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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 isequivalent to the Kolmogorov complexity density almost everywhere for anergodic shift-invariant measure.

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