  We study atomic measures on $[0,1]$ which are invariant both under
multiplication by $2\mod 1$ and by $3\mod 1$, since such measures play an
important role in deciding Furstenberg's $\times 2, \times 3$ conjecture. Our
specific focus was finding atomic measures whose supports are far from being
uniformly distributed, and we used computer software to discover a number of
such measures (which we call \emph{outlier measures}). The structure of these
measures indicates the possibility that a sequence of atomic measures may
converge to a non-Lebesgue measure; likely one which is a combination of the
Lebesgue measure and one or more atomic measures.
