Admissible Reversing and Extended Symmetries for Bijective Substitutions

Abstract In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic dynamical properties these recall known results regarding their symmetry groups. provide equivalent conditions for a permutation on the alphabet to generate reversing/extended symmetry, algorithms how compute them. Moreover, any finite group H subgroup P d -dimensional hyperoctahedral group, construct substitution which generates an aperiodic subshift $${\mathbb {Z}}^{d}\times H$$ Z d × H $$({\mathbb {Z}}^{d} \rtimes P)\times ( ⋊ P ) . A similar construction same but \times H) P$$ is also provided under mild assumption dimension.