On some properties of doubly-periodic words

Baiocchi, Claudio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 8 (1997) no. 1, pp. 39-47

Résumé

We study the functional equation: \( (1) \, ABC = CDA \) where \(A, B, C\) and \(D\) are words over an alphabet \( \mathcal{A} \). In particular we prove a «structure result» for the inner factors \(B, D\): for suitably chosen words \(X, Y, Z\) one has: \( (2) \, B =XYZ\), \(D = ZYX\)\( (2) \, B =XYZ\), \(D = ZYX\)\( (2) \, B =XYZ\), \(D = ZYX\)\( (2) \, B =XYZ\), \(D = ZYX\). It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) \(A\) or \(C\) is empty, formula (2) holds true with one among \(X, Y, Z\) which can be chosen empty.