The automorphism group of a shift of subquadratic growth
Van Cyr, Bryna Kra · Proceedings of the American Mathematical Society · 2015
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.