Sturmian measures can be sublinearly approximated by periodic measures

To study the approximation rate of an ergodic measure by periodic measures with respect to the Wasserstein distance, we introduce the concept of τ-uniquely ergodic measures, with $\tau \ge 0$. We demonstrate that a τ-uniquely ergodic Borel probability measure on a subshift of finite type can be approximated by periodic measures at a rate of $o({\log }_2^{-\tau }N)$. In particular, we show that a Sturmian measure, which is τ-uniquely ergodic for any $\tau \in [0,1)$, can be approximated by periodic measures with a sublinear rate.