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

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