Standard words and solutions of the word equation [formula omitted

Abstract

We consider solutions of the word equation X12⋯Xn2=(X1⋯Xn)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.