On spatial entropy of multi-dimensional symbolic dynamical systems

Wen Guei Hu, Song-Sun Lin

Research output: Contribution to journalArticlepeer-review


The commonly used topological entropy htop(U) of the multidimensional shift space U is the rectangular spatial entropy hr(U) which 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 = {(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 d-dimensional Ω. Furthermore, when is genuinely d-dimensional and satisfies certain conditions, then hΩ (U) = hr(U). On the contrary, when Ω (n) contains a lower-dimensional part which is comparable to its d-dimensional part, then hr(U) < hΩ (U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

Original languageEnglish
Pages (from-to)3705-3718
Number of pages14
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number7
StatePublished - 1 Jul 2016


  • Block gluing
  • Shift space
  • Spatial entropy
  • Symbolic dynamical system
  • Topological entropy


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