A characterization of Sturmian sequences by indistinguishable asymptotic pairs

We give a new characterization of Sturmian configurations in terms of indistinguishable asymptotic pairs. Two asymptotic configurations on a full Z-shift are indistinguishable if the sets of occurrences of every pattern in each configuration 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 characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.