Topological properties of Rauzy fractals

Abstract. — Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geo-metric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions. To be more precise, let σ be a substitution over the finite alphabet A. We assume that the incidence matrix of σ is primitive and its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to σ a set T which is now called central tile or Rauzy fractal of σ. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n = #A subtiles T (1),..., T (n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary. Pisot substitutions and central tiles are of high relevance in several branches of mathemat-ics like tiling theory, spectral theory, Diophantine approximation, the construction of discret