Automorphisms of \(\mathbb Z^ d\)-subshifts of finite type (Q1842552)

Let \((\Sigma, \sigma)\) be a \(\mathbb Z^ d\)-subshift of finite type. The automorphism group \(\Aut(\Sigma)\) is the group of homeomorphisms of \(\Sigma\) commuting with \(\sigma\). This paper extends to dimensions \(d>1\) results of \textit{M. Boyle}, \textit{D. Lind} and \textit{D. Rudolph} [Trans. Am. Math. Soc. 306, No. 1, 71--114 (1988; Zbl 0664.28006)] and Curtis, Hedlund and Lyndon [see \textit{G. A. Hedlund}, Topological Dynamics, Int. Symp. Colorado State Univ. 1967, 259--289 (1968; Zbl 0195.52702)]. In particular, it is shown that under a strong irreducibility condition, \(\Aut(\Sigma)\) contains any finite group. For \(\mathbb Z^ d\)-subshifts of finite type without this irreducibility condition, it is shown using examples that topological mixing is not sufficient to give any finite group in the automorphism group, in general. \(\Aut(\Sigma)\) is explicitly calculated for a particular topological mixing \(\mathbb Z^ 2\)-subshift of finite type and some open questions are posed.