure-theoretic entropy for a free semigroup action. Taiwanese Journal of Mathematics, 21(2), 429–440 (2017)

    Article  MathSciNet  Google Scholar 

  13. Ju, Y., Ma, D., Wang, Y.: Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems A, 39(2), 995–1017 (2019)

    Article  MathSciNet  Google Scholar 

  14. Lin, X., Ma, D., Wang, Y.: On the measure-theoretic entropy and topological pressure of free semigroup actions. Ergodic Theory and Dynamical Systems, 38(2), 686–716 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ma, D., Wu, M.: Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems A, 31(2), 545–556 (2011)

    Article  MathSciNet  Google Scholar 

  16. Ma, J. H., Wen, Z. Y.: A Billingsley type theorem for Bowen entropy. Comptes Rendus Mathematique, 346(9), 503–507 (2008)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  18. Pesin, Y. B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997

    Book  Google Scholar 

  19. Pesin, Y. B., Pitskel, B. S.: Topological pressure and the variational principle for noncompact sets. Functional Analysis and Its Applications, 18(4), 307–318 (1984)

    Article  MathSciNet  Google Scholar 

  20. Ren, Y., He, L., Lü, J., et al. Topological r-entropy and measure-theoretic r-entropy of a continuous map. Science China Mathematics, 54(6), 1197–1205 (2011)

    Article  MathSciNet  Google Scholar 

  21. Ruelle, D.: Statistical mechanics on a compact set with Zv action satisfying expansiveness and specification. Transactions of the American Mathematical Society, 185, 237–251 (1973)

    Article  Google Scholar 

  22. Tang, J., Li, B., Cheng, W. C.: Some properties on topological entropy of free semigroup action. Dynamical Systems, 33(1), 54–71 (2018)

    Article  MathSciNet  Google Scholar 

  23. Tang, X., Cheng, W. C., Zhao, Y.: Variational principle for topological pressures on subsets. Journal of Mathematical Analysis and Applications, 424(2), 1272–1285 (2015)

    Article  MathSciNet  Google Scholar 

  24. Walters, P.: An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982

    Book  Google Scholar 

  25. Wang, C., Chen, E.: Variational principles for BS dimension of subsets. Dynamical Systems: An International Journal, 27(3), 359–385 (2012)

    Article  MathSciNet  Google Scholar 

  26. Wang, Y., Ma, D.: On the topological entropy of a semigroup of continuous maps. Journal of Mathematical Analysis and Applications, 427(2), 1084–1100 (2015)

    Article  MathSciNet  Google Scholar 

  27. Wang, Y., Ma, D., Lin, X.: On the topological entropy of free semigroup actions. Journal of Mathematical Analysis and Applications, 435(2), 1573–1590 (2016)

    Article  MathSciNet  Google Scholar 

  28. Zhu, L., Ma, D.: Topological R-entropy and topological entropy of free semigroup actions. Journal of Mathematical Analysis and Applications, 470(2), 1056–1069 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, P. R. China

    Xing Fu Zhong

  2. School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, P. R. China

    Zhi Jing Chen

Authors
  1. Xing Fu Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Jing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi Jing Chen.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11771459, 11701584 and 11871228), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110932), and the Natural Science Research Project of Guangdong Province (Grant No. 2018KTSCX122)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhong, X.F., Chen, Z.J. Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions. Acta. Math. Sin.-English Ser. 37, 1401–1414 (2021). https://doi.org/10.1007/s10114-021-0403-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-021-0403-9

Keywords

MR(2010) Subject Classification

Search

Navigation