Low complexity subshifts have discrete spectrum

Abstract

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, that is, 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 $C < \frac {4}{3}$ .