Atomic surfaces, tilings and coincidence I. irreducible case

An irreducible Pisot substitution defines 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’sβ-tiling as a subclass, is constructed. Related tiling and dynamical properties are studied. Based on the coincidence condition defined by Dekking [Dek], we introduce thesuper-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical properties of atomic surfaces. We conjecture that every Pisot substitution satisfies the super-coincidence condition. The second author is supported by a JSPS Postdoc Fellowship.