$C^{\ast}$-Algebras Associated with Mauldin--Williams Graphs

A Mauldin–Williams graph mathcal{M} is a generalization of an iterated function system by a directed graph. Its invariant set K plays the role of the self-similar set. We associate a C^*-algebra mathcal{O} _{mathcal{M}(K) with a Mauldin–Williams graph mathcal{M} and the invariant set K, laying emphasis on the singular points. We assume that the underlying graph G has no sinks and no sources. If mathcal{M} satisfies the open set condition in K, and G is irreducible and is not a cyclic permutation, then the associated C^*-algebra mathcal{O} _{mathcal{M}(K) is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.