A natural bijection for contiguous pattern avoidance in words

Two words \(p\) and \(q\) are avoided by the same number of length-\(n\) words, for all \(n\), precisely when \(p\) and \(q\) have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit bijection. We give a bijective proof for all pairs \(p, q\) that have the same set of proper borders, establishing a natural bijection from the set of words avoiding \(p\) to the set of words avoiding \(q\).