Admissible reversing and extended symmetries for bijective substitutions

TL;DR

This paper investigates reversing and extended symmetries in shift spaces generated by bijective substitutions, providing conditions, algorithms, and constructions for symmetry groups in higher dimensions.

Contribution

It introduces criteria and algorithms for identifying symmetries in bijective substitutions and constructs examples with prescribed symmetry groups in multiple dimensions.

Findings

Conditions for permutations to generate symmetries

Algorithms to verify symmetries

Existence of substitutions with specified symmetry groups

Abstract

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to check them. Moreover, we show that, for any finite group $G$ and any subgroup $P$ of the $d$-dimensional hyperoctahedral group, there is a bijective substitution which generates an aperiodic hull with symmetry group $\mathbb{Z}^{d}\times G$ and extended symmetry group $(\mathbb{Z}^{d} \rtimes P)\times G$.