Bratteli-Vershik representations of some one-sided substitution subshifts

TL;DR

This paper explores the representation of one-sided substitution subshifts using Bratteli-Vershik systems, focusing on quasi-invertible substitutions and their properties, including an algorithm for recognition and conditions for Bratteli-Vershik representation.

Contribution

It introduces an algorithm to identify quasi-invertible substitutions and proves that such subshifts can be represented by Bratteli-Vershik systems under certain conditions.

Findings

Quasi-invertible substitutions with rational Perron values are orbit equivalent.

Left proper quasi-invertible substitution subshifts have a specific fixed point structure.

Examples show differences between 1-sided and 2-sided subshifts in conjugacy.

Abstract

We study one-sided substitution subshifts, and how they can be represented using Bratteli-Vershik systems. In particular we focus on minimal recognizable substitutions such that the generated one-sided substitution subshift contains only one non-shift-invertible element (a branch point), and we call these substitutions quasi-invertible. We give an algorithm to check whether a substitution is quasi-invertible, and show that any substitution with a rational Perron value is orbit equivalent to a quasi-invertible substitution. If the quasi-invertible substitution is left proper, then its subshift is equal to a substitution subshift where the original branch point is the new substitution fixed point. We use these results to prove that any reasonable quasi-invertible substitution subshift has a Bratteli-Vershik representation. We also give an example of a pair of substitutions whose 2-sided subshifts are topologically conjugate, while their 1-sided subshifts are not.