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On substitution tilings and Delone sets without finite local complexity

  • * Corresponding author: Jeong-Yup Lee

    * Corresponding author: Jeong-Yup Lee 
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  • 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.

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

    Citation:

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

    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

    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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