C^*-algebras associated with self-similar sets

Let $\gamma = (\gamma_1,...,\gamma_N)$, $N \geq 2$, 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 ${\mathcal G} = \cup_{i=1}^N \{(x,y) \in K^2 ; x = \gamma_i(y)\}$ of the cographs of \gamma _i$. Then$X = C({\mathcal G})$is a Hilbert bimodule over$A = C(K)$. We associate a$C^*$-algebra${\mathcal O}_{\gamma}(K)$with them as a Cuntz-Pimsner algebra${\mathcal O}_X$. We show that if a system of proper contractions satisfies the open set condition in$K$, then the$C^*$-algebra${\mathcal O}_{\gamma}(K)$ is simple and purely infinite, which is not isomorphic to a Cuntz algebra in general.