Abelian periods of factors of Sturmian words

Abstract

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 α with continued fraction expansion [0;a1,a2,…] is either tqk with 1≤t≤ak+1 (a multiple of a denominator qk of a convergent of α) or qk,ℓ (a denominator qk,ℓ of a semiconvergent of α). 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.