Abstract : 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.