A Minimal Subsystem of the Kari-Culik Tilings

TL;DR

This paper studies a specific subset of Kari-Culik tilings, demonstrating minimality of the translation action on those with Sturmian sequence rows, and provides explicit bounds on pattern recurrence times.

Contribution

It characterizes a minimal subsystem of Kari-Culik tilings with Sturmian rows and describes its structure as a skew product, including recurrence bounds.

Findings

The translation action is minimal on the subset with Sturmian rows.

The space is characterized as a skew product.

Explicit bounds on recurrence times are provided.

Abstract

The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that the $\mathbb{Z}^2$ action by translation on a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of $m\times n$ configurations.