Subshifts with sparse traces

TL;DR

This paper classifies two-dimensional subshifts with sparse horizontal traces, providing new insights into their structure and implications for cellular automata, nilpotency, and related areas.

Contribution

It offers a classification of minimal subshifts with sparse traces and new proofs for known results, along with constructing illustrative examples.

Findings

Classification of minimal subshifts with sparse traces

New proofs for trace-related properties of SFTs and cellular automata

Construction of illustrative sofic examples

Abstract

We study two-dimensional subshifts whose horizontal trace (a.k.a. projective subdynamics) contains only points of finite support. Our main result is a classification result for such subshifts satisfying a minimality property. As corollaries, we obtain new proofs for various known results on traces of SFTs, nilpotency and decidability of cellular automata, topological full groups and the subshift of prime numbers. We also construct various (sofic) examples illustrating the concepts.