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

2020/09/16 by Ayşe Şahı̇n, Ayse A. Sahin, Michael Schraudner +4
Computer Science · Materials Science · Mathematics · #37B10 #37B50 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Quasicrystal Structures and Properties #math.DS #msc:37B10 #msc:37B50

paper · pdf · doi:10.48550/arxiv.2009.07751

28 pages, 8 figures; revised version accepted to Illinois Journal of Mathematics

openalex publication_date 2020/09/16 · arxiv created 2021/07/16 · arxiv updated 2021/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Z² 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.

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