Topological Methods in Nonlinear Analysis

A note on dimensional entropy for amenable group actions

Dou Dou and Ruifeng Zhang

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Abstract

In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered Følner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some Følner sequence) equals its topological entropy. This answers questions by Zheng and Chen [10] and Simpson [9].

Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 2 (2018), 599-608.

Dates
First available in Project Euclid: 24 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1524535231

Digital Object Identifier
doi:10.12775/TMNA.2017.056

Mathematical Reviews number (MathSciNet)
MR3829046

Zentralblatt MATH identifier
06928850

Citation

Dou, Dou; Zhang, Ruifeng. A note on dimensional entropy for amenable group actions. Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 599--608. doi:10.12775/TMNA.2017.056. https://projecteuclid.org/euclid.tmna/1524535231


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References

  • R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136.
  • M. Coornaert, Topological Dimension and Dynamical Systems, Springer, Cham, 2015.
  • D.J. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal. 263 (2012), no. 8, 2228–2254.
  • H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Syst. Theory 1 (1967), 1–49.
  • D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer Monographs in Mathematics, Springer, Cham, 2016.
  • E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), 259–295.
  • D.S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141.
  • Y.B. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, University of Chicago Press, Chicago, IL, 1997.
  • S.G. Simpson, Symbolic dynamics: Entropy,$=$Dimension,$=$Complexity, Theory Comput. Syst. 56 (2015), 527–543.
  • D. Zheng and E. Chen, Bowen entropy for actions of amenable groups, Israel J. Math. 212 (2016), no. 2, 895–911.