C*-algebras associated with self-similar sets

Let γ = (γ₁,..., γ_N), N ≥ 2, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset K. We consider the union G = ∪_i=1^N {(x,y) ∈ K²;x = γ_i(y)} of the cographs of γ_i. Then X = C(G) is a Hilbert bimodule over A = C(K). We associate a C*-algebra script O sign_γ(K) with them as a Cuntz-Pimsner algebra script O sign_X. We show that if a system of proper contractions satisfies the open set condition in K, then the C*-algebra script O sign _γ(K) is simple, purely infinite and, in general, not isomorphic to a Cuntz algebra.