The authors consider the system of iterates of a finite number of similitudes with the same contraction ratio in the Euclidean space. Overlappings in the images of the similitudes are allowed, but still verifying a weak separation assumption, generalizing the results for systems under the strong open set condition.
The paper focusses on the analysis of the self-similar measure supported in the invariant subset associated with the iterated function system, which is either absolutely continuous or singular. The theorems state several conditions to determine that dichotomy: mainly a sufficient assumption for the singularity of the measure, and an equivalent condition for its absolute continuity.