Inventiones mathematicae

, Volume 184, Issue 3, pp 567-589

First online:

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

  • Tom MeyerovitchAffiliated withPacific Institute for the Mathematical Science Email author 

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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 Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, 1992), we show that growth complexities of the form exp (n α+o(1)) are possible for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).

Mathematics Subject Classification (2000)

37B10 37B40 37B50