Symbolic dynamics in mean dimension theory

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1 (1967), 1-49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to -subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of -subshifts with respect to a subaction of. The resulting formula is quite analogous to Furstenberg's theorem. We also calculate the rate distortion dimension of -subshifts in terms of Kolmogorov-Sinai entropy.