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

We revisit the problem of extending the notion of a balanced circular word and focus on the case of a ternary alphabet. Basing on the fact that the upper bound for the abelian complexity of balanced ternary words is 3 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.