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

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.