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

Inventiones mathematicae volume 184, pages 567–589 (2011)Cite this article

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 Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, 1992), we show that growth complexities of the form exp (n α+o(1)) are possible for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

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  1. Pacific Institute for the Mathematical Science, 107-1933 West Mall, Vancouver, British Columbia, V6T 1Z2, Canada

    Tom Meyerovitch

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  1. Tom Meyerovitch
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Meyerovitch, T. Growth-type invariants for ℤd subshifts of finite type and arithmetical classes of real numbers. Invent. math. 184, 567–589 (2011). https://doi.org/10.1007/s00222-010-0296-1

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