On balanced and abelian properties of circular words over a ternary alphabet

TL;DR

This paper classifies balanced circular words over a ternary alphabet and constructs uncountably many aperiodic words with minimal abelian complexity, advancing understanding of their combinatorial properties.

Contribution

It introduces a 3D generalization of Christoffel words and classifies all ternary circular words with minimal abelian complexity, also constructing uncountably many aperiodic examples.

Findings

Classification of balanced ternary circular words with abelian complexity 3

Introduction of a 3D generalization of Christoffel words

Construction of uncountably many aperiodic words with abelian complexity 3

Abstract

We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering the minimal bound $3$ for abelian complexity of balanced circular words over a ternary alphabet, we provide a classification of all circular words over a ternary alphabet with abelian complexity subject to this bound. This result also allows us to construct an uncountable set of bi-infinite aperiodic words with abelian complexity equal to $3$.