On spatial entropy of multi-dimensional symbolic dynamical systems

Wen-Guei Hu Sichuan University Song-Sun Lin National Chiao Tung University

Dynamical Systems mathscidoc:1609.11004

Discrete and Continuous Dynamical systems -A, 36, 3705-3718, 2016
The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.
Spatial entropy; shift space; topological entropy
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  title={On spatial entropy of multi-dimensional symbolic dynamical systems},
  author={Wen-Guei Hu, and Song-Sun Lin},
  booktitle={Discrete and Continuous Dynamical systems -A},
Wen-Guei Hu, and Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. 2016. Vol. 36. In Discrete and Continuous Dynamical systems -A. pp.3705-3718. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914161142351112006.
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