A Three-Dimensional SFT with Sparse Columns

TL;DR

This paper constructs a 3D subshift of finite type with a 2-sparse projective subdynamics, demonstrating a novel connection to cellular automata and exploring variants with binary alphabets.

Contribution

It introduces a new 3D subshift with specific sparsity properties and establishes its relation to cellular automata, including variants with simplified alphabets.

Findings

Constructed a 3D subshift with 2-sparse projective subdynamics.

Proved the subshift is topologically conjugate to spacetime diagrams of a partial cellular automaton.

Presented variants defined by Wang cubes and with binary alphabets.

Abstract

We construct a nontrivial three-dimensional subshift of finite type whose projective $\Z$-subdynamics, or $\Z$-trace, is 2-sparse, meaning that there are at most two nonzero symbols in any vertical column. The subshift is deterministic in the direction of the subdynamics, so it is topologically conjugate to the set of spacetime diagrams of a partial cellular automaton. We also present a variant of the subshift that is defined by Wang cubes, and one whose alphabet is binary.