Speed of convergence for the realization of an effective subshift by a multidimensional SFT or Sofic

TL;DR

This paper investigates how quickly effective subshifts can be realized as projective sub-actions of higher-dimensional sofic subshifts, focusing on the minimal strip width needed to capture finite portions of the subshift's language.

Contribution

It introduces a topological conjugacy invariant measuring the convergence speed of such realizations, revealing algorithmic properties of effective subshifts.

Findings

Defined a function measuring minimal strip width for finite language approximation

Analyzed the invariant's properties and implications for effective subshifts

Provided insights into the algorithmic complexity of realizing effective subshifts

Abstract

Realization of $d$-dimensional effective subshifts as projective sub-actions of $d+d'$-dimensional sofic subshifts for $d'\geq 1$ is now well know~\cite{Hochman-2009,Durand-Romashchenko-Shen-2010,Aubrun-Sablik-2010}. In this paper we are interested in the speed of convergence of this realization. That is to say given an effective subshift $\Sigma$ realized as projective sub-action of a sofic $\T$, we study the function which on input an integer $k$ returns the smallest width of the strip which verify the local rules of $\T$ necessary to obtain exclusively the language of size $k$ of $\Sigma$ in the central row of the strip. We study this topological conjugacy invariant for effective subshifts in order to exhibit algorithmic properties of these subshifts.