The automorphism group of a shift of slow growth is amenable

Abstract

Suppose $(X,\sigma )$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and $\mathrm{Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/{\log }^{2}n)$, then $\mathrm{Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then $\mathrm{Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup.

Cited by in corpus (3)

• Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts
• Slow entropy and variational dynamical systems
• Large normalizers of ${\mathbb{Z}}^d$-odometers systems and realization on substitutive subshifts