  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
[Lee-Solomyak (2012)] 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.
