Linearly recurrent subshifts have a finite number of non-periodic subshift factors

TL;DR

This paper proves that linearly recurrent subshifts have only finitely many non-periodic factors up to isomorphism and provides a constructive way to characterize these subshifts.

Contribution

It establishes finiteness of non-periodic subshift factors for linearly recurrent subshifts and offers a constructive characterization of such subshifts.

Findings

Finiteness of non-periodic subshift factors for linearly recurrent subshifts.

Constructive characterization of linearly recurrent subshifts.

Bounded return times related to the generating words.

Abstract

A minimal subshift $(X,T)$ is linearly recurrent if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word $u$ the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a linearly recurrent subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.