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

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.