Abelian periods of factors of Sturmian words

Abstract

<p>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; <em>a</em>₁, <em>a</em>₂, ...] is either <em>tq</em>_k with 1 ≤ t ≤ <em>a</em>_k+1 (a multiple of a denominator <em>q</em>_k of a convergent of α) or <em>q</em>_k,l (a denominator <em>q</em>_k,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.</p>