Toeplitz subshifts with trivial centralizers and positive entropy

Abstract

Given a dynamical system $\left(X,G\right)$, the centralizer $C\left(G\right)$ denotes the group of all homeomorphisms of $X$ which commute with the action of $G$. This group is sometimes called the automorphism group of the dynamical system $\left(X,G\right)$. We generalize the construction of Bułatek and Kwiatkowski (1992) to ${\mathbb{Z}}^d$-Toeplitz systems by identifying a class of ${\mathbb{Z}}^d$-Toeplitz systems that have trivial centralizers. We show that this class of ${\mathbb{Z}}^d$-Toeplitz systems with trivial centralizers contains systems with positive topological entropy.