**Juhani Karhum aki,
Aleksi Saarela,
and Luca Q. Zamboni**

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by $k \geq 0$. Two finite words $u$ and $v$ are said to be $k$-abelian equivalent if for all words $x$ 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, bridging the gap between the usual notion of abelian equivalence (when $k = 1$) and equality (when $k = \infty$). Given an infinite word $w$, we consider the associated complexity function 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.

Available electronic editions: PDF.

10.14232/actacyb.23.1.2017.11

DOI link: https://doi.org/10.14232/actacyb.23.1.2017.11

author = {Juhani Karhum\" aki and Aleksi Saarela and Luca Q. Zamboni},

title = {Variations of the Morse-Hedlund Theorem for $k$-Abelian Equivalence},

journal = {Acta Cybernetica},

year = {2017},

volume = {23},

pages = {175--189},

number = {1},

abstract = {In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by $k \geq 0$. Two finite words $u$ and $v$ are said to be $k$-abelian equivalent if for all words $x$ 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, bridging the gap between the usual notion of abelian equivalence (when $k = 1$) and equality (when $k = \infty$). Given an infinite word $w$, we consider the associated complexity function 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.},

doi = {10.14232/actacyb.23.1.2017.11}

}