A strongly aperiodic shift of finite type on the discrete Heisenberg group using Robinson tilings

Abstract

We explicitly construct a strongly aperiodic subshift of finite type for the discrete Heisenberg group. Our example builds on the classical aperiodic tilings of the plane due to Raphael Robinson. Extending those tilings to the Heisenberg group by exploiting the group’s structure and posing additional local rules to prune out remaining periodic behavior, we maintain a rich projective subdynamics on $\mathbb{Z}2$ cosets. In addition, the obtained subshift factors onto a strongly aperiodic, minimal sofic shift via a map that is invertible on a dense set of configurations.