We study Cuntz-Pimsner algebras naturally associated with self-similar groups (like iterated monodromy groups of expanding dynamical systems). In particular, we show how to reconstruct the Julia set of an expanding map from the Cuntz-Pimsner algebra of the associated iterated monodromy group and the gauge action on it. We compute K-theory of algebras associated with complex hyperbolic rational functions. It is proved that under some natural conditions the Cuntz-Pimsner algebra of a self-similar group is purely infinite, simple and nuclear. We also show a relation of our algebras with Ruelle algebras of the associated solenoids.
© Walter de Gruyter Berlin · New York 2009