Computable F{\o}lner monotilings and a theorem of Brudno II

Abstract

A theorem of A.A. Brudno says that the Kolmogorov-Sinai entropy of a subshift X over $\mathbb{N}$ with respect to an ergodic measure $\mu$ equals the asymptotic Kolmogorov complexity of almost every word $\omega$ in X. The purpose of this article is to extend this result to subshifts over computable groups that admit computable regular symmetric F{\o}lner monotilings, which we introduce in this work. These monotilings are a special type of computable F{\o}lner monotilings, which we defined earlier in order to extend the initial results of Brudno. For every $d \in \mathbb{N}$, the groups $\mathbb{Z}^d$ and the groups of unipotent upper-triangular matrices of dimension $d+1$ with integer entries admit particularly nice computable regular symmetric F{\o}lner monotilings for which we can provide the required computing algorithms `explicitly'.