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

TL;DR

This paper studies the structure of extended symmetries in $bZ^d$-substitution shifts, providing algorithms and new insights into irregular fibres and hierarchical structures within these complex dynamical systems.

Contribution

It introduces an algorithm to compute extended symmetries preserving hierarchical structure and extends the understanding of irregular fibres using derived substitutions.

Findings

Algorithm for computing extended symmetries

Complete description of irregular fibres in higher dimensions

Extension of one-dimensional results to multi-dimensional shifts

Abstract

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 Lema\'{n}czyk, M\"ullner 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.