Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

TL;DR

This paper demonstrates that any effective subshift in dimension d can be simulated by a two-dimensional subshift of finite type through specific dynamical operations, extending Hochman's earlier results.

Contribution

It proves that effective subshifts can be realized as images of finite type subshifts in one higher dimension using dynamical operations.

Findings

Effective subshifts are simulatable by finite type subshifts in higher dimension.

The result extends Hochman's theorem to a broader class of subshifts.

Simulation is achieved via operations inspired by dynamical systems theory.

Abstract

In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction...). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d -- that is a subshift whose set of forbidden patterns can be generated by a Turing machine -- can be obtained by applying dynamical operations on a subshift of finite type of dimension d + 1 -- a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman's [Hoc09].