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The automorphism group of a shift of subquadratic growth


Authors: Van Cyr and Bryna Kra
Journal: Proc. Amer. Math. Soc. 144 (2016), 613-621
MSC (2010): Primary 37B50; Secondary 68R15, 37B10
DOI: https://doi.org/10.1090/proc12719
Published electronically: June 9, 2015
MathSciNet review: 3430839
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Abstract: 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
Email: van.cyr@bucknell.edu

Bryna Kra
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: kra@math.northwestern.edu

DOI: https://doi.org/10.1090/proc12719
Keywords: Subshift, automorphism, block complexity
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
Article copyright: © Copyright 2015 American Mathematical Society

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