Symbolic substitution systems beyond abelian groups
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 -step nilpotent Lie groups with rational structure constants such as the Heisenberg group. More generally, we consider the class of -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 -step nilpotent case, it is even strongly aperiodic.
Keywords
Cite
@article{arxiv.2109.15210,
title = {Symbolic substitution systems beyond abelian groups},
author = {Siegfried Beckus and Tobias Hartnick and Felix Pogorzelski},
journal= {arXiv preprint arXiv:2109.15210},
year = {2025}
}
Comments
Restructured in order to make it accessible to a wider audience. Sections 1 to 7 do not require prior knowledge of Lie group theory, and all Lie theoretic arguments are collected in Section 8. The appendix now contains a complete classification of 7-dimensional substitution groups. The criterion for sufficiently large stretch factors has been relaxed to apply to larger classes of examples