In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z + ∪ {+∞}. Let k ∈ Z + ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A * are said to be k-Abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations ∼ k on A * , bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +∞). We show that the number of k-Abelian equivalence classes of words of length n grows polynomially, although the degree is exponential in k. Given an infinite word ω ∈ A N , we consider the associated complexity function P (k) ω : N → N which counts the number of k-Abelian equivalence classes of factors of ω of length n. We show that the complexity function P (k) is intimately linked with periodicity. More precisely we define an auxiliary function q k : N → N and show that if P (k) ω (n) < q k (n) for some k ∈ Z + ∪ {+∞} and n ≥ 0, the ω is ultimately periodic. Moreover if ω is aperiodic, then P (k) ω (n) = q k (n) if and only if ω is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemerédi's theorem, we show that if ω has bounded k-Abelian complexity, then for every D ⊂ N with positive upper density and for every positive integer N, there exists a k-Abelian N power occurring in ω at some position j ∈ D.

Our objects of study are finite state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words.We use methods from linear algebra and analysis to show that there is an infinite number of atoms in the transducer degrees, that is, minimal non-trivial degrees.

In this paper we investigate local to global phenomena for a new family of complexity functions of infinite words indexed by k ∈ N1∪{+∞} where N1 denotes the set of positive integers. Two finite words u and v in A * are said to be k-Abelian equivalent if for all x ∈ A * of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations ∼ k on A * , bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +∞). Given an infinite word w ∈ A ω , we consider the associated complexity function P (k) w : N1 → N1 which counts the number of k-Abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: Each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper.

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