Abstract
AbstractFurstenberg [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.
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