Empirical approach to the x2, x3 conjecture

Abstract

We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\mathrm{mod}1$ and by $3\mathrm{mod}1$, 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.