Vertical representation of $C^{\infty}$-words

We present a new framework for dealing with $C^{\infty}$-words, based on their left and right frontiers. This allows us to give a compact representation of them, and to describe the set of $C^{\infty}$-words through an infinite directed acyclic graph $G$. This graph is defined by a map acting on the frontiers of $C^{\infty}$-words. We show that this map can be defined recursively and with no explicit references to $C^{\infty}$-words. We then show that some important conjectures on $C^{\infty}$-words follow from analogous statements on the structure of the graph $G$.