“…Our study indicates a unifying perspective among the subjects of interval maps, (relative) branching systems, and relative graph algebras, whenever the underlying graphs are finite. This is built starting from previous work on the construction of branching systems and the associated representations of graph algebras by the second author [3,10,15,16,18], and by the three other authors in relating open dynamical systems and representations of relative graph algebras (underlying the transition matrices) on the orbit spaces [5,6,7,8].…”