Vertical representation of C∞C∞-words

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