On substitution tilings and delone sets without finite local complexity

Jeong Yup Lee; Boris Solomyak

doi:10.3934/dcds.2019130

On substitution tilings and delone sets without finite local complexity

Jeong Yup Lee, Boris Solomyak

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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.

Original language	English
Pages (from-to)	3149-3177
Number of pages	29
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	39
Issue number	6
DOIs	https://doi.org/10.3934/dcds.2019130
State	Published - Jun 2019

Bibliographical note

Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All Rights Reserved.

Funding

Acknowledgments. We would like to thank to M. Baake, D. Frettlöh, C. Richard, and N. P. Frank for valuable comments and discussions. We are grateful to the anonymous referee for many constructive suggestions which helped to improve the exposition. J.-Y. Lee would like to acknowledge the support by Local University Excellent Researcher Supporting Project through the Ministry of Education of the Republic of Korea and National Research Foundation of Korea (NRF) (2017078374). She is also grateful for the support of the Korea Institute for Advanced Study (KIAS). The research of B. Solomyak was supported in part by the Israel Science Foundation (Grant 396/15). He is grateful to the KIAS, where part of this work was done, for hospitality.

Funders	Funder number
Local University
Institute for Advanced Study
Ministry of Education
National Research Foundation of Korea	2017078374
Israel Science Foundation	396/15

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On substitution tilings and delone sets without finite local complexity

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