Order Types of Shifts of Morphic Words

The shifts of an infinite word $W=a_0a_1\cdots$ are the words $W_i=a_ia_{i+1}\cdots$. As a measure of the complexity of a word $W$, we consider the order type of the set of shifts, ordered lexicographically. We consider morphic words (fixed points of a morphism under a coding) that are not ultimately periodic. Our main result in this setting is that if the first letter of $W$ appears at least twice in $W$, then the shifts of the aperiodic image of $W$ under a coding are dense in the sense that there is a shift strictly between any two shifts. In particular, any purely morphic binary word is either ultimately periodic or its shifts are dense. As a concrete example, we give an explicit order-preserving bijection between the shifts of the Thue--Morse word and $(0,1]\cap {\mathbb Q}$. We then give special consideration to morphisms on 3 letters whose shifts do not contain an infinite decreasing sequence.