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Standard words and solutions of the word equation X12⋯Xn2=(X1⋯Xn)2
Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2020-10-07 , DOI: 10.1016/j.jcta.2020.105340
Jarkko Peltomäki , Aleksi Saarela

We consider solutions of the word equation X12Xn2=(X1Xn)2 such that the squares Xi2 are minimal squares found in optimal squareful infinite words. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. A particular and remarkable consequence is that a word w is a standard word if and only if its reversal is a solution to the word equation and gcd(|w|,|w|1)=1. This result can be interpreted as a yet another characterization for standard Sturmian words.

We apply our results to the symbolic square root map studied by the first author and M. A. Whiteland. We prove that if the language of a minimal subshift Ω contains infinitely many solutions to the word equation, then either Ω is Sturmian and -invariant or Ω is a so-called SL-subshift and not -invariant. This result is progress towards proving the conjecture that a minimal and -invariant subshift is necessarily Sturmian.



中文翻译:

标准词和词方程的解 X1个2Xñ2=X1个Xñ2

我们考虑方程的解 X1个2Xñ2=X1个Xñ2 这样的正方形 X一世2是在最佳平方无穷词中找到的最小平方。我们应用第二作者开发的一种方法来研究单词方程,并证明存在两种完全正确的解决方案系列:反向标准单词和通过简单替换方案从反向标准单词获得的单词。一个特殊且引人注目的结果是,当且仅当单词w的反转是单词方程的解并且光盘|w||w|1个=1个。这个结果可以解释为标准Sturmian单词的又一个表征。

我们将结果应用于符号平方根图 由第一作者和MA Whiteland研究。我们证明,如果最小子移位Ω的语言包含对单词方程式的无限多个解,则Ω是Sturmian且-不变或Ω是所谓的SL-subshift,而不是 -不变。这个结果正在朝证明最小和最小-不变亚变位必然是Sturmian。

更新日期:2020-12-14
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