Effective closed subshifts in 1D can be implemented in 2D

TL;DR

This paper demonstrates that any one-dimensional effectively closed subshift can be realized in two dimensions using fixed point tilings, leveraging aperiodic tile sets and their extensions.

Contribution

It introduces a method to implement 1D effectively closed subshifts in 2D via fixed point tilings, answering a question by Hochman.

Findings

Every 1D effectively closed subshift can be implemented in 2D

Uses fixed point tilings and aperiodic tile sets

Provides a constructive approach for 2D implementation

Abstract

In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions.