Atomic surfaces, tilings and coincidences II. Reducible case

An irreducible Pisot substitution deflnes a graph-directed iterated function system. The invariant sets of this iterated function system are called the atomic surfaces. In this paper, a new tiling of atomic surfaces, which contains Thurston's fl-tiling as a subclass, are constructed. Related tiling and dynamical properties are stud- ied. Based on the coincidence condition deflned by Dekking (Dek), we introduce the super-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical proper- ties of atomic surfaces. We conjecture that every Pisot substitution satisfles the super-coincidence condition.