Brudno's theorem for Z^d (or Z^d_+) subshifts

TL;DR

This paper extends Brudno's theorem, originally for 1D systems, to higher-dimensional lattice subshifts, establishing the equivalence of entropy and complexity density in these systems.

Contribution

It generalizes Brudno's theorem to multi-dimensional subshifts, connecting entropy and Kolmogorov complexity in higher dimensions.

Findings

Kolmogorov-Sinai entropy equals complexity density almost everywhere

Extension of Brudno's theorem to $bZ^d$ and $bZ_+^d$ subshifts

Applicable to ergodic shift-invariant measures

Abstract

We generalize Brudno's theorem of $1$-dimensional shift dynamical system to $\mathbb{Z}^d$ (or $\mathbb{Z}_+^d$) subshifts. That is to say, in $\mathbb{Z}^d$ (or $\mathbb{Z}^d_+$) subshift, the Kolmogorov-Sinai entropy is equivalent to the Kolmogorov complexity density almost everywhere for an ergodic shift-invariant measure.