Kth power-free codes

Reduction in non-(k+ 1)-power-free morphisms

“…Obviously, taking c = b, and s = ε in a first time and p = ε in a second time, we obtain that a ps-morphism is a bifix morphism. Lemma 1.6 [6,7] If f is not a ps-morphism then f is not a k-power-free morphism for every integer k ≥ 2.…”

“…Thus a 3 2 -power-free word is a 3-anti-power word (but the converse does not hold). Thus a Dejean's word [4,3,8] over a four-letter alphabet, which does not contain any ℓ-power with ℓ > 7 5 -power-free, is a 3-anti-power word. More generally, a non-k-anti-power word (among k consecutive factors of the same length of this word, at least two of them are equal) contains at least one fractionnal ℓ-power with ℓ ≥ k k−1 .…”

“…Lemma 1.2 [6,7] If a non-empty word v is an internal factor of vv, i.e., if there exist two non-empty words x and y such that vv = xvy, then there exist a non-empty word t and two integers i, j ≥ 1 such that x = t i , y = t j , and v = t i+j .…”