Skip to main content
SpringerLink
Log in

Bowen entropy for actions of amenable groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, Vol. 1007, Springer, Berlin, 1983, pp. 30–38.

    MATH  Google Scholar 

  3. H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Vol. 153, Springer-Verlag, New York, 1969.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Lindenstrauss, Pointwise theorems for amenable groups, Inventiones Mathematicae 146 (2001), 259–295.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Institut des Hautes Études Sientifiques. Publications Mathématiques 51 (1980), 137–173.

    Article  MathSciNet  Google Scholar 

  7. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  8. J. M. Ollagnier and D. Pinchon, The variational principle, Studia Mathematica 72 (1982), 151–159.

    MathSciNet  MATH  Google Scholar 

  9. D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d’Analyse Mathématique 48 (1987), 1–141.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ya. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.

    Book  MATH  Google Scholar 

  11. A. Shulman, Maximal ergodic theorems on groups, Dep. Lit. NIINTI, No.2184, 1988.

    Google Scholar 

  12. A. M. Stepin and A. T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Doklady Akademii Nauk SSSR 254 (3) (1980), 545–549 (in Russian).

    MathSciNet  MATH  Google Scholar 

  13. B. Weiss, Actions of amenable groups, in Topics in Dynamics and Ergodic Theory, London Mathematical Society Lecture Note Series, Vol. 310, Cambridge University Press, Cambridge, 2003, pp. 226–262.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University, Nanjing, 210023, Jiangsu, P. R. China

    Dongmei Zheng

  2. Department of Applied Mathematics, College of Science Nanjing Tech University, Nanjing, 211816, Jiangsu, P. R. China

    Dongmei Zheng

  3. School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University, Nanjing, 210023, Jiangsu, P. R. China

    Ercai Chen

Authors
  1. Dongmei Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ercai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Dongmei Zheng or Ercai Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, D., Chen, E. Bowen entropy for actions of amenable groups. Isr. J. Math. 212, 895–911 (2016). https://doi.org/10.1007/s11856-016-1312-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-016-1312-y

Search

Navigation