On the Unavoidability of k-Abelian Squares in Pure Morphic Words

We consider a recently defined notion of k-abelian equivalence of words by concentrating on avoidability problems. The equivalence class of a word depends on the number of occurrences of different factors of length k for a fixed natural number k and the prefix of the word. We show that over a ternary alphabet, k-abelian squares cannot be avoided in pure morphic words for any natural number k. Nevertheless, computational experiments support the conjecture that even 3-abelian squares can be avoided over a ternary alphabet. This illustrates that the simple but widely used method to produce infinite words by iterating a single morphism is not powerful enough with k-abelian avoidability questions.