Regularity of aperiodic minimal subshifts

Research output: Contribution to journalArticlepeer-review


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

Colleges, School and Institutes

External organisations

  • University of Bremen


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