A characterization of Sturmian sequences by indistinguishable asymptotic pairs

TL;DR

This paper characterizes biinfinite Sturmian sequences through indistinguishable asymptotic pairs, extending Pirillo's theorem and using substitutions and derived sequences for a comprehensive understanding.

Contribution

It introduces a new characterization of biinfinite Sturmian sequences via indistinguishable asymptotic pairs, extending classical results to broader contexts.

Findings

Characterization of biinfinite Sturmian sequences using indistinguishable asymptotic pairs

Extension of Pirillo's theorem to biinfinite sequences

Full characterization of indistinguishable pairs on arbitrary alphabets

Abstract

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.