Brudno's theorem for \(\mathbb{Z}^d\) (or \(\mathbb{Z}_+^d\)) subshifts (Q515582)

In [Trans. Mosc. Math. Soc. 1983, No. 2, 127--151 (1983; Zbl 0532.28019)], \textit{A. A. Brudno} defined a complexity of the trajectory of a point in the space by using the notion of Kolmogorov complexity and showed that for an ergodic shift-invariant measure for a \(1\)-dimensional subshift the Kolmogorov complexity density is equal to the Kolmogorov-Sinai entropy almost everywhere. The authors of this paper generalize Brudno's results to the \(d\)-dimensional case (\(\mathbb{Z}^d\) or \(\mathbb{Z}_+\) subshifts).