  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.

Operator Algebras;Dynamical Systems