Regularity of aperiodic minimal subshifts

Fabian Dreher, Marc Kesseböhmer, Arne Mosbach, Tony Samuel, Malte Steffens

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2 Citations (Scopus)
122 Downloads (Pure)


At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely α-repetitive, α-repulsive and α-finite (α≥1), have been introduced and studied. We establish the equivalence of α-repulsive and α-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk’s infinite 2-group G. In particular, we show that these subshifts provide examples that demonstrate α-repulsive (and hence α-finite) is not equivalent to α-repetitive, for α>1. We also give necessary and sufficient conditions for these subshifts to be α-repetitive, and α-repulsive (and hence α-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.

Original languageEnglish
Pages (from-to)413–434
Number of pages22
JournalBulletin of Mathematical Sciences
Issue number3
Early online date29 Mar 2017
Publication statusPublished - 2018


  • Aperiodic order
  • Complexity
  • Subshifts
  • Grigorchuk group
  • Unique ergodicity


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