Slopes of 3-dimensional Subshifts of Finite Type

TL;DR

This paper investigates the slopes of periodicity in 3D subshifts of finite type, demonstrating that any $\,\Sigma^0_2$ set can be represented as a set of slopes, revealing complex relationships between computability and periodic structures.

Contribution

It establishes that all $\,\Sigma^0_2$ sets can be realized as slope sets of 3D SFTs, linking computability classes to geometric periodicity properties.

Findings

Any $\,\Sigma^0_2$ set can be realized as a set of slopes of an SFT.

The study characterizes the possible slopes in 3D SFTs in terms of computability.

It advances understanding of the relationship between computability theory and symbolic dynamics.

Abstract

In this paper we study the directions of periodicity of three-dimensional subshifts of finite type (SFTs) and in particular their slopes. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope being the angle of the periodicity vectors. In this paper, we prove that any $\Sigma^0_2$ set may be realized as a a set of slopes of an SFT.