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

TL;DR

This paper extends Brudno's theorem to computable groups with special F{ }olner monotilings, linking entropy and Kolmogorov complexity in a broader class of dynamical systems.

Contribution

It introduces computable regular symmetric F{ }olner monotilings and proves an extension of Brudno's theorem for these structures.

Findings

Groups like ^d and unipotent upper-triangular matrices admit explicit computable Ff6lner monotilings.

The extended theorem relates entropy to asymptotic Kolmogorov complexity in these groups.

Algorithms for computing the monotilings are explicitly provided.

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'.