Symbolic substitution systems beyond abelian groups

TL;DR

This paper constructs the first examples of strongly aperiodic, linearly repetitive Delone sets in non-abelian Lie groups using symbolic substitutions, expanding the understanding of aperiodic order beyond abelian groups.

Contribution

It introduces a method to create strongly aperiodic Delone sets in non-abelian Lie groups via symbolic substitutions, a novel extension from abelian cases.

Findings

Constructed strongly aperiodic Delone sets in 2-step nilpotent Lie groups.

Proved the associated dynamical systems are minimal, uniquely ergodic, and weakly aperiodic.

In the 2-step case, the sets are even strongly aperiodic.

Abstract

In this article we construct the first examples of strongly aperiodic linearly repetitive Delone sets in non-abelian Lie groups by means of symbolic substitutions. In particular, we find such sets in all $2$-step nilpotent Lie groups with rational structure constants such as the Heisenberg group. More generally, we consider the class of $1$-connected nilpotent Lie groups whose Lie algebras admit a rational form and a derivation with positive eigenvalues. Any group in this class admits a lattice which is invariant under a natural family of dilations, and this allows us to construct primitive non-periodic symbolic substitutions. We show that, as in the abelian case, the associated subshift (and hence the induced Delone dynamical system) is minimal, uniquely ergodic and weakly aperiodic and consists of linearly repetitive configurations. In the $2$-step nilpotent case, it is even strongly aperiodic.