On the asymptotic abelian complexity of morphic words

The subword complexity of an infinite word counts the number of subwords of a given length, while the abelian complexity counts this number up to letter permutation. Although a lot of research has been done on the subword complexity of morphic words, i.e., words obtained as fixed points of iterated morphisms, little is known on their abelian complexity. In this paper, we undertake the classification of the asymptotic growths of the abelian complexities of fixed points of binary morphisms. Some general results we obtain stem from the concept of factorization of morphisms. We give an algorithm that yields all canonical factorizations of a given morphism, describe how to use it to check quickly whether a binary morphism is Sturmian, discuss how to fully factorize the Parry morphisms, and finally derive a complete classification of the abelian complexities of fixed points of uniform binary morphisms.