Standard words and solutions of the word equation X_1^2 ··· X_n^2 = (X_1 ··· X_n)^2

Abstract

<p><br></p><p>We consider solutions of the word equation <em>X</em>₁² ··· <em>X</em>_n² = (<em>X</em>₁ ··· <em>X</em>_n)² such that the squares <em>X</em>_i² 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 <i>w</i> 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.<br><br>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.</p>