Corrigendum and addendum to: Linearly recurrent subshifts have a finite number of non-periodic factors

TL;DR

This paper establishes a precise characterization of linearly recurrent subshifts as primitive and proper S-adic systems, correcting a previous proposition and deepening understanding of their structure.

Contribution

It provides a corrected and complete proof that linearly recurrent subshifts are exactly primitive and proper S-adic systems, refining prior results.

Findings

Characterization of linearly recurrent subshifts as primitive and proper S-adic systems

Correction of Proposition 6 from Durand (2000)

Enhanced understanding of subshift structures

Abstract

We prove that a subshift $(X,T)$ is linearly recurrent if and only if it is a primitive and proper $S$-adic subshift. This corrects Proposition 6 in F. Durand ({\it Ergod. Th. & Dynam. Sys. {\bf 20}} (2000), 1061--1078).