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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The automorphism group of a shift of subquadratic growth
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by Van Cyr and Bryna Kra PDF
Proc. Amer. Math. Soc. 144 (2016), 613-621 Request permission


For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift $X$, the automorphism group $\operatorname {Aut}(X)$ is small: if $H$ is the subgroup of $\operatorname {Aut}(X)$ generated by the shift, then $\operatorname {Aut}(X)/H$ is periodic. For linear growth, we show the stronger result that $\operatorname {Aut}(X)/H$ is a group of finite exponent.
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Additional Information
  • Van Cyr
  • Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
  • MR Author ID: 883244
  • Email:
  • Bryna Kra
  • Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
  • MR Author ID: 363208
  • ORCID: 0000-0002-5301-3839
  • Email:
  • Received by editor(s): March 2, 2014
  • Received by editor(s) in revised form: January 8, 2015
  • Published electronically: June 9, 2015
  • Additional Notes: The second author was partially supported by NSF grant $1200971$.
  • Communicated by: Nimish Shah
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 613-621
  • MSC (2010): Primary 37B50; Secondary 68R15, 37B10
  • DOI:
  • MathSciNet review: 3430839