ON κ-ABELIAN AVOIDABILITY

摘要

We consider a recently defined notion of the k-abelian equivalence of words, giving some basic results and concentrating on avoidability problems. This equivalence relation counts the numbers of factors of length k for a fixed natural number k. We ask for the size of a smallest alphabet for which k-abelian squares or cubes can be avoided. For 2-abelian squares, the answer is four, as in the case of abelian words; while for 2-abelian cubes, we only have strong evidence that the answer is two, as it is in the case of words. In addition, we point out a few properties of morphisms supporting the view that it might be difficult to find solutions to our questions by simply iterating a morphism. Bibliography: 17 titles.