Abelian bordered factors and periodicity

TL;DR

This paper explores the relationship between abelian bordered factors and periodicity in infinite words, extending classical border concepts to abelian and weak abelian contexts, and providing new insights into their periodic structures.

Contribution

It introduces abelian and weak abelian analogues of a classical periodicity characterization, answering open questions and extending the abelian critical factorization theorem.

Findings

Infinite words with sufficiently long (weakly) abelian bordered factors are (weakly) abelian periodic.

Resolved an open question regarding the abelian critical factorization theorem.

Extended classical border and periodicity results to abelian and weak abelian settings.

Abstract

A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.