Admissible reversing and extended symmetries for bijective substitutions

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 $({\mathbb{Z}}^d⋊P)\times G$.

29 pages

References in corpus (1)

• Number-theoretic positive entropy shifts with small centraliser and large normaliser

Cited by in corpus (1)

• Large normalizers of ${\mathbb{Z}}^d$-odometers systems and realization on substitutive subshifts