  We examine the topological dynamics of the automorphism groups of
omega-categorical sparse graphs resulting from Hrushovski constructions.
Specifically, we consider the fixed points on type spaces property, which a
structure M has if, for each positive integer n, every Aut(M)-subflow of the
space of n-types has a fixed point. Extending a result of Evans, Hubicka and
Nesetril, we show that there exists an omega-categorical structure M, resulting
from a Hrushovski construction, such that no omega-categorical expansion of M
has the fixed points on type spaces property.
