The automorphism group of a minimal shift of stretched exponential growth

Van Cyr; Bryna Kra

doi:10.3934/jmd.2016.10.483

Article Contents

2016, Volume 10: 483-495. Doi: 10.3934/jmd.2016.10.483

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The automorphism group of a minimal shift of stretched exponential growth

Van Cyr^1, and
Bryna Kra^2,

1.
Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837
2.
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208

Received: August 31, 2015

Revised: July 31, 2016

Published: September 2016

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Abstract

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 43A07, 54H20, 68R15.

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