Quantified block gluing, aperiodicity and entropy of multidimensional SFT

TL;DR

This paper explores the relationship between mixing properties, aperiodicity, and entropy in multidimensional subshifts of finite type, demonstrating the existence of linearly block gluing subshifts with various entropy characteristics.

Contribution

It introduces the concept of linearly block gluing subshifts of finite type and shows their aperiodicity and the realization of all right-recursively enumerable positive entropies.

Findings

Existence of linearly block gluing aperiodic subshifts of finite type.

All right-recursively enumerable positive numbers can be realized as entropy.

Answer to the question about entropy characterization of transitive subshifts.

Abstract

It is possible to define mixing properties for subshifts according to the intensity which allows to concatenate two rectangular blocks. We study the interplay between this intensity and computational properties. In particular we prove that there exists linearly block gluing subshift of finite type which are aperiodic and that all right-recursively enumerable positive number can be realized as entropy of linearly block gluing Z 2-subshift of finite type. Like linearly block gluing imply transitivity, this last point answer a question asked in [HM10] about the characterization of the entropy of transitive subshift of finite type.