Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.