Abelian properties of Parry words

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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 finite-state automaton as well.