Irreducible fractal structures for Moran's type theorems

In this paper, we characterize a novel separation property for IFS-attractors on complete metric spaces. Such a separation property is weaker than the strong open set condition (SOSC) and becomes necessary to reach the equality between the similarity and the Hausdorff dimensions of strict self-similar sets. We also investigate the size of the overlaps from the viewpoint of that separation property. In addition, we contribute some equivalent conditions to reach the equality between the similarity dimension and a new Hausdorff type dimension for IFS-attractors introduced by the authors in terms of finite coverings.