论文标题
单词方程的标准单词和解决方案$ x_1^2 \ dotsm x_n^2 =（x_1 \ dotsm x_n）^2 $
Standard words and solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$
论文作者
论文摘要
我们考虑单词方程的解决方案$ x_1^2 \ dotsm x_n^2 =（x_1 \ dotsm x_n）^2 $，使得正方形$ x_i^2 $是最小的正方形，以最佳的方形无限单词找到。我们应用了第二作者开发的一种方法来研究单词方程，并证明有两个解决方案家族：通过简单的替换方案从反向的标准单词获得的反向标准单词和单词。一个特殊而显着的后果是，当且仅当它的反转是对单词方程的解决方案，而$ \ gcd（| w |，| w | _1）= 1 $时，$ w $是标准单词。该结果可以解释为标准斯特里派单词的另一个表征。 我们将结果应用于第一作者和M. A. Whiteland所研究的符号平方根地图$ \ sqrt {\ cdot} $。我们证明，如果最小亚平移$ω$的语言包含无限多个单词方程的解决方案，那么$ω$是sturmian和$ \ sqrt {\ cdot} $ - 不变式或$ω$是$ \ sl-subshift，而不是$ \ \ sqrt $ \ sqrt {\ cdot} $ variant。该结果是证明最小值和$ \ sqrt {\ cdot} $ - 不变的次要shift必然是Sturmian的猜想的进展。
We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$ such that the squares $X_i^2$ 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 $\sqrt{\cdot}$ 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 $\sqrt{\cdot}$-invariant or $Ω$ is a so-called SL-subshift and not $\sqrt{\cdot}$-invariant. This result is progress towards proving the conjecture that a minimal and $\sqrt{\cdot}$-invariant subshift is necessarily Sturmian.
