Growth-type invariants for $\mathbb{Z}^d$ subshifts of finite type and classes arithmetical of real numbers

Details Growth-type invariants for $\mathbb{Z}^d$ subshifts of finite type and classes arithmetical of real numbers Growth-type invariants for $\mathbb{Z}^d$ subshifts of finite type and classes arithmetical of real numbers

2009-02-02

arxiv.org/abs/0902.0223v1

Mathematics

Dynamical Systems

17 pages, 2 figures

Scientific paper

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 Extending an unpublished example of Tsirelson, we show that growth complexities of the form $\exp(n^\alpha)$ are possible for non-integer $\alpha$'s. In terminology of Carvalho, such subshifts have entropy dimension $\alpha$. The class of possible $\alpha$'s are identified in terms of arithmetical classes of real numbers of Weihrauch and Zheng.

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