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

We generalize Brudno's theorem of 1-dimensional shift dynamical system to Z(d) (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. (C) 2017 Elsevier Inc. All rights reserved