A simple approach to substitution minimal subshifts

TL;DR

This paper introduces the concepts of 'tame' and '$l$-primitive' substitutions, providing a simple, checkable characterization of minimality in substitution subshifts, unifying previous results and enabling finite algorithmic verification.

Contribution

It establishes that all minimal substitution subshifts can be generated by substitutions satisfying both 'tame' and '$l$-primitive' conditions, offering a unified and computationally feasible framework.

Findings

Characterization of minimality via 'tame' and '$l$-primitive' conditions

Equivalence of these conditions with minimality in substitution subshifts

Validation of linear repetitivity criterion for all substitution subshifts

Abstract

In the study of substitution minimal subshifts, some complicated trivialities have hindered simple and general approaches. Recently, Maloney and Rust introduced the term "tame," simplifying the study. We introduce another term "$l$-primitive" for the substitutions and show that the combination of these two conditions can characterize the minimality of substitution subshifts. We shall show that all substitution minimal subshifts can be generated by substitutions that satisfy both conditions; conversely, all substitutions that satisfy the two conditions always generate minimal subshifts. As an application, we show that the result by Damanik and Lenz that an admissible substitution subshift is minimal if and only if it is linearly repetitive is valid for all substitution subshifts. The above set of conditions can be checked by finite calculations (algorithms).