1D Effectively Closed Subshifts and 2D Tilings

TL;DR

This paper demonstrates that 1D effectively closed subshifts can be simulated by 2D subshifts of finite type, extending Hochman's 3D results and exploring two alternative methods rooted in tilings theory.

Contribution

The paper proves the possibility of simulating 1D effectively closed subshifts in 2D, providing two different approaches based on existing tilings theory tools.

Findings

2D subshifts of finite type can simulate 1D effectively closed subshifts.

Two alternative methods for simulation are discussed, one based on Levin's work and the other on Gacs' approach.

The results extend Hochman's 3D findings to 2D, using established tilings techniques.

Abstract

Michael Hochman showed that every 1D effectively closed subshift can be simulated by a 3D subshift of finite type and asked whether the same can be done in 2D. It turned out that the answer is positive and necessary tools were already developed in tilings theory. We discuss two alternative approaches: first, developed by N. Aubrun and M. Sablik, goes back to Leonid Levin; the second one, developed by the authors, goes back to Peter Gacs.