Abelian periods of factors of Sturmian words

We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $\alpha$ with continued fraction expansion $[0; a_1, a_2, \ldots]$ is either $tq_k$ with $1 \leq t \leq a_{k+1}$ (a multiple of a denominator $q_k$ of a convergent of $\alpha$) or $q_{k,\ell}$ (a denominator $q_{k,\ell}$ of a semiconvergent of $\alpha$). This result generalizes a result of Fici et. al stating that the abelian period set of the Fibonacci word is the set of Fibonacci numbers. A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary.

💡 Research Summary

The paper investigates the set of minimum abelian periods of factors of Sturmian words, which are infinite binary sequences obtained by coding irrational rotations on the unit circle. An abelian period of a finite word w is defined as the length m of a block such that w appears as a factor of an abelian power—i.e., a concatenation of blocks of length m that are pairwise abelian equivalent (they have the same multiset of letters). The abelian period set of an infinite word is the collection of the smallest abelian periods of all its non‑empty factors.

The authors begin by recalling classical results on ordinary periods of Sturmian words. Currie and Saari showed that the ordinary period set of a Sturmian word with slope α (continued‑fraction expansion