On abelian periodicity of purely morphic words

Deciding periodicity of infinite words generated by morphisms is a classical result in combinatorics on words from 80’s by Harju, Linna and Pansiot. In this paper, we are interested in this question in the abelian setting. Two words are called $\mathit{\text{abelian}}\mathit{\text{equivalent}}$ if they contain the same numbers of occurrences of each letter. An infinite word $s$ is called \emph{ultimately abelian periodic} if it can be factorized as $s=u{v}_1{v}_2{v}_3\cdots$, where $v_i$’s are abelian equivalent words. If $u$ is empty, then $s$ is called \emph{purely abelian periodic}. We provide the following characterization of binary morphisms generating abelian periodic words: A word generated by a binary morphism $f$ is abelian periodic if and only if either it is periodic or there exist an integer $K$ and words $u$, $v$, $u^{\mathrm{\prime }}$, $v^{\mathrm{\prime }}$ such that $f^K(a)=uv$, $f^K(b)=u^{\mathrm{\prime }}{v}^{\mathrm{\prime }}$, $u{\sim }_{ab}{u}^{\mathrm{\prime }}$, and $vu$ and $v^{\mathrm{\prime }}{u}^{\mathrm{\prime }}$ are abelian periodic with abelian equivalent periods. For the case of the purely abelian periodic words, we also provide an upper bound on $K$ which makes the obtained characterization algorithmic.