The Jewett-Krieger Construction for Tilings

TL;DR

This paper introduces a tiling construction that models impurity distributions in crystals, creating a uniquely ergodic tiling space with properties similar to the original impurity distribution, extending the Jewett-Krieger theorem to tilings.

Contribution

It develops a new tiling construction analogous to the Jewett-Krieger theorem, capturing impurity distributions in aperiodic, repetitive tilings with finite local complexity.

Findings

Constructed a tiling space with entropy close to impurity distribution

Produced aperiodic, repetitive tilings with finite local complexity

Extended Jewett-Krieger theorem to tiling spaces

Abstract

Given a random distribution of impurities on a periodic crystal, an equivalent uniquely ergodic tiling space is built, made of aperiodic, repetitive tilings with finite local complexity, and with configurational entropy close to the entropy of the impurity distribution. The construction is the tiling analog of the Jewett-Kreger theorem.