The automorphism group of a shift of subquadratic growth (Q2789860)

Let \(\mathcal{A}\) be a finite alphabet and \(X\subset\mathcal{A}^\mathbb{Z}\) a one-dimensional topologically transitive shift. The automorphism group \(\text{Aut}(X)\) is the set of homeomorphisms of \(X\) commuting with the shift map \(\sigma\). The authors show that if the complexity of the shift grows slowly, then \(\text{Aut}(X)\) is small. More precisely, let \(H\) be the subgroup generated by \(\sigma\). Then \(\text{Aut}(X)/H\) is a periodic group if the growth is subquadratic, and even a group of finite exponent if the growth is linear. The proof of these one-dimensional results relies on a theorem by \textit{A. Quas} and \textit{L. Zamboni} [Theor. Comput. Sci. 319, No. 1--3, 229--240 (2004; Zbl 1068.68117)] on colorings of \(\mathbb{Z}^2\), that can be produced here via \(\text{Aut}(X)\). These results contrast with previous studies of \textit{G. A. Hedlund} [Math. Syst. Theory 3, 320--375 (1969; Zbl 0182.56901)] or of \textit{M. Boyle} et al. [Trans. Am. Math. Soc. 306, No. 1, 71--114 (1988; Zbl 0664.28006)] on classes of shifts having an exponential complexity growth, and for which \(\text{Aut}(X)\) is proved to be large.