Standard words and solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$

Abstract: 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 $\Omega$ contains infinitely many solutions to the word equation, then either $\Omega$ is Sturmian and $\sqrt{\cdot}$-invariant or $\Omega$ 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.