We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full $\mathbb{Z}$-shift are indistinguishable if the sets of occurrences of every pattern in each sequence coincide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillo’s theorem which characterizes Christoffel words. Furthermore, we provide a full characterization of indistinguishable asymptotic pairs on arbitrary alphabets using substitutions and biinfinite characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.

我们用不可区分的渐近对给出了双无限Sturmian序列的新特征。完整的两个渐近序列$\mathbb{ž}$如果每个序列中每个模式的出现集与有限支持的排列重合，则-shift是无法区分的。这种表征可以看作是对Pirillo定理的双无限序列的扩展，该定理描述了Christoffel单词。此外，我们提供了使用替换和双无限特征Sturmian序列在任意字母上无法区分的渐近对的完整表征。该证明基于派生序列的众所周知的概念。