Let $\varphi$ be a homeomorphism on a totally disconnected, compact metric space $X$. We introduce a binary relation on the family of clopen subsets of $X$, which is described in terms of the $\varphi$-invariant probability measures. We show that $\varphi$ is uniquely ergodic if and only if any two clopen subsets of $X$ are comparable with respect to the binary relation.
"Comparability of clopen sets in a zero-dimensional dynamical system." Proc. Japan Acad. Ser. A Math. Sci. 87 (8) 123 - 127, October 2011. https://doi.org/10.3792/pjaa.87.123