On substitution tilings and delone sets without finite local complexity

Jeong Yup Lee, Boris Solomyak

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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.

Original languageEnglish
Pages (from-to)3149-3177
Number of pages29
JournalDiscrete and Continuous Dynamical Systems
Volume39
Issue number6
DOIs
StatePublished - Jun 2019

Bibliographical note

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© 2019 American Institute of Mathematical Sciences. All Rights Reserved.

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