Abelian periods of factors of Sturmian words

JOURNAL OF NUMBER THEORY

卷:214

Article

Peltomaki, Jarkko^1,2,3 

[1] Univ Turku, Turku Coll Sci & Med TCSM, Turku, Finland

[2] Turku Ctr Comp Sci TUCS, Turku, Finland

[3] Univ Turku, Dept Math & Stat, Turku, Finland

关键词: Sturmian word;    Continued fraction;    Abelian equivalence;    Abelian period;    Singular word;   

DOI  :  10.1016/j.jnt.2020.04.007

来源: Elsevier

【 摘 要 】

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; alpha(1), alpha(2), ...] is either tq(k) with 1 <= t <= a(k)(+1) (a multiple of a denominator q(k) of a convergent of alpha) or q(k)(,l) (a denominator q(k,l) 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. (C) 2020 Elsevier Inc. All rights reserved.

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