On minimal subshifts of linear word complexity with slope less than 3/2

TL;DR

This paper proves that minimal subshifts with word complexity growth less than 1.5 times linear are measure-theoretically isomorphic to their maximal equicontinuous factors, confirming Sarnak's conjecture for such systems.

Contribution

It establishes a structural characterization of low-complexity minimal subshifts using S-adic decompositions and describes their maximal equicontinuous factors explicitly.

Findings

Subshifts with $ ext{limsup } p(q)/q < 3/2$ have discrete spectrum.

All such subshifts are measure-theoretically isomorphic to their maximal equicontinuous factors.

The structure of these factors can be explicitly described in terms of substitutions.

Abstract

We prove that every infinite minimal subshift with word complexity $p(q)$ satisfying $\limsup p(q)/q < 3/2$ is measure-theoretically isomorphic to its maximal equicontinuous factor; in particular, it has measurably discrete spectrum. Among other applications, this provides a proof of Sarnak's conjecture for all subshifts with $\limsup p(q)/q < 3/2$ (which can be thought of as a much stronger version of zero entropy). As in \cite{creutzpavlov}, our main technique is proving that all low-complexity minimal subshifts have a specific type of representation via a sequence $\{\tau_k\}$ of substitutions, usually called an S-adic decomposition. The maximal equicontinuous factor is the product of an odometer with a rotation on a compact abelian connected one-dimensional group, for which we can give an explicit description in terms of the substitutions $\tau_k$. We also prove that all such odometers and groups may appear for minimal subshifts with $\limsup p(q)/q = 1$, demonstrating that lower complexity thresholds do not further restrict the possible structure.