Low Complexity Subshifts have Discrete Spectrum

Darren Creutz; Ronnie Pavlov

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arXiv · 2302.10336

Low Complexity Subshifts have Discrete Spectrum

Darren Creutz·Ronnie Pavlov

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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 $C < \frac{4}{3}$.

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Darren Creutz, Ronnie Pavlov. 2023-02-20. Low Complexity Subshifts have Discrete Spectrum. https://arxiv.org/abs/2302.10336

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