An extension of the Lyndon-Schützenberger result to pseudoperiodic words

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form u^l=vⁿw^m, to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l?5,n,m?3, then all three words involved can be expressed in terms of a common word t and its complement θ(t). Moreover, if l?5, then n=m=3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper.