Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

Abstract

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 $\mathcal P^{(k)}_w : \mathbb N_1 \rightarrow \mathbb N_1$ 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.