Growth-type invariants for ℤd subshifts of finite type and arithmetical classes of real numbers

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Abstract

We discuss some numerical invariants of multidimensional shifts of finite type (SFTs) which are associated with the growth rates of the number of admissible finite configurations. Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, 1992), we show that growth complexities of the form exp (nα+o(1)) are possible for non-integer α's. In terminology of de Carvalho (Port. Math. 54(1):19-40, 1997), such subshifts have entropy dimension α. The class of possible α's are identified in terms of arithmetical classes of real numbers of Weihrauch and Zheng (Math. Log. Q. 47(1):51-65, 2001).

Original languageEnglish
Pages (from-to)567-589
Number of pages23
JournalInventiones Mathematicae
Volume184
Issue number3
DOIs
StatePublished - 1 Jun 2011
Externally publishedYes

Keywords

  • 37B10
  • 37B40
  • 37B50

ASJC Scopus subject areas

  • Mathematics (all)

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