On some properties of doubly-periodic words (Q1373477)

Summary: We study the functional equation: \[ ABC= CDA,\tag{1} \] 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: \[ B= XYZ,\quad D= ZYX.\tag{2} \] It is a generalization of the Lyndon-Schützenberg's Theorem: if in (1) \(A\) or \(C\) is empty, formula (2) holds true with one among \(X\), \(Y\), \(Z\) which can be chosen empty.