Symbolic dynamics in mean dimension theory

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.