Symbolic dynamics in mean dimension theory

<jats:title>Abstract</jats:title><jats:p>Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.<jats:italic>Math. Syst. Theory</jats:italic><jats:bold>1</jats:bold>(1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385720000474_inline1.png" /><jats:tex-math>$\mathbb{Z}^{2}$</jats:tex-math></jats:alternatives></jats:inline-formula>-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385720000474_inline2.png" /><jats:tex-math>$\mathbb{Z}^{2}$</jats:tex-math></jats:alternatives></jats:inline-formula>-subshifts with respect to a subaction of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385720000474_inline3.png" /><jats:tex-math>$\mathbb{Z}$</jats:tex-math></jats:alternatives></jats:inline-formula>. The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385720000474_inline4.png" /><jats:tex-math>$\mathbb{Z}^{2}$</jats:tex-math></jats:alternatives></jats:inline-formula>-subshifts in terms of Kolmogorov–Sinai entropy.</jats:p>