Abelian properties of Parry words

Abelian complexity of a word u is a function that counts the number of pairwise non-abelian-equivalent factors of u of length n. We prove that for any c-balanced Parry word u, the values of the abelian complexity function can be computed by a finite-state automaton. The proof is based on the notion of relative Parikh vectors. The approach works for any function F (n) that can be expressed in terms of the set of relative Parikh vectors corresponding to the length n. For example, we show that the balance function of a c-balanced Parry word is computable by a finitestate automaton as well.