# Abelian periods of factors of Sturmian words

## TL;DR

This paper investigates the abelian period sets of Sturmian words, revealing their structure in relation to continued fraction expansions and generalizing known results about Fibonacci words.

## Contribution

It provides a detailed characterization of the minimum abelian periods of factors in Sturmian words based on their continued fraction expansion.

## Key findings

- Minimum abelian period is either a multiple of a convergent denominator or a semiconvergent denominator.
- Generalizes the Fibonacci word abelian period set to all Sturmian words.
- Characterizes Fibonacci words through their abelian period sets.

## 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 $\alpha$ 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.