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arXiv · 1910.00844

Symbolic dynamics in mean dimension theory

Mao Shinoda·Masaki Tsukamoto
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Abstract

Furstenberg (1967) calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^2$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and mean Hausdorff dimension of $\mathbb{Z}^2$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg's theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^2$-subshifts in terms of Kolmogorov-Sinai entropy.

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Mao Shinoda, Masaki Tsukamoto. 2019-10-02. Symbolic dynamics in mean dimension theory. https://doi.org/10.1017/etds.2020.47

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