On homeomorphic Bernoulli measures on the Cantor space

Let $\mu(r)$ be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights $r$ and $1-r$. It is a long-standing open problem to characterize those $r$ and $s$ such that $\mu(r)$ and $\mu(s)$ are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending $\mu(r)$ to $\mu(s)$). The (possibly) weaker property of $\mu(r)$ and $\mu(s)$ being continuously reducible to each other is equivalent to a property of $r$ and $s$ called binomial equivalence. In this paper we define an algebraic property called ``refinability'' and show that, if $r$ and $s$ are refinable and binomially equivalent, then $\mu(r)$ and $\mu(s)$ are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers $r$ and $s$ such that $s = r^2$ and $r = 1-s^2$ are refinable, so the corresponding measures are topologically equivalent.