  We show that Kellendonk's tiling semigroup of an FLC substitution tiling is
self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend
the notion of the limit space of a self-similar group to the setting of
self-similar semigroups, and show that it is homeomorphic to the
Anderson--Putnam complex for such substitution tilings, with natural self-map
induced by the substitution. Thus, the inverse limit of the limit space, given
by the limit solenoid of the self-similar semigroup, is homeomorphic to the
translational hull of the tiling.
