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We consider substitution tilings and Delone sets without the assumption offinite local complexity (FLC). We first give a sufficient condition for tilingdynamical systems to be uniquely ergodic and a formula for the measure ofcylinder sets. We then obtain several results on their ergodic-theoreticproperties, notably absence of strong mixing and conditions for existence ofeigenvalues, which have number-theoretic consequences. In particular, if theset of eigenvalues of the expansion matrix is totally non-Pisot, then thetiling dynamical system is weakly mixing. Further, we define the notion ofrigidity for substitution tilings and demonstrate that the result of[Lee-Solomyak (2012)] on the equivalence of four properties: relatively densediscrete spectrum, being not weakly mixing, the Pisot family, and the Meyer setproperty, extends to the non-FLC case, if we assume rigidity instead.

On substitution tilings and Delone sets without finite local complexity