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.

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Sturmian 词因子的阿贝尔周期

我们研究了 Sturmian 词的阿贝尔周期集，它们是一维环面上无理旋转的编码。主要结果表明角 α 的 Sturmian 词的因子的最小阿贝尔周期具有连分数展开式 [0; a1, a2, . . .] 是 tqk，其中 1 ≤ t ≤ ak+1（α 的收敛的分母 qk 的倍数）或 qk,l（α 的半收敛的分母 qk,l）。该结果概括了 Fici 等人的结果。al 说明斐波那契词的阿贝尔周期集是斐波那契数集。就其阿贝尔周期集而言，斐波那契词的特征是作为推论获得的。