Journal of Modern Dynamics (JMD)

The automorphism group of a minimal shift of stretched exponential growth

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

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Van Cyr - Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837, United States (email)
Bryna Kra - Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, United States (email)

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.

Keywords:  Subshift, automorphism, block complexity, zero entropy, amenable.
Mathematics Subject Classification:  Primary: 37B10; Secondary: 43A07, 54H20, 68R15.

Received: September 2015;      Revised: August 2016;      Available Online: October 2016.