Vertical representation of C ∞ -words

We present a new framework for dealing with C ∞ -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 ∞ -words through an infinite directed acyclic graph G. This graph is defined by a map acting on the frontiers of C ∞ -words. We show that this map can be defined recursively and with no explicit reference to C ∞ -words. We then show that some important conjectures on C ∞ -words follow from analogous statements on the structure of the graph G.