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

TL;DR

This paper explores growth invariants of multidimensional shifts of finite type, demonstrating that their complexity can follow non-integer power laws and linking these to arithmetical classes of real numbers.

Contribution

It extends previous examples to show growth complexities of the form exp(n^α) for non-integer α, connecting these to arithmetical classifications.

Findings

Growth complexities of the form exp(n^α) are possible for non-integer α.

Such subshifts have entropy dimension α.

The class of α's is characterized by arithmetical classes of real numbers.

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 Extending an unpublished example of Tsirelson, we show that growth complexities of the form $\exp(n^\alpha)$ are possible for non-integer $\alpha$'s. In terminology of Carvalho, such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.