Computable Følner 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 $μ$ equals the asymptotic Kolmogorov complexity of almost every word $ω$ in X. The purpose of this article is to extend this result to subshifts over computable groups that admit computable regular symmetric Følner monotilings, which we introduce in this work. These monotilings are a special type of computable Følner 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 Følner monotilings for which we can provide the required computing algorithms `explicitly'.