Automatic Sequences of Rank Two

Abstract

Given a right-infinite word $\mathbf{x}$ over a finite alphabet $A$, the rank of $\mathbf{x}$ is the size of the smallest set $S$ of words over $A$ such that $\mathbf{x}$ can be realized as an infinite concatenation of words in $S$. We show that the property of having rank two is decidable for the class of $k$-automatic words for each integer $k\ge 2$.