Radius-zero Extended Symmetries and Irregular Fibres of ${\mathbb{Z}}^d$-Substitution Subshifts

In this work, we consider ${\mathbb{Z}}^d$-shifts generated by digit substitutions. For such a shift $\mathbb{X}$, we study the elements of the normaliser of ${\mathbb{Z}}^d$ in the group of self homeomorphisms (called extended symmetries) whose local maps guaranteed by the generalised Curtis–Hedlund–Lyndon theorem have radius-zero. Using the formalism of minimal sets developed by Lemańczyk, Müllner and Yassawi, we provide an algorithm to compute elements of $\mathcal{N}(\mathbb{X})$ that preserve the hierarchical structure. We also investigate the interaction of extended symmetries with (i) the height lattice and (ii) the irregular fibres over the maximal equicontinuous factor. Towards (ii), we introduce the notion of derived substitutions to provide a complete description of the irregular fibres, extending a result by Coven, Quas and Yassawi in the one-dimensional case.