Sturmian measures can be sublinearly approximated by periodic measures

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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 τ≥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(log2−τ⁡N). In particular, we show that a Sturmian measure, which is τ-uniquely ergodic for any τ∈[0,1), can be approximated by periodic measures with a sublinear rate.