Advertisement

Remarks on Metric Entropy of Random \({\mathbb {Z}}^2\) Actions on a Non-compact Space

  • Zhiming Li
  • Dingxuan Tang
Article
  • 8 Downloads

Abstract

In this paper, we establish Brin–Katok local entropy and Katok \(\delta \) entropy formula of random \({\mathbb {Z}}^2\) transformations on a non-compact space.

Keywords

Metric entropy Local entropy Random dynamical system 

Mathematics Subject Classification

37A35 37H99 

Notes

Acknowledgements

The first author was supported by National Natural Science Foundation of China (No. 11871394), Israel Science Foundation (No. 1289/17) and Natural Science Foundation of Shaanxi Provincial Department of Education (No. 17JK0755). He would also like to thank Professor Jon Aaronson and the School of mathematical sciences of Tel Aviv University for hospitality during his visit there.

References

  1. 1.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)CrossRefGoogle Scholar
  2. 2.
    Bogenschutz, T.: Entropy, pressure and a variational principle for random dynamical systems. Random Comput. Dyn. 1, 99–116 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bogenschutz, T.: Equilibrium states for random dynamical system, PhD. Thesis, Bremen University (1993)Google Scholar
  4. 4.
    Brin, M., Katok, A.: On Local Entropy. Lecture Notes in Mathematics, vol. 1007, pp. 30–38. Springer, Berlin (1983)Google Scholar
  5. 5.
    Crauel, H.: Random probability measures on polish spaces, Habilitationsschrift. Universitat, Bremen (1995)Google Scholar
  6. 6.
    Dooley, A.H., Zhang, G.: Local Entropy Theory of a Random Dynamical System, vol. 233, p. 1099. Memoirs of the American Mathematical Society, Providence (2015)Google Scholar
  7. 7.
    Katok, A.: Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dyn. 1, 545–596 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Katok, A.: Lyapunov exponents, entroy and periodic orbits for diffeomorphisms. Inst. Hautes Etudes Sci. Publ. Math. 51, 137–173 (1980)CrossRefGoogle Scholar
  9. 9.
    Kifer, Y.: Ergodic Theory of Random Transformations. Birkhauser, Basel (1986)CrossRefGoogle Scholar
  10. 10.
    Kifer, Y.: Multidimensional random subshifts of finite type and their large deviations. Probab. Theory Relat. Fields 103, 223–248 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kifer, Y., Liu, P.-D.: Random dynamical systems. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, pp. 379–499. Elsevier, New York City (2006)Google Scholar
  12. 12.
    Li, Z., Shu, L.: The metric entropy of random dynamical Systems in a Banach space: Ruelle inequality. Ergod. Theory Dyn. Syst. 34(02), 594–615 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li, Z., Ding, Z.: Remarks on topological entropy of random dynamical systems. Qual. Theory Dyn. Syst. 17(03), 609–616 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Liu, P.-D., Qian, M.: Smooth Ergodic Theory of Random Dynamical Systems. Lecture Notes in Mathematics, vol. 1606. Springer, Berlin (1995)CrossRefGoogle Scholar
  15. 15.
    Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)CrossRefGoogle Scholar
  16. 16.
    Yujun, Z.: Two notes on measure-theoretic entropy of random dynamical systems. Acta Math. Sinica Engl. Ser. 25(6), 961–970 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhu, Y.: Entropy formula for random \({\mathbb{Z}}^k\)-actions. Trans. Am. Math. Soc. 369, 4517–4544 (2017)CrossRefGoogle Scholar
  18. 18.
    Zhu, Y.: A note on two types of Lyapunov exponents and entropies for \({\mathbb{Z}}^k\)-actions. J. Math. Anal. Appl. 461, 38–50 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

Personalised recommendations