On substitution tilings and Delone sets without finite local complexity

Jeong-Yup Lee; Boris Solomyak

doi:10.3934/dcds.2019130

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2019, Volume 39, Issue 6: 3149-3177. Doi: 10.3934/dcds.2019130 open access

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On substitution tilings and Delone sets without finite local complexity

Jeong-Yup Lee^1,2, , and
Boris Solomyak^3,

1.
Department of Mathematics Education, Catholic Kwandong University, Gangneung, Gangwon 210-701, Korea
2.
KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea
3.
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

^* Corresponding author: Jeong-Yup Lee
^* Corresponding author: Jeong-Yup Lee

Received: March 31, 2018

Revised: October 31, 2018

Published: February 2019

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Abstract

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

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Mathematics Subject Classification: Primary: 37B50; Secondary: 52C23.

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Figure 1. Prototiles of the Frank-Robinson substitution tiling without FLC

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Figure 2. A patch of the tiling from Example 6.4 for $a = 2-\sqrt{2}$. The dots in the figure indicate the representative points of tiles

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Figure 3. Modification of Kenyon's example. The figure shows a patch of the substitution tiling in the case of $a = 2-\sqrt{2}$. The dots in the figure indicate the representative points of tiles

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