Low Complexity Subshifts have Discrete Spectrum

TL;DR

This paper investigates subshifts with linear complexity, proving that those with complexity growth rate below 4/3 have discrete spectrum, and providing examples with higher complexity exhibiting weak mixing, thus partially answering an open question.

Contribution

It establishes a threshold for complexity below which subshifts have discrete spectrum and introduces an example with higher complexity showing weak mixing.

Findings

Subshifts with complexity rate less than 4/3 have discrete spectrum.

An example with complexity rate 3/2 exhibits weak mixing.

The infimum complexity for discrete spectrum lies between 4/3 and 3/2.

Abstract

We prove results about subshifts with linear (word) complexity, meaning that $\limsup \frac{p(n)}{n} < \infty$, where for every $n$, $p(n)$ is the number of $n$-letter words appearing in sequences in the subshift. Denoting this limsup by $C$, we show that when $C < \frac{4}{3}$, the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with $C = \frac{3}{2}$ which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether $C = \frac{5}{3}$ was the minimum possible among such subshifts; our results show that the infimum in fact lies in $[\frac{4}{3}, \frac{3}{2}]$. All results are consequences of a general S-adic/substitutive structure proved when $C < \frac{4}{3}$.