Low Complexity Subshifts have Discrete Spectrum

Abstract

We prove results about subshifts with linear (word) complexity, meaning that $\mathrm{limsup}\frac{p(n)}{n}<\infty$, 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 $C<\frac{4}{3}$.

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• Slow entropy and variational dynamical systems