Symbolic dynamics in mean dimension theory

TL;DR

This paper extends Furstenberg's dimension calculations from one-sided subshifts to two-dimensional subshifts using mean dimension theory, providing formulas for various dimensions and entropy-related measures.

Contribution

It introduces a generalization of dimension formulas to $ ext{Z}^2$-subshifts, connecting mean dimension, Hausdorff dimension, and entropy.

Findings

Calculated metric mean dimension and mean Hausdorff dimension for $ ext{Z}^2$-subshifts.

Derived formulas for rate distortion dimension in terms of Kolmogorov-Sinai entropy.

Established an analogy to Furstenberg's theorem in higher dimensions.

Abstract

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.