C^*-algebras associated with self-similar sets

Let $\gamma =({\gamma }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\gamma }_N)$, $N\ge 2$, 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$\mathrm{.}Then$X = C({\mathcal G})$isaHilbertbimoduleover$A = C(K)$. We associate a $C^{\ast }$-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 $C^{\ast }$-algebra ${\mathcal{O}}_{\gamma }(K)$ is simple and purely infinite, which is not isomorphic to a Cuntz algebra in general.