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

TL;DR

This paper constructs a strongly aperiodic subshift of finite type on the discrete Heisenberg group by extending Robinson tilings, resulting in a rich, minimal, and invertible-factor subshift with applications in symbolic dynamics.

Contribution

It introduces the first explicit strongly aperiodic subshift of finite type on the discrete Heisenberg group, extending classical tilings to a non-abelian setting with novel local rules.

Findings

Successfully constructs a strongly aperiodic subshift on the Heisenberg group.

The subshift factors onto a minimal, strongly aperiodic sofic shift.

Maintains a rich projective subdynamics on $\

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.