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Symbolic dynamics in mean dimension theory
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 8 / August 2021
- Published online by Cambridge University Press:
- 15 June 2020, pp. 2542-2560
- Print publication:
- August 2021
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- Article
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Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] 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 the 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.
